GIFT OF Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/elementaryalgebrOOtannrich THE MODERN MATHEMATICAL SERIES LUCIEN AUGUSTUS WAIT . . . General Editok (senior PKOFE8SOB OF MATUEMATIC3 IN COKNELL UNIVEESITY) The Modern Mathematical Series, lucien augustus wait, {Senior Professor of Mathematics tn Cornell University,) GENERAL EDITOR. This series includes the following works : ANALYTIC GEOMETRY. By J. H. Tanner and Joseph Allen. DIFFERENTIAL CALCULUS. By James McMahon and Virgil Snyder. INTEGRAL CALCULUS. By D. A. Murray. DIFFERENTIAL AND INTEGRAL CALCULUS. By Virgil Snyber and J. I. Hutchinson. ELEMENTARY ALGEBRA. By J. H. Tanner. ELEMENTARY GEOMETRY. By James McMahon. The Analytic Geometry, Differential Calculus, and Integral Calculus (pub- lished in September of 1898) were written primarily to meet the needs of college students pursuing courses in Engineering and Architecture ; accordingly, prac- tical problems, in illustration of general principles under discussion, play an important part in each book. These three books, treating their subjects in a way that is simple and practi- cal, yet thoroughly rigorous, and attractive to both teaclier and student, received such general and hearty approval of teachers, and have been so widely adopted in the best colleges and universities of the country, that other books, written on the same general plan, are being added to the series. The Differential and Integral Calculus in one volume was written especially for those institutions where the time given to these subjects is not su£ficient to use advantageously the two separate books. The more elementary books of this series are designed to implant the spirit of the other books into the secondary schools. This will make the work, from the schools up through the university, continuous and harmonious, and free from the abrupt transition which the student so often experiences in changing from his preparatory to his college mathematics. ELEMENTARY ALGEBRA I BY J. H. TANNER, Ph.D. ASSISTANT PROFESSOR OF MATHEMATICS IN CORNELL UNIVERSITY NEW YORK-:. CINCINNATI.:. CHICAGO AMERICAN BOOK COMPANY T-3 COPTEIGHT, 1904, BY J. H. TANNEE. Entered at Stationers' Hall, London, tanner's elem. alg. (^A/'i,^ P.. PREFACE In writing this book one of the chief aims of the author has been to make the transition from arithmetic to algebra as easy and natural as possible, and at the same time to arouse and sustain the student's interest in the new field of work. Accordingly the first few pages are devoted to a restatement and slight extension of the meaning of the ordinary arithmetical operations. Then the literal notation is introduced, and the innovation immediately justified by showing that, among other advantages, it enables the student to solve with ease a class of problems which, by unaided arithmetical analysis, had previously been very difficult for him. In Chapter II negative numbers are introduced, but only after it has been shown, by concrete examples, that these numbers are essential to man's needs, and that they arise naturally from positive numbers. Moreover, to make this extension of the number system seem less startling, it is pointed out that an altogether similar extension has already been made in arithmetic by the introduction of fractions. And so on throughout the book, wherever an essentially new step is to be taken, its naturalness and advantages are presented with it, and it is thereafter freely employed until it becomes a useful tool in the student's hands. Moreover, in order to avoid every unnecessary discouragement to the student, the proofs of the various principles involved in his work are deferred, not only until after he has correctly apprehended and freely employed those principles, but also until after he has been convinced of fjfce necessity of a proof; compare §§ 49, 62 (note), 95, 146 (footnote), 176, etc. Another important object of this book is to teach the student to think clearly. "There is considerable danger of the true educational value of arithmetic and algebra being seriously im- paired by reason of a tendency to sacrifice clear understanding to mere mechanical skill." * The mere manipulation of algebraic * From the report of a Committee of the London Mathematical Society ap- pointed to consider the subject of the teaching of elementary mathematics. 137919 VI PREFACE symbols, however cleverly performed, is of no advantage what- ever in after life to the vast majority of those who study algebra in the schools ; but the training in correct reasoning and in an appreciation of the validity of conclusions that may be drawn from given data, which algebra wheh rightly taught affords, is of vast importance to every one. Accordingly, although the early part of each new topic has been presented as concretely and simply as possible, and although the student has been led, often without conclusive proofs, to infer correctly the principles involved and to perform the various operations freely, his attention has always been called to the fact that results obtained in this way must be regarded as tenta- tive until after the proofs have been given; and the discussion of no topic has been finally closed without a rigorous demonstra- tion of all the principles involved therein. New topics have always been brought in where they were needed, and this has made it necessary in some cases to defer the final proofs considerably (cf. Chapters VI, XVIII, and the Appendices) ; this arrangement has the further advantage, how- ever, of making it possible, if the teacher prefers, to omit the harder proofs altogether on a first reading, without breaking the continuity of the subject. While this book is designed to meet the most exacting entrance examination requirements in Elementary Algebra of any college or university in this country, and especially the excellent revised requirements of the College Entrance Board, yet the arrangement of the book will be found to be peculiarly suited to a briefer course where that should be desired. The author takes pleasure in acknowledging his indebtedness to his colleagues in Cornell University for valuable suggestions, especially to Professors Wait and McMahon, who have read both the manuscript and the proof-sheets; to Miss Lelia J. Harvie, formerly of the Virginia State Normal School, who assisted in preparing and grading the exercises in a large part of the book ; to Dr. William J. Milne of the State Normal College, Albany, N.Y., for his kind permission to make free use of the exercises in his books ; to Professor H. W. Kuhn of the Ohio State Univer- sity, and to several colleagues in the secondary schools, whose advice has been helpful. CONTENTS [See also Index at the end of the book.] ARTICLES I. INTRODUCTION page 1- 4. Number. Arithmetical processes 1 5- 8. Literal notation ; operations witli literal numbers ... 5 9-10. Advantages of literal notation. Recapitulation ... 16 II. POSITIVE AND NEGATIVE NUMBERS 11-15. General remarks ; negative numbers defined and interpreted . 18 16-20. Operations with negative numbers . . . , . .23 21-22. Algebraic expressions. Recapitulation 30 III. EQUATIONS AND PROBLEMS 23-25. Definitions. Directions for solving equations .... 32 26. Problems leading to equations 36 IV. ADDITION AND SUBTRACTION — PARENTHESES 27-28. Definitions : monomials, polynomials, positive and negative terms, etc 42 29-30. Addition of monomials and of polynomials .... 44 31-32. Subtraction of monomials and of polynomials .... 46 33-35. Parentheses ; removing and inserting parentheses ... 49 V. MULTIPLICATION AND DIVISION 36-38. Multiplication. Law of exponents. Product of monomials . 52 39-40. Product of polynomials 55 41-42. Multiplication with arranged polynomials ; detached coefficients 58 43-44. Division. Law of exponents. Negative and zero exponents . 62 45-47. Division with monomials and with polynomials ... 64 48. Remainder theorem 71 Review questions on Chapters I-V 72 VL COMBINATORY PROPERTIES OF NUMBERS 49-51. Commutative and associative laws of addition . . . . 74 52-53. Commutative and associative laws of multiplication ... 77 54. Fundamental principles in operations with fractions ... 80 55. Zero ; operations involving zero 84 VII. TYPE FORMS IN MULTIPLICATION — FACTORING 56-61. Various type forms of products 87 62. Binomial theorem 92 vii Ylll CONTENTS 63- 66. 67. 68. 69- 71. 72. VIII. 73- 75. 76- 78. 79. 80- 82. Various type forms in factoring 94 Factc^ring by means of the remainder theorem . . . 100 Binomial factors of x" ± a** . 102 Other devices for factoring 105 Solving equations by factoring 109 HIGHEST COMMON FACTORS — LOWEST COMMON MULTIPLES H. C. F. by means of factoring 112 H. C. F. of expressions which can not be readily factored ; demonstration of principles involved . . . .114 An expression can be factored into primes in but one way . 120 L. C. M. of two or more algebraic expressions . . . 122 IX. ALGEBRAIC FRACTIONS 83- 88. Transformation of fractions 126 89- 93. Operations with fractions ....... 131 Review questions on Chapters VI-IX 139 94- 95. 96- 97. 99. 100. 101-103. 104-107. 108-109. 110-111. 112-113. 114-116. X. SIMPLE EQUATIONS Introductory remarks. Equivalent equations ^ 117-118. 119. 120-121. 122-124. 125. 126. 127-128. 129. 141 Literal equations. A simple equation has one and only one root 145 Fractional equations 147 Demonstration of principles involved in § 98. Problems . 149 General problems. Interpretation of results .... 157 XL SIMULTANEOUS SIMPLE EQUATIONS Indeterminate equations . Simultaneous equations. Elimination .... Principles involved in elimination Fractional equations ; literal equations .... Systems of equations containing three or more unknowns Graphic representation and solution of equations . XII. INEQUALITIES Definitions and general principles Conditional and unconditional inequalities Review questions on Chapters X-XII XIII. INVOLUTION AND EVOLUTION Involution. Exponent laws . Evolution. Roots extracted by inspection Square roots of polynomials . Sciuare roots of arithmetical numbers Cube roots of polynomials and of numbers Higher roots of polynomials and of numbers 162 165 170 174 183 189 193 197 200 201 205 209 213 216 221 CONTENTS IX XIV. IRRATIONAL AND IMAGINARY NUMBERS — ARTICLES FRACTIONAL EXPONENTS 130-132. Irrational numbers ; preliminary remarks and definitions 133-135. Product and quotient of radicals of the same order 136-139. Transformation of radicals 140-144. Operations with radicals 145. Important property of quadratic surds 146-150. Imaginary numbers, and operations with them 151. Important property of complex numbers . . . . 152. Complex factors. Solving equations by factoring . 153-154. Fractional exponents 155-159. Demonstration of exponent laws with fractional exponents ; summary of these laws 160. Operations involving fractional exponents . . . . 161. Rationalizing factors of binomial surds XV. QUADRATIC EQUATIONS 162-163. Introductory remarks and definitions 164-166. Solution of quadratic equations, — by "completing the square," by factoring, and by formula . 167-168. Character of the roots ; their sum and product 169-170. Fractional and irrational quadratic equations 171. Problems which lead to quadratic equations . 172. Equations solved like quadratics 173. Maxima and minima values .... 174-179. Quadratic equations in two unknowns ; various devices for solving 180. Systems containing three or more unknowns . 181-182. Square roots of quadratic surds and of complex numbers 183-185. Graphic representation and solution of quadratic equations XVL RATIO, PROPORTION, AND VARIATION 186-187. Ratio. Incommensurable numbers .... 188-189. Proportion; definitions and principles . . . . 190. Variation. Constants and variables .... PAGE 223 228 233 237 243 244 250 251 252 255 261 264 268 277 282 286 291 294 297 308 310 314 318 320 327 191-194. 195-198. 199. 200. XVIL SERIES. THE PROGRESSIONS Series. Arithmetical progression . . . , Geometric progression ...... Aritlimetico-geometric series Harmonic progression 331 336 342 342 XVin. MATHEMATICAL INDUCTION— BINOMIAL THEOREM 201. Proof by induction 344 202-204. The binomial theorem 346 205. The square of a polynomial 350 Appendix A. Irrational Numbers 351 Appendix B. Complex Numbers 365 NOTICE It is not expected that pupils will be asked to solve all of the very large number of exercises and problems, but rather that the teacher will make such selections as will best suit the needs of his or her classes. If the teacher desires a briefer course than that provided in the book, or prefers to omit the proofs on a first reading, the following articles, together with their attached exercises, may- be omitted without breaking the continuity of the work : Articles 50-54 Pages 74-83 Omit exercises 1-14, pp. 84- u 77-79 116-122 (I 95 143-144 Take exercises 5-15, p. 145 a 99 149-150 " " 3-6, p. 151 t( 103 163-164 (C 108-109 170-172 Take exercises on p. 173 (( 114-116 189-192 (( 127-129 216-222 Notes 1-2 285 Omit exercises 17-22, p. 28 Articles 173 294-297 (I 176 298-299 (C 183-185 314-317. The teacher will also find it easy to abbreviate somewhat the work of Chapters XIV and XV. If the above omissions are made, it will be necessary to pass over a few isolated exercises and notes such as Ex. 3, p. 184, and note 1, p. 301, and also to change slightly the headings to some sets of exercises such as those on p. 145. / OrTHf ELEMENTARY ALGEBRA CHAPTER I INTRODUCTION 1. Algebra may be regarded as, in a certain sense, a continuation and extension of arithmetic ; it may be best, therefore, to recall briefly the subject matter and some of the processes of arithmetic before taking up the study of algebra. It will presently appear (§ 6) that algebra abbreviates and greatly simplifies the solution of certain kinds of problems. It will also be shown that the meaning hitherto attached to num- ber, as well as its mode of representation, is greatly extended in algebra; and that the "equation," which plays a very minor part in arithmetic, is of great importance in algebraic investigations. 2. Number. The first numbers that present themselves are those which arise from counting and from measuring things;* they are usually called whole numbers, and also integers, but may quite appropriately be called the natural numbers. These numbers are always definite, and are represented by one or more of the Arabic characters 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Out of combinations of these natural numbers have grown other kinds of numbers, such as fractions, which have already been studied in arithmetic, and still other kinds which will "be pre- sented in later chapters of this book. * Numbers themselves are not found ready made in nature ; there are, how- ever, everywhere things, and the counting or the measuring of these gives rise to numbers. Since much of the intercourse of life is concerned with the things about us, and with their relations to one another, and since these relations are expressed by means of numbers, it is for this reason alone — to say nothing of other excellent reasons — of fundamental importance that numbers and their combinations be carefully studied. It will be found advantageous, and will add clearness of view, if in our reasoning about numbers we frequently go back to the things themselves from which these numbers may have arisen. 1 2 ELEMENTARY ALGEBRA [Ch. I 3. Arithmetical processes, (i) Addition. Fundamentally, addi- tion of natural numbers is merely counting. E.g., to add 4 to 7, means to find that number which is four greater than seven ; we begin therefore with 7 and count four, forward, which gives 11. Similarly in general. The sign of addition is an upright cross (+), which is read plus (meaning more) ; when written between two numbers, it means that the second is to be added to the first. E.g., 7 + 4 is read " seven plus four," and means that 4 is to be added to 7. The result of adding two or more numbers is called their sum; the numbers to be added are called the summands. It is evident that addition, in the case of natural numbers, is always a possible arithmetical operation ; that this is not true of subtraction will be seen in (ii) below. Two short parallel horizontal lines (=) are used to express that one of two numbers is equal to, i.e., is the same as, the other ; e.g., 7 + 4 = 11. This expression is called an equation, and is read " seven plus four equals eleven." (ii) Subtraction. Subtraction is tlie inverse* of addition; with natural numbers it is a counting off. E.g., to subtract 3 from 15, we begin with 15 and count off (or backward) 3 units, thus: 14, 13, 12; and 12 is the result of the subtraction. In other words, to subtract the first of two munbers froiYi the second is to find a third number such that this third number plus the first number equals the second number. The 'sign of subtraction is a short horizontal line (— ), which is read minus (meaning less) ; when written between two numbers, this sign means that the second number is to be subtracted from the first. E.g.y 7 — 4 is read " seven minus four," and means that 4 is to be subtracted from 7. * An inverse operation may be defined as one whose effect is neutralized by the corresponding direct operation. Addition and multiplication are direct opera- tions; their inverses are subtraction and division. 3] INTEOBUCTION ' 3 The result of subtracting one number from another is called their difference, and also the remainder; the number which is sub- tracted is called the subtrahend, and the one from which the subtraction is made is called the minuend. In the above example, 7 is the minuend, 4 the subtrahend, and 3 the remainder, all of which is expressed by the equation 7—4 = 3, which is read " seven minus four equals three." From the above definition it follows that subtraction is a possible arithmetical operation only when the minuend is at least as great as the subtrahend. (iii) Multiplication is usually defined as the process (or operation) of taking one of two numbers, called the multiplicand, as many times as there are units in the other, which is called the multiplier. In this sense multiplication is, fundamentally, the same as addition. E.ff., 8 multiplied by 5 means that 8 is to be used 5 times as a summand; i.e., the product of 8 multiplied by 5 is 8+8 + 8 + 8 + 8. The sign of multiplication is an oblique cross ( x ), which is read multiplied by ; when written between two numbers, it means that the first is to be multiplied by the second. The result of multi- plying one number by another is called their product. Note. The definition of multiplication just given applies only when the mul- tiplier is an integer. Under it, multiplication by a fraction or by a mixed number has, strictly speaking, no meaning. For example, let it be required to multiply 8by 5|; to do this under the definition just given, it is necessary to take 8 as many times as there are units in 5|, but manifestly, while 8 may be taken addi- tively five times, it can not be taken tioo thirds of a time* and the proposed problem, therefore, does not admit of solution under this definition. A far more useful definition of multiplication than that given above, and one that will serve all future needs, may be stated thus : The product of two numhers is the result obtained hy performing upon the first of these numhers {the multi- plicand) tl%e same operation that must he perfonned upon the unit to obtain the second {th^, inultiplier) . This definition not only includes the former one, but it also gives an intelligible meaning to multiplication when the multiplier is a fraction or a mixed number. * This is as meaningless as " to fire a gun two- thirds of a time." 4 ELEMENTARY ALGEBRA [Ch. I E.g., consider again the question of multiplying 8 by 5f ; the multiplier, 5|, is obtained from the unit by taking the unit five times, and ^ of the unit twice, as summands ; i.e., 5| = l + l + l + l + l + i + i and, therefore, by this new definition of multiplication, 8x51 = 8 + 8 + 8 + 8 + 8 + 1 + ! = 40 + J# = 45i (iv) Division. In algebra as in arithmetic, to divide one of two given numbers by another is to find a number which, being multiplied by the second of the given numbers, will produce the first ; the symbol of division is -h, and is read divided by. E.g., 15 -^ 5 = 3, because 3 X 5 = 15 ; the first of these equations is read " fifteen divided by five equals three." The operation of dividing one number by another is called division, the first of the given numbers is called the dividend, the second is the divisor, and the result, i.e., the number sought, is the quotient. E.g., in 15 ^ 6 = 3 the dividend, divisor, and quotient are 15, 5, and 3, respec- tively. Note 1. Observe that, under the above definition, the test of the correctness of a quotient is quotient x divisor = dividend. Division is therefore the inverse of multiplication (cf. footnote, p. 2). NoTJB 2. Observe also that while the sum, the difference, and also the product of any two integers is an integer, their quotient may or may not be an integer ; for instance, 6 -f- 3 is an integer, but 7-^3 and 5 -7-9 are called fractions [cf. § 7 (V)]. 4. Symbols of continuation and deduction. The symbol of con- tinuation is •••; it is read "and so on," or "and so on to," and is used to denote that a given succession of numbers is to con- tinue, either without end or up to a given number. E.g., 1, 2, 3, ••• is read "one, two, three, and so on" ; while 1, 2, 3, ••• 27 is read "one, two, three, and so on to twenty-seven." The symbols of deduction are ••• and .-. , and are read " since " and " therefore," respectively. E.g., ■.•3x5 = 15, .-. 15 -4- 5 = 3; this expression is read " since three multi- plied by five equals fifteen, therefore fifteen divided by five equals three." The symbols explained in this section are, like all other signs and symbols, merely, abbreviations for longer, expressions. 3-5] INTBODUCTION EXERCISES Read the following expressions, and give the names of their parts : 1. 3 + 7 = 10. 3. 15 - 3 = 5. 2. 13 - 8 = 5. 4. 4 X 6 = 24. 5. State the definitions of the operations indicated in exercises 1-4. Show that your definition of multiplication applies also to cases in which the multiplier is a fraction or a mixed number. 6. Which of the operations in exercises 1-4 are direct, and what are their respective inverse operations? Explain your answer. 7. How is the correctness of an inverse operation to be tested? Illus- trate your answer by testing the correctness of 15 -^ 3 = 5. Read the following expressions : 8. •.•5x3= 15, .•. 15 - 3 = 5. 9. •.• 5 + 8 = 13, .•. 13 - 8 = 5. 10. The numbers 1, 3, 5, ••• are called odd numbers. The sum of the numbers 1, 3, 5, ••• 13 is 49. 5. Literal notation. The Arabic characters of arithmetic, viz., 0, 1, 2, 8, '" 9, and also the signs +, — , x, -^, and =, are all retained in algebra, and each with its precise arithmetical mean- ing ; but algebra also frequently employs some of the letters of the alphabet to stand for, or represent, numbers.* E.g., in a certain problem it may be agreed (possibly merely for brevity) to let n stand for a particular number, say 786 ; in that case — (i.e., one half of n) would, in the same problem, stand for 393, while 3 n {i.e.,n + n + n) would stand for 2358, etc. In another problem, however, n may be employed to represent any other desired number. One advantage of representing numbers by letters is explained in § 6 below; others will appear later. For the present it is perhaps sufficient to say that, just as in arithmetic we speak of 4 books, 7 bicycles, 85 pounds, 3 men, etc., so in algebra we shall frequently, in addition to these expressions, use such expressions as a books, n bicycles, x pounds, y men, etc. When it is necessary to distinguish between numbers which are represented by the Arabic characters 0, 1, 2, •••, and numbers which are represented by letters, the latter will be called literal numbers. * This way of representing numbers is, however, not entirely new to the stu- dent because, even in arithmetic, in problems concerning " interest," the princi- pal, amount, rate, interest, and time are often represented by the letters p, a, r, i, and t, respectively. 6 ELEMENTARY ALGEBRA [Ch. I The properties of numbers are, of course, precisely the same whether these numbers are represented by the Arabic characters, by letters, by words, or in any other way. E.g., just as 3 books + 8 books = 11 books, so m books + ?i books = (m+n) books ; and if k stands for 20, then Sk-i ■2yfc==25. 4 Again, just as 7 — 3 means that 3 units are to be subtracted from 7 units, so a~b means that b units are to be subtracted from a units. EXERCISES 1. If 5 represents 16, what number is represented by 2 s? by | of s, i.e., by i? by2s + -?* 4 4 2. If a, b, and c represent, respectively, 2, 5, and 8, what is the value of 3 a- 6? of a + & + c? of ^^^^? 3. If a: represents the number of panes of glass in a window, how may the number of panes of glass in 3 such windows be repi-esented ? — _ 4. If a suit of clothes costs 8 times as much as a hat, and if d stands for the number of dollars which the hat costs, what will represent the cost of the suit? How may the combined cost of the suit and hat be represented ? 5. Since ^ of any number is the same as /g of that number, and i of a number is the same as y\ of that number, what is the remainder when ^ of n is subtracted from | of n, where n represents any number what- ever? i.e., ^ - ^ = ? ' 3 4 6. Just as 37 may be represented by 10 x 3 + 7, so 10 ^ + w represents a number whose tens' digit is t and whose units' digit is u. If the units', tens', and hundreds' digits of a number are represented by x, y, and z, respectively, how may the number itself be represented? 7. If X represents the number of years in a man's present age, how may his age 5 years ago be represented ? What will represent his age 12 years hence? 8. If X represents any integer, how may the next higher integer be represented? The next above that? If n represents any integer, does 2 n represent an even or an odd number? How may the next higher even number be represented? Show that 2w — 3, 2n— 1, 2n + l, 2n + 3, •••, represent consecutive odd numbers. * In these exercises, and throughout the first five chapters of this book, a knowledge of the ordinary arithmetical processes is assumed ; the fundamental principles involved will be studied in Chapter VI. 5-6] INTRODUCTION 7 9. A thermometer reads 80*^ at noon and falls y° during the next 6 hours. What is its reading at 6 o'clock? 10. What number multiplied by 8 gives the product 40? If 8 a: = 40, what is the value of a:? li 3 y + b y — 2 y = bi, what is the value of ?/ ? 6. One advantage of literal notation. The use of letters to repre- sent numbers greatly simplifies the solution of certain kinds of arithmetical problems. This is illustrated in the examples that follow. ^ Prob. 1. A gentleman paid $45 for a suit of clothes and a hat. If the clothes cost 8 times as much as the hat, what was the cost of each? Arithmetical Solution The hat cost "some number of dollars," and since the clothes cost 8 times as much as the hat, therefore the clothes cost 8 times "that num- ber of dollars," and therefore the two together cost 9 times "that number of dollars"; hence 9 times "that number of dollars" is $45, therefore "that number of dollars" is $5, and 8 times "that number of dollars" is $40; i.e., the hat cost $5, and the clothes cost $40. This solution may be put into the following more systematic form, still retaining its arithmetical character. Some number of dollars = the cost of the hat ; then 8 times that number of dollars = the cost of the clothes, 9 times that number of dollars = the cost of both, i.e., 9 times that number of dollars = $45, that number of dollars = $5, the cost of the hat, and 8 times that number of dollars = $40, the cost of the clothes. Algebraic Solution The solution just given becomes very much simplified by letting a single letter, say x, stand for "some number" and "that num- ber " which occur so often above ; thus : Let X = the number of dollars* the hat cost. Then 8 x = the number of dollars the clothes cost, and X + 8 a: = the number of dollars both cost, i.e.f 9 a; = 45, x= 5, and 8 a: = 40; i.e., the hat cost $5, and the clothes cost $40. * The letter x here stands for a number, not for the cost of the hat ; the equa- tions are numerical. 8 ELEMENTARY ALGEBRA [Ch. I Prob. 2. Three men, A, B, and C, form a business partnership with a capital of $30,000 ; if A furnishes twice as much of this capital as B, and C furnishes as much as A and B together, how much does each furnish? Solution Let X = the number of dollars furnished by B. Then 2 x = the number of dollars furnished by A, and 3 X = the number of dollars furnished by C ; and the algebraic statement of the conditions of the problem becomes a: + 2x + 3a;= 30,000, i.e., Qx = 30,000, whence x = 5000, 2x = 10,000, and Sx = 15,000 ; i.e., A furnishes |10,000, B |5000, and C |15,000 of the capital. Prob. 3. Of three numbers the second is 5 times, and the third 2 times, the first, and the sum of these numbers exceeds the third number by 42 ; what are the numbers ? Solution Let X = the first of the three numbers. Then 5 x = the second of the three numbers, and 2 X = the third of the three numbers ; and the algebraic statement of the conditions of the problem becomes x + 5x-h2x = 2x + i^2, i.e., 8 a: = 2 a: + 42, hence 6^=42, TSubtract 2 x from [_each member therefore x=7, 5 a: = 35, and 2 x = 14 ; and the required numbers are, respectively, 7, 35, and 14. Note. Observe that the plan of each of the foregoing solutions is to let some letter, say x, stand for one of the unknown jiumbers (preferably the smallest), then to express the other unknown numbers in terms of x, and finally to trans- late into algebraic language the conditions which are verbally stated in the prob- lem ; this last statement is an equation, and from it the required numbers are easily found. Observe also that while the above problems can be solved by arithmetical analysis, the algebraic solution is much simpler. 6] INTRODUCTION PROBLEMS 4. In a room containing 45 pupils there are twice as many boys as girls. How many boys are there in the room ? 5. If a horse and saddle together cost $ 90, and the horse cost 5 times as much as the saddle, how nmch did each cost ? 6. In a business enterprise, the combined capital of A, B, and C is 121,000. A's capital is twice B's, and B's is twice C's. What is the capital of each ? 7. The difference between two numbers is 8, and their sum is 30. What are the numbers? 8. Divide 98 into three parts such that the second is twice the first and the third is twice the second. 9. A number, plus twice itself, plus 4 times itself, is equal to 56. What is the number? 10. The sum of three numbers is 160 ; two of these numbers are equal, and the third is twice either of the others. Find the numbers. 11. In a fishing party consisting of 4 boys, 2 of the boys caught the same number of fish, another caught 2 more than this number, and another 1 less ; if the total number of fish caught was 29, how many did each catch? 12. If a locomotive weighs 3 times as much as a car, and the difference between their weights is 50 tons, what does the locomotive weigh ? 13. Of two numbers, twice the first is seven times the second, and their difference is 75 ; find the numbers. Suggestion. Let 1 x= the first number, then 2x = the second. 14. An estate of $ 19,600 was so divided between two heirs that 5 times what one received was equal to 9 times what the other received; what was the share of each ? 15. A horse, harness, and carriage together cost |340; if the horse cost 3 times as much as the harness, and the carriage cost \\ times as much as the horse and harness together, what was the cost of each ? 16. A, B, C, and D together buy % 16,000 worth of railroad stock. B buys three times as much as A, C twice as much as A and B together, and D one third as much as A, B, and C together. How much does each 10 ELEMENTARY ALGEBRA [Ch. 1 17. What number added to I of itself equals 20 ? Solution Let X = the number. Then x + ^ x = 20, i.e., |x = 20, .-. ' cc = 20-^1 = 15. 18. If I of a number is added to the number, the sum is 120; what is the nural)er? 19. If I of a number is added to twice the number, the sum is 35; what is the number? 20. 'I'he difference between 4 times a certain number and | of that number is 30; what is the number ? 21. Three times A's age is four times B's, and the sum of their ages exceeds | of A's age by 24 years; what is the difference between their ages? 22. A merchant owes a certain sum of money to A, | as much to B, and twice as much to C as he owes A; various persons owe him 12 times as much as he owes B, and if all these debts were paid, the mer- chant would have $4000. What is the total amount that the merchant owes ? 23. A boy found that he had the same number of 5, 10, and 25 cent pieces, and that the total amount of his money was $3.20; how many coins of each kind had h3? 24. Of a family of seven children each child is 2 years older than the next younger; if the sum of their ages is 81 years, how old is the youngest child? 25. In a number consisting of two digits, the digit in units' place is 3 times that in tens' place, and if these digits be interchanged, the num- ber will be increased by 36 ; w^hat is tlie number (cf. Ex. 6, § 5) ? 26. The president of a stock company owns 3 times as many shares as the vice president, and the secretary owns 6 shares less than the vice presi- dent ; if these three men together own 539 shares, how many shares does each own? 27. Three newsboys sold a total of 191 papers in an afternoon; if the second sold 5 more than twice as many as the first, and the third sold three times as many as the second, how many did each sell? 28. A tree, whose height was 150 feet, was broken off by the wind, and it is found that 3 times the length of the part left st-anding is the same as 7 times that of the part broken off ; how long is each part ? i ] • INTRODUCTION 11 29. In a yachting party consisting of 36 persons, the number of chil- dren is 3 times the number of men, and the number of women is one half that of the men and children combined; how many women are there in this party ? 30. If two boys together solved 65 problems, and if 8 times the num- ber solved by the first boy equals 5 times the number solved by the second boy, how many did each boy solve ? 31. An estate valued at $ 24,780 is to be divided among a family con- sisting of the mother, 2 sons, and 3 daughters ; if the daughters are to receive equal shares, each son twice as much as a daughter, and the mother twice as much as all the children together, what will be the share of each? 32. A library contains 17 times as many scientific books, and 6 times as many historical books, as books of fiction; if the books of fiction number 220 less than the scientific and historical books together, how many books are there in this library ? 33. A, B, and C enter into a business partnership in which A furnishes 6 times as much capital as C, and B furnishes | as much as A and C together; if the total capital is |13,500, how much is furnished by each partner ? 7. Operations with literal numbers. As is pointed out in § 5, the reasoning employed with numbers represented by letters is pre- cisely the same as if those numbers were represented by the Arabic characters. It may be worth while, however, to examine the fun- damental operations a little more closely. (i) AdditioTi. Just as 3 + 7 means that 7 is to be added to 3, so too, if a and h stand for any two numbers whatever, a + h means that h is to be added to a. Similarly, a-\-x-{-p means that x is to be added to a, and that p is then to be added to that sum ; and so in general. (ii) Subtraction. Just as 15 — 9 means that 9 is to be sub- tracted from 15, %o x — y means that y is to be subtracted from a;, whatever the numbers represented by x and y. Note. Observe that, while addition is always possible, the indicated subtrac- tion a; — ?/ is arithmetically possible only when the number represented by x is at least as great as that represented by ?/. This restriction upon the relative values of the two numbers in such an expres- sion as x — ?/ is often very inconvenient; in Chapter II the meaning of number is so extended as to make this subtraction possible even when y is greater than x. 12 ELEMENTARY ALGEBRA [Ch. I (iii) Multiplication. Just as 6 x 5 means that 6 is to be multiplied by 5, so 6x3 means that h is to be multiplied by 3. Again, a x y X n means that a is to be multiplied by y, and that their product is then to be multiplied by n ; and so in other cases. Instead of the oblique cross ( x ), a center point (•) placed be- tween two numbers (a little above the line to distinguish it from a decimal point) is frequently used as a sign of multiplication. E.g., instead of 4x6, 3xn, axk, etc., it is usual to write 4 • 6, 3 • n, a • A;, etc. And even the center point is usually omitted in cases where its omission causes no misunderstanding. E.g., 3 Xn = 3-n = 3n, and aX k = a - k= ak; but, while 4 X 6 = 4 • 6, it can not be written "46," for in that case it would be confused with 40 + 6. (iv) Powers, exponents, etc. Products in which all the fac- tors are identical with one another are usually written in an abbre- viated form. This form consists of the repeated factor written only once and having attached to it (at the right and slightly above) the number which tells how many times the given factor is to be repeated. E.g., 2 • 2 • 2 is written 2^, a ' a ' a • a ' a is written a^, and the product of n factors each of which is a is written x^. The expression a?" is called the nth power of x, and is usually read " x nth. " ; the number n is called the exponent of the power, and X is called the base. In particular, 2^ is the third power of 2, the exponent is 3, and the base is 2. A power is called odd or even according as its exponent is odd or even. Similarly, a product in which the factor 2 is repeated 3 times, and the factor 5 is repeated 2 times, is written 2'' • 5^. And, more generally, the expression a^'b^'c^ is the product of a repeated m times, b repeated n times, and c repeated p times; it is read "the mth power of a, multiplied by the nth. power of b, multiplied by the pth power of c." Note. Under the definition of a power given above, it is evident that a^ has the same meaning as a, and the exponent 1, therefore, need not be written. The second and third powers of numbers are, for geometric reasons, often called by the special names of square and cube, respectively; thus, a!^ is known as the "second power of a," the "square of a," and also as "a squared"; and x^ is known as the " third power of x," the " cube of x," and also as " x cubed." Cor- responding to the other powers there are no such special names. 7-8] INTRODUCTION 13 (v) Division. Just as 40 -h 5 indicates that 40 is to be divided hj 5, so a-v-b indicates that a is to be divided by b, whatever the numbers represented by a and b ; that is, (a-i-b) •b = a for all values of a and b [cf. § 3 (iv), note 1]. Other forms of writing a-i-b are : -, a:b, and a/b. b In algebra, as in arithmetic, if the divisor is not exactly contained in the dividend, the indicated division is called a fraction.* ^'9', I, — . -» and ^-±^ are fractions. 3 5 n y It is to be remarked, in passing, that literal numbers may be fractional in form and yet have integral values, and vice versa. E.g., — , though fractional in form, has the integral value 3 if a = 12 and 6 = 4; b and m + 3 7i, though integral in form, has the value j| if m = j and n = j. 8. The order in which arithmetical operations are to be performed. Signs of aggregation. When there is no express statement to the contrary, a succession of multiplications and divisions is under- stood to mean that these operations are to be performed in the order in which they are written from left to right. The same rule applies in the case of a succession of additions and sub- tractions. E.g., 9 • 8 -^ 6 • 2 means that 9 is to he multiplied hy 8, that product to be divided by 6, and the resulting quotient to be multiplied by 2; it does not moan that the product of 9 by 8 is to be divided by the product of 6 by 2 : the result is 24, and note. So, too, 7 + 9 — 6 + 3 means that 9 is to be added to 7, then 6 subtracted from that sum, and finally 3 added to this remainder ; it does not mean that 6 + 3 is to be subtracted from 7 + 9 : the result is 13, and not 7. Again, by a succession of the operations of addition; subtraction, multiplication, and division, when the contrary is not expressly stated, it is customary to mean that all the operations of multi- plication and division are to be performed in the order in which * A fraction is usually defined as " one or more of the equal parts into w^hich a unit has been divided," but this definition is only a special case of the one given above ; it is meaningless when the denominator is not an integer. 14 ELEMENTARY ALGEBRA [Ch. I they are written from left to right, before any of those of addi- tion and subtraction are performed ; the resulting expression will then contain only the operations of addition and subtraction, and these operations are then to be performed in the order in which they occur. E.g., the expression 2 + 6-5 — 8-^2 means 2 + 30 — 4, which is 28. Should the writer of such an expression desire that a different meaning be given to the expression (e.g., that one or more of the additions and subtractions be performed before some of the mul- tiplications and divisions are performed), he would indicate his meaning by employing one or more of the so-called signs of aggregation; among these are the parenthesis ( ), the brace | \, the bracket [ ], and the vinculum '. An expression, included in the parenthesis, brace, or bracket, or under the vinculum, is to be regarded as a whole, and is to be treated as though it were repre- sented by a single symbol. E.f/., (2 + 6) .5-4-3-(7 + 8-^2)=8-5-=-3 — 11, i.e., 2^. So, too, (4 + 6)-^2 = 5. while without the parenthesis its value would be 7. It may even be useful sometimes to employ one sign of aggre- gation within another. E.g., 72 ^ {252 - (24 • 4 + 6)}. In such a case the innermost sign of aggregation is, of course, to be attended to first ; the value of the above expression is 6. EXERCISES Find the value of each of the following expressions : 1. 38-6 + 14-12-2. 2. 38 - (6 + 14) -(12- 2). 3. 9. 6 -4(36 -3 -2) +54 -(17 -12 -5). 4. 12 . 3 - (9 + 3 - G) . 18 - 6^=^. 5. {4 . 9 - 16 ^ 2 - (12 - 8) ^ (4 + 6 -^ 3)} - (6 - 2). 6. Give a definition of a fraction that will include cases in which the denominator is such a number as 3|. 7. May an expression be fractional in /orm, but integral in value? Give three examples of this kind. 8-9] INTRODUCTION 15 Read each of the following expressions, then tell in what order the indicated operations are to be performed, and finally find the numerical values of these expressions when a = 8, 6 = 3, c = 12, and d = \: 8. 12. c A ba ' d hd ah cd 9. 4a +36-C-' 10. (a + 6)2- (a- by -4 aft. 11. {ahc + h) - - (4 cc/ + d) -i -[^ -(a + 4rf)], cd-. C2- rUd' h'^c 13. . a( ;c- 6) + 6(a o6c d (^l)' 14. {6 a - 2 c - 2 (/-'} + J^iii - (2 ft • ^/). 4 c/3 9. Advantages of using letters to represent numbers. Attention has already been called (§ 6) to one of the many advantages which result from the use of letters to represent numbers ; two further advantages will now be considered. (i) Suppose it to have been noticed, in a few particular cases, that half the sum of two numbers plus half their difference equals the greater of these numbers, and suppose that it is required to ascertain whether this is true for a certain few pairs of numbers only, or whether it is true for all possible pairs of numbers. For any particular pair of numbers that may be under con- sideration, 15 and 7 for example, its correctness is easily verified, thus AK\7 ip; 7 but after having made this verification one is still in doubt about every untried pair of numbers. If, on the other hand, letters are employed, it may be proved, once for all, that the above property belongs to every pair of numbers, and no further verifications are needed. Thus, let a and h represent any two numbers whatever, and let a be greater than 6; then x a4-5 ■ a — h _a h i^_^_^,^i^_^_^i^_^ 2 2 "2 2 2 2~2 2 2 2~2 2~ ' which proves that half the sum of any two numbers ivJiatever, plus half their difference, equals the greater of these numbers. The literal notation has here served to prove a general law. 16 ELEMENTARY ALGEBRA [Ch. I (ii) Another advantage of the literal notation may be illustrated by comparing the solutions of the two following problems. Prdb. 1. If A can do a piece of work in 15 days, and B can do it in 10 days, in how many days can both working together do it? Prob. 2. If A can do a piece of work in a days, and B can do it in b days, in how many days can both working together do it? Solution of Problem 1 Since A can do all of the work in 15 days, therefore he can do j\ of it in one day ; similarly, B can do j\ of it in one day, and both together can therefore do ^^ + Jj, that is, I, of it in one day ; hence it will take both together 1 -f- ^, i.e., 6, days to do the work. Solution of Problem 2 Since A can do the work in a days, therefore he can do - of it in 1 ' " one day ; similarly, B can do - of the work in one day, and both together can do - + 7, i.e., — ;— , of it in one day; hence it will a h ah take both together 1 -^ ^ , that is, ^ . days to do the work. ab a + b The reasoning in the two solutions just given is exactly the same ; it is to be observed, however, that while in the course of the first solution the numbers given in that problem (viz., 15 and 10) have, by combining, completely lost their identity before the result is reached, yet the numbers given in the second problem (viz., a and b) preserve their identity to the end. Because of this fact the answer to the. second problem may be used as 2i, formula by means of which the answer to any other like problem may be immediately written down. Thus, if a = 15 and h = 10, then the second problem becomes exactly like the first, and its answer, viz., , becomes - — '- , which is 6 as before. a + h 15 + 10 In other words, the solution of the second problem includes the solution of every other similar problem ; numerical problems like the first are merely particular cases of the second. 9-10] INTBOBUCTION 17 10. Recapitulation. Two things mentioned in this chapter must be carefully kept in mind when reading the following pages; they are : (1) the somewhat broader, and at the same time more precise, definitions * of the fundamental arithmetical operations ; and (2) the advantages connected with the use of letters to repre- sent numbers. While the Arabic characters, 1, 2, 3, • • •, always represent the same numbers, wherever they occur, a letter may be chosen to represent one number in one problem, and a different number in another problem ; a letter may also represent a number to which no specific value is assigned (cf. § 9), as well as a number whose value is at first unknown and is to be found in the course of the solution of the problem (cf. § 6). EXERCISES 1. Express the following indicated products by means of the expo- nent notation : 3 • .3 • 3 • 3 • 3 ; a - a ■ a • a; x • a: • a: ••• to 12 factors ; 5 • 5 • 5 ... to n factors; ax > ax - ax '•• to k factors. 2. Define the expressions: power, base, and exponent, and illustrate your meaning by means of exercise 1. . 3. Express the following numbers as products of powers of prime numbers: 48, 200, 972, and 1183. When a = f and 6 = |, verify the following statements : 4. a(a + 26) =a2+2a&. 6. (a - ft)3 = a^ - 3 a2^, + 3 ^^2 _ 53. 5. (a + 6)2 = a2 4- 2 a& + 6=. 7. (a + 6) (a - &) = a^ - h\ Find the numerical value of each of the following expressions when a = 3, & = I, c = i, a: = 4, y = 2, m = 5, and n = 2 : o a^ft" — c^x e.a-^-x^c-^mc'^ o. • ». 1 xy'^ — ax** ■{■ y x +n n" * These definitions pave the way for the proofs of some fundamental laws to be given later. CHAPTER II POSITIVE AND NEGATIVE NUMBERS 11. General remarks. As already pointed out, a.n important use of numbers is to enable man to express, in a brief and simple way, the relations of the things which are everywhere round about him. At first he used only the natural numbers, i.e., the integers, to express these relations, but as his need and desire for precision and conciseness increased, he found it necessary to extend his number system so as to include in it, not only fractions, but also other kinds of numbers, some of which will presently be studied. E.g., when he wished to express even so simple a relation as that between the lengths of two lines, he found that integers alone are not sufficient unless the lengths of these lines happen to be such that the longer can be divided into parts each of which will be just as long as the shorter; thus, if the given lines are 12 ft. and 5 ft. long, respectively, then the relation between their lengths can not be exactly expressed by an integer, because 12 -^ 5 is not an integer. In order to meet this and other like needs, man extended his number system so as to make the arithmetical operation of division always possible, i.e., he included common fractions in his number system (§ 3, note 2). Before fractions were introduced, division was possible only in the comparatively few cases in which the dividend happened to be a multiple of the divisor. 12. Need of negative numbers. In § 11 it is shown that a number system consisting of integers only is not sufficient for man's needs, but that if the system be so enlarged as to make division always possible, i.e., so as to include fractions also, this enlarged system will serve him far better — indeed this enlarged system serves all the purposes of ordinary arithmetic. In the study of algebra, however, there are many considera- tions which make it very advantageous to enlarge the number system still further. To illustrate : on every hand there are found things which stand in a relation of opposition to each other — e.gf., assets and liabilities in business, latitude north and latitude south of the equator, temperature above zero and temperature below zero, etc. — and if the relations between these opposite things are to be expressed in the simplest possible way, then there must be numbers which stand in this same relation of opposition to each other. 18 11-13] POSITIVE AND NEGATIVE NUMBERS 19 How to enlarge the number system — which now consists of integers and fractions (§ 11) — so that it will meet the require- ments just now pointed out, becomes evident if it be observed that all such cases of opposition as those mentioned on- the pre- ceding page, may be arrived at by subtracting a number from one that is less than itself. E.g., if a business man whose assets are ^5000 loses ^6000, i.e., if $6000 be subtracted from his $5000 of assets, it leaves him not only without any assets, but with $ 1000 of liabilities, i.e., he has $ 1000 less than nothing; if from latitude 40° north 50° be subtracted (counted off), the result is latitude 10° south; if the thermometer records 5° above zero and the temperature falls 8°, it will then record 3° below zero ; etc. Hence, if the number system be so enlarged as to make subtrac- tion always possible, even when the subtrahend is greater than the minuend, this enlarged system of numbers will provide for all such cases of opposition as those above mentioned. The nature of these new numbers will be more closely examined in the next article. Note. The considerations mentioned in §§ 11 and 12 demand, respectively, that the natural number system be extended so as to make division and subtrac-' tion always possible, i.e., so as to give a meaning to the expressions a -r- 6 and a—b, whatever the relative values of a and 6. There are, however, other important considerations which lead to the same conclusions; e.f/., algebra makes extensive use of letters to represent numbers, and it often happens, as in the problems of § 0, that the number represented by a given letter may be unknown until after the problem is solved ; if then the num- ber system consists of integers only, and if a and b represent two numbers whose values are not yet known, then, should the combination a -r- & present itself in a problem, one would not know whether or not it could be treated as a number (because it would be a number of the given system only if a happened to be a multiple of 6), and further progress with the problem must necessarily cease. A much wiser plan is, of course, to extend the number system so as to make a -4- 6 represent a number, whatever the relative values of a and 6 {i.e., to include frac- tions in the number system) ; then the solution may be continued and the proper interpretation given at the end. A similar argument applies to such an expression as a — 6. 13. Negative numbers introduced. The natural numbers arranged in a series increasing by one from left to right, and therefore decreasing by one from right to left, are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...; 20 ELEMENTARY ALGEBRA [Ch. II addition is performed by counting toward the right (cf. § 3), and subtraction by counting toward the left, in this series. More- over, addition is always possible because this series extends with- out end toward the right, and subtraction is arithmetically possible only when the subtrahend is not greater than the minuend because this series is limited at the left. What has just been said shows that to make subtraction with natural numbers always possible, it is only necessary to add to the present number system such numbers as will continue the above series indefinitely toward the left. Let the result of subtracting 1 from 1 be designated by ; of subtracting 1 from 0, by ~1 ; of subtracting 1 from "1, by ~2 ; of subtracting 1 from ~2, by ~3, etc. ; with these new numbers in- cluded, and arranged as before, the series becomes -., -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, •.., which extends without end toward the left as well as toward the right. Since in this enlarged series each number is less by one than the next number at its right (and therefore greater by one than the next number at its left), therefore addition and subtraction with natural numbers may, as before, be performed by counting toward the right and left respectively. E.g., to subtract 8 from 5, i.e., to find the number which is 8 less than 5, we begin at 5 and count 8 toward the left, arriving at -3; hence, 5 — 8 = -3. Similarly, 4 — 6 = -2, 4— 9 = -5, — 2 — 3 = -5, etc. ; hence, besides indicating a particular place in the enlarged number series, -5 also indicates that the subtra- hend is 5 greater than the minuend.* Similarly in general. Again, to add 7 to -4, i.e., to find the number which is 7 greater than —4, we begin at -4 and count 7 toward the right, arriving at 3. Similarly in general. 14. Negative numbers defined. Numbers less than are called negative numbers, and are written thus: ~1, ~2, ~3, ••• ; while num- bers greater than are, for distinction, called positive numbers, * Such an expression as 4 — 9 = -5 is, of course, not to be understood to mean that 9 actual units of any kind can be subtracted from 4 such units ; 4 of the 9 units may be immediately subtracted, leaving the other 5 units to be subtracted later if there is anything from which to subtract ; in this sense the number -5 may be said to indicate a postponed subtraction, and thus to have a suhtractive quality ; hence the appropriateness of attaching the minus sign to such numbers. 13-15] POSITIVE AND NEGATIVE NUMBERS 21 and are written either ''"1, +2, "'"3, •••, or, when there is no danger of confusion, simply 1, 2, 3, •••. Positive and negative numbers taken together are sometimes called algebraic numbers, while positive numbers alone are called arithmetical numbers. The signs ^ and ~ employed in the alge- braic numbers above are called signs of quality, or simply the signs, of these numbers. Two algebraic numbers, one of which is positive and the other negative, are said to be of opposite quality, or to have unlike signs, while if both numbers are positive, or both negative, they are of the same quality, i.e., they have like signs. A number written without a sign is understood to be positive ; the negative sign, however, is never omitted. The numbers ~1, ~2, ~3, •••, are read: negative one, negative tivo, negative three, etc., and also minus one, minus two, etc. ; and the numbers +1, +2, +3, •••, i.e., 1, 2, 3, •••, are read: positive one, posi- tive two, etc., also plus one, plus two, plus three, etc., or simply one, two, three, etc. By the absolute value of a number is meant its mere magnitude irrespective of its quality ; thus, ~2 and +2 have the same abso- lute value, so too in general have ~a and """a, whatever the number represented by a. Two numbers which have the same absolute value, but which are of opposite quality, are called opposite numbers ; thus, 5 and ~5 are opposite numbers, so too are "^a and ~a, whatever the number represented by a. 15. Interpretation of negative numbers. The interpretation of a negative number depends upon the nature of the problem which gives rise to it. E.g., a lady with S15 in her purse goes shopping and makes purchases amounting to $12 ; how much money has she left? Here the answer is clearly 15 — 12 dollars, that is, 3 dollars. Had the pur- chases amounted to $ 19, the answer would have been 15 — 19 dollars, that is, -4 dollars ; and the -4 dollars would mean that she not only had no money left, but that she was 4 dollars in debt. In this case then, when possessions are under consideration, the negative num- ber means indebtedness. The student should now re-read § 12 ; he should also show that if in a certain problem temperature above zero is under considera- 22 ELEMENTARY ALGEBRA [Ch. II tion, then a negative number means temperature below zero ; simi- larly, if positive numbers are used to represent degrees of north latitude, then negative numbers will mean degrees of south lati- tude, etc. ; in other words, negative numbers must in all cases be interpreted as representing things opposite in character to those dealt with in the problem. EXERCISES [The following questions should be supplemented by others asked by the teacher.] 1. If temperature above zero be regarded as positive, interpret the following temperature record taken from a U. S. Weather Bureau report : Albany, +8"^; Bismarck (S.D.), -11°; Buffalo, -2'; Chattanooga, +26^; Denver, "5° ; Galveston, +34'' ; Marquette, "9° ; Oswego, +1''. 2. How much warmer is it at Albany than at Bismarck in the above record? at Buffalo than at Denver? at Buffalo than at Chattanooga? 3. Answer the questions in Ex. 2 if the word "colder" be put in place of " warmer." 4. The value of all the available property of a merchant is a dollars, and his total indebtedness is b dollars, hence the value of his estate is (a — b) dollars. In such a case is it possible that h is greater than a? If so, what kind of a number is a — 6? In this case how should this negative number be interpreted? Can one actually pay out more money than he has? 5. If assets are represented by positive numbers, how may indebted- ness be represented? Interpret the financial conditions represented by the following numbers: $+783; $"2568; $'374.20; and 1.(856 - 1232). Also interpret these conditions if indebtedness be represented by posi- tive numbers. 6. A boy who weighs 54 lb. is playing with a toy balloon which pulls upward with a force of 6 lb. ; if the boy were weighed while holding the balloon, what would be the combined weight? If +54 lb. represents the weight of the boy, what w^ould represent the tceight of the balloon ? 7. In Ex. 6 the combined weight of the boy and the balloon may be represented as (+54 -f "6) lb., hence adding the ne,2:ative number cancels part of the positive number ; is this true in general for additions of posi- tive and negative rmmbers? Illustrate your answer. 15-16] POSITIVE AND NEGATIVE NUMBERS 23 8. If distances upstream on a river be indicated by positive numbers, what would "5 miles along this stream mean V Indicate by a number and sign the distance and direction that a boat would Jioat on this stream, in 1| hours, if ttie river flows 2-J- miles an hour. 9. An oarsman who can row 4 miles an hour in still water is rowing upstream on the river in Ex. 8 ; show tliat the distance he will go in one hour is (4 4- "2^) miles. Here too adding a negative number to a posi- tive number cancels it in part. How far upstream can he row in 7 hours ? 10. An ocean steamer is in 12° east longitude; if east longitude be indicated by positive numbers, and if the vessel moves westward through 7° of longitude per day, indicate by a number and sign the longitude of the vessel 4 days hence; 1| days hence; 2 days ago. 11. If the vessel in Ex. 10 sails westward for 2 days and then, being disabled, drifts 1^° eastward, what is its longitude? 12. What is meant by the absolute value of a number ? Which is the greater, 8 or -12 ? Why?* Which of these numbers has the greater ab- solute value? 16. Addition of negative numbers. In order to understand just what is meant by adding a negative number to any given number, one has only to recall the essential meaning of a negative num- ber. The symbol ~3, for example, means (and may always be replaced by) a subtraction in which the subtrahend exceeds the minuend by 3 units, i.e., it is equivalent to an unperformed (post- poned) subtraction of 3 units. t Hence, to add ~3 to any number whatever means to subtract +3 from that number. E.g., 8+-3 = 8 — 3 = 5; 4 + -10 = 4- 10 =-6; -9 + -5 = -9-5 = -14; etc. Manifestly the above reasoning applies to any negative num- bers whatever, hence the sum of two or more negative num- bers is a negative number whose absolute value is the sum of the absolute values of the given numbers ; And the suin of a negative and a positive number is a number whose absolute value is the difference of the abso- lute values of the two given numbers, and whose sign is that of the larger of these numbers. * Compare § 117. t Compare footnote, p. 20. 24 ELEMENTARY ALGEBRA [Ch. II EXERCISES Find the value of each of the following expressions : 1. 13 + -4. 3. -6 + 10 +-7. 5. -6t + 10+-ll|. , y 2. -8 +-3. 4. 3^ + -9i+-5|. 6. "2 + -13 + 8 + -4 + 6. 7. Regarding a negative number as a postponed subtraction, show that the result in Ex. 6, and in all others like it, might be found by adding the positive numbers separately, and the negative numbers sepa- rately, and then uniting these two sums. 8. If money in hand, or to be received, is represented by a positive number, then how should money owed (a postponed subtraction), or to be paid out, be represented ? Indicate by a sum of positive and negative numbers that a man had $20 and received f 15 more, and that he paid out for various things $8, $3, and $7.50; also show in two ways that he then had $16.50 left. 9. If distances westward from a certain point be indicated by posi- tive numbers, how should distances to the eastward be indicated? A wheelman after riding 37 miles westward from a certain point rides back 12 miles; show that 37 + "12 miles indicates both his direction and distance from the starting point. 10. Indicate by a sum of positive and negative numbers what tempera- ture is now registered by a thermometer which stood at 4° above zero, then rose 2°, later fell 9°, and then rose 2i° (cf. Ex. 9). 11. Make up exercises similar to 8, 9, and 10 to illustrate exercises 1-6 ; observe, however, that the demonstration given in § 16 relies wholly upon the definition of a negative number, and is in no way dependent upon any illustration. 12. From the reasoning in § 16 it follows that in adding a positive and a negative number, negative units and positive units cancel each other ; show that this is true in aU the illustrations above. 17. Subtraction of negative numbers. Since subtraction is the inverse of addition, i.e., since to subtract any number, a, from another number, 6, means to find the number to which a must be added to produce h,* therefore the results of § 16 may be used to show how to subtract negative numbers. * Definition of subtraction, § 3 (iii). 16-17] POSITIVE AND NEGATIVE NUMBERS 25 Thus, to subtract ~3 from 8 means to find the number to which -3 must be added to produce 8, and by § 16 this number is 11, hence 8 -"3 = 11; but 8 + 3 = 11, 8--3 = 8 + 3. Similarly, 15 - "2 = 15 + 2; 4 - "9 = 4 + 9; "8 -"3 ="8 + 3; and, in general, +a—~h= +a ++6, and -a—~h — ~a ++&, whatever the numbers represented by a and h ; i.e., subtracting a negative number from any given number {positive or negative) gives tl%e same result as adding a positive num- ber of the same absolute value to the given number. Note. If three or more algebraic numbers are to be combined by addition and subtraction, the order in which these operations are to be performed, when there is no express indication to the contrary (parenthesis, bracket, etc.). is understood to be from left to right as in § 8. E.g., +11 -+4 +-2 =+7 +-2 =+5. Moreover, since the subtraction of an algebraic number is equivalent to the addition of its opposite, such an expression as +11— +4 +-2 (above) is usually spoken of as an algebraic sum. EXERCISES 1. To what number must "5 be added to produce 12 ? What then is the value of 12 — -5 ? Answer these questions if 12 is replaced by 3 ; by -2 ; by a; ; by 4 + n. Find the value of each of the following expressions : 2. 9 --6. 3. -4 --12. ■ 4. 26§ - -41- 5. A " rule " for subtracting one number from another is often stated thus : " reverse the sign of the subtrahend and proceed as in addition." By means of § 17 establish the correctness of this rule when the subtra- hend is a negative number. 6. Using positive numbers to represent money in hand or receivable, illustrate the fact that subtracting a negative number from a positive number increases that number. Does subtracting a negative number always enlarge the minuend? Is it so in -7 —"3? 7. In the extended number series of § 13, viz., •••, "3, -2, -1, 0, 1, 2, 3, 4, •••, how by counting may we add 5 to 3? to "2? to -8? Do we count forward or backward when adding a positive integer? Since sub- traction is the inverse of addition, which way should we count when subtracting a positive integer? State and explain the corresponding facts for adding and subtracting negative integers. 26 ELEMENTARY ALGEBRA [Ch. II Simplify each of the following expressions, that is, find the value of each of these algebraic sums : 8. 137 +-86 --7 +-26 -8. 10. 4p --54^ +-38| - 28. 9. -54 +-864 + 732 -"413 - 36. 11. 18 --4' - 13^ +"6 --17^. 12. Mount Washington is 6290 feet above the sea level. Pikes Peak is 14,083 feet above the sea level, and a place near Haarlem, in Holland, is 16^ feet below the sea level. Find by subtraction how much higher Pikes Peak is than Mount Washington ; and also how much higher Mount Washington is than the place near Haariem. 13. An engineer when making measurements for the grade of a street indicates the distances of points above a certain horizontal reference plane by positive numbers, and those that are below this plane by negative numbers. Show that the difference of level between any two points may always be found by subtraction. Also draw figures to illustrate several different cases. 18. Product of two algebraic numbers. Rule of signs. The prod- uct of any two algebraic numbers is readily obtained from the definition of a product, which is given in § 3 (iii), viz., the product of any two numbers is the result obtained by performing upon the multiplicand the same operation that must be performed upon the positive unit to get the multiplier. E.g., since 3 = 1 + 1 + 1, therefore 8- 3 = 8 + 8+8 = 24;' and -8.3=-8+-8+-8=-24. Again, to get "3 from 1, this positive unit must be increased 3-fold and then have its quality sign reversed; 'therefore, to multiply any number by ~3, first increase that number 3-fold and then reverse the quality sign. E.g., since -3 ="(1 + 1 + 1), therefore 8 • "3 = "(8 + 8 + 8) = ^24 ; similarly, "8 • "3 means that ~8 is to be increased 3-fold and then have its quality sign reversed, but ~8 increased 3-fold is ~24, therefore -g . -3 _+24 From what has just been said, "8 • 3 =-(8 • 3), 8 • "3 ="(8 • 3), and ~8- -3=+(8-3); by the same reasoning as that employed -18] POSITIVE AND NEGATIVE NUMBERS 27 in these particular cases, it follows that, whatever the numbers represented by a and b, +a'+b = +(a . b), -a-'^b = ~(a • b), +a '-b = -(a -b), and ~a . "6 = +(a • b). These results may be formulated in words thus : the absolute value of the product of any two numbers is equal to the product of their absolute values, and this product is posi- tive if the factoids have like quality signs, otherwise it is negative. Note 1. Since a succession of multiplications* is to be performed by first getting the product of the first two numbers, then multiplying this product by the next number, and so on (cf. § 8), tlierefore, by the successive application of the principle established for the product of two numbers, it follows that the abso- lute value of a continued product is the product of the absolute values of the factors, and this product is negative if it contains an odd number of negative factors, otherwise it is positive. E.fj., 5 • -3 • -2 . 7 = -15 • -2 • 7 = 30 . 7 = 210= +(5 .3.2. 7). Note 2. From Note 1 it follows that odd powers {i.e., powers whose expo- nents are odd numbers) of negative numbers are negative, while even powers of negative numbers are positive, and all powers of positive numbers are positive. E.g., (-2)2 = +4, (-2)3 = -8, (-2)* = +16, etc. EXERCISES Find the value of each of the following indicated products : 1. 5-3. 5. -7f--6. 9. -2c. 3c. 2. -6 . 4. 6. -m ' -5. 10. "3 • 4 • -6 • 2. 3. -7.-2. 7. -4 a- 3. 11. 3.-A;.-x-4a. 4. 12 • 9. 8. -12 . -3 X. 12. (-3)2 .5.-2. 13. In the above products, how does the absolute value of the product compare with the product of the absolute values of the factors? What is meant by the absolute value of a number? * A succession of multiplications such as 3 • 5 • 9 • 4 ••• is often called a con- tinued product. 28 ELEMENTARY ALGEBRA [Ch. II 14. If two numbers have like signs (both plus, or both minus), what is the sign of their product? If they have unlike signs, what is the sign of their product ? 15. In the continued product of Ex. 10 above, what is the sign of the product of the first two factors? of this product multiplied by the next factor ? of this product by the next factor ? 16. Can the sign of a continued product be ascertained without actu- ally performing the multiplication? How? What is the sign of the result in Ex. 10 above? in Ex. 11? in Ex. 12? If a continued product has five negative factors, what is the sign of the result? 17. Define the product of two numbers, and on the basis of your definition prove that the sign of the product -4-7 is negative. Also that the sign of the product -4 • "7 is positive. 18. How is -5 obtained from the positive unit? How then is the product 8 • -5 obtained ? the product "8 • -5 ? Show that -2.-2.-2- -2, i.e., (-2)4, is 16; also that (-2)5 = -32. What is the sign of ("6)8? of (-2)4. (-3)2? of (-1)10? 19. Define a continued product, and state the order in which its multiplications are to be performed. What is an odd power of a number (cf . § 7) ? an even power ? Find the value of (a + b) - (x — y) : 20. When a = 2, & = -3, x = -4, and y = 6. 21. When a = |, b = -2a, x = -6, and y = -10. 22. When a = -4, & = 6, x = ah, and y = "12. 23. When a = -4, b = a'^, x= Sa, and y = 2a\ 19. Division of algebraic numbers. Since division is the inverse of multiplication [cf. § 3 (iv)], therefore the results of § 18 may be used to show how to divide algebraic numbers. For example, to divide +24 by ~3 means to find the number which being multiplied by ~3 will produce +24 ; but, by § 18, this number is "8 ; hence +24 -f- -3 = -8. And, in general, whatever the numbers represented by a and h, +(a .6)- - +6 = +a. +(« .6)- --h = ~a, -(« .6)- -+h = -a, and -(a .6)- --h = +a. 18-20] POSITIVE AND NEGATIVE NUMBERS 29 These results may be formulated in words thus : the absolute value of the quotient of two numhers is the quotient of their absolute values, and this quotient is positive if the dividend and divisor have like signs, otherwise it is negative. EXERCISES Find the value of each of the following indicated quotients : 1. 14 -r- 2. 4. -31 --If. 7. 15-i--l. 2. 14 H- -2. 5. -24 - 9. 8. -365 - -9^. 3. -18 -=-4^. 6. (-6)2 -(-2)3. 9. "63 a^ - -7. 10. Of what operation is division the inverse? What is an inverse operation ? In an exercise in division, what is it that corresponds to the product in multiplication ? How may the correctness of an exercise in division be tested ? 11. If the dividend is positive, and the divisor negative, what is the sign of the quotient? If the dividend is positive, how do the signs of divisor and quotient compare ? if the dividend is negative ? Find the value of each of the following expressions : 12. 24 - 28 ^ -7 + -16 -^ -4 --3. 13. -8 • -6 ->■ 24 - 27 -^ -6 ^ 3. 14. {28 -f- -7 - 2 . (-4 - 2) + 24} -j- (-2)3. Verify that «±^ . ^Ln^ ^ ^^ : x-\-y x-y x^-y^ 15. When a = 6, 6 = 2, x = 10, and ?/ = 6. 16. When a = -8, 6 = 12, a; = -9, and y = 'l. 20. Small quality signs (+ and -) dispensed with. To distinguish sharply between the positive and the negative quality of numbers, and at the same time to avoid confusing signs of quality with the signs of the operations of addition and subtraction, the small plus and minus signs (+ and ~) have thus far been employed. In order to simplify this notation, which is manifestly some- what cumbersome, the larger plus and minus signs (+ and — ) may in future be employed to indicate both the quality of numbers, and also the operations of addition and subtraction. A number without a quality sign attached to it will continue to mean a 30 ELEMENTARY ALGEBRA [Ch. II positive number, while a negative number will be indicated by writing the minus sign before the numeral, and inclosing both the numeral and its sign in a parenthesis when the parenthesis is necessary to avoid ambiguity : the quality sign — is never omitted. With this simpler notation : 5 means the same as +5 ; a the same as +a ; — 8, or (—8), the same as -8; 9 — 5— (—3)* the same as +9 — +5 — -3, etc. In general it may be said that the sign prefixed to a number indicates an opera- tion unless that number stands alone, or stands first among several which are to be united, or is inclosed, together with its sign, in a parenthesis. EXERCISES 1. In the expression +5 + '*"3 — +4, which are signs of quality and which are signs of operation? . 2. Rewrite the expression in Ex. 1, omitting the quality signs. Has this change in the writing really made any change in the quality of the numbers ? 3. Answer questions 1 and 2 with regard to the expression +5 — +3 + +4. 4. Could all the quality signs in the expression +15 — +3 + -8 be omitted without changing the meaning of the expression? Which of these signs might be omitted? When no quality sign is written, what is the quality of the number? 5. If the expression in Ex. 4 be written so as to use only the larger signs, is a parenthesis necessary to preserve the meaning? Write the expression so. Also answer the same questions with regard to the expression a; — "5 + ~8. 6. Show that the expression a: — "5 + -8 is equal to ar + 5 — 8, wherein both 5 and 8 are positive numbers, and the signs + and — indicate operations. 21. Algebraic expressions. Terms. In the course of operations with algebraic numbers, it often happens that the expression for a number does not consist of a single symbol, but rather of a combination of such symbols. E.g., if a and h represent numbers, then ab, a + b, and a^ — 3 aft^ also represent numbers. * By §§ 16 and 17 this expression equals 9 — 5 + 3, which is 7. In this connec- tion attention may also be called to the fact that since a-\- ( — b) = a — b (§16), therefore such an expression as a — 6 may be understood as meaning either that b is subtracted from a, or that — 6 is added to a. 20-22] POSITIVE AND NEGATIVE NUMBERS 31 Such expressions for numbers as a + b, 3xy, m' + 27i'-5x, dax" + — -10^^ +Saxy', etc., are called algebraic expressions.* The parts of an algebraic expression which are connected by the signs + and — (or, rather, these parts together with the signs preceding them) are called the terms of the expression. Terms preceded by the plus sign are called positive terms, while those preceded by the minus sign are called negative terms. E.g., in the expression 5cfi-\-Zh — lQc^x'^, there are three terms, viz.: 5a2, + 36, and — 10 c^x^ ; the first two are positive, and the third is negative. EXERCISES 1. How many terms are there in the expression 5 a% + 2 axif - 7 mx^ - 26 ? What are they ? Which are positive V Which negative ? 2. Answer the same questions as in Ex. 1 with regard to the expression -12 + 7 rrfix^ - 5 a?/^ - 3 a;2 _ § ahiA 3. The sum of two times a number and three times the same num- ber is how many times that number? Unite the two terms 3a: + 5a: into one. What single term is equal to \ x — \ x'l Is5a:+13a: — 9x equal to (5 + 13 - 9) a: ? Why ? 22. Recapitulation. In this chapter it has been shown that, in order to express in a simple way the relations between assets and liabilities, latitude north and latitude south of the equator, tem- perature above zero and temperature below zero, in fact, between any of the things which bear a relation of opposition to each other, and which are everywhere met with in one's daily intercourse, it is advantageous to extend the number system so as to make subtraction always possible. Further considerations have shown that the numbers needed to make subtraction always possible are the so-called negative num- bers, and in §§ 15-19 it has been shown how to interpret these numbers, and also how to operate with and upon them. A rapid re-reading of these paragraphs is recommended. * An algebraic expression is spoken of as an expression, or as a wwrrtfter accord- ing as the thought is of the combined symbol, or of the numerical value which that symbol represents. CHAPTER III THE EQUATION 23. Definitions. Although a discussion of the fundamental principles relating to equations must be postponed until more of the theory connected with algebraic expressions has been developed (see Chapter X), yet the importance of the equation as an instrument of investigation demands that it be presented as early as possible. An equation has already been defined [§ 3 (i)] as a statement that each of two expressions has the same value as the other, i.e., it is a statement that each of these expressions represents the same number. These two expressions are called the members of the equation, and that expression which is written at the left of the sign of equality is known as the first member, while the other is known as the second member. E.g., 8 « — 21 = 3 a; + 4 is an equation of which 8 a; — 21is the first member, and 3 a; + 4, the second member. Manifestly the two members of the equation just written do not represent equal numbers for all values that may be assigned to the unknown number represented by x : indeed there is only one value of X for which they are equal ; viz., for a? = 5. Hence such an equation is called a conditional equation; it is an equation only on condition that a; = 5. An equation which is true for all values that may be assigned to its letters is called an identical equation or, more briefly, an identity. To indicate that an equation is an identity, rather than a condi- tional equation, the sign = may be used instead of = to connect the two members. E.g., 3a; + 5 — x = 2x4-7 — 2 and ax"^ + & — ax"^ = b are identities. Many other examples of identities will present themselves in the following pages. 82 23-24] THE EQUATION 33 The process of deducing from any conditional equation the values that must be substituted for the unknown number to make the two members equal, is called solving the equation, and these values themselves are called the solutions or roots of the equation. Note. The final test as to whether a number is or is not a root of a given equation is to substitute that number for the letter representing the unknown number in the equation ; if this substitution satisfies the equation, i.e., if it makes the two members reduce to the same number, then it is a root, otherwise it is not. E.g., 5 is a root of the equation 8a; — 21 = 3x+4, because substituting 5 for x satisfies this equation. 24. Some axioms and their use. The following principles, usually called axioms, are useful in solving equations. (1) If equals he added to or subtracted from equals, the results will he equal.* (2) If equals he multiplied or divided hy equals, the results will he equal A The application of these axioms to the solution of equations is illustrated by the following examples : X Ex. 1. If 8 a; — 21 = 3 X + 4, find the value of x ; i.e., solve this equation. Solution Since 8a:-21 = 3a: + 4, therefore 8 x - 21 + 21 = 3 a; + 4 + 21, [Axiom (1) i.e., 8 a: = 3 a: + 25, and therefore 8a; — 3a; = 3a: + 25 — 3x, [Axiom (1) i.e., 5 a: = 25, whence a: = 5. [Axiom (2) Verification. Substituting 5 for x in the original equation, each member reduces to 19 ; that is, the substitution of 5 for x satisfies this equation, and 5 is therefore a root of it. * Equal numbers are really the same number ; such numbers may, of course, be expressed in different ways (e.g., 19 + 5, 3 • 8, and 5-5 — 1 each express 24), but they are, nevertheless, the same number, and the self-evidence of these axioms rests upon that fact. t It is not permissible, however, to divide by zero. J See footnote, p. 6. 34 ELEMENTARY ALGEBRA [Ch. Ill Ex. 2. Solve the equation ^ x + 12 -\- 7 x = \x - l()\ - 4:X. Solution Since |a:+12 + 7x = |x-10i-4a:, therefore, multiplying each member by 6, , 4 a; + 72 + 42 a: = 3 a; - 62 - 24 X, |; Axiom (2) i.e., 46 a: + 72 = - 21 a; - 62, and therefore, subtracting 72 from each member, 46 a; = - 21 x - 62 - 72 [Axiom (1) zz - 21 a; - 134, and, adding 21 x to each member, 67 a; = - 21 a: - 134 + 21 a: = - 134, whence a: = — 2. [Axiom (2) Verification. Since the substitution of — 2 for x satisfies the origi- nal equation, therefore — 2 is a root of that equation. EXERCISES 3. Define an equation. Also distinguish between a conditional equation and an identity. Give an illustrative example of each of these two kinds of equations. Is 2 ax + 3a = a(4ar + 3)— 2 aa; a con- ditional equation or an identity? 4. What are the members of an equation? Which is called the first member? What is the other member called? What is meant by a root of an equation? Illustrate your answers by suitable examples. 5. What is meant by solving an equation? Describe briefly the process of solving an equation. State the axioms which have thus far been employed in solving equations. Illustrate your answers by suitable examples. 6. How may the correctness of a solution (root) be verified ? Show that 4 is a root of 7 a: - 10 = 4 x + 2. Is 2 a root of a:2-5a: + 6 = 0? Is 3 also a root of this last equation? Solve the following equations, give the reasons for each step of the work, and test the correctness of the roots : 7. 3 a; -F 2 = a; + 30. 9. 2 a: + - = — • 3 6 8. 7 a: - 55 = 18 - 2 a: - 1. 10. 5 a: - 3j a: = 17 - a:. 24-25] THE EQUATION 35 11. If the second member of an equation be multiplied by any num- ber, say 4, what must be done to the first member in order to preserve the equality? If any given number be added to either member, what must be done to the other member? Why? 12. If 2 a be subtracted from each member of the equation 5 a: + 2 a = 3 a; + 4 &, what is the resulting equation ? What does this show with reference to removing a term from the first to the second member of an equation? Is the same thing true when a term is removed from the second member to the first? Show this by adding -3 x to each member of the given equation. 25. Transposition ; directions for solving equations. Eemoving a term from one member of an equation to the other is spoken of as transposing that term. It has doubtless been observed, in the solutions of the equations of § 24, that a term may be trans- posed froin one member of an equation to the other by merely reversing its sign. This fact may be formally proved as follows : let any term of either member {e.g., the first) of any given equation be represented by k, — this term may be positive or negative, and may contain any number of letters, — and let the remain- ing terms of the first member of this equation be represented hy M, and its second member by N ; then the equation is M-\-k = N. Subtracting k from each member of this equation, it becomes, by axiom (1), M=N-k, i.e., the term k has disappeared from the first member of the given equation, but has reappeared, with its sign reversed, in the second member. The following simple directions may now be given for solving such equations as those considered in § 24. (1) If the equation contains fractions, multiply both of its mcTnbers by the legist common multiple of the denomina- tors of these fractions (axiom 2); this is usually spoken of as clearing the equation of fractions. (2) Transpose all the terms containing the unknown number to the first member of the equation, and all other terms to the second member. (3) Unite the terms of each member, and then divide both members by the coefficient '^ of the unknown number. * The coefficient of the unknown number is the factor which multiplies it. 36 ELEMENTARY ALGEBRA [Ch. Ill (4) Substitute the value thus found for the unhnown number in the given equation; if this satisfies the equa- tion, then it is a root of the equation, otherwise it is not. These directions may be illustrated by solving again Ex. 2 of § 24, thus : Given lx-^12-^1 x=lx-l0}-4:x\^ multiplying the given equation by 6 to clear it of fractions, it becomes 4 a: + 72 + 42 a: = 3 a; - 62 - 24 X, [Axiom (2) whence, transposing, 4 x+42 x—^x + 24 x = — 62 — 72, i.e., 67 a; = — 134 ; [Uniting terms therefore, dividing by 67, x = — 2 ; and this value of x proves, on substitution, to be a root of the given equation. EXERCISES Solve the following conditional equations, and verify the results : 1. 12a:+5x + 20-8a: = 48 + 3a:-4. 5. §^ + 5 = 91 - lOar. 2. 3(x-5)*+4a; + 8 = 5(4ar-20). ^ 7^+2-^=17 3. 5(2a:-10)+7ar-15 = 20a:. „ o„ ^ 7. 8 + 2v + '^=l| + ^- 3 7 ' Q. ^k-lQ=.2k-\-ll. 9. Uk-20 + lk-2 = Qk + V- 10. 2t;+^-^ + 14 = 7.-^ + ^-i^. 2 4 4 7 2 26. Problems leading to equations. A problem is a question pro- posed for solution; it always asks to find one or more numbers which at the beginning are unknown, and it states certain relations (conditions) between these numbers, by means of which their values may be determined. The process of solving problems has already been illustrated in § 6, — which should now be re-read. The important steps are : (1) Represent one of the unhnown numbers involved in the problein by some letter, as x. (2) From the verbal conditions of the problem find alge- braic expressions for the other unhnown numbers, and form two such expressions that are equal to each otlwr. * That 3(a; — 5) = 3 a; — 15 may for the present be assumed ; it is proved in § 39. 25-26] PROBLEMS 37 (3) With these two equal expressions, form an equation, ~ called the equation of the problem. (4) Solve this equation and verify the correctness of the result. These steps are illustrated in the solutions of the following problems : Prob. 1. The sum of the ages of a father and son is 54 years, and the father is 24 years older than the son. How old is each? Solution The conditions of this problem, stated in verbal language, are : (1) The number of years in the father's age plus the number of years in the son's age is 54. / (2) The number of years in the son's age plus 24 equals the number of years in the father's age. To translate these conditions into symbolic language, let x represent the number of years in the son's age,* then by the second condition the number of years in the father's age is a: + 24, and by the first condition a; + 24 + a; = 54, which is the equation of the problem. From this equation it is found that x = 15, which is the number of years in the son's age, and a: + 24 = 39, the number of years in the father's age. By substituting these numbers it is found that they satisfy the two given conditions of the problem and are, therefore, its solution. Note. It maybe worth remarking that it was not necessary, but only con- venient, to let z stand for the number of years in the son's age. Thus, if X represents the number of years in the father's instead of in the son's age, then the given conditions translated into algebraic language become : (1) 54 — cc = the number of years in the son's age, and (2) 54 — a; + 24 = X', which is the equation of the problem. From this equation it is found that x = 39, whence 54 — x = 15 ; these are the same numbers as obtained before. Again, if 3x were chosen to represent the number of years in the son's age, then the equation of the problem would be 3 X + 24 -h 3 X = 54, whence x = 5 and 3 x = 15, the son's age, and 3 x + 24 = 39, the father's age. * It is to be carefully noted that x represents a number ; it does not represent the son's age, but represents the number of years in the son's age. 38 ELEMENTARY ALGEBRA [Ch. Ill Prob. 2. A boy was given 39 cents with which to purchase 3-ceiit and 5-cent postage stamps, and was told to purchase 5 ixiore of the former than of the latter. How many of each kind should he purchase ? Solution The conditions of this problem, stated in verbal language, are : (1) The total expenditure is 39 cents. (2) There are to be 5 more 3-cent stamps than 5-cent stamps. To translate these conditions into symbolic language, let x stand for the number of 5-cent stamps purchased ; their cost is then 5 x cents : then, by the second condition, the number of 3-cent stamps is x + 5, and their cost is (3a:-j-15) cents; hence, by the first condition, 5 a; -1- 3 a; 4- 15 = 39, which is the equatjon of this problem. The solution of this equation gives a; = 3, the number of 5-cent stamps, and X -{- D = S, the number of 3-cent stamps ; and it is easily verified by substitution that these two numbers do, in fact, satisfy both the condi- tions of the problem ; hence they are the numbers sought. Prob. 3. If a certain number be diminished by 6, and 2 times this difference be added to 5 times the number, the result will equal 88 minus 3 times the number. What is the number ? Solution To form the equation of this problem, let x represent the given number ; then 5 times the number is 5 x, the number diminished by 6 is a: — 6, etc., and the given condition becomes 5 ar -}- 2(a: - 6) = 88 - 3 a:, whence 5 a; -|- 2 a; - 12 = 88 - 3 a:, and, transposing, 5a: + 2a:-F3a:=88 4-12, » i.e., 10 a: = 100, and, therefore, x = 10, which, on verification, proves to be the required number. Prob. 4. A number consists of two digits whose sum is 5 ; if the digits be interchanged, the number will be diminished by 9. What is the number ? Solution To form the equation of this problem, let x represent the digit in units' place ; then, by the first condition, 5 — a; will represent the digit in 26] PROBLEMS 39 tens' place ; therefore, the number is 10(5 — x) + x, — compare Ex. 6, § 5, — and the number formed by interchanging the digits is 10 a; + (5 — x). The second condition then gives 10 X + (5 - x) = 10(5 - x) + a: - 9, whence x = 2, the digit in units' place, and 5 — X = 3, the digit in tens' place. These two digits are found to satisfy both the conditions of the prob- lem, hence the number sought is 32. PROBLEMS 5. Divide 28 into two parts whose difference is 4. 6. The sum of two numbers is 63, and the larger exceeds the smaller by 17. What are the numbers ? 7. If I of a certain number exceeds I- of that number by 8, what is the number ? 8. Divide 48 into two parts such that twice the larger part equals 5 times the smaller part. 9. A man who is 32 years old has a son who is 8 years old ; how many years hence will the father be 3 times as old as his son ? 10. On being asked his age, a gentleman replies that liis age 5 years hence will be twice as great as it was 20 years ago ; how old is he? 11. How old is a person if 20 years hence his age will be less by 5 years than twice his present age ? 12. If 16 be added to a certain number, the result will be the same as it would be if 7 times the number were subtracted from 56 ; what is the number ? 13. If 6 times a certain number is as much less than 62 as 3 times this number exceeds 19, what is the number? 14. Of four given numbers each exceeds the next below it by 3, and the sum of these numbers is 58 ; find the numbers. 15. Mary is 25 years younger than her mother, but if she were one year older than she is she would be i as old as her mother ; what is the age of each ? 16. The sum of three numbers is 25; the first of these numbers is gi-eater by 5 than the third, but only -^ as great as the second ; find the numbers. 17. Divide f 2200 among A, B, and C in such a way that B shall have twice as much as A, and C $200 more than B. 40 ELEMENTARY ALGEBRA [Cii. Ill 18. Divide $351 among three persons in such a way that for every dime the first receives, the second shall receive 25 cents, and the third a dollar. 19. Three boys together have 140 marbles ; if the second has twice as many as the first, but only half as many as the third, how many marbles has each boy ? 20. After taking 3 times a certain number from 11 times that number, and then adding 12 to the remainder, the result is less than 117 by 7 times the number ; what is the number ? 21. A number consists of two digits whose sum is 8, and if 36 be sub- tracted from this number the order of its digits will be reversed ; what is the number? 22. In a certain two-digit number the tens' digit is twice the units' digit, and the number formed by interchanging the digits equals the given number diminished by 18 ; w^hat is the number ? 23. In a three-digit number the tens' digit exceeds the hundreds' digit by 3, the units' digit is 4 less than twice the hundreds' digit, and interchanging the units' and tens' digits decreases the number by 45 ; what is the number ? 24. A two-digit number is equal to 7 times the sum of its digits, and the tens' digit exceeds the units' digit by 3 ; what is the number? 25. A merchant owes A three times as much as he owes B, he owes C twice as much as he owes A, and he owes D as much as he owes A and B together ; if the sum of his indebtedness to A, B, C, and D is $28,000, how much does he owe each? 26. Two clerks, A and B, have the same salary ; A saves i of his, but B, by spending $150 more than A each year, saves only $350 in 7 years ; what is the salary of each? 27. A merchant bought some eggs at the rate of 2 for 3 cents, he then bought J as many more at the rate of 6 for 5 cents, and later sold them all at the rate of 3 for 4 cents, thereby losing 6 cents ; how many did he buy? 28. If I of a number is as much less than the number itself as | of the number is less than 65, what is the number ? 29. The sum of three consecutive integers is 51 ; what are these thiee numbers (cf. Ex. 8, § 5)? Show that the sum of any three consecutive integers is 3 times the second of these integers. 30. The sum of four consecutive odd integers is 80; what are these four numbers? Prove that the sum of any four odd integers is an even integer. 26] PROBLEMS 41 31. M can do a certain piece of work in 8 days, and N can do it in 12 days; iu how many days can both do it when working together [cf. §9(ii)]? 32. If M begins the work mentioned in Prob. 31, and, after working a certain number of days at it, turns it over to N to finish, and the entire piece of work is done in 10 days, how long did each work at it ? 33. A country club consisting of 200 members, having decided to build a new club house, assessed each of its members a certain sum for that purpose ; meanwhile the membership was increased by 50, and it was then found that the assessment could be reduced by $10; what was the cost of the proposed house? 34. A real estate dealer purchased three houses, paying 1| times as much for the second as for the first, and If times as much for the third as for the first ; if the difference between the cost of the second and third was 11500, what was the cost of each? 35. A gentleman left his property, valued at $800,000, to be divided among three colleges; if the first was to receive $30,000 more than the second, and the third half as much as the other two together, how much was each to receive? 36. Five boys had agreed to purchase a pleasure-boat, but one of them withdrew, and it was then found that each of the remaining boys had to pay $2 more than would have been necessary under the original plan; how much did the boat cost? 37. A lady having already spent $10 more than | of her money made further purchases amounting to $10 more than f of what then remained, and found that she had only $2 left; how much had she at first? 38. A laborer was engaged to do a certain piece of work on condition that he was to receive $2 for every day that he worked, and to forfeit 50 cents for every day that he was idle ; at the end of 18 days he received $28.50. How many days did he work? 39. A certain number being subtracted from 50, and also from 84, it is found that f of the first of these remainders exceeds | of the second by 47 ; what is the number ? CHAPTER IV ADDITION AND SUBTRACTION OF ALGEBRAIC EXPRES- SIONS — PARENTHESES I. ADDITION 27. Monomials, binomials, etc. ; coefficients. An algebraic expres- sion consisting of but one term* is called a monomial, while one consisting of two or more terms is called a polynomial. A poly- nomial consisting of only two terms is usually called a binomial, and one consisting of three terms, a trinomial ; but to polynomials consisting of more than three terms it is not customary to give special names corresponding to binomial and trinomial. E.g., 2ax^,—7m^p^, and Sbx^i/^ are monomials; x^ + Sy, 5 m — 2 z^, and — 3 a62 — f ^Sy4 are binomials ; 'and 2 a;8 + 4 ay — 5 62, 2 s* — 6 ?/ + 3 m^x^, and x + 'dt — j abx'^ are trinomials. If a term is composed of several factors, any one of its factors, or the product of two or more of them, is called the coefficient of the product of the remaining factors. E.g., in the term 5 axy^, the coefficient of axy^ is 5, the coefficient of xy^ is 5 a, the coefficient of 5 xy'^ is a, etc. A coefficient consisting of Arabic characters only is a numerical coefficient, while one that contains one or more literal factors is a literal coefficient. E.g., in the term ~3ax^y^, the numerical coefficient of ax^y* is — 3, but —3a and 3 ay^ are literal coefficients of x^y^ and — x^ respectively. Note. The word "coefficient" is usually understood to mean "numerical coefficient," and the sign (+ or — ) written before a term is usually regarded as belonging to the numerical coefficient. When no numerical coefficient is written, the term is understood to have the coefficient 1. * For the definition of an " algebraic expression," and of a *' term," see § 21. 42 27-28] ADDITION 43 28. Positive and negative terms ; like and unlike terms. A term whose sign is + is called a positive term, and one whose sign is — is called a negative term. If the first term of an algebraic expres- sion is positive, its sign is usually omitted, but the sign of a nega- tive term is never omitted. Note. As has already been pointed out, the letters in an algebraic expression may represent any numbers whatever, — they may be positive or negative, even or odd, integers or fractions, — and therefore an algebraic expression which is fractional in appearance may have an integral value, and vice versa ; so too a term which is positive in appearance may still, for certain values of the letters involved in it, have a negative value, and vice versa. Terms which either do not differ at all, or which differ only in their numerical coefficients, or in their quality signs, are called like terms, and also similar terms; terms which differ in other respects are called unlike terms, and also dissimilar terms. E.g., Zxhj, hx^ij, and —Ix^y are like terms, while 2 ax, —ob^x^y, and Sxy^ are unlike terms. Like terms must contain the same letters, and these letters must be affected with the same exponents, but they may differ in their signs and also in their coefficients. EXERCISES 1. What is the coefficient of a^x in each of the following expressions : 3a%, - 6a^x, a% 4 a^&x, - 4a%, ^" " "^ , and -9a^x? 1 m 2. Which of the above coefficients are literal and which numerical? Which of the terms in Ex. 1 are positive and which negative? 3. Do the positive terms in Ex. 1 necessarily represent positive num- bers for all values that may be assigned to the letters involved ? Try a = 3 and x ■=—2. 4. What is the coefficient of x—yin each of the following expressions : \^{x - y), — a(x — y), f m(x - y), and (4 - a^)(x — y) ? Which of these coefficients are numerical? Which literal? Which of these expres- sions are positive and which negative ? Try various values for the letters and see whether the negative expressions necessarily represent negative numbers. 5. Consult a good dictionary for the derivation of the words " mono- mial," "binomial," "trinomial," and "polynomial." Write three mono- mials, three binomials, three trinomials, and three polynomials. 6. Distinguish carefully between the meanings of 5 in the expressions 5 X and x^. What name is given to the 5 in each of these expressions? 44 ELEMENTARY ALGEBRA [Cn. IV 7. What are like terms? By what other name are they known? In what respects may they differ and still be like terms ? Are 3 x^y, —2x^i/, and I a;2^ similar? Are 4: ax^ and — Gbx^ similar? Are these last two terms similar if 4a and —Qb are regarded as their respective coefficients? 8. Write three sets of like terms, some terms being positive and some negative, and each set containing at least four terms. 29. Addition of monomials. That the sum of several similar monomials may be united into a single term has already been illustrated in some of the exercises and problems in the preceding pages ; this subject will now be considered in greater detail. Since 5 times any given number, plus 2 times that number, is 7 times the given number, i.e., (5 + 2) times the given number, therefore o a -^2 a = (o-i-2) a = 7 a, whatever the number repre- sented by a. So too 3 mxhj + 8 mx-y = (3 + 8) mx^y = 11 mx^y. Observe that this reasoning applies to any two similar monomials whatever. Since the sum of three or more numbers is obtained by adding the third to the sum of the first and second, the fourth to the sum of the first three, etc. , therefore, to add any nuinber of similar monomials, add their coefficients, and to this result annex the common literal factors. It is usually most convenient to write the terms to be added under one another, as in arithmetic, thus : 3x?/2 153a2ma;8 l%ak^8 8 xy^ 74 a2mx8 _ 7 ak'^s 11 xy"^ 227 a^mx^ 11 ak'^s*' If the monomials to be added are dissimilar, they cannot be united into a single term, but their sum may be indicated in the usual way ; e.g., the sum of 5 a and 2 car' is 5 a -f 2 ca^. EXERCISES 1. If 6 times any number whatever be added to 13 times that number, the result is how many times the given number ? 2. To 6 times any given number add 13 times that number, and to this sum add -8 times the given number ; what is the result ? * Since 18 + (- 7) = 11 ; compare § 16. 28-30] ADDITION 45 3. State in words a convenient rule for adding any number of like terms. Does your rule apply to cases in which some of these terms are negative ? 4. Find the sum of 6 n, 7 n, — 3 n, 18 n, and — 11 n. 5. Find the sum of 4 a^x^, 5 a^x% - 2 a^x% and - 6 a^x^. Simplify the following expressions, i.e., unite similar terms : 6. 3 mxy^ + ( - 4 mxy^) + ( - 12 mxy^) + 5 mxy^ 7. 14 ahx^ + 32 abx^ + ( - 19 abx^) + 5 abx^. 8. 3 mp^ + 7 mp^ + 13 a^x - inip'^ + (- 6 a%) - 2 a%. 9. 4:(a - b) + '6 (a - b) - 2{a - b) + (a - b). 10. 4 (aa:)^ + 11 (axy - 3 (axy + [- 6 («x)2]. 11. 7 (:r + 2/ + 2) + 19 (x + y + z) + 4 (a; + ^ + 2:) - 8 (a: + r/ + 2). 12. - 15 (ax^ + 3) + 27 ^ax^ + 3) - 9 (ax^ + 3). Add the following terms, uniting as far as possible, and indicating the addition where necessary : 13. Smp% - 8 mp% 5 a% - 4 mp% - Sa% and 2 a^x. 14. 23 a\ 5 &2, _ 8 a^b% - 13 &2, 24 02^2, and - 19 a^. 15. - 5 (a - &), 2 (ax) 2, - 8 (ax)^, 12 (a - J), and - 4 (aa:)2. 16. 16 X, — y, 4:x, — X, 4 z, 5 ?/, x, 2 x, and —32. 17. 7nxy + nxz/ equals how many times xy ? 18. ax2 + &x2 — cx2 — Zx2 is how many times x2 ? 30. Addition of polynomials. The explanation given in § 29 for the addition of monomials is easily extended so as to apply to the addition of polynomials also. E.g., 7 62^3 _ 3 ax2 + 6 abc and 4 bhj^ + 5 aa-2 — 12 ahc may be added thus • 7 62?/3_3ax2+ 6 abc 4 b'^ij^ + 5 0x2 _ 12 abc 1162«/8 + 2ax2— 6a6c Similarly in general, hence: To add two or more polynoTnials, uurite theirv under one anotlier so that similar terms shall stand in the same column, and then add each column separately as in § 29. 46 ELEMENTARY ALGEBRA [Ch. IV EXERCISES Find the sum of each of the following groups of polynomials : 1. 6a-5& + 3c, 7a + 106-6c, 8a-9&-10c, and 19a+8&+2c. 2. 2c-l d+Qn, 8c?-3n-9c, 4(/+16n-4c, and 3c-4n + (i. 3. 2 c -1 d - X + Qn, 8^-14n-3z, 18z + 10n + 8c?+3a, 4n-18c-5x + 6rf, 19c + 4a; + 8n-6fi?, and 5c + 2^-10c-4^. 4. 2 x8 + 7 6a;2 - 4 fi^x + 3 &3, 8 ft^^ - 15 hx^ _ 5 63 _ 10 a-^, 3 a;^ - 6 fix^, 46a;2 - 6 ?>3 + 10x8, and -hx'^-\- x^-^ b». Simplify the following polynomials, i.e., unite their similar terms : 5. 8 ma; - 5 x2 + 3 m2 + 2 a:2 - 8 m2 + 13 m2 - 18 W2X + 6 a;2 - 9 m^. 6. 3a2-6a6-862 + 7a2_3a2 + 2a6-14&2_6a6 + 862. 7. 4x2^-0:?/+ 10x3-4i/2_8x8-4a:3-f 3?/2_ 15xy + 2'dxy. a 4 a2 - 6 a + 4 - 3 a2 + a + 1.5 a2 - 2 + 5 a - 3.4 a2 _ 3.75 - 2 a. 9. ax^ - 4 a;2 + &?y3 _ ^^2 + 14 x^ - by^ +ay^ -3 y^. [Collect all the x^ terms and all the y^ terms.] II. SUBTRACTION 31. Subtraction of monomials. Since 5 times any given number, minus 2 times that number, is 3 times the given number, i.e., (5—2) times the given number, therefore 5 a— 2 a = (5 —2) a =3 a, whatever the number represented by a. So too 13 mx^y^ — Smci^y^ = (13 — 8) mx^y^ = 5 mx^y^. Observe that the reasoning just now given applies to any two similar monomials whatever, hence: To subtract one of two sivxilar monomials from tl%e other, subtract the coefficient of the subtrahend from that of tl%e minuend, and to this reinainder annex the com^mon literal faxitors. Here, as in arithmetic, it is usually most convenient to write the subtrahend under the minuend, thus : 126 a^s 13ma;2?/8 53 6ex8 92fl22 8 rna;2ji/8 — 9 hcx^ 34 a 22 5mx^y^ 62 6car8* * Since 63 — (— 9) = 53 + 9 = 02 ; compare § 17. 30-31] SUBTRACTION 47 Note. Since algebraic expressions represent numbers, the rule just now given may be stated thus : To siibtract one of tioo similar monomials fro7n the other, reverse the quality sig7i of the subtrahend and proceed as i7i addition (ef . § 17, Ex. 5). In order to avoid confusion when reviewing one's work, it is usually best not actually to change the sign of the subtrahend, but only to conceive it to be changed, or at most to write the changed sign helovo the term, thus: 13 77ixhj^ 53 6cx3 8 mxhj^ — 9 6cx8 + 5 mx2«/8 62 6ca;8 EXERCISES In the following exercises subtract the number written below from tlie one above it : I" 1. 2. ^8^ I. 1 7a 4a locx^ ^cx- -18 5 16 6^2 -3 6x2 6 m'^p'^ — 5 m-^9* -18 -5 - 18 m^ b7n^ 18 -9 -5 9 - 18 rH^ -5r2a;8 - 34.7 k^Y 6.8 k'^xY 9 -9 26 vY - 7 vh/ 3. - 5| a2w?* - 2\ a"'m^ 4. Are the signs written in the above exercises signs of operation or signs of quality ? 5. Define subtraction, and from your definition show how to verify the correctness of the above exercises. 6. Show that " changing the sign of the subtrahend and proceeding as in addition " will give the remainder in each of the above exercises. 7. From 5 (a — 2 6^) subtract —11 (a — 2 6-^) ; also subtract 15 m^(x—y) from — 23 7n^(x — y) ; and — 2 x{l + 5 a2y/) from 14 x(l + 5 a2^). 8. From the sum of Q ax^, — 3 ax^, and 11 ax^, subtract the sum of — 4 ax^, 9 ax^, and — 7 ax^. 9. Re-read §§ 16 and 17, and then pi-ove that, in any subtraction, the remainder may be obtained by adding the subtrahend, with its sign changed, to the minuend. 48 ELEMENTARY ALGEBBA [Ch. IV 32. Subtraction of polynomials. From the reasoning already given, it is evident that one polynomial may be subtracted from another by writing the subtrahend under the minuend, similar terms under one another, and subtracting term by term, thus : 4 62y8 + 5 aa.2 _ 12 ahc - - + 352y3_8aa;2 + i8a6c EXERCISES ^1. From 12 a — 3 & subtract 6 a — 5 &. —2. From 3a:— 2y+53 subtract 5 y — 2 — 8 ar. ^-Q. From 4 a^xi/ -9x'^y + 10 a^y'^ take 7 a32/2 _ 3 a^xy^ - 12 x2?^. --4. From 8 ^2 _ 7 y^iS _ 13 aa;2 take 4 m^ - 8 aa;2. -5. From 5 a;2 + 4 a%^ take 13 a^"^ -2x^+5 abx. 6. From a;^ + 1 take 1 - 2 x + a;^ + 3 a;2 - 4 a;3. 7. From 2 a — 3 x+ z take the sum of9a; + z — 4a and 10 z— 5 x+ a. 8. From 7.42 a:2 - 3} xy + 10 y'^ take 2.5 xy-7f-\- 3.02 a,-2. 9. From 34 a2x8 - 10 mV take 15 if + 10 a2a:3 + m*y\ 10. Subtract - 7 c^r^ + 3 a2 _ ^s from 5 a2 + 2 r2 + .s* - 3 ch^ 11. Subtract 1 - 3 a: + 10 a:2 from 2 a;2 + 5 ; from 4 ; from 0. 12. Subtract -8a + 3 &-13x2 from 5 6; from -6x2 + 2 a; from -7. 13. Subtract 3 &2 __ 10 aa: + 5 a:2 from the sum of 5 aa: — 2 a:2 and 10 62 _ 13 ax. 14. Subtract the sum of6a — 46 + 3c and 6h — 2a — c from 8 a — 3 6. 15. Subtract 3 a: - 10 ay^ — 2a^ minus x — 6ay+a^ from 4 a;2 — a; plus 5 a3 - 3 ay\ 16. From fa:8-|a;2+|a;-3 subtract | a:^ - 2^ - ^ a: - f x^. 17. Subtract 1 + 3(x - y) - 5(a2 + b) from a^ + 2(a^+b)- 8(x - y). 18. Subtract the sum of 5 a - 3 &2 + 2 x and - 4 x + 2 ^2 from 3 x2 + 4 a - 12 52 minus 3 a - 7 62 + 2 x2. 19. Subtract 4.5 m - 1.3 y' + 10 a^c^ from 1.4 y - 8 a^c^ plus 6.3 y - 181. ^2^4. 20. From the sum of x2 - 1 , 3 x + 2, and - 8 x2 - 5 x, subtract 4 x - 3 x2 plus 4 — 2 X minus 6 x + 3 x2 — 8. > 32-34] ALGEBliAlG EXFUESlSlONS — PARENTHESES 49 III. PARENTHESES 33. Removal of parentheses. That one expression is to be sub- tracted from another may be indicated by inclosing the subtra- hend in a parenthesis and writing the minus sign before it. E.g., 6 X — (2 a — ?/) means that 2 a; — r/ is to be subtracted from 6 x. Moreover, since a subtraction is performed by changing the sign of each term of the subtrahend and then adding it to the minuend (§§ 32 and 31), therefore a parenthesis preceded by the minus sign may he removed by simply changing the sign of each term inclosed by it* E.g., a—{— be + mp) = a + bc — mp;3kx2~i2by — 7a2)=3kz^ — 2by + 7a^', a;2 + 2 6x — (&3 ~ bx + S x^) = x^ -\- 2 bx — b''^ -\- bx — 3 x^ =— 2x'^-^Sbx — b^; and — (- 4 A;2 + 5 ax - 8 6?/3) = 4 ^^ — 5 ax + 8 by^. Note. If a parenthesis is preceded by the plus sign, it may be removed without changing the signs of the terms inclosed by it, because the expression within such a parenthesis is to be added to whatever precedes it. 34. Parenthesis within parenthesis. It often happens that a sign of aggregation may inclose one or more other signs of aggre- gation, thus : 3a''x-l2 mb + [a^x _ (- 4 s^^ -f 5 mb) -f sH'] J . In such cases it may be best for the beginner, after removing all those signs of aggregation which are preceded by the plus sign (§ 33, note), to remove the innermost of those signs of aggre- gation which are preceded by the minus sign, then the next inner- most, and so on until all are removed. E.c/., omitting the square bracket in the above expression, since it is preceded by the plus sign, that expression becomes 3 a2x -{2mb + a^x - (- 4 s2i + 5 mb) + sH}; now removing the parenthesis, this expression becomes 3 a2x - (2 m6 + a2x -h 4 s^t — 5 m& 4- s^] ; and, removing the brace, we obtain 3 a2x — 2 mb — a2x — 4 ^2^ + 5 mb — s% i.e., 2a2x + 3m6 — 5s2<. * Compare also § 39, Ex. 19. 50 ELEMENTARY ALGEBRA [Ch. IV Note. The work of removing parentheses in such expressions as that just given may be somewliat shortened by removing the outermost negative paren- thesis first, then the next outermost, and so on, instead of beginning with the innermost. The expression witliin an inner parenthesis is, of course, to be regarded as a single term of the next outer parenthesis. Parentheses preceded by the sign + should be dropped whenever they occur. The student may simplify the above expression by this method and then compare his work with that above. The essential thing in both plans is that on removing a negative parenthesis the sign of every term inclosed by it must be reversed. EXERCISES Simplify the following expressions : 1. 7a;-3ac + (a:-2ac). 2. 1 X — Zac —{x — 2ac). 3. 4 a - 2 & - (c + 3 a) - (2 c + 3 & - 2 a). 4. 5 a:2 + (7 aa: - 10 ^/) + 3 ?/ - (4 ao; - 5 y + 3 a;2). 5. ^xy+^if-^-x^ + y'^ + xy). 6. mx2-[8?/+(6a-wa:)-2a]. 7. _ (a + 5 - c) + 4 a - (c + 3 h). 8. ^x-2y+f-g-{2x-{^y + ^z-2V)^-2f-2g]. 9. a-y-{a-{-y-^^r^)}. 10. 15 - (6 - a:)- [13 - {x-{y + 2) + 2 ?/} + 2 x]. 11. x-{Zx-l-{-^x + 2y)+by^-^y]. 12. -{-[-(x-y)]}. 13. 8 a - 2 & - {(3 6- - ite each of these expressions in its simplest form. 24. Without actually performing the following indicated operations, tell by inspection what the sign of the result is in each case, and why : (_3)4. (-2)9; (-11)40; 5^ 724; (_5)«when n is an even positive integer, and when n is an odd positive integer; (— 3)2«and (— 3)2« + i, when n is any positive integer. 25. As in Ex. 24, determine the sign of the result in each of the fol- lowing indicated operations if a = 2 and 6 =—4: (a — 6)^; (a — 6)*; (a+by-, {ab'^Y] (a-4:by; {o.^'^y-, and {a%y^\ 26. Tell what is meant by the commutative and associative laws of addition and multiplication. Illustrate your answer in each case. 39. Product of a polynomial by a monomial. Since the product of two numbers is obtained from the multiplicand in the same way as the multiplier is obtained from the positive unit [§ 3 (iii)], therefore 5 • (2 + 6) = 5 • 2 -f 5 • 6, because the multiplier 2 -f 6 is obtained by first taking the unit 2 times, then 6 times, and adding the two results. Similarly, whatever the numbers or expressions represented by a, b, c, d, ", a{b-\- c + d -\- •••) = ab -\- ac -\- ad + ••• ; and, applying the commutative law to each member of this equa- tion, it becomes (b-\-c + d-\ )• a = ba + ca-\-da + ••♦. These last two equations state what is known as the distributive law * of multiplication as to addition ; it may be put into words * The multiplication of a sum is " distributed " over the parts of that sum. 56 ELEMENTARY ALGEBRA [Ch. V thus: the product of a polynomial hy a jnonomial is obtained hy multiplying each term, of the polynomial hy the monomial and adding the partial products. E.g., 5a;(3a2-2 6 + c2) = (3a2-26 + c2) . 5x = 16 a2a;-10 &x + 5 c2a;. The actual work may be conveniently arranged thus: 3 a2 - 2 6 + c2 5x 15 a2x — 10 6x + 5 c^x. each term of the multiplicand being multiplied by the multiplier, and the partial products added. EXERCISES 1. How is a + 1 — c obtained from + 1 ? How then is the product 3 • (a + 6 - e) to be obtained from 3 ? 2. Is 3 • (a + & - c) equal to (a + & - c) • 3 ? Why? 3. What is the product of 365 by 2? of (300 + 60 + 5). 2? Show that this illustrates the distributive law. 4. Since a(6 ■{■ c ■\- d -^ •••) = (6 + c + c? H — ) ' a = ah + ac + ad-\ , whatever the numbers represented by a, b, c, d, •••, what is the product of 2 ax and 3 a:2-4 a2a;8 + 5 aa;*? i> 5. Multiply 3 a'^b^—7 ax by 2 abx. Also 5 mx^ -7ay^-4: aHi by - 2 am^. Write a rule for multiplying a polynomial by a monomial. 6. When an indicated multiplication has been performed, and the result is expressed by an equation, is that equation an identity or merely a conditional equation ? E.g., is (3 a%^ — 7 ax) • 2 abx = 6 a%^x — 14 a^x^ a conditional equation or an identity ? 7. The fact that the equation in Ex. 6 is an identity may be used as a partial check upon the correctness of the multiplication. Are the two members equal when a = & = a; = l? If they were not equal when these special values are assigned to the letters, could the multiplication be correct ? Does the equality of the two members for this set of values prove that the multiplication is correct, or does it merely increase the probability of its correctness? Is it then a " complete " or only a " partial " check ? 8. 9. 10. Multiply 8a2-4aa: + 3m2 -3 xh -5 x^ + 4:xz^ 2a-3J + c by — 4 am^ —2xz^ ' — abc o'J-40] MULTIPLICATION 67 11. Check Exs. 8, 9, and 10 by the method of Ex. 7. Could other special values for the letters than those there given be employed for such a check? Why? Multiply (and check the work) : 12. 5 m2 - 2 P by 3 mF. 13. - 8.5 h^x'^y + 5f hy^ by ^V ^y- 14. 25 «» - 17 a^ _ a6 by - 3 a\ 15. a;^6 _ 2 x^y^ — 15 a:*?/^ + 4 x^y by — x^-'^y'^-^. Perform the following multiplications and check the work: 16. - 2 a:2 . {x^ -bxhj- 16 x'^y'^ + 24 xy^ -y^-xy-^). 17. (a362c3 _ 3 ah^c^ - 4 a^fes^ + aj^) • 2 a&c^. 18. -1. (3r«a;-4m2-2a;2). 19. Since -1.(3 mx - 4 m^ - 2 a:^) = - (3 mx - 4 m^ - 2 a:2), derive from P]x. 18 a new proof that a parenthesis preceded by the minus sign may be removed if the sign of each term inclosed by it be reversed (cf. § 33). 40. Product of two polynomials. Since m + n is obtained from the positive nnit by adding n times this unit to m times the unit, therefore, by the definition of multiplication, (a 4- 6 + c) • (m + n) = (a + 6 + c)m + (a + 6 + c)n = am + 6m + cm -\-an-\-hn + en. [§ 39. Similarly for any polynomials whatever; i.e., the product of two polynomials is obtained hy multiplying each term of the multiplicand hy each term of the multiplier, and add- ing the partial products. If any two or more terms of a product are similar, they should, of course, be united. The actual work of such a multiplication, and its check, may be conveniently arranged thus: Check a2 + 2 a& - 62 = + 2, when a = 6 = 1 a+6 = +2, when a = 6 = 1 (a2 + 2a& — 62) .a= a3 + 2a26-a62 (a2 + 2 a6 - 62) . 6= a26 + 2a62— 6^ a8 + 3a26 + a62-68 = +4, when a = 6 = 1 Note. The product of three or more polynomials may be obtained by multiplying the product of the first two by the third, this product by the fourth, and 80 on. 58 ELEMENT ABY ALGEBRA [CJh. V EXERCISES Multiply (and check the work) : 1. 4 az + 5 a2 - 2 a;2 by 3 a - 4 a:. 2. 2x^-7 xy + Sa^x hj - 5 x + 3 y. 3. 4 m2 - 3 mp by 3 p"^ - 2 m + m^. V4. 5s-3^ by 2s -Zr + t. 5. ax"^ — hij'^ by hx + ay. 6. a^ — 2 «:c + a;^ by a — x. 7. 2 a^ - 6 a6 + 3 &-2 by a + 6 + aft. 8. X - 5 x^ +^10 by 2 - 7 X + a;"^- 9. ai* - 2 a^r + 5 by a - X - 3. 10. w^ + 2 m.n + n^ by m + n — mn. 11. a + 6 — c + c? by a — 6 + c — rf. 12. 3 a - 5 ^2 _^ a/x by - f + 2 a - 3 x^. 13. a^ + b'^ + c"^ - 2 ah - 2 ac + 2hc hy a - h - c. f 14. xn + j,H by X - y. 15. a:n + yn by :c2 _ ^2. 16. X" + r by X" - 2/«. 17. X" + ?/» by x'- - J/*-. 18. 3 a8 - 4 a26 + 2 ab^ 19. 1.8x2-2x^-2.3 5 20. 2.5a2a;2_i,4a:ry + ft3 by 5 a2 - 3 aft + ft2. by lix-3|3/. 1 3/2 by - 3 ax -42/ - 1.2 a. 41. Integral expressions, degree and arrangement of expressions, etc. In multiplications with polynomials, and elsewhere, it is often advantageous to arrange the terms of a polynom.ial in a particular order; such arrangements will now be explained. A term is said to be integral if it contains no letters in its denominator ; * it is integral imth regard to a particular one of its letters if that letter does not appear in its denominator. A polynomial is integral, or integral with regard to a particular letter, if each of its terms is so. E.g., 3ax'^+-—^ — oct^y jg integral with regard to b, w, x, and y, it is fractional with regard to a; its first and last terms are altogether integral, while its second term is integral only with regard to 6, m, and y. * It may contain numerical denominators and still be called integral. 40-42] MULTIPLICATION 59 By the degree of an integral term is meant the number of literal factors which that term contains, i.e., it is the sum of the exponents of all the letters of that term. E.g., 5 ax is of the 2d degree, and 32 a^cy^ is of the 8th degree. An integral polynomial is said to be of the same degree as its highest term ; if all of its terms are of the same degree, it is said to be homogeneous. E.g., 6 aby- — 2 bmx + 5 ciH^y is of degree 6, and 2 ax^ — 6 xyz + 5 abx — y^ is homogeneous, and of degree 3. One is often concerned with the degree of a polynomial (or of a term) with regard to some rather than all of its letters ; in such a case only those letters are considered in determining the degree. E.g., 5 a^x^y — 3 ab^xy"^ + 2 x^ is homogeneous, and of degree 3, with regard to the letters x and y; it is of degree 2 in y alone, and of degree 3 in x alone, and non-homogeneous; its degree in all the letters is 7. A polynomial is said to be arranged according to ascending powers of some one of its letters if the exponents of that letter, in going from term to term toward the right, increase, and that letter is then called the letter of arrangement; it is arranged according to descending powers of the letter of arrangement if taken in the reverse order. E.g., 2 x3 — 5 ax^y — 7 b^y^ + 3 m^y^ is arranged according to descending powers of x, and ascending powers of y. 42. Multiplication in which the polynomials are arranged. If each of two polynomials be arranged according to powers of some letter which is contained in each, then their product will arrange itself according to powers of that letter, and the actual multipli- cation will take on an orderly appearance. E.g., to get the product of 7x — 2x^ + 5 + x^hYSx + 4:X^ — 2, arrange the work thus: Check x3-2a;2+ Tx + 5 =11, whenx=l 4.r2 + 3 3; — 2 = 5 , when x = l 4 x5 — 8 a;4 + 28 xS + 20 3-2 3x*- 3;3 + 21x2+15x —2x3+ 4a-2— 14 a; — 10 4x5 — 5x* + 20x3 +45x2 -1-x— 10 =55, when x = 1 60 ELEMENTARY ALGEBRA [Ch. V EXERCISES 1. Is the monomial fa^z* integral or fractional? With regard to what letters is " ^ -^ integral? With regard to what letters is it fractional ? 2. What is meant by the degree of an integral algebraic expression? When is such an expression said to be homogeneous? 3. Arrange the expression 4 ax^ — 7 x^ A- b x^ — 2hx — ^ a^ according to descending powers of x. Also according to ascending powers of x. Of what degree is its present first term? 4. Arrange the expression 3 x'^y^ + xy^ — 8 x^y^ — 6 x^y"^ + x^y accord- ing to descending powers of x. How is it then arranged with reference to y ? Of what degree is this expression ? Is it homogeneous ? In the following exercises arrange both multiplier and multiplicand according to some letter contained in each, then multiply and observe that the product has a corresponding arrangement. Multiply : 5. 6 a;2 - 2 + 5 a: + 3 a:8 by a;2 + 5 - a:. 6. 2a + a8 - a^ - 1 by 4 - a2 + a. 7. 3 a^x - 4 ax"^ ■{■ x^ - a^ by a'^-ax-\- x\ 8. 3 xy^- 2/8-3 x'^y + x^ hj -2xy-\-x^ + y\ 9. a:2^2 _ y.yz + 2/4 _ ^.s^ ^ a:* by x'^-\-xy- y\ 10. 4A2r-/ir2-A8+2r« by ^-2r. IJ.. In the product of two homogeneous polynomials, one of degree 5 and the other of degree 2, what is the degree of each term? Why? Is then this product homogeneous? Show that this consideration may be used as a partial check upon the correctness of such a product. Compare also Exs. 7-10. 12. Find the product of ax"^ + h'^x + a%, a-\-h+ x, and a — x. Should this product be homogeneous ? Why ? 13. Find the product of 2 m^ — 5 mn + 3 n^, 3 m — 2 n, and 1 — m — n. Should this product be homogeneous? Expand,* and check, the following indicated multiplications : 14. (Za + 2h)(2ax-a'^-x'^)(hx-2a). 15. {x^ _ 3 a:8^ + 2/4 - 3 xy^) {x'^ -2xy+ y^). * An indicated product is said to have been expanded when the multiplication has been performed. 42] MULTIPLICATION 61 16. (3r2_5r + 25)(s-2f + r)(3-s- 0- 17. [3 X + 2 2/ - 3 (y + 2 a;) - 2] [2 - 5 (a: - 2 + 3y)] (a; + y - 1). 18. (x + yy, i.e., (x + 7/)(x + y) (x -\- y). 19. (^x - yy(x -h yy. 20. (a-2 6)8(2a-&)(2a + 6). 21. (x2 + xy + ?/2)(a:-2/). 22. (a8 + a% + a6'^ + 63) (a-b). 23. (z2 + a:?/ + ?/2) (x-2 _ a;e, + ^2) (^ _ y) (a; 4. 2,). 24. If the multiplier and multiplicand are each arranged according to the descending powers of some particular letter, how will the product arrange itself ? From what two terms is the highest term in the product obtained? The lowest term in the product? 25. The results in Exs. 21-23 show that some of the terms of a product may cancel each other, and that the number of terms in a product of polynomials may be as small as two. Show that there must be at least two terms in such a product (cf. Ex. 24). 26. When both multiplier and multiplicand are arranged according to the powers of some letter, the actual work of multiplying may be some- what shortened, thus : Multiply 3a;4-2a;3_5x2 + (Ja:-4 by 7a:2 — 3a; + 2. Ordinary Process Shorter Process 3 X*— 2 a;8- 5 x^+ 6 a; - 4 3 x^— 2 x^- 5 x^+ 6 a; — 4 7 a;2_ 3 a; + 2 7 a;^— 3 a; + 2 21a;6- -14 x^—'So a;4+42 a;3- -28 .i;2 21x6- -14 a:5-35 a;4+42 a;8- -28x2 - - 9 a;5+ 6 x^-\-15 x^- -18a;2+12a; - - 9 + 6 +15 - -18 +12 a; + 6 a;4- 4 a-3- -l()a;2+12a;- -8 •+ 6 - 4 - -10 +13 - -8 21 a;6— 23 x6-23 a;4+53 a:3-5(i x2+24 a;— 8 21 x6-23 a;5-23 x^+ss a;8-56 x2-(-24 x— 8 Perform Exs. 5-9 by this shorter process, and check the work. 27. Since the powers of the letter of arrangement in the multiplication in Ex. 26 follow one another in regular order, in each partial product, the process may be still further abridged by omitting the letters until the very end. This is known as the method of detached coefBlcients. Thus, to multiply 3 x* - 2 x^ — 5 x2 + 6 x — 4 by 7 x2 - 3 x + 2, write only the coefficients : 3 2 5+6 4 7- 3+ 2 21 — 14 — 35 + 42 — 28 - 9+ 6 + 15-18 + 12 + 6- 4-10 + 12-8 21 — 23 — 23 + 53-56 + 24 — 8 i.e., the product is 21 xS — 23 xS — 23 x^ + 53 x3 — 56 x2 + 24 x — 8. Perform Exs. 5-9 by the method of detached coefficients. 62 ELEMENTARY ALGEBRA [Ch. V 28. Since, for example, 7325= 7(10)3+ 3(10)2 + 2 (10) + 5, jg not ordinary arithmetical multiplication performed by means of detached coefficients? Only the coefficients of the various powers of 10 are used. 29. Any absent term, in the regular order of arrangement of a polynomial to be multiplied by using detached coefficients, should be inserted, with zero for its coefficient. Thus, multiply 3 a;4 - 2 xs + 6x-i, i.e., dx^ - 2x^ + Ox^+ 6x - i,hj bx-2. Compare this with such multiplications in arithmetic (see Ex. 28). 30. Multiply 2 a^ — 5 a + 1 by 4 a — 2, using detached coefficients. 31. Multiply 6 x* — 2 a;2 — 5 by 3 x^ + 5 a;, using detached coefficients. II. DIVISION 43. Law of exponents in division. Assuming for the present, as in arithmetic, that the quotient is not changed if equal factors be cancelled from dividend and divisor, the law of exponents in division is easily discovered. For example, —^ = — — '- — '- — '— [Definition of exponent i.e.f a'^-i-a^ = a^'^. Similarly, x^ -h x^ = x^~'^ = oc^ ; and s^ _i. g8 _ — _ — In precisely the same way, it follows that if m and n are any two positive integers, then a"* -f- a" = a"*"", when m > ri,* . a*" -J- a"* = 1 , when m = n, and a"* -7- a" = , when in < n. * The symbols > and < stand, respectively, for " is greater than " and " is less than " ; thus, m > n is read : " m is greater than n." 42-44] DIVISION 63 44. Zero and negative exponents defined. Thus far the symbol a" has been defined only when n is a positive integer ; we are there- fore still free to say what we shall mean by such symbols as a~^ and a". It will be found advantageous to agree that, when such symbols present themselves in any operation, a" shall be inter- 1 * preted to mean 1, and a~* shall mean — • Under this definition of a" and a~*, the three expressions for the quotient of a*" -r- a", which are given in § 43, may be replaced by the single expression ^m _^ ^» _ ^m-n whether m "> n, m = n, ov m < n. For, when m = n, then a^'-i-a'' = a"*"", because then a"* h- a^ is manifestly 1, and a™"" is a", which is also 1. Again, when m)2. 11. (4a + 7a;)2. ^2a 3a;/ 5. (a - py. 12. (3 m4 - 2 n)2. 17. (9 aJc + hcdy. 6. (c-A)2. 13. (|a:2-|)2. 18. {{a + h)+cf. 7. (a; + 3)2. 14. (2 a^x + 3 hy^y. 19. {(a + &) - c}2. 20. Compare the fully expanded form of Ex. 18 with (a + & + c)2, and state, if you can, a general rule for writing down the square of any trinomial (see also § 61). 21. Expand (x-y + z^- sy. Suggestion. x--y + z + s={x — y) + {z + s). 22. Since a — h=a-\-{ — h), show that case (ii), p. 87, is included under case (i). 23. Expand (x^ + y") 2. Also (3 a« - 2 s'«)2. 24. What must be added to a;2 + 6 a; to make it the square of a; + 3 ? 25. What must be added to f^ + H to make it the square of < + |? 26. What must be added to a* + 0.%"^ + ft* to make it the square of a2 + 62? 27. What must be added to x^-\-2 x^y^ + 4 y^ to make it the square of a;4 + 2 2/3 ? 28. Find what must be added to each of the following expressions to make them exact squares; also give the expressions of which they are then the squares : ?n4-8m2n2 + 4n4; a^-^ah; x'^y^ + 12 xyz^ ', x'^ + ax\ ?^ndi A^ + -AB. n 29. Find, by the method of § 57, the square of 53, i.e., of 50 + 3. 30. Write down the squares of the following numbers : 18 (i.e., 20—2), 39, 71, 83, and 34. 57-58] TYPE FORMS IN MULTIPLICATION 89 58. Product of sum and difference. If a and b represent any two numbers whatever, then, by actual multiplication, (a ^b)(a-b) = a' - b\ i.e., the product of tTie sum of any two numhers, hy the difference of these numhers,* is tlxe square of the first num- ber minus the square of the second. E.g., (a; + 3)(a; — 3) = a'2-9; (5 + ?>i) (5 — m) = 25 — m2 ; etc. Note. Here, too, as in § 67, complex terms may be iticlosed in parentheses, thus : (3x2 + 5^/) (3x2-5?/) = (3 a:2)2 - (5 y)2 = 9x4-25^2. / EXERCISES Without actually performing the following indicated multiplications, write down the products by inspection : 1. ix^y-){x-y). 8. (x^ + ?/2) (^3 _ ^^2) . 2. (m + n)(m-n). 9. {\^lmn-^'p\^){\^lmn^-^'p\^). 3. (3 a; + 2/) (3 a;-?/). 10. {^x - y^ ^ z}{{x - y) - z\. 4. (J x-2y)Qx^1y). 11. {{a? J^y^) - ab^iia^ ^V') ^ ah}. 5. (14a+15&)(14a-15&). 12. (« + 6 + c)(a + &- c). 6. (6jo-5^)(6/>+59). 13. {a-h^c^ia-h-c). 7. (4m2-3n3)(4m2+3n3). 14. (a - &+ a:)(a + J - x). 15. (m - 2 n + s - (w - < + 2 n - s). 16. Show that a:2 + 2 x?/ + ?/^ — 2^ is the product of the sum and differ- ence of X + 2/ and z. 17. Show that a^ -{■ 2 ah -^ V^ - c^ — 1 cd - d^ \s, the product of the sum and difference of a + 6 and c-\- d. 18. (9a;2-42/2)-(3a;-2?/) = ? Why? 19. (16a2-25&2)-^(4a + 5J)=? Why? 20. (a;4 _ ^/4) ^ (a;2 _ ^2) ^ ? Why? 21. (a:6 _ 1/4) ^ (^8 _ 2/2^ ^ ? 22. (xis - 3/8) ^ (a:9 + i/4) = ? 23. Find, by the above method, the product of 22 by 18. Suggestion. 22 = 20 + 2 and 18 = 20 — 2. * The order in which these numbers are written being the same in both factors. 90 ELEMENTARY ALGEBRA [Ch. VII 24. By this method find the following products : 63 by 57 ; 48 by 52 ; 34 by 26. Note. The identity (a + 6) (a - 6) = cfi - 62, i.e., a2 = (a + &) («-&) + 6^, furnishes a very practical device for mentally squaring any number consisting of two digits. E.g., to square 73 mentally, let a = 73 and 6 = 3; then the last formula above ^^*^°°^®^ (73)2 = 76 . 70 + 9 = 5329. Similarly, to square 58, let a = 58 and 6 = 2; then the formula becomes (58)2 = 60.56 + 4 = 3364. 25. By the method given in the above note, write down the square of 47 ; of 82 ; of 29 ; of 53 ; of 98 ; and of 61. 59. Product of binomials having common term. By actual mul- tiplication, (a; + 3)(a; + 5) = a;2_^8a; + 15 = a;2_|_(3^5)^_|.15. and (a; + 3)(a;-5) = a^-2a;-15 = ar^+(3-5>-15. So, too, in general, (x -{• a) (x -\- h) = x^ -\- {a -{- h)x -|- ah ; i.e. the product of two binomials having a term in common equals the square of the common term, plus the algebraic sum of the unlike terms multiplied by the common term, plus the product of the unlike terms. EXERCISES Without actually performing the following multiplications, write down the products by inspection : 1. (a+5)(a + 7). 10. {a + h){a + c). 2. (a-5)(a-7). 11. {a-h){a + c). 3. (a+5)(a-7). 12. (2 a; + 3)(2 x- 5). 4. (a-5)(a + 7). 13. (3 a + 4)(3 a- 6). 5. (y_c)(2/ + 2c). 14. (4rt2_5)(4a2+ 1). 6. (a:2 + 4)(a:2+5). 15. (xy- 4)(a:7/+ 16). 7. (a:2+4)(a:2-5). 16. Q^mH^ + 2)(lhnhi^ -d>). 8. (a:2-4)(a;2_5). 17. {Q ^ m) - 2}{{l + m) - b}. 9. (a;2-4)(a;2+5). 18. {(/ + ,«)+ 8}{(^ + m) - 15}. 58-61] TYPE FORMS IN MULTIPLICATION 91 60. Product of two binomials which contain the same letters. The product of two binomials containing the same letters is a trinomial which, by a little practice, may be written down without writing the intermediate steps. E.g., the product of 3 a; + 5 and 2 cc — 7 may be arranged as in the margin : the term (5 x'^ is the product of the first terms of the binomials, the term — 11 a; is the algebraic sum of the " cross products " (2 cc by 5 and 3 a; by — 7) , and —35 is the product of the last terms of the binomials. 3x +5 This final product may, with a little practice, be easily written 2 a; — 7 down, omitting the intermediate steps. i-UAn Similarly, in the product of3x + 4?/ by 5a — 2 y, the prod- _ _ uct of the first terms is 15 x^, the algebraic sum of the cross products is 14 xy, and the product of the last terms is — 8 2/2 ; 6 a;2 — 11 a: — 35 hence (3 a; + 4 ?/) (5 a; — 2 ?/) = 15 x^ + 14 x?/ — 8 y^. So, too, (ax + 6) (ex + (^) = acx2 + (ad + 6c)x + 6(^. EXERCISES Write down the following products by inspection : 1. (3a;+2)(4a;-3). 5. (7 0^+ 62)(3 ^2+ 8 &2). 2. {^x + 2y){4.x + Zy). 6. (Q x -2 y){x+ y). 3. (a:-3y)(5a:+6y). 7. (x + d){x+h). 4. (2a-4&2)(5a-6 62). 61. The square of any polynomial. By actual multiplication it is found that (ct + 6 + c)2 = a2 + 62_,_c2_p2a6 + 2ac + 2&c, {a + h ■\- G-\- df = a? + h^ + (? -\- d} + 2 ab ^2 ac + 2 ad + 2hc + 2hd + 2cd, (a.+ 6 4- c + d+e)2 = a2 _,_ 52 _|_c2 ^^2 ^ g2^ 2 a6 + 2 ac + 2 ac^ + 2 ae + 2 6c + 2 &d 4- 2 &e + 2 cd + 2 ce + 2 de, etc. This may be formulated into words, thus : the square of any polynomial whatever equals the sum of the squares of all the terms of the polynoinial, plus twice the product of each term hy all the term^s that follow it* * The formal proof of this theorem is given in Chapter XVIII. 92 ELEMENTARY ALGEBRA [Ch. Vll EXERCISES Expand the following expressions by inspection : 1. (m+n-.s)2. a (rt_6 + c-rf)2 2. (a-&-c)2. 9. (ax + hy + czy. 3. (2x + y + 2)2. 10. {ahx - acy - hczy\ 4. (2 a; + 3 ?/ - 2;)2. 11. (^ + /^j + n + p + (^ + r + s)2. 5. {2x-^y + zy. ^ 12. (2x-3 2/ + 4 2-5a+3 6-4)2. 6. (3 a + 4 & + c)2. 13. (a;4 + 2 a:8 - 3 a:2 + 4 a; - 5)2. 7. (3a-46-2c)2. 62. Higher powers of binomials — binomial theorem. By actual multiplication it is found that (x -\- yy = x^ -\- 4.:»?y + Q ^y"^ ^^xf-\- y\ (a; + ?/)« = a^ + 5 x'^y + 10 a^?/^ + 10 x^ + 6xy^-\- f, {x + yf = x^ + Qx^y + l^xy -\-20 ^f + 15xY + Qxf -^y"^, etc. ; and that {x - 7jy>= a? - ^x'y + ^xy'- f, {x — yy = x^ — 4:a:^y-\-6 a?y'^ — 4 a^?/^ + y^, {x — 7jy = x^~5 x^y + 10 a^2/^ — 10 x^y^ -{-5xy* — f', etc. A careful study of the second members of the above equations will show that they all follow the same laws, and that they may, therefore, be written down by the same rules. In fact, such a study will show that : (1) The number of terms in the expansion is in every case greater by 1 than the exponent of the binomial. (2) The X * appears in every term of the expansion except the last, and the y appears in every term ^ the expan- sion except the first. (3) The exponent of x in the first term of the expansion is the same as the exponent of the binomial, and it decreases by 1 from term to term in passing to the right, while the * In applying these rules to other hinomials, observe that cc is here used for " the first term of the binomial " and y for " the second term of the binomial." \ 61-62] TYPE FORMS IN MULTIPLICATION 93 exponent of y in the second term of the expansion is 1, and it increases hy 1 from term to term in passing toward the right. (4) The coefficient of the first term of the expansion is 1 ; the coefficient of the second term is the same as the exponent of the binomial ; and if the coefficient of any term he mul- tiplied hy the exponent of x in that term, and this product he divided hy the /rtumher of the term (i.e., hy this term's exponent of y increased hy/T), the result will he the coeffi- cient of the next term. (5) The signs of the terms of the expansion are all posi- tive if each term of the hinomial is positive, hut if the second term of the hinomial is negative, then the terms of the expansion are alternately positive and negative — the first term heing positive. Note. It is proved later (Chap. XVIII) that the above laws apply to all positive integral powers of any binomial whatever ; hence such powers may be expanded without actually performing the multiplications. Ex. 1. Expand (a — hy. Solution. By (1), (2), and (3) above, the letters and exponents in the several terms of this expansion are : a^ a'b a^b^ a%^ a^b* a%^ • a%^ ab'^ b^; by (4), the coefficients are : 1 8 28 56 70 56 28 8 1; and by (5), the signs are: + -.+ - + - + -+; hence, combining these results, (a _ J) 8 zz: a8 - 8 a^ft + 28 a662 _ 56 a^b^ + 70 a^b^ - 56 a%^ + 28 a^js _ 8 a&H i^ Ex. 2. Expand (2 x - a^y. Solution. Letters and exponents, (2 xy (2 xy (a2) (2 x) (a2)2 (a^)^ ; [Cf . (1), (2), (3) coefficients, 1 3 3 1 ; [Cf. (4) signs, + - + - ; [Cf. (5) combined result, (2 x - a^y = (2 xy - 3 (2 xy(a^) +3(2x) (a^y - (a^y ; simplified result, (2 x - a'^y= 8 x^ - 12 x^a^ + 6 xa* _ a^. With a little practice the combined result may be written down at once instead of making several steps of the work. 94 ELEMENTARY ALGEBRA [Ch. VII EXERCISES Expand the following expressions : 3. (a + 6)3. 6. (u-vy. 9. (a:-y)io. 4. (a~xy. 7. (a: + -)*- 10. (x-2ay. 5. (m-ty. 8. (3 a2- 2 65)3. n. (;n2 + 3n)6. 12. Write the first 4 terms of (a + x)^. 13. Write the first 3 terms, and also the 7th term, of (x — yY^. 14. Write the first 5 terms of (2 ax - 3 k^y, II. FACTORING 63. Definitions. In a broad sense, any two or more numbers whose product is a given number are factors of that number. Thus, since -i- • |- • 10 = 4, therefore \, f , and 10 are factors of 4; so also are j^, 18, and j%. In this sense, however, the problem of finding the factors of any given number, or algebraic expression, is manifestly inde- terminate; it is therefore customary, when speaking of factors, to mean only the rational * and integral exact divisors of a given number or expression. E.g., ±l,t ±2, ±3, ±4, ±6, and ±12 are factors of 12; and ±1, ±5, ± (2 a; + ?/) , ± (2 a; — z/) , as well as products of any two or more of these, are factors of 20 x^ — 5 y^. Every number is a factor of itself, and 1 is a factor of every number. A number, or an algebraic expression, is said to be prime if it has no exact divisor {i.e., factor) except itself and unity; other- wise it is composite. A factor is prime or composite according as the expression for it is prime or composite ; and it is integral with regard to any given * An expression is rational with regard to a particular letter if it cpntains no indicated root of that letter (see § 130) . t The sign db is called the double sign, and is read " plus or minus " ; it is used to combine two expressions into one : thus the expression ± a means both + « and also — a. 62-66] FACTORING 95 letter if the algebraic expression for it is integral with regard to that letter (cf. § 41). It will appear later that the writing of an expression as the product of its prime factors often greatly simplifies algebraic work ; and it is therefore important that the student should early master those cases of factoring which present themselves most frequently. Some of these cases will now be considered. 64. Factors of a monomial. This is the simplest of all the exer- cises in factoring, and can be done by inspection. E.g., 30 ax'^y = 2 -^ -^ • a- x -x -y, which exhibits the given monomial as the product of its prime factors; the product of any two or more of these prime factors is a composite factor of the given monomial (cf . § 63)\ A rule for this kind of factoring may be stated thus : by inspec- tion, or hy trial, find the prime factors of the numerical coefficient of the given rnonomial, and to their indicated product annex, each of the literal factors as many times as there are units in its exponent, EXERCISES Separate the following monomials into their prime factors : 1. Qa^x^ 2. 15 mY^^- 3. 36 sHK 4. 420 m%V- 5. 572 a^c^uv^. 65. Monomial and polynomial factors of a polynomial. If a poly- nomial contains a monomial factor, the latter can usually be dis- covered by mere inspection. E.g., in 12 a^^ + 4 abx'^ — 8 axhj'^, it is seen that each term contains the factor ^ ' 12 a'hfi + 4 ahxhj — 8 axhj^ = 4 oa;2 . {^ax-\-by — 2y^. In order to factor a polynomial completely, it is then only necessary to consider further how to factor a polynomial which contains no monomial factor. This problem, however, is in general very difficult, and only its simplest cases will at present be con- sidered. Fortunately it is these simpler cases which present themselves most frequently in practice. 1. 5a-106. 2. 17 x^ - 289 xK 3. 4 a:3 - 8 x'^y. 4. 10 m%2 _ 15 ,w8„8. 5. lQx'^-2ahx. 6. 4 a%'^ - 24 a258. 7. 15 a:* _ 10 a;3 + 5 a;2. 8. 3 a^ - 6 a-ife + a4^,2. 9. :ci2^i2 + a;iYi + 3,103,8 _ .0. 3 7715 _ 12 mH'^ + 6 mn*. 96 ELEMENTARY ALGEBRA [Ch. VII EXERCISES Separate the following expressions into their monomial and poly- nomial factors : 11. ac — be — cd — abed. 12. 13 xY - 13 xY + 12 xy. 13. 14 xYz^ - 7 xYz^ + 8 xy^z^. 14. 60m2nV2-45m%V + 90m%V2. 15. 12 x'^b^y - 18 xy% + 24x4&4^4. 16. 14 ahnn^- 21 a^m^^-^9 a^mn^. 17. 25 c'^dx^ + 35 03^2^:4 _ 55 c2rt s+t The student may make a verbal statement of this case of factoring similar to the last paragraph in (i) above. (iii) Again, x-\-a, i.e., » — (— a), is a factor of a;" + a" when n is odd, for in that case (— a)" + a" = — a" + a" = (§ 18, note 2). Hence, the suuv of like odd positive powers of two num- bers is exactly divisible by the sum, of these numbers. By actual division, it is found that ai±X^=a;2-a;?/ + ?/2; ^^^^^^ = x^ - x^y + x^^ - xy^ -\- y^ • x+y x+y 2^i^ = x6 — x5y + a;4y2 _ x^yB + xhj^ — xy^ + ?/8 ; etc. x-\-y The student may formulate this principle into words, — see last paragraph in (i) above. (iv) Finally, a; — a is never a factor of a?" + a" ; for if a be sub- stituted for X in this expression it becomes a" + a", which is not either when n is even or when n is odd, and therefore x^ -f a" is not exactly divisible by a; — a (§ 48). Note. Principles (i) to (iv), above, may be briefly recapitulated thus: xn — an is always divisible by a; — a, xn — a" is divisible by x + a only when n is even, x" + a" is divisible by x + a only when n is odd, x" + a« is never divisible by x — a. EXERCISES / 1. 'Show by means of the remainder theorem that x^ — a^ is exactly divisible by a: — a ; also that x^ + a^ is exactly divisible by a: + a. 2. Prove that a; — a is a factor of x" — a^ for every positive integral value of n. 3. Prove that ar + a is a factor of a:" + a" for odd positive integi-al values of n, and of a;" — a" for even positive integral values of n. 104 ELEMENTARY ALGEBRA [Ch. VII 4. Prove that neither x — a nor x + a is a factor of x"* + a" when n is an even positive integer. Write out the following quotients by inspection, and then verify them by actual division : 21. 22. 23. 24. 25. 26. 27. 5 x'^-y\ x-y 6. x^-y^ x-y 7. a^-b* a-b 8. w8 _ ^,B U — V 9. v^-w^ V -\- w 10. m^ — n* m + n 11. uS-yS w + u 19 x^ + y^ 13. a:5 + 2/5 x + 2/ 14. m^ + s^ m + s 15. a^ + b^ a-\-b 16. (x^y + (^/2)^ x'^ + y'^ 17 (2 a)4 - a:* 2a-x 18. m6 - 32 m-2 19. 4P-9 2ifc-3 9n 16p*-81 28. 2:2 + 3,2 81a^-16 3a + 2 64 -rg r + 2 ' 27 a:8 + 64 gS 3x + 4a 32x5 + 1 2a: + l ' x^ + yg x2 + 3/2* a2 + 62 ' 32 xio + ?/i6 a; + 2/ 2jt) + 3 2 a;2 _|- yS 29. Compare the quotients in Exs. 5-15 with the corresponding powers of a binomial (§ 62), with reference to coefficients, exponents, signs, etc. 30. Of what is x^ the square ? Of what is it the cube ? Write x^ — y^ as the difference of two squares ; of two cubes. Is a;2 - 2/2 a factor of x^ - y^l Why? Is x^ - y^2 Is x^ + /? Why? Find the prime factors of x^ — y^. 31. When seeking the prime factors of x^ - y^ show that it is better not to divide out the factor x — y at once, but rather to separate x^ — y^ first into the factors x^ — y^ and x^ + y^, and then to separate each of these factors further. Is a similar plan advisable in general, e.g., with a;8-y8 and p"^ - 720? 32. Find the prime factors of mi2 _ ni2 ; compare Ex. 31. 33. Find the prime factors of x^ - 3/^ ; also of 64 a^ _ 1. 34. Prove that jt)« — r" is exactly divisible by p^ - r2, if n is an even positive integer. 35. For what positive integral values of n between 1 and 9 has a;" + «/" no binomial factor ? Is ic2 + 3/2 a factor of x^ -\- y^t 68-69] FACTORING 105 Resolve the following expressions into their prime factors : 36. x*-y\ 40. aioxio-yio. 44. S as^^ - S at^^. 37. a^-b^ 41. p9 + l. 45. jfi + y^. 38. a8-68. 42. 16 a*^^ - 81 xV- 46. 64 x^ + y«. 39. m8 - 1. 43. ai2a:i3 _ b^^xy^-2, 69. Factoring by rearranging and grouping terms. A rearrange- ment and grouping of the terms of an expression will often reveal a factor which could not be easily seen before. Ex. 1. Find the factors of ax — 3 &y + 6x — 3 ay. Solution, ax — dby + bx — Say = ax + bx — Sby — day = x(a + 5) - 3 y(a + b) = (a + 6)(x-3y). Ex. 2. Find the factors of x(x + 4) - ^(^^ + 4). Solution. x(x + 4) — y(y + 4) =x^-\-4:Z — y^ — iy = x-2~y^ + 4(x - y) = (^ -y)(^ + Z/ + 4). Note. The factor z — y could also have been detected by means of § 67, because the given expression is zero when x = y. EXERCISES Find the factors of the following expressions : 3. ax8 + 1 + a + X. 7. m^ - n^ - (m - n)^. 4. a2J2 + a2 + 62 ^ 1. 8. x^ + x2 _ 4 a- _ 4. 5. ac ■\- bd — ad — he. 9. 5 x^ + 1 — x^ — 5 x. 6. ac"^ + bd"^ - ad^ - bc^. 10. a^ - 9 x2 + 4 c2 - 4 ac. Suggestion. The first, third, and fourth terms of the expression in Ex. 10 are together (a — 2c)2, i.e., the given expression equals (a — 2c)2 — 9^2^ of which the factors are obvious. 11. X* — xy^ — ax^ + ay^- 15. ab + bx"* — x''^™ — ay^. 12. 1 +bx- (a^ + ab)x^. 16. 3 xy(x + y) + 16 x^ + 16 y^ 13. a^c^ + acd + abc + bd. 17. (p^ - q^y - (p^ - pgy. 14. x4 - 4 xY + 2 x8 - 16 y^ 18. (x + yy + 12 (x + y) - 85. 19. a^x 4- abx + ac + b^y + aby + be. 20. (x2 + 6 X + 9)2 - (x2 + 5 X + 6)2. 21. x^+ (a + b - c)x2 + (ab-ac - bc)x - abc. 106 ELEMENTARY ALGEBRA [Ch. VII 22. m^ + n^ -f m + 77Jn + n + run. 23. 14 a{x - ?/) + 49 a2 + (x - yy. 24. a;2 - a2 + y2 _ ^,2 4. 2 arz/ - 2 a6. 25. A2 _ „i2 + 10 m + F - 25 - 2 ^^. 26. 9 a2 + 12 a6 + 4 &2 _ 15 a - 10 & - 24. 27. a2 + &2 + c2 + 2 (rt6 + «c + &c) + 5 (a + & + c). 28. a:2 + 3^2 + ^2 + 2 (a:?/ + X2: + 3/z) + 5 (a; + 3/ + z) + 6. ' 29. 4 a;2 + 10 X + 6 - 5 a - 4 aa; + a2. 70. Factoring by means of other devices. It often happens that the factors of an expression will become apparent by adding a certain number to, and subtracting the same number from, the given expression ; this, of course, leaves the value of the expres- sion unchanged. Ex. 1. Find the factors of a;* + a;2 + 1. Solution. If the second term in this expression were 2 x^ instead of a;2, then [§ 66 (i)] the expression could be written (x2 + 1)2; this suggests that x^ be both added and subtracted, which gives a:* + a;2 + 1 = x* + 2 a;2 + 1 - a;2 = (:c2 + 1)2 _ a-a = (:c2 + 1 4. a.) (3,2 4. 1 _ a-)^ [■§ 66 (ii) i.e., a;4 + a:2 + 1 = (a:2 + x + 1) (a:2 - a: + 1). Ex. 2. Find the factors of a"^ + a^h"^ + &*. Solution. This expression may be treated in the same way as Ex. 1, *^^^ • a* + a2j2 + ft4 ^ ^4 .|. 2 a252 ^. 54 _ ^2^2 = (a2 + &2')2 _ ^cihy = (a2 -\-ah + 62) (^^2 - ab + b^). Ex. 3. Find the factors of a;^ - 4 a; - 32. Solution. Here the first two terms, plus 4, are an exact square, and this suggests the following arrangement : a;2 - 4 X - 32 = a;2 - 4 a; + 4 - 32 - 4 = (a; _ 2)2 _ 36 = (a: _ 2 + 6) (a: - 2 - 6), I.C., a;2 - 4 a: - 32 = (a: + 4)(a; - 8). Note. Observe the superiority of the method of Ex. 3 over the method of § 66 (iii) for factoring the same expressiou. 69-70] FACTORING 107 EXERCISES Factor the following expressions : 4. m^ + m2n2 + n*. 13. 5 a;* - 70 xY + 5 y*- 5. p^+4: qK 14. 9 a* + 26 a%'^ + 25 6*. 6. a:2 + 6 a; + 5. 15. a^ + 2 a6 - rf^ _ 2 6rf. 7. 9 s2 + 30 s< + 16 A 16. x^ + 64 y^. 8. a;4 + a2x2+a4. 17. 4a4+81. 9. a-8 + a:4//4 + ?/8. 18. x^y^ + 4 ar^/*. 10. 4 a8 - 21 a4&4 + 9 68. 19. m6 + 4mn4. 11. a4j4 + ^262^2^2 ^ ^4^4. 20. a^ + 8 a2 _ 128. 12. 9 x^ + 8 a:22/2 + 4 3/*. " 21. 5 na;* - 70 na;2 + 200 n. 22. What must be added to a:* + 3 x2 + 4 to make it an exact square? What must then be subtracted to leave the result unchanged? Factor this expression. 23. What must be added to a;*— 3 x2 + 4 to make it an exact square, and what must then be subtracted so as not to change the value of the given expression ? If the given expression is written in the form (a:2 — 2)2 + a:2, can it be factored by any of the preceding methods (cf. § 68) ? 24. Can the sum of two squares be factored (cf. § 68) ? Is not the expression in Ex. 5 above the sum of two squares? Could this expres- sion be written (/)2 + 2 ^2)2 _ (2^9^)2? 25. Factor the expression 3 a:2 + z - 10 [cf. Ex. 1, § 66 (iv)]. Solution. Zx^ + x-\0 = 12(3a;2 + cc-10) 12 ^ 36a;2 + ]2a;-120 12 36x2 + 12 a; + 1 — 121 12 _ (6 a; + 1)2 -(11)2 12 _ (6 a; + 12) (6 a; -10) 12 = (a; + 2)(3x-5). Note. The above method is more direct than that given in § 66 (iv) ; it con- sists in multiplying the given expression by such a number as will make its highest term an exact square, and the next highest term exactly divisible by tivice the square root of the highest term, then factoring the resulting expression as explained in § 70, and finally dividing the whole by the number first used as a multiplier, so as not to change the value of the expression. 108 ELEMENTARY ALGEBRA [Ch.VII Factor the following expressions : 26. 5 ,n2 - 2 m - 3. 30. 8.12 + 23^5- 352. 27. 6«2_n«_35. 31. 4iV2+16iViM3 + 15 M6. 28. 18a:2- 3 a; -36. 32. 2 a;2 + 5 ar«/ + 2 ?/2. 29. 6 i?2 _ 2 72 _ 20. 33. 3 a:2 - 10 x?/ + 3 ?/2. 71. General plan for factoring a polynomial. Based upon § § 65-70, the following suggestions for separating a polynomial into its prime factors may be made, ^j inspection find the monomial factors of the given polynomial, if there are any such, and then write this polynomial as the indicated product of the monomial and the corresponding polynomial factor ; then, by rearrangement of the terms, or by some one of the other methods given above, separate this polynomial factor into two factors, and replace it by their indicated product; then further separate each of these factors into two others, if possible, and so continue until all of the factors are prime. EXERCISES Factor the following expressions : 1. m^x^ + 7n2^5. 4. x^ ■\- ax — ay — yx. 2. c2 - 5 c - 14. 5. a;4 - 8 a:3 + 15 x^. 3. 21 m^-ma- 10 a\ 6. m%4 - 5 m'^n'^ + 4. 7. 25 a2 + 2/2 + 10 a;2 + 10 a?/ - 35 aa; - 7 xy. 8. 7n2 + 6 m - a:2 + 9 - 4 x?/ - 4 y2^ 9. 2 (a2&2 _ a2c2 + 52^.2) _ ^^^4 _^ J4 _^ ^4). 10. ari2 _ 2,12. 16. ai6 + 1. 11. ai2a:i22^i2 + ri2si2. 17. (a2 + 5a + 4)2-(a2_5a-6)2. 12. 4 ax2 4- 4 ay\ 18. a;2« 2 + ^,2^2 _^ 2 a;»-%. 13. a%H'^ + 4 ah'^xy + 4 h'^y\ 19. x^ - y^ - ^ x^ + 3 a:22/4. 14. 32 a - ax^. 20. a;^!/ - 15 x'^y + 38 xy - 24 ?/. 15. a9 + 4 a. 21. js 4. ^,4^,2 ^ ^4. 22. w%3 4. 2 mhi''rh^ + m%iV%«. 23. a;2 + 9 y2 + 25^2 _ 6 a:?/ - 10 xs + 30 3^2. 24. a:5 + 5 x^az^ +10 a;^^^* + 10 0:20826 4. 5 xaH^ + a^sio. 25. a2 - 2 a6 + &2 _ 2 ac + 2 6c + c2 - 2 a(/ + 2 Jrf + 2 cd/ + tf^. 70-72] FACTORING 109 72. Solving equations by factoring. If all the terms of an equation be transposed to its first member, factoring that member will always simplify the finding of the roots of the given equa- tion ; this is illustrated by the following examples. Ex. 1. Given x'^ — 5 x + Q = 0; to find its roots, i.e., to find those values of x for which this equation is satisfied (cf. § 23). Solution. By § 66 (iii) the first member of this equation is the prod- uct of a; — 3 and x — 2, and the given equation may, therefore, be written (x-2)(x- 3) =0. It is manifest, moreover, that a product is if, and only if, at least one of its factors is ; hence (x — 2)(x— 3) = if, and only if, a:-2 = 0ora;-3 = 0, i.e., if, and only if, a: = 2 or re = 3 ; hence the roots of the given equation are 2 and 3. Ex. 2. Given x^ = ^ x + i; to find its roots. Solution. On transposing, this equation becomes a;2 _ 3 a; - 4 = 0, i.e., -, (x - 4) (a: + 1) = ; [§ 66 (iii) hence either a; — 4 = or x -f 1 = 0, i.e., a; = 4 or a: = — 1, and therefore the roots are 4 and — 1 . Ex. 3. Solve the equation 6 x^ — 11 a; = 35. Solution. Transposing and factoring [§ 66 (iv)], this equation may be written (3x+5)(2x-7) = 0; hence 3a; + 5=:0 or 2x — 7 = 0, i.e., x = — f or x = |, and therefore the roots are — | and |. Note. Since the roots of the equation {x — a)(x—b)—0 are a and 6, therefore an equation which shall have any given numbers as roots may be immediately written down ; thus the equation whose roots are 3 and 8 is (x -3){x-8)= 0, i.e., cc2— 11 a; + 24 = 0. Similarly, the equation whose roots are 2,-1, and 5 is (a; - 2) (a; + 1) (x — 5) = 0, i.e., a:8 — 6 a;2 + 3 a; + 10 = 0. 110 ELEMENTARY ALGEBRA [Ch. VII EXERCISES 4. What is meant by a root of an equation ? May an equation have more than one root? 5. Find the roots of x^ — 4 a: — 21 = 0. Verify their correctness by substituting them, in turn, for x in the given equation. 6. Solve the equation ^^ _ g ^ ^ 5 — o, and verify the solution. 7. What values of x will satisfy the equation (a; — 2) (x — 3) = 0? If X =fz 2* will a: - 2 be ? If a: ^ 3, will ar - 3 be ? If, then, x is neither 2 nor 3, can the given equation be satisfied ? This equation has then how many roots? 8. Write the equation whose roots are 5 and 1. Also one whose roots are 3, 2, and 7. 9. W^rite the equation whose roots are : 1 and — 5 ; | and 6 ; a and 6 ; 3, — 1, and 5 ; a, — a, and 2a; 1, 2, 3, and 4. Solve the following equations, and verify the correctness in each case : 10. a:2-2a: = 15. 13. Sy^ + 15 = - 26 3/. 16. 2a:3 + 5a:2= 2 a; + 5. 11. 6 a;2 _ a: - 1 = 0. 14. 5 a;^ - 7 a; = 0. 17. a:^ - 4 = 0. 12. ^if+y= 10. 15. 12 22 = 4 2. 18. x^ - 13 a:2 + 36 = 0. 19. x3 + a;2-x = l. 20. (a: - l)(a;+ l)(a: - 2) = 0. 21. Can the roots of the equation in Ex. 20 be determined by mere inspection ? Can the roots of the equation (3a:-2)(a;+l)=2 be so determined ? What are these roots ? 22. Write out a rule for solving such equations as those given in the above examples. PROBLEMS By the meth6d of § 26 f solve the following problems : 1. If the product of the two remainders obtained by first subtracting 3 from a certain number, and then 5 from the same number, is 24, what is that number? How many solutions has this problem? Explain. 2. If the sum of two numbers is 12 and one of these numbers is x, what is the other number? Find two numbers whose sum is 12 and of which the square of the larger is 1 less than 10 times the smaller. * The expression a; ::^2 is read " x is not equal to 2." t § 26 should now be re-read. 72] FACTORING 111 3. The difference between two numbers is 2, and the sum of their squares is 130. What are these numbers? 4. One side of a rectangle is 3 feet longer than the other. If the longer side be diminished by 1 foot and the shorter side increased by 1 foot, the area of the rectangle will then be 30 square feet. How long is this rectangle ? 5. A rectangular orchard contains 2800 trees, and the number of trees in a row is 10 less than twice the number of rows. How many trees are there in a row? 6. If the dimensions of a certain rectangular box, which contains 120 cubic inches, were increased by 2, 3, and 4 inches, respectively, the new box would be cubical in form. Find the dimensions of this box. 7. Plow may $128 be divided equally among a certain number of persons so that the number of dollars received by each person shall exceed the number of persons by 8 ? 8. A certain club banquet is to cost $75, and it is found that this will require each member of the club to pay 50 cents more than ^q as many dollars as there are members in the club. How much must each pay, and how many members are there in the club ? CHAPTER VIII HIGHEST COMMON FACTORS — LOWEST COMMON MULTIPLES I. HIGHEST COMMON FACTORS 73. Definitions. A factor of each of two or more numbers or algebraic expressions is called a common factor of these numbers or expressions; the highest common factor — usually designated by the letters H. C. F. — of two or more numbers or expressions is the product of all the prime factors (§ 63) that are common to these numbers or expressions. E.g., the H. C. F. of 12 a^h'^cx^ and 6 ah^x.^y is 6 ah^x'^, because when this factor is removed from the given expressions they have no comrnon factor left ; 6 ab^^ is then the product of all the common prime factors of the given expressions. Similarly, 3a(a;-l)2(a; — 2) is the H.C.F. of 6 a2x(x — l)4(a; — 2)(a — ?/) and 15 ab{x — y){x — l)2(a; — 2)3. Note. It is evident from the above definition that no common factor of two or more expressions is of higher degree in any letter than their H. C. F. Two or more numbers or algebraic expressions which have no common factor except unity are said to be prime to each other. 74. Highest common factor of two or more monomials. From the definition and illustration given above, it is clear that the H. C. ¥. of two or more monomials can be found by inspection. E.g., to find the H. C. F. of 12 a%^xy, 6 ab^z^, and 9 ab^^. Inspection shows that these monomials have the prime factors 3, a, b, b, and x in common, and that, when these are removed, there are no other factors common to the given monomials ; hence their H. C. F. is 3 • a • 6 • 6 • x, i.e., 3 ab^x. A rule for writing down the H. C. F. of several monomial ex- pressions may be formulated thus: to the H.C.F. of the nu- merical coefficients annex those letters that are found in each one of the given monomials, and give to each of these letters the lowest exponent which it has in any of the monomials. 112 73-75] HIGWEST COMMON FACTORS 113 EXERCISES Find the H. C. F. of the following sets of monomials: 1. 3 a2j8c(^ and 6 a6 Vrf3. 2. 15 x^z, 24 xYz\ and 18 x^. 3. 16 x'^yHhn% 169 y^z^, and 39 x'^y^m*. 4. 2041 a^i^cT and 8476 a%c^d. 5. 292 x^y'^z^, 1022 x^^^^ and 1095 x^^*. 6. 364 x-'^if'^z^ and 455 x'^y'^^'z^. 7. Is the H. C. F., as above defined, the same as the greatest common divisor (G. C. D.) in the arithmetical sense? What is the H. C. F. of aH^y and a^xy^'i Is this H. C. F also the G. C. D. when a = ^,x=Q, and y — ^'i Note. Observe that highest refers to degree, while greatest refers to value. If c is any proper fraction, then c > c^ > c^ • ••, but c" is always higher than c^. Find the H. C. F. of the following sets of expressions : 8. 24 aH{y - zy{w + 3) and 56 a%x^{y - zy(w + 3)2. 9. 473 hhH(^x - 1)2(3 - 2 yy and 319 a^hs\x - l)(x - 2)2(3 - 2 yy. 75. H. C. F. of two or more polynomials whose prime factors are known. The H. C. F. of several polynomials whose prime factors are known may be written down by inspection as is done for monomials in § 74. EXERCISES Find the H. C. F. of each of the following sets of expressions: 1. 4(rt + J)3(a-J) and J(a + &)2(a - &)2. 2. Q(a + by{a-hy and 15(a- &)2(a + &). 3. 4 aa;2 - 20 ax + 24 a and 6 06^2 + 24 ahx - 126 ah. Solution Since 4 a3:2- 20 ax + 24 a = 4 «(a;2- 5 a; + 6) = 2 • 2 a(x- 2) (x - 3), and 6 a&x2 + 24 ahx - 126 a6 = 6 a6 (a;2 + 4 a; - 21) = 3 . 2 a& (x + 7) (a; - 3), therefore the H. C. F. is 2 a (a; — 3) . 4. a2-&2, a{a + h), and a2 + 2a& + 62. 5. 5-19iP-4a:2 and 2^2 + 7 a: -15. 114 ELEMENTARY ALGEBRA [Ch. VIII 6. z2 + 5 :c + 6, x2 + 7 a; + 10, and x^ + 12 a; + 20. 7. a^-a-12 and a^ - 4 a - 21. 8. 15(^2 - z) and 35(?/% - ?/2;). 9. X* + a:^^'^ + ?/'' and (x^ - a:?/ + ?/2)2, 10. Of what is the H. C. F. of two or more expressions composed? State a rule for finding the H. C. F. of two or more expressions which are already separated into their prime factors, or which may be easily so separated. 11. What is the H. C. F. of x^x-iy and a:(a;2-l)? Is this also the G. C. D. of these expressions for all values of x V Try 05 = 3, and also x = i. Compare Ex. 7, § 74. Find the H. C. F. of the following sets of expressions: 12. 4 afeV -f 12 ab^x - 40 ah\ 6 aH'^y - 6 a^xy - 12 o?-y, and 18 a%x2 - 54 a'^mx + 36 ahn. 13. 15 a4a;2 + 15 a%H^ + 15 }fix'^ and 3(a2 _ aW- + h% 14. a:8 + aS and 3 a^ + 3 a^ - 5 ax^ - 5 x\ 15. 2 a;2 - X - 3 and 2 x^ + 11 a;^ - a: - 30.* 16. (a;+3)(a:2-4), x^H- 4a:3 + 2 a;2 - x+ 6, and 2 a:8 + 9 a;2+ 7 x - 6. 17. a3 + l, 3a3-4a2 + 4a-l, and 2a3+a2-a + 3. 76. H. C. F. of two polynomials neither of which can be readily factored. Although, it is only in exceptional cases that the factors of a polynomial can be found (such cases were examined in Chap- ter VII), yet the common factors of any two given polynomials can always be found. The method for finding the H. C. F. of two polynomials neither of which can be readily factored, is precisely the same as that used in arithmetic for finding the G. C. D. of two numbers, neither of which can be easily factored. * Since the second of these expressions is not easily factored, — although the first is, — find by trial whether the factors of the first expression are also factors of the second. This method may be employed whenever any one of a given set of expressions is easily separated into its prime factors. 76-76] HIGHEST COMMON FACTORS 115 To illustrate, let it be required to find the G. C. D. of 1183 and 2639. 1183)2039(2 2366 273)1183(4 1092 91)273(3 273 The last divisor, 91, is the G. C. D. of the given numbers. This work may be more compactly arranged thus : Quotients 1183 1092 91 2639 2366 273 273 Similarly, the H. C. F. of x4_{_3x3 + 222_3a;_3 and x^ + x^ — 2 may be found thus: Quotients a* + 3 a;3 + 2x2- 3a;- 2cc 2x3 + 2x2— X — 3 2 xs + 2 x2 — 4 x + 1 x + 2 x2-2x-2 X8 + X2 ■ X3 — X2 2x2- 2x2- 2 2x 2x-2 2x-2 and — x + 1, which is the last divisor, is the H.C.F. of the given polynomials.* The procedure illustrated above may be formulated in words thus: Arrange the given polynomials according to the descend- ing powers of some common letter, and divide the higher expression hy the lower, continuing the division until the remainder is of lower degree than the divisor; then using this rejnainder as a divisor, with the preceding divisor as a dividend {and with the same letter of arrangement), divide as before; continue this process until the remainder is either zero, or free from the letter of arrangement : — if ' it is zero, the last divisor is the H. C. F. sought ; and (cf . § 77) if it is free from the letter of arrangement, the given ex- pressions have no common factor containing that letter. * The H. C. F. of these polynomials may also be regarded as x — 1. Why ? 116 ELEMENTARY ALGEBRA [Ch. VIII EXERCISES By the above method, find the H. C. F. of the following pairs of expressions : 1. a:2 + 5 a: + 6 and 4 a;3 + 21 a:2 4- 30 ar + 8. 2. 12 a;4 - 8 a;3 - 55 a;2 - 2 a; + 5 and 6 a;^ - a;^ - 29 x - 15. 3. 6 a2 - 13 a - 5 and 18 a^ - 51 a^ + 13 a + 5. 4. 5n4-10n3+lln2-6n+l and 10n5_5n4_7w34-19n2-14n + 2. 77. Fundamental principle. The success of the method em- ploj^ed in § 76 for finding the H. C. F., whether in arithmetic or algebra, is due to the following principle: // an integral algebraic expression * he divided hy another such expression which is of the same or of a lower degree in the letter of arrangement, and if there he a remainder, then the H. C. F. of this remainder and the divisor is also the H. C. F. of the two given expressions. To prove the correctness of this principle, let Ei and E2 repre- sent any two given integral expressions, and let the degree of E2, in the letter of arrangement, be at least as low as that of Ei ; also let Qi and Ri represent, respectively, the quotient and remainder when El is divided by E2 ; then (§ 47, Ex. 11), E, = q,E2 + Bi, (1) whence, Ri= E^ — q^E^. (2) Now since any factor of each term of an expression is a factor of the whole expression, therefore any factor common to E2 and Ri is also a factor of gi^/a + R\, and therefore, by equation (1), of Ex ; i.e., all the factors common to R^ and E2 are also factors of jEJi, and therefore common to E2 and Ey But, by exactly the same reasoning, equation (2) shows that all the factors common to E^ and E^ are also common to E.^ and R^ ; * " Integral expression " as here used includes arithmetical numbers also. 76-77] HIGHEST COMMON FACTORS 117 i.e., the factors common to Ri and E2 are precisely those which are common to E^ and Eo. Hence the H. C. F. of R^ and E2 is also the H. C. F. of El and E^. From the proof just given it follows : (1) that if E2 be now divided by R^, giving a remainder R2, then the H. C. F. of R2 and i?i is also the H. C. F. of ^2 and R^, and therefore of E^ and E2. So, too, if J?i be divided by R2, giving a remainder R^, then the H. C. F. of R2 and R^ is also the H. C. F. of E^ and E^, and so on ; i.e., the H. C. F. of E^ and E2 is also the H. C. F. of any two consecutive remainders in this succession of divisions. But these successive remainders are of lower and lower degrees,* hence a remainder i?„ which is either 0, or free from the letter of arrangement, must finally be reached ; if i?„ = 0, then Rn-i is the H. C. F. of i?„_i and i?n_2, and therefore of E^ and E2, but if Rn is merely free from the letter of arrangement, then Rn-\ and R^-^ can have no common factor containing this letter, and therefore E^ and E2 have no common factor which contains that letter. Note. It follows directly from the definition (§ 73) that the H. C. F. of two entire expressions is not altered by multiplying or dividing either of them by any number which is not a factor of the other. By introducing and suppressing suitable factors during the divisions above described, fractional coefficients, which might otherwise arise, may always be avoided. To illustrate, let it be required to find the H.C.F. of 3x3 + 8x2 + 3x — 2 and a3_2a;24-a; + 4. Since these expressions are of the same degree, either one may be used as divisor; the work may be arranged thus: Before beginning the second division the factor 14 is sup- pressed (see note \ above), and later 2 is also suppressed; fractional coeffi- cients are thus avoided. 3a;8 + 8x2 + 3x- 2 3a;3-6a;2 + 3x + 12 3 x-2 x-1 x8-2x2 + x + 4 X3 -X 14)14x2-14 x2-l -2x2 + 2x + 4 -2x2 +2 X2 + X 2x + 2(2 -x-1 -x-1 x + 1 and x + 1, which is the last divisor, is the H. C. F. of the given expressions. * If El and E^ represent arithmetical numbers, then i?i, R.2, and R2, sent smaller and smaller numbers. repre- 118 ELEMENTARY ALGEBRA [Ch. VIII As a further illustration, let us find the H. C. F. of a;4 + 4a;3 + 2a;2 — x + 6 and 2a;8 + 9a;2 + 7a; — 6. Before beginning the division the fac- tor 2 is introduced so as to avoid frac- tional coefficients in the quotient (cf . note above) ; later — 2 is introduced for the same purpose; and finally —3 is rejected. cb2 + 5 a; -f 6, which is the last divisor, is the H. C. F. of the given expressions. 2 2a;-l 2a:3 + 9x2 + 7a;-6 2a:3 + 10a;2 + 12a; 2a;44-8a;8 + 4a;2-2x + 12 — a;2 — 5 a; — 6 — a;2 — 5a; — 6 -a;3-3a;2 + 4a; + 12 -2 2a;8 + 6a;2-8a;-24 2a:3 + 9:r2 + 7a!- 6 -3)-3a;2-15a;-18 x^ + 5x + 6 EXERCISES By the above method find the H. CF. of the following pairs of expressions : 1. x^-dx^+dx-1 and x* -2 x3 + 2 a:2 -2 a: + 1. 2. 8 x3 - 22 a:2 -F 17 a; - 3 and 6 a;8 - 17 a;^ + 14 x - 3. 3. a:6-4x4-f 5a:3-3a:2 + 3a:-2 and 2 x^ - o x^- + x + 2. 4. a:6-4a;4 + 5a:2_2 and Sx^ + 5x + 2. 5. a;5_ 2x4 -2x8-11x2 -a:- 15 and 2 x5-7 xH4x8-15x2+x-10. 6. x6 + x4 + x8-x-2 and 3x6 + x6-x2-l. 7. a8 + 3a2-2a-6 and 4a2- a + 0^+ 4a4 - 12-}- 4a8. 8. 1 - 4 m8 -f 3 m* and 1 - 5 w3 -f 4 m* -h m - tw^. 9. x5-3x4-3x8-15-19x and 3 x^ - 3 x^ -K x^ - 15 -f 9 x2- x. 10. What is meant by the H.C.F. of two expressions E^ and iJg^ If a is not a factor of E^, how does the H. C. F. of jEJ^ and a • E^ compare with the H. C. F. of E^ and E^f Why? Compare § 77, note. 11. If a is a factor of E^, but not of E2, how does the H. C. F. of E^ and a • E^ compare with the H. C. F. of E^ and JEJg? In introducing and suppressing factors during the process of division (§ 77), what special precaution must be exercised, and why ? 12. Suppose that, at some stage of the work in an exercise like those above, the divisor is 2 x2 — 4 x -{- 2, and the dividend x8 — 3x2 + 3x-fl; what would be the eifect on the final result if the factor 2 were intro- duced into the dividend to avoid fractional coefficients? What should be done in this case instead of introducing the factor 2 ? Why? 77-78] HIGHEST COMMON FACTOBS 119 13. Show that every factor common to A and B is also common to A — B and A + B; and also to 7nA + nB and mA — nB. Is the H. C. F. of A and B necessarily the H. C. F. of ^ - 5 and ^ + -B? 78. Supplementary to §§ 76 and 77. (i) H.C.F. of poly- nomials which contain monoTnial factors. The problem of finding the H. C. F. of a pair of polynomials, either of which contains monomial factors, is usually much simplified by setting aside these monomial factors before the division process is begun. Factors which are common to the given polynomials must, of course, be reserved as factors of their H. C. F. ; all others may be rejected. Thus, to find the H. C. F. of 6x5 + 18a;4 + 12x3-18a;2— 18x and Sax^ + 3ax^ — 6ax, remove the monomial factors Hx and 3 ax from the given expressions, and the remaining polynomial factors are, respectively, x^-{-3x^-{-2x^ — Sx — S and a;8-f-a;2_2 ; the H.C.F. of the monomial factors is Sx, and the H.C.F. of the polynomial factors is a; — 1 (see illustrative example, § 76) ; hence the H. C. F. of the given polynomials is 3 x(a; — 1). (ii) H. C. F. of polynomials zvhich involve several letters. Although the examples given in § 77 involve only one letter, yet it should be especially observed that the demonstration there given applies to expressions involving any number of letters. Thus, if the given expressions involve several letters, then, to find whether they have a common factor containing any particular one of these letters, they need only be arranged according to the descending powers of that letter, and divided as above described. If, therefore, the given expressions be successively arranged according to each of the several letters which they have in common, and divided as above, then all their common factors {i.e., their H.C.F.) will be found. Manifestly, however, any common factor which contains two or more letters will be found when the given expressions are arranged according to any one of these letters. (iii) H. C. F. of three or more polynomials. Since the H. C. F. of three polynomials is a factor of each of them, it is also a factor of the H. C. F. of any two of them ; therefore the H. C. Y. of three polynomials is found by first finding the H. C. F. of any two of them, and then the H. C. F. of that result and the third polynomial. By continuing this process the H. C. F. of any num- ber of polynomials may be found. 120 ELEMENTARY ALGEBRA [Ch. VIII EXERCISES Find the H. C. F. of : 1. 21 ax — 17 ax^ — 5 ax^ + ax^ and 5 ax^ — 34 ax^ — 1 ax. 2. 7 m^^pS — 49 in^x + 42 m^ and 14 ahnx^ + 14 a^mx^— 56 a%a: — 56 a'hn. 3. 48s3te4-162s3fxH54s3^ and 18 s^^a^^- 9 s%%a;- 48 s2f%a:2+ 2452^2^^x3. 4. 6 ca:3(l + 2/2) - 18 cx^z + 2 cy^ - 4: cyH + 12 cz^ - 2 C2(3 z/+ 2) + 2 cy and 2 a?/4 + 2 ax2(3/2 _ 3 ;2) _ 6 03/2^ + 2 a(a:2 _ 2/2') ^ 4 ^(3 ^ _ i). 5. 4 a;4 - 12 x^^ + 5 x22/2 + 12 xi/^ -^y^ and 12 a;4 - 36 a:^^/ + 11 a;22/2+ 48 a:3/3 _ 36 3^*. 6. 7?2n(a:2 + 2/2)+ x?/(?n2 _|_ ^2) ^nd 'mn{x^ + 3/^)+ xy{m^y + n2a:). 7. 3 ax2 - 6 a2x + 9 a3 _ 3 a:2 + 6 ax - 9 a2 and 6 a2x2 + 24 aH + 6 «*_ 6 x2 - 24 ax - 6 a\ 8. Show that the proof given in § 77 applies to expressions contain- ing any number of letters. 9. Explain fully the method of finding the H. C. F. of more than two expressions. 10. Why must the H. C. F. of any number of expressions be a factor of the H. C. F. of any two of these expressions ? Must it be the H. C. F. itself of any two of the given expressions ? Explain. FindtheH.C.F. of: 11. a4 + 4 a3 + 4 a\ a% - 4 ah, and a% + 5 a^^ 4. 6 a%. 12. x8 - 6 x2 + 11 X - 6, x8 - 9 x2 + 26 X - 24, and x^ - 8 x2 + 19 x - 12. 13. a3 + a2x-2x3, a8+3a2a;+4ax2+2x3, and 2 a8 + 3 02-^+2 ax2-2x8. 14. ax + }p-x + cH - acy - h'^cy - c^y, a2 4. 2 a& + a62 ^ 2 i^ + 00^+ 2 hc\ and 2 a2 + 2 alP- + c^h + 2 ac^ + 6^ + ah. 79.* Other important consequences of § 77. Some further im- portant conclusions may be easily drawn from such a series of divisions as that described in §§ 76 and 77 ; thus, if iHf and N are any two integers, of which M is the greater, and if M be divided by JV, giving a quotient Qi and a remainder i?i, and if -^be then divided by iJi, giving a quotient Q2 and a remainder i?^? ^iid so on, — sub- sequent quotients and remainders, all of which are, of course, * This article may be omitted on a lirst reading. 78-79] HIGHEST COMMON FACTORS 121 integers, being designated by Q^, Q^, Q^, "-, and R^, R^, R^, ••, respectively, — then (§ 47, Ex. 11) • M=Q,N'-\-R„ N=q,Ri+R2, Ri=QsR2+R3, ^2=9^+^4, etc. From this series of equations it is easy to express the several remainders Ri, Ro, R^, •••in terms of M, N, and the quotients Qi, Q,2j ^3? "•• Thus, by transposing, the first equation becomes Ri=M—QiN., transposing in the second equation, and then substituting this value of Ri, gives R, = N-q,R^=N-Q,{M-Q,N) = -q,M+(X+QiQ2)N) similarly, from the third equation, R^=R^-Q,R,={M-q,N)-q,\{i + q,q.:)N- q,M] = (1 + Q2Q3) M- (Qi + 4 + q,q,q,) n; and so on for the later remainders; i.e., the successive re- mainders may each he expressed in the form aM+ hN, wherein a and b are integers (one positive and the other nega- tive), which involve the successive quotients, hut not the given numhers, nor the remainders. Again, if M and N are prime to each other, then (§ 77) the last remainder is 1, and therefore, by what has just been said, two integers a and h can be found such that aM-\-hN=l. From this last equation it is easy to establish the following important principle : if M is a factor of NL, hut is prime to N, then it is a factor of L. To prove this it is only necessary to multiply the above equa- tion by L ; this gives aML + hNL = L, wherein the first member is manifestly divisible by M (M being a factor of NL by hypothesis) ; therefore the second member, viz., Z, is also divisible by J/, which was to be proved. 122 ELEMENTARY ALGEBRA [Ch. VIII EXERCISES The following direct consequences of the principle just now established may be proved by the student: 1. Ji M is prime to N and also to L, then it is prime to the product NL. 2. ]f ilf is prime to N, L, P, •••, then it is prime to the product NLP •••. 3. A number can be separated into but one set of prime factors. 4. If M is a prime to N, then it is prime to any integral power of N. 5. Show that, with slight verbal modifications, the principles proved above apply also to integral expressions involving one or more letters. II. LOWEST COMMON MULTIPLES 80. Multiples of algebraic expressions. A multiple of an alge- braic expression* is another algebraic expression that is exactly divisible by the given one, i.e., it is an algebraic expression that contains all the prime factors of the given expression. A common multiple of two or more algebraic expressions is a multiple of each of these expressions. E.g., 12 a4a;3 (2/2—1) is a common multiple of.3a'^x^{i/-{-l) and 2a^z{y — l). The lowest common multiple — usually v^rritten L. C. M. — of two or more algebraic expressions is that algebraic expression of lowest degree which is exactly divisible by each of the given ex- pressions; it is that expression which contains all the prime factors of each of the given expressions, but no superfluous factors. From these definitions, it is easy to find a common multiple of any two or more algebraic expressions whose prime factors are known. E.g., a common multiple of a%'^x^ and a^x^fj* may be found thus : Since a^ is the highest power of a that is found in either of these expressions, therefore any common multiple of the given expressions must contain the factor aS; it mar/, of course, contain a still higher power of a. Similarly, a common multiple of these two expressions must contain &2, x^, and y^ as factors. More- over, any expression which contains among its factors a^, b^, x^, and y^, is exactly divisible by each of the given expressions, and is, therefore, a common multiple of them. The L. C. M. of these expressions is that one of their common multiples which contains no factor that is superfluous; it is a%^x^y^. Similarly, 6 a^x^{x — 2)^(x — 1)3 is a common multiple of a^x{x — 2)^(x — 1) and x^ix — 2){x — 1)3, but it is not their L. C. M. , because it contains the factor (x—2)^ when only (x— 2)2 is needed, and it contains the further superfluous factor 6; the L. C. M. of these given expressions is a^x^{x — 2)^(x — 1)3. * "Algebraic expressioas " as here used include arithmeticalnumbers also. 79-80] LOWEST COMMON MULTIPLES 123 A rule for writing down the L. C. M. of two or more monomials, or of any two or more entire algebraic expressions ichose prime factors are either known, or can easily be found, may be formulated thus: write down the indicated product of the different prime factors that enter into any of the given expressions, giving to each of these factors the highest exponent which that factor has in any of the given expressions. EXERCISES Find the L. C. M. and H. C. F. of 1. 8 a%'^, 24 a'^b'^c^, and 18 ahcK 5. x"^ - if- and x^ + 2 a:y + y\ 2. 15 a%\ 20 a%2c2, and 30 ac^. 6. 21 a;^ and 7 x'^{x + 1) . 3. 16 aWc, 24 aMc, and 36 a^W^. 7. x^ - 1 and x2 + x. 4. V^a%r\ V^pYr, and 54 a&y. 8. ^x'^y-y and 2 x'^ + x. Find the L. CM. of: 9. a + i, a - &, a2 + h"^, and a* + h\ 10. 3 + a, 9 - a2, 3 - a, and 5 a + 15. 11. x^ — y% x'^ + xy + y^, and x'^ — xy. 12. 4 a + 4 ^>, 6 a2 - 24 &2, and a2 - 3 a6 + 2 h\ 13. x^ + y^, x^y — ?/*, and x^ — y^. 14. 2/2 _ 5 y _^ 6 and ?/2 - 7 ?/ + 10. 15. x'^ ~ {a -\- b)x + ab and x'^ — (a — b)x — ab. 16. Is 12 a%^(x'^ — if) a common multiple of 2 a%{x — y) and 3 ab\x - y) ? Is it their L. CM.? 17. What is the essential requirement in order that one expression may be a common multiple of two or more others? that it may be their L. CM.? Find the L. C M. of 18. 3 x2 + 7 x + 2 and x"^ - x - Q. 19. a2 + 4 a + 4, a^ - 4, and «4 _ 16. 20. (« + 6)2 _ e2 and (a + & + c)2. 21. a:'-" - ?/2n and (a:" - ?/")2. 22. a;3 + 6 a:2 + 5 a: - 12 and a,-3 - 8 a;2 + 19 a: - 12. Suggestion. Use § 67 to find one factor of each of these expressions. 23. a;3 - 6 a;2 + 11 X - 6 and a;3 - 9 x2 + 26 a: - 24. 24. o3 + 2 a2 - 4 a - 8, a8 _ ^2 _ 8 a + 12, and a^ + 4 rt2 - 3 a - 18. 124 ELEMENTARY ALGEBRA [Ch. VIII 81. The L. C. M. of two entire algebraic expressions found by means of their H. C. F. The use of the H. C. F. in finding the L. C. M. may be better understood if a particular example be first worked out before the general discussion is given. Let it be required to find the L. C. M. of Sx^ — x^ — a?-\-x — 2 and 2 ar^ - 3 or - 2 a; + 3. By § 76 it is found that the H. C. F. of these expressions is a? — 1] they may, therefore, be written thus : ^x"^ - a? - x" + x -2 = {x^ -l){^o? - X + 2), and 2 x^ -^ x'' - 2 X + ^ = (x" -1){2 X - 3), wherein Za? — x + 2 and 2 a? — 3 have no common factor. Hence the L. C. M. of the given expressions is (a^ - 1) (3 a^ - .T + 2) (2 a; - 3). * Similarly, in general, let Ei and E2 be any two entire algebraic expressions, and let their H. C. F. be F) then they may be written : and E2 = FQ2, wherein Qi and Q2 have no common factor, since F is the H. C. F. of El and E2. Hence the L. C. M. of Ei and E2 is the product of Ff Qi, and Q2, i.e., it is FQ1Q2. Moreover, since E^- E2 = FQi • FQ2 = FiFQ^Q^, therefore the product of any two entire algebraic expressions is equal to the product of their H. C. F by their L. C. M. Hence : to find the L. C. M. of any two entire algebraic expres- sions, divide the product of the given expressions hy their H. C. F. ■r.. , , . ^ ,, . EXERCISES Find the L.C.M. of: 1. a:3 - 6 a:2 + 11 a: - 6 and x^ - 9 a:^ + 26 a; - 24. 2. a;8-5 a:2 - 4 z + 20 and x^ ■\-2x'^-2ox- 50. 3. 2 !/3 - 11 3/2 ^ 18 ?/ - 14 and 2 ?/3 + 3 .y2 _ 10 2/ + 14. 4. 6 a^a; - 5 a^x - 18 ax - 8 ar and 6 a% - 13 a'-b - 6 ab + 8 b. 5. 4 x* - 17 xY + 4 ?/* and 2 x* - xhj - S x^y^ -5xy^-2 y\ 6. 2 x4 - 9 a;3 + 18 x2 - 18 a: + 9 and 3 a:* - 11 a;^ + 17 a;2 - 12 a: + 6. * This is the L. C. M. because it contains all the necessary factors, and none that are superfluous. 81-82] LOWEST COMMON MULTIPLES 125 82. The L. C. M. of three or more expressions. The L. C. M. of three or more entire algebraic expressions, whose factors are not easily determined, may be found by first finding the L. C. M. of two of the given expressions (§ 81), then the L. C. M. of that result and another of the given expressions, and so on. T.. , , . ^ ,, . EXERCISES Find the L. C. M. of : 1. a;4 - 2 a;3 + x2 - 1, a:* - a;2 + 2 a: - 1, and a:* - 3 z^ + 1. 2. a:8 + 3 x^ - G ;r - 8, x3 - 2 a:2 - X + 2, and x^ + x - Q. 3. a;2 - 4 a^, a:3 + 2 ax^ + 4 cfix + 8 a^, and x^-2ax'^^-^ a^x - 8 a^. 4. If A, B, and C stand for any tliree given expressions, and if 31^ is the L. C. M. of A and B, while M.j, is the L. C. M. of M^ and C, prove that M^ is the L. C. M. of A, B, and C. Find the L. C. M. of : 5. a3 + 7 a2 + 14 rt + 8, a3 + 3 rt2 _ 6 n - 8, and a^ + a^ - 10 a + 8. 6. ^3 _ 9 ^2 + 23 ^ - 15, P + ^.-2 _ 17 ^ + 15, and P + 7 ^•2 + 7 ^ - 15. CHAPTER IX ALGEBRAIC FRACTIONS 83. Definitions. An algebraic fraction is an indicated division in which the divisor, or both dividend and divisor, are algebraic expressions, and the dividend is not a multiple of the divisor. E.g., 5^ ix — 2y), x^-^y, and Sax-i- {a^ — x^) are algebraic fractions. Fractions in algebra are written in the same form as that used in arithmetic, and the parts are called by the same names, i.e., the dividend is called the numerator, the divisor is called the denominator, the numerator and denominator taken together are called the terms of the fraction, and the numerator is usually written above the denominator, from which it is separated by a line. E.g., the fractions 5-^ (x — 2?/), x^-^y, and 3aa;-^ {a^ — x^ are usually written ^ ^^ and f "^ ■ respectively. x — 2y y a^ — x^ An algebraic fraction is called a proper fraction if its numerator is of lower degree than its denominator, otherwise it is called an improper fraction. E.g., ^ is a proper fraction, while _^' is an improper fraction. An expression which consists of a part that is fractional and a part that is integral is called a mixed expression. E.g., m +-, a-\ —^-, and x-\-y — are mixed expressions. p a-\- c x — y Observe the difference in writing a mixed number in arithmetic and a mixed expression in algebra: 5| means 5 + | in arithmetic, while in algebra m- means m ' -, and not m + -• P P It is sometimes desirable to write an integral expression in the form of a fraction; this is done by using 1 as the denominator; e.g., a? — 2x, in the form of a fraction, is ^" ~ ^ ' 126 83-86] ALGEBRAIC FBACTIONS 127 Attention is again called to the fact that algebraic expressions may be frac- tional in form and yet, for certain values of the letters involved, represent integers, and vice versa [cf. § 7, (v)]. 84. Operations with algebraic fractions. As in arithmetic, so in algebra, it is often necessary to reduce fractions to their " lowest terms" and to a "common denominator," and also to change mixed expressions to improper fractions, and vice versa. The operations of addition, subtraction, multiplication, and division must also often be performed with algebraic fractions. Moreover, since algebraic expressions represent numbers, there- fore the principles which were demonstrated in § 54 apply to algebraic as well as to arithmetical fractions, and all of the above operations are therefore essentially the same in algebra as in arithmetic ; the student should carefully observe this similarity in the next few articles. 85. Converting an improper fraction into a mixed expression. This change in form is made in precisely the same way as the corresponding case was treated in arithmetic. E.g., just as V" = 3^> *-^-» 3 + J, so, too, since a fraction is an indicated divi- z^ + z + 1 x2 4-a; + i EXERCISES Keduce each of the following improper fractions to an equivalent mixed expression, and explain your procedure : ^ a^-2ab + c ^ a^ + a ^ + 1 3a: ' 3 2 a:^ + ax - 3 a^ g 4. -"-I-IL^. 9. X + 2 5 x^- a:8 -23;''-2a;- 1 ^q x'^-x-l ' ' a:2-3x+l 11. Is ^ ~ -^ + — a proper or an improper fraction ? Why ? 5 a^ — 8 a + 3 a 3 a:2 + 9 a; + 2 3a: 2 a:2 + aa: - 3 a2 x-\-a t + 16 a+1 . 8x3- I0a;2-3a; + 5 4 a2 - 3 3 x6 + 2 X - 5 x8 + 2x+l 7x6-1 X3 + X + 1 18 x4 - X3 - 2 X2 -7 128 ELEMENTARY ALGEBRA [Ch. IX Reduce the following mixed expressions to equivalent improper frac- tions, and check the correctness of your work (cf. Ex. 7, § 39) : 12. 2x + ^^^1^^. 14. :r + y + e - V^^^^- x^ + >> // x^ — y — z 13. Qy-x-\- — — 15. 3a-26 + c- 4 + *"^ 4:y^ + X a — 5 b + 2 c 86. Reduction of fractions to lowest terms. In § 54 (v), it was shown that any factor which is found in both terms of a fraction -may be rejected (canceled) without changing the value of the fraction. 3a^_3ax, , x^-1 _ {x-\-i) (x-1) ^x + 1^ '^'' 4.bxy^4:by' x'^ — 2x-i-l {x — l){x — l) x — 1 In algebra, as in arithmetic, a fraction is said to be in its lowest terms when the numerator and denominator have no com- mon factor; hence, a fraction may always he reduced to an '^equivalent fraction in its loiuest terms by dividing both its numerator and denojjiinator by their H. C. F. fi a'^x7/^ E.g., to reduce - — z'-- to its lowest terms, divide both numerator and denomi- 9 ax^i/^ nator by 3 axy^, which (§ 73) is their H. C. F. Instead of dividing both terms of a fraction by their H. C. F., and thus redu- cing the fraction to its lowest terms in a single operation, the same result may, of course, be accomplished by canceling any common factor as soon as it is dis- covered, and continuing this process until the resulting numerator and denomi- nator are prime to each other. Recourse to the H. C. F. is necessary only when no common factors can be found by other methods. Observe that it is only equal factors, and not equal parts, that may be canceled. E.g., 5^"t^ - is not equal to |f ; nor is f^^^i^ equal to -^^^^. ' "^ 5&C 6s — 5^2^ 3 s — 5 n^ EXERCISES Reduce each of the following fractions to its lowest terms : 1. ^^JZJI^. 4 a^ + 2ab + b^ ^ a^ + b» &2 a8 + 63 a^ + a2b' + b^ 2 34 a^^c* g 2 x^ + 3 a: + 1 g 3 a^ _ 2 a - 1 51 a^b^c ' ' x'^+5x + 4: ' ' l + a~a^-a^ ^ ap. - yi ^ a:« + y8 ^ ^ a* - ^2 - 20 (a -6)2 y^-x^ a*-9a2 + 20 85-87] ALGEBRAIC FRACTIONS 129 10 a:^ + 2 xy + if - z^ ^5 Sfl^-f OQa^ - q - 2 z^ + x^ + 7f-^2xij-^2xz + 2yz ' Sa^ + 17 a'-^ + 21 a - 9* j_j_ a^-h^ ^g a:5- 2x4 -2x3 -11x2-3;- 15 12. a^ + a^'ft + a^6"^ + aW + b^ 2 x5-7x4 + 4 x^- 15 x^-f x-lO x3+3x2 + 4x+2 ^.y a6(x2 -}- y^) .j. a,-y(a-2 + /j2-) x3-3x2-8x-10' * a&(x-'2-2/=2)+x^(a^-62)' j_3 x8 + x2 - 22 X - 40 j_Q x8-6x2v + 2x?/2 + 3?/3 8-7x2+ 2x + 40 x=^+0x2?/-2x?/'^-53/S j_^ 1 - 2 X - 5 x2 + 6 x3 ^g a2 ^ 52 _^ 2 c2 + 2 a5 + 3 ac + 3 ^»c * l + 5x+2x2-8x8' ' a2 + 6^ + c2+2a6 + 2«c + 26c ■ 20. May the factor 5 ax be canceled from the first two terms of the numerator and denominator of 5ax2- 10fl2^- + 3 6(x -2 a) ^-^^Yxout 15 a3x4 - 30 a4x3 + 6 6(x - 2 a) changing the value of this fraction? Why? 21. Is the value of a fi-action changed by canceling equal /ac/ors from both numerator and denominator? Is it changed by canceling equal parts or equal factors of parts of the numerator and denominator? 87. Changing fractions to equivalent fractions having given denomi- nators. Since multiplying both terms of a fraction by the same number does not change its value, therefore any given fraction may be reduced to an equivalent fraction whose denominator is any desired multiple of the given denominator. E.g., to reduce j—^ to an equivalent fraction whose denominator shall he 12 cx^y, multiply both terms of the given fraction by 3 cy. EXERCISES 1. If the denominator of a fraction be multiplied by any given ex- pression, what must be done to the numerator in order to preserve the value of the fraction ? 2. How find the expression by which it is necessary to multiply both terms of a given fraction in order that the new equivalent fraction shall have a given denominator? A given numerator? 3 a — 5 3. Reduce - — — ^ to an equivalent fraction whose denominator is 2 x(3 + ax) 18 X — 2 a^x^. Also to one whose numerator is 12 ax + 18 a?/ — 20 x — 30 y. 130 ELEMENTARY ALGEBRA [Ch. IX Find the required numerator in each of the following equations : 4. 3x-2a ? 5. x^-Sax+2a^ 2x^- 7 ax^+7 a^x -2a^ 4 ? (y - a) (a _ x) (3 - 4 i/) (a - y) (a - a:) (4 ?/ - 3) (3 - 7 y) g 3m -8 ^ ? ^ 3x ? 2x-5 -2a:+5 1 7a;2-3a; + 5 88. Reduction of fractions to common denominators. In § 87 it is shown that any given fraction may be reduced to an equiva- lent fraction whose denominator is any desired multiple of the given fraction ; if then any common multiple of the denominators of two or more given fractions be chosen as the new denominator, it is clear that these fractions may be reduced to equivalent frac- tions having this denominator in common. E.g., since 12 a^x^ is a common multiple of the denominators of — , — ^, and 2x 3 .t2 -— , therefore these fractions may be reduced to the equivalent fractions " •^ . o a 12 a2a;2 8 a^ 10 amx^ ^o 9 o ' ^^^ .,„ o o > which have the common denominator 12 a^x^. Similarly 12 a^x^ 12 a^x^ "^ for any given fractions whatever. In practice it is usually desirable to keep the denominators of fractions as small as possible, and therefore, instead of choosing any common multiple, as above, it is best to choose the L. C. M. of the given denominators. 2a „„■, 5x E.g., the L. C. M. of the denominators of — and (a;-l)(a; + l) (x-\-l){x + 3) is (x — l){x-h 1) (x + 3) , and these fractious are respectively equal to ix-l)T+~l)l + B) ^"^ i:c-l]T+l)l+S) ' ^"''°^^'.' *^^ ^^^^^ ^^"^^'""^ can not be reduced to equivalent fractions having a lower common denominator. To reduce tvs^o or more given fractions to equivalent fractions having the lowest possible common denominator, divide the L. C. M. of the ^iven denominators hy the denotninator of one of the given fractions, and then multiply both terms of that fraction by the resulting quotient; do the same with each of the given fra/itions. 87-89] ALGEBRAIC FRACTIONS 131 EXERCISES Reduce the following fractions to equivalent fractions having the lowest possible common denominator : 1. 3a+i and ^^±i. 5. ^ and 4 6 (m-l)(m-2) (2 - wO(m - 3) 2. ^-^^ and ^ + ^^. 6. ^ + y and ^ " -^ • 16 6 20 62 x'^ + xy^y^ x^-xy + y^ 3. ±±^ and ^Lll^. 7. £^ll, ^±i^, and ^' + ^^ a — b a + h x + y x — y x^ — y'^ 4. -^^1- and ^ + y . 8. ^ " -^ and ^' " ^ + ^ • X3 - 3/3 3;3 + 2^3 a-S _ 2/3 3.4.^ 3.22/2 _^ y* 9. , -, and 1 + X 1 — x^ x^ — 1 10. ^(--^'^ , ^(^^-^) , and ^« 2 rt6 + />' a^ + 2ab+ b^ a^ - b^ 11. 6x 3a ^^^^ 3a- 6x 12. 7 a:, 15 _ 13 a: + 2 0:2 a;^ - 8 a: + 15 ^2 - 2 a: - 15 b — X a — X _ J 3 :i.-2_62' x2 -(a + 6)a; + a6 13. ^--'^ , ^_, and ^(^ + ^) . a:2+7a: + 10 a;2 + a: - 2 a:2 + 4 a: - 5 14. , ^ + ^ , . "-^ , and « + 1 a2 - 4 a + 3' a2 - 8 a + 15' a2 - 6 a + 5 ^5^ 5(u - 3 .) ^ 8 and ^^-^^) . u - 2 V w2 — 5 My + 6 y2 ^^ _ 3 y 89. Addition and subtraction of fractions. As in arithmetic, so in algebra, the sum (or difference) of two given fractions which have a cormnon denominator is a fraction whose numerator is the sum (or difference) of the given numera- tors, and whose denominator is the common denominator of the §iven fractions [of. § 54 (vii)]. E.g. a^2 + a; + 5 a;2 + 3 ^ a;2 + a; +5-(a;2 + 3) Note 1. The minus sign before the second fraction means that all of that fraction is to be subtracted, hence the necessity for the parenthesis in the numera- tor of the next fraction. 132 ELEMENTARY ALGEBRA [Ch. IX Note 2. Since a fraction is a quotient, therefore its sign (i.e., the sign written before the dividing line) is governed by the laws of signs in division. Thus, if Ei and E2 are any algebraic expressions whatever, then — — l = + — ^ = + -^^. Hence the above example may also be arranged thus : x'^ + x + 5 a;2 + 3 ^ x'^ + x + b , -a;2-3 ^ a; + 2 a;2-2a; + l x^-2x + l a;2_2a; + i x'^-'lx + l x^-2x + i If the given fractions have 7iot a common denominator, they must be reduced to equivalent fractions which have a common denominator (§ 88) before they can be added or subtracted. ^ _1 3, 2 ^ xix + \) 3(a;-l)(a; + l) I 2x(x-l) ■•^■' x-\ X x + 1 a:(x-l)Ca; + l) a;(a;-l)(a; + l) x{x-l){x + l) _ x{x + l)—Z{x — \){x + \) + 2x{x — l) x{x-l){x + l) ^-x x{x-\){x + \)' 3 3. and this result, viz., ; — -, is called the "algebraic sum" of the given fractions. x{x-l){x + l) EXERCISES , Simplify the following expressions : 1 ^-^ ^ 4. ±±A. 9 ^ + "^ ^ + ^ a;2 - 3 x - 10 x2 + 2 X - 35 1 ^ re + 3 J a: + 7 , ^^ 1 3 2 5 10 3. a-\-x ^ a-x ^ j_3__ a — a; a -\- x 4. 1 + ^ + y— . 12. 2 a; - 3 y/ 4 a;2 - 9 2/^ g 2 a; — 3 g 2 x - a je -j^g a; — 2 a x — a 6. -3- + -1-. 14. a; + y a; — y a'-^ + ax + a;'-^ a + a.- n ^ 1^_ ,g a -\-h + c a — h -\- c 1 + X - 2 a;2 6 x2 - a; -2 ^ 1 + 1 . - 1 X + 2 - x2 1 1 (X -yy {x + yy a A-h a ■ -b cfi - 2 a6 + 62 a2 + 2 ab + b^ a2 — ax + x^ a — X x-e x-2 a-2_(/;+c)2 (a-6)2-c'-2 X X - - a — X , a + X 0.2 _ 2-2 8. ~t ± 16. " ~ ^ + 1 — x2 1 -f x2 X X 2 ax * Compare example under Note 2, above. 89] ALGEBRAIC FRACTIONS 133 17 ^ ~ ^ 1 ^ — c c — a ah be ac 18. 2 3 ?/2 -x^ ^ XI/ + f xy xy^ x'^y^ 19. h 3 a:. X — 2 a x — a Suggestion. 3 a: = — • 1 20. ^ 1 ^-y 3:2-a:.y X -\- y x^ — xy -\- y^ x^ -\- y^ 21. UL^ + lzi^ + a:. 1 — a; 1 + a: 22. 23. 24. 25. 26. 37. a; — a a;^ — a" a:(a: 1 -y) ^Ri 1 a:?/ 1 1 a:2- ■7x- + 12 a:2- 5 a:-F 6 2a;2 1 — X -1^ 1 3-a;- 2a:2 1 1 2a 27. 28. 29. 30. 31. 32. 33. 34. 35. a- 1 a (a — 1) 1 2 a a + l"^a + 2 1 1 x-1 2(x + 1) a a- 1 2 a + 1 a a 1 + 2 a 1 1 a: + 2(x2 + :} 1-x 2(a: + 1) 1) 7h- :. ^' 1-x 2 a:-3 a.-3 x + 4. + 64 b ai a62 a + 6 a — b b^ — a^ -1 1 a + 6 (a + />)2 (a + ft) 3 a:^ + ax^ _ x(x — a) _ 2 aa: aa;2 — a^ a{x -\- a) x^ — a'^ x(a -b) b -2i b^-x^ x + b 36 ^^ -^ _ 3 x(a - b) b -2a X — b 1 (a - ft) (a - c) (ft - c) (ft - a) (c - a) (c - ft) Is (rt — 6)(a — (^(^ — c) a common multiple of these three denomiuators? Is (a - 6) (6 — c) (c - a) ? 38. 1 1 + ^ X — 1 a: + 1 ' a- — 2 a; + 2 x^ -\- y^ x^ — y^ x^ — y^ 3 a; ^ 4 - 13 a: l + 2a; l-2a: 4a:2-l 42. 41. ^-+ 4« 1 a 1 + 2 a; + a x^ — cfi a — x x^ + a''^ 3 a;2-5a; + 6 3a:-2-a:2 4a;-3-a:2 43. 44. a: - 1 2 (a: - 2) a:-3 (x-2)(a;-3) (3-a:)(x-l) (a: - 1)(2 - x) a2 , ft2 c2 (a-ft)(a-c) (ft-a)(ft-0 (c-a)(c-ft) 134 ELEMENTARY ALGEBBA [Ca IX 45. -^-^ + ^« + «* 46. (a - c) (a -b) (6 - c)(b -a) (c - a)(c - b) 1 2 , L x^ - 5 a:^/ + 6 2/2 x"^ - ^ xy -\- 3 y^ x'^-3xy+2 ^^ g^ + 2 g + 1 _ 2 g^ - 2 g + 1 ^g a:« + ar^ + a: + 1 3 ^ ■g2-2g + l g2 + 2g + l' * x^-ar + l x-1 90. Reducing mixed expressions to improper fractions. Since an entire expression may be written in the fractional form with the denominator 1, therefore reducing mixed expressions to im- proper fractions is merely a special case of addition. '^'' x — 1 1 x — 1 z — 1 x — 1 x — 1 EXERCISES By the above method simplify the following expressions : 1. .-l+^!_. 6. 3a-Sb 186^-5o« a;2 — g + 2^> J.3 x*-\-x^-x + l 1 + a; + a.-2 ^ 4 + 4y2+y3 l-22/ + 2^2 4 g2 + 9 62 2a + 36 7 gx + 6a: + g6 2. x + 1-^^. 7. a;-a;2-: x — 1 a:2-2a: + l ^ ^ 4. l-y-yi-tuJl. 9. 2g-36 1 - 2/4 5. g2 _ aft + ft2 ^ . 10. 1 - ga: - ... rt + 6 1 - g6 + a:2 11. Prove that any mixed expression may be reduced to an improper fraction by multiplying the integral part by the denominator of the frac- tion, adding or subtracting the numerator as the case may be, and placing this result over the denominator. Also compare § 47, Ex. 11. 91. The product of two or more fractions. In algebra, as in arithmetic, the product of two or more fractions is a frac- tion whose numerator is tlie product of the numerators of the given fractions, and whose denominator is the product of their denominators [cf. § 54 (ii)]. 3x2 2a2?/ 5rt62 _ ma^h'^T^y Sa^b E.g., ^by^ 3x2 2x-Zy 12 bx^y^2 x - 3 y) 2yC2x-3y) ALGEBRAIC FRACTIONS 135 i'in d the product of : EXERCISES 1. «^^ and ^'< b^dh-^ ab-2 3. f ": and *"'"^ 6'»+2 a-" 2. Sxy ^^^ 16 ,V 8yz 9xY 4. "and * . a + 6 a - 6 5. «'-«^ and ^'+^-^. x^ — xy a^ + ab 6. ! and — ' - xy + y'^ X* + x-y'^ + y^ (a - by 7. Simplify (x -\-2y — ) ^ , making use of the distributive law. \ yl a + a; 8. Simplify (x + 2y — ] — ^ by first reducing the multiplicand \ y/a + X to an improper fraction (cf. Ex. 7). 9. Simplify f ?/ + 3 ^] f 2 ?/ + 3 - — -^ — ] by the method of \ y — 0/ \ 2 y — '6/ Ex. 7, and also by that of Ex. 8, and compare results. 10. Give a convenient rule for multiplication when one or more fac- tors are mixed expressions. p r pr 11. Prove that —.- = —;-, and show that the proof is still valid when some or all of the letters represent algebraic expressions (cf. § 54 and § 84). 12. How may an integral expression be multiplied by a fraction (cf. § 54) ? Is n . - equal to — ? Is it equal to -^— ? 13. How does the identity I^Y =^ follow from § 54 (ii)? 14. Prove that ^S.= P^, and thus prove that El . iL = P/: . A = 2L. q qn qs rw q^ /w qio 15. Based upon Ex. 14, give a convenient rule for multiplying two or more fractions together by cancellation. Find the product of : 16. ^«-^^'and ^'+-y^y' 19. ^'^y' and «*-«^'^^ + ^\ x^ — y^ a^ — "2 ax + x^ a6 + 66 x^+2 a;23/2 + / 17. i^-^y-^ and « + ^+l. (a + 6)2_i a -6-1 20. ^' + ^^ and f ^ - y ' a:2+y2 \x-y x^y 18. «'-! and («^+l)(^ + l)^ 21. a+ «* and b "^ • a-\-b 136 ELEMENTARY ALGEBRA [Ch. IX x-^-9x + 20 a;2 - 6 X + 9 (ga-DCaS+l) ^^^ (a-DHa + iy , x^ + x-^?/=2 + y^ (a2 - 1) (a4 _ 2 a2 + 1) 24 «iz^8^ and « + -^ - ' -^ 4 62 a2 + 2 a6 + 4 62 25. 5!^ll?, ££:L + 1^, and -liii- 2 xy XT/ + 2/2 x^ — xy xy + 2/2 aj2 _ 2:^ 26 " ^ + 6'^ - c2 + 2 aft ^^^ ^2 _ ^,2 ^ ^2 _ o ^^ • (,2 _ j-2 _ c2 _ 2 he c^-a^+}p-- 2 6c* 27. ^^JLi + ^LZL^ and ^L±i _ ^^IL:^. a — 6 a + 6 a — 6 a + 6 28. « -A-l--? and 1 ^-^—' he ac ah a a + 6 + c 92. Division of fractions. In algebra, as in arithmetic, to divide hy any fraction gives the same result as to multiply hy the reciprocal of that fraction [cf. § 54 (vi)]. ■^*' b^y^ ' bijlf^' c»s bey Note. If the divisor is an integral expression, it should be first written in a fractional form, and if it is a mixed expression, it should be first reduced to an improper fraction, before proceeding as above. EXERCISES 1. Prove that -2 - ^ = ^ . ! (cf. § 54 and § 84). q s q 7- Perform the following indicated operations, and simplify the results : 2 Qx^y . 2x^ - / 3x2W ^2 A . a 14a364 ■ 2a262" ' V a JVSx^ ) ' x'' (q _ />)2 _ p q _ ;, + 3 (a + 6)2 - 9 ■ a + 6 + '6 x2- 1 .y2- 12 X + 8.5 x2- ;}x - 10 ■ x2 + 3x +2 ' <72 -I- .r-2 _ 1 4. 9 ax . g + 1 + X x2 + y- — d -{- 2 xy x + 3 + y 3:3 _ n y2 4. 30 y x^ + 216x x2 - 49 ' x2 - X - 42' 3. " - ^-^^ ^^ -r ^-^, Q 14 a364 ■ 2a262 a2 _ 121 . a + 11 a2-4 a + 2 X3 - «8 (x-«)2 x3 + a3 x2-a2 14x2- 7x 2x- - 1 12 x3 + 24 x2 ■ x2 + 2x a*- 64 . (« - -6)« a4 + a26' 2 + ¥ «« -6« 5. "^-^ Lj^ ^ ^uf: i.. 10. 12 x3 + 24 x2 x2 + 2 X 6. «^-^^ ^ (^^ - ^>)" 11. 91-93] ALGEBRAIC FRACTIONS 137 i m^n — 5 n"^ m^ — mn + n^ IS- 14. 2 x^ + 13 a: -f 15 . 2 a:^ + 11 r + 5 4 x2 - 9 4 a;-^ - 1 a* - 84 a"'^ + 25 8 a-^ + 8 a + 5 15 4 g-^ + />2 _ ^2 ^ 4 ,,ft 2 g + ^ + c ^ 4 rt--^ - //2 _ c-^ - 2 6c ■ 2 a - 6 - c' ■ 16. 17. a^ -\- ah -]- ac + he ^ a^ — ax + ay — xy a^ — a(y — h^ — hi) ax — ay — x'^ -^ xy d^ + ac + ax -\- ex x"^ — x(y — a) — ay a:4 - .3 a:3 - 28 x^ + 75 x - 50 . 3:3 - 12 a:^ + 45 -g _ 50 a;4 _ 5 a;3 _ 21 a;2 + 125 a: - 100 " a:^ _ 10 a:^ + 29 a; - 20' 18. P^-^* ^ P^ + P^ + P^ . p--^P9 + g^ iP - Q)^ P - Q P^ + (f P^ + 2/'^ + )2|| (a + 6)2 I 1 - rrr + n' n m 1 + (a + by m2- n2 m^ — n^ 1 + ^) 6. X - y X + y - 2 - a - 3 i) J I 1 + 26-a ^ + X + 1 xy x^ + x^ — y' X + y } 10. a — 6 a + b 11. X — ^ - a: -4 ^ 4 - 1 ' a: -4 fl?;2 4 rr/>(a + ft) 3 + &3 a2 - - a6 + 62 a2 + ft2 a2 -62 a2-ft2 a2 + 62 a + 6 a -6 12. m — n m-^ + n' m + n m2 — 71^ 13 6 a + 6 ^8 _ yZ a — b a + 6 7n2n + n* m — n (m — ny x^ — xy -\- ^2 X x^ + y^ . (l ^ y \ '^ + xy + y^ \ X - yl 98] ALGEBRAIC FRACTIONS 139 9 a:2 - 64 14. 1 + 17. I + X + O -r^ ' „ 1 1 X- 1 1-a; 1 ^ 4 + a: 1 18. ,^ 3x-2 Sx + 2 15. _ 1 ^ ^^ 1 - o: ^ + Ui 19 £±iL 1^ « ^ <^ , , 16. J a; + 3/ + a , b , c ,1 H h- X - y + b c a X -\- y 20. Recalling the meaning of a negative exponent (§ 44), show that ,4.1 cr- ^2 - ^,2 -^2^3* 21. As in Ex. 20, show that _'?!r^lV ^ a^ffi, gj^o^ ^jgo l^h^t ^ ^mjy-n m^s-^w^ m'^wH^ by ~ b^s-"' 22. Prove that any factor whatever of the numerator of a fraction may be transferred to the denominator by merely reversing the sign of the exponent of that factor. Also show how a factor may be transferred from denominator to numerator. 23. Is ^^ + ^~^^^ equal to 9^+^'? Why? Observe carefully that 4 X 4 xb^ a factor, but not a part, may be transferred as in Ex. 22. Clear the following expressions of negative exponents, and simplify them as far as possible; in any case of doubt employ the definition of § 44, viz., a~* = — • a* 24 3 m-^n^ 25 ^'^^ 26 3a + 2ftc-8 ' 2 (a + a:)* ' 3 • 2-^b^y-^' ' 5 x-^ - H y REVIEW QUESTIONS-CHAPTERS VI-IX 1. Define and illustrate : even numbers; odd numbers; prime num- bers; composite numbers; finite numbers; and infinite numbers. 2. What is the value of g? Of§? Explain. 3. Show that the absurdity in Ex. 17, § 55, arises from dividing zero by zero. 140 ELEMENTARY ALGEBRA [Ch. IX 4. By applying the distributive law show that — (a + x — 5) = — a - a: + 5. 5. State the binomial theorem. Apply this theorem to expand (2 a - 3 x^y. 6. If Ax"^ + Bx"^-^ + ••• + Hx -\- K is divided by x — a, prove that the remainder is Aa"^ -\- Ba"-'^ + ■- -\- Ha -{■ K. 7. By means of Ex. 6, and without actually performing the division, show that a: — 1 and x + 2 are factors of x* + 2 x^ + 7 x - 10. 8. As in Ex. 7, show that x" - ?/", wherein n is any positive integer is exactly divisible hy x — y. 9. By means of factoring, find the roots of x^ — 7 x + 12 = 0, and explain. 10. Form the equation whose roots are 3 and — 7, and explain. 11. What is meant by the L. C. M. of two or more expressions? How may it, in general, be found? 12. How may the L. C. M. of three or more given expressions be found? 1+ ^ 13. Simplify l + x^C^-D-^. ^ ^ , 1 x^ + x + l X + ^ ^ a: + 1 14. Is 2rtx5^ g J ^^ a^ — J Explain. 3 63 ^ 3-2-163^-2 ^ CHAPTER X SIMPLE EQUATIONS I. INTEGRAL EQUATIONS 94. Introductory remarks and definitions. Some preliminary work in simple equations has already been given in Chapter III ; the text of that chapter should now be rapidly reread. In the present chapter it is proposed to treat this subject in a somewhat more careful and rigorous manner. Every algebraic problem involves one or more numbers whose values are at first unknown, and which are to be found from given relations which they bear to other numbers whose values are known ; to distinguish between these two kinds of numbers the first are called unknown numbers, and are usually represented by some of the later letters of the alphabet, as x, y, and z (cf. § 26), while the others are called known numbers, and are repre- sented either by the Arabic characters, 1, 2, 3, •••, or by some of the early letters of the alphabet, as a, b, and c. If any of the known numbers in an equation are represented by letters, then it is called a literal equation, otherwise it is called a numerical equation. If its members are integral expressions so far as the unknown numbers are concerned (§ 41), then it is called an integral equation ; known numbers may appear as divisors and the equation still be integral. E.g., 3x2 + 5x1/ -10 2/2 = 8, 4-^ = 7x, and 5(x2 + 2/2)=^ are integral equations; the first two are numerical, while the third is literal. By the degree of an integral algebraic equation is meant the highest number of unknown factors which it contains in any one term. If all of its terms are of the same degree, the equation is homogeneous. 141 142 ELEMENTARY ALGEBRA [Ch. X E.g., 3x4-7 = 13 and 2 + 4?/ — 5x = are numerical equations of the first degree, while z^ + 10x = ^x — ^, 4 xi/2 = 3 ax^ — 7 ?/3, and azy^ — x = 3y are of the third degree ; of these last three equations the first is numerical, the second and third are literal, and the second is homogeneous. Special niimes are often given to equatioils of the lower degrees; thus an equation of the first degre6 is known as a simple equation and also as a linear equation ; * one of the second degree is also called a quadratic equation ; one of the third degree, a cubic equation; etc. EXERCISES 1. What is meant by a root (or solution) of an equation? Is 2 a root of a:2 - 7 a: + 10 = ? What then are the factors of x'^ - 7 x + 10 (of. § 67)? What other root has this equation? 2. Verify that x = 4: and y = 3 constitute a solution of the equation 7 X + 2 y = M. If a: = 2 in this equation, what must be the corresponding value of y? If a: = a, what is y ? If y = 6, what is x? Find four other .solutions of this equation. 3. How many solutions has the equation in Ex. 1 ? How many solutions has the equation in Ex. 2? 4. Is the equation in Ex. 1 homogeneous? integral? literal? numeri- cal? simple ? Define each of these kinds of equations. 5. Show that x^ + lOx'^y + 8^^ = dxy^ is a homogeneous equation. What is its degree ? Can a homogeneous equation have a term free from the unknown number ? 6. Is 3x2 — 5?/2 =:2a2 homogeneous? Why? Write a homogeneous linear equation in two unknown numbers ; also an integral, literal, quad- ratic, non-homogeneous equation in two unknown numbers. Solve the following equations, using the methods of Chapter III, and also § 72 : 7. ^-'^^^Ji^ + 5 = 0. 10. 2ax = 2c-3bx. 8. x-Sx + 4:-(sx+2-'f\ = 0. 11. -^--^ = c. \ 4:1 2a 46 9. ^^-=i-^^;^+2=0. 12. {a-x){a-h)-a{h-x) = (). 13. x^-x = Q. 14. x2 + (a - 6)r = «&. 15. x^ ^2 x"- = x ^2. 16. Find three solutions of .5 a: — 3 y = 7. * The appropriateness of this name will be seen in § 115. 94-95] SIMPLE EQUATIONS 143 95. Equivalent equations. Two equations are said to be equiva- lent if every root' of eitlier is also a root of the other. The methods thus far employed for solving equations (in Chap- ter III, and elsewhere) consist in clearing equations of fractions, transposing and collecting terms, etc., i.e., these methods consist in deducing from any given equation a succession of iiew equations whose roots are more and more easily found, and then finding the root of the simplest of these new equations, — compare Exs. 1 and 2, § 24. That the root of this final simplest equation happens also to be a root of the given equation depends upon the following prin- ciples : , (1) Adding* the saine numher to each member of any given equation, forins a new equation which is equiva- lent to tJie first (cf. § 24, Ax. 1). (2) Multiplying* each member of an equation by the same number or algebraic expression, which does not involve the unhnown number, and which has a finite value different from zero, forms a new equation which is equivalent to tlze first (cf. § 24, Ax. 2). To prove Principle (1) let the member^fVny fei\^eu equation be represented by El and E^ respectively, i.e., let the equ^iott be ^ Ei^E^ \ \ (1) This does not mean that Ei and E.2 represent the same number for every value that may be substituted for the unknown nurnbet^ but that they represent the same number only when a root of the equation if substituted for the unknown number. But manifestly, if N represents any nuntfipn whatever, then Ei + N=B^^N I ^ (2) whenever Ei = E^; i.e., every root of Eq. (1) is also'a root of Eq. (2). By precisely the same reasoning, every rootr^ Equation (2) is also a root of (El -{-N) + (-N) = (E2 +% + (- -ZVT), (3) i.e., of Ei = E6? Ifn = 6? 14. A fathei- is m times as old as his son, and in p years he will be n times as old. Find their respective ages. Interpret your result when m<,n. Is jo positive or negative in this case? 15. A merchant has two kinds of sugar worth, respectively, a and h cents a pound. How many pounds of each kind must be taken to make a mixture of n pounds worth c cents a pound? Interpret the result if a = ft, and c is less than a ; also when a = h = c. Do these interpretations of the results agree with the conditions of the problem under the same suppositions ? 16. An alloy of two metals is composed of m parts (by weight) of one to n parts of the other. How many pounds of each of the metals are there in a pounds of the alloy? Show that the problem just stated is the generalization of such a prob- lem as this: Bell metal is an alloy of 5 parts (by weight) of tin to IG of copper; how many pounds of tin and of copper in a bell weighing 4200 lb.? 17. A wheelman sets out from a certain place at m miles an hour, and is pursued by a second wheelman, who starts from the same place a hours later, and rides p miles an hour. How far from the starting point will the second wheelman overtake the first? What does this result become if m = 10, jt? = 12, and a = 4? 18. Two wheelmen, A and B, are observed passing a certain point, A being c hours in advance of B, and traveling at the rate of a miles in h hours, while B travels p miles in q hours. How far will A travel before he is overtaken by B ? Under what conditions is this solution positive? Negative? Zero? Infinite ? Interpret the result in each case. CHAPTER XI SIMULTANEOUS SIMPLE EQUATIONS I. TWO UNKNOWN NUMBERS 101. Indeterminate equations. Although a simple equation in one unknown number has one and but one solution (of. § 97), yet it is easy to see that an equation which involves two or more unknown numbers has an infinite number of solutions. E.g.f in the equation x + 3 y = 5, which is equivalent to y = ^, [§95 there is a perfectly definite value of y corresponding to every value that one may- choose to assign to x ; thus, if x = l, then ?/ = f , if a; = 2, ?/ = 1, if a; = 3, ?/ = f , if x=—l, y = 2, and so on indefinitely; i.e., each of these pairs of numhers, viz., 1 and f , 2 and 1, 3 and f , etc., constitutes a solution of the given equation, because, when substituted for x and y respectively, they satisfy that equation. An equation, such as the one just now considered, which has an infinite number of solutions, is, for that reason, called an indeterminate equation. 102. Positive integral solutions of indeterminate equations. Al- though the number of solutions of an indeterminate equation, as has just been illustrated, is unlimited, yet it often happens that only solutions of a particular kind are sought, — e.g., those that are positive integers, — and the number of these may be finite. In practice the positive integral solutions of an indeterminate equation can usually be found by mere inspection, or by trial. E.g., to find the positive integral solutions of the equation 2 a; + 3 y = 7, it is only necessary to assign to one of the unknown numbers, say x, the values 1, 2, 3, ••• in turn, and to find the corresponding values of the other unknown number, which are f, 1, ^, ••• ; moreover, if a; = 4, or. any greater number, then y is nega- tive, hence the only positive integral solution of the given equation is x = 2 and y = l. 162 101-103] SIMULTANEOUS SIMPLE EQUATIONS 163 Many problems lead to indeterminate equations which, from the nature of the things involved, demand solutions that are positive integers. E.g., a farmer spent S22 purchasing two kinds of lambs, the first kind costing him $ 3 each, and the second kind $ 5 each. How many of each kind did he buy ? Solution. Let x = the number of the first kind, and y = the number of the second kind. Then one condition of the problem is that Sx+5tj = 22, and the other condition is that z and y shall be positive integers.* By § 95, this equation is equivalent to a;= — ^, o and, if the values 1, 2, 3, and 4 be assigned to y, the corresponding values of z are found to be V> 4, I, and |; moreover, if y = 5, or more, then z is negative, and therefore the ojily positive integral solution of the above equation is a; = 4 and y = 2; i.e., 4 and 2 are, respectively, the numbers of lambs purchased. 103. Positive integral solutions : another method. Another method of finding the positive integral solutions of an indetermi- nate equation will now be illustrated. Given the equation 7 x +4 ?/ = 46 ; to find its positive integral solutions. By transposing and dividipg, this equation becomes 4 4 i.e., y-ll-^z = ^^, and, since z and y are integers, therefore the first member of this equation repre- 2 3 2; sents an integer, and therefore the second member, viz., — - — , also represents an integer. Again, since = represents an integer, therefore the product obtained by 4 multiplying it by any integer whatever also represents an integer ; moreover, if this multiplier be so chosen that the new coefiScient of z shall exceed some multi- ple of the denominator by 1 (cf. § 79), then the integral values of z and y may be easily determined as follows : 2 3 -J. 3/2 3 2;) 5 9 X 2 z Since — : — represents an integer, therefore -^ — - — ^ = — 7 — = 1—2 z-] — — 4 2—x 4 4 4 represents an integer, and therefore represents an integer. If this last 2 X integer be designated by p, then — — =p, - * Although this condition is not expressible by means of an equation, yet it is none the less vital on that account. 164 ELEMENTARY ALGEBRA [Ch. XI whence cc = 2 — 4 p, and, on substituting this value of z in the given equation, it becomes y = 8 + 7p. In these last two equations x and ?/ are positive integers, and p is an integer, though not necessarily positive. This shows that p is either — 1 or (in order that X and y may be positive), whence x = 6 and y = 1, ov x = 2 and y = S; and there are no other positive integral values of x and y which satisfy the given equation. EXERCISES Find five solutions to each of the following equations : 1. Sx-4y=8. 2. 2w = 5 + 3z. 3. 3r + 6s = 20. 4. How many solutions has each of the above equations? Why? What are such equations called? 5. If possible solve the equations in Exs. 1, 2, and 3 above, in posi- tive integers. How many such solutions has each? Find the positive integral solutions of the following equations : 6. ? + ^ = 5. 7. 6x + 7y = 52. 8. 13 w + 5 y = 229. Show that the following equations have no positive integral solutions : 9. 2x -4:y = l. 10. dx + 6y = 5. 11. 9 x + 3 y = 17. 12. Sliow that the indeterminate equation ax + ly = c can not be solved in positive integers when a + ft > c ; nor when a and h have a common factor which is not a factor of c. 13. Find three solutions of the equation 2a: — 5?/ + 32 = 6. 14. If a man spends $300 for cows and sheep, which cost respectively $ 45 and $ 6 a head, how many of each does he purchase ? 15. In how many and what ways may a 19-pound package be weighed with 5-pound and 2-pound weights ? 16. How many pineapples, at 25 cents each, and watermelons, at 15 cents each, can be purchased for $2.15? 17. Divide a line which is 100 feet long into two parts, one of which shall be a multiple of 11, and the other of 6. 18. Find the least number which when divided by 4 gives a remainder of 3, but when divided by 5 gives a remainder of 4. 19. A man selling eggs to a grocer counted them out of his basket 4 at a time and had 1 egg left over, and the grocer counted them into his box 5 at a time and there were 3 left over. If the man had between () and 7 dozen eggs, how many must there have been ? 103-105] SIMULTANEOUS SIMPLE EQUATIONS 165 104. Definitions: simultaneous equations, etc. Although a single equation which involves two unknown numbers has just been shown to be indeterminate, i.e., to have an indefinite number of solutions, yet if two such simple equations be given, it usually happens that one, and only one, pair of numbers can be found which will satisfy each of them, i.e., be a solution of each. E.g., the equations 4 a; + 3 y = 5 and 2 x — 5 y = 9 are each satisfied by x = 2 andy =—1, and by no other pair of numbers. Two or more equations which are satisfied by the same set (or sets) of numbers are called simultaneous equations (also called consistent equations), while two equations which have no solu- tion whatever in common are called inconsistent equations (also called incompatible equations) ; e.g., x + y = 4: and 2 x-\-2y = 9 are inconsistent equations. Two or more equations which express different relations be- tween the unknow^n numbers, and therefore can not be reduced to the same form, are called independent equations. Two or more equations taken together are often called a system of equations ; and any set of numbers which satisfies every equa- tion of the system is called a solution of the system. 105. Solving simultaneous equations. The process of finding a solution of a system of simultaneous equations is called solving the equations; this process will now be illustrated by some easy examples. f X + 7/ ■= 4:, (1) Ex. 1. Solve the equations -; ^ lx->j = 2. (2) Solution. Adding these two equations, member to member, gives 2 a: =6, whence x = d. Substituting this value of x in Eq. (1) gives 3 + 2/ = 4, whence 2/ = !• That these numbers, viz., x = S and ?/ = 1, really constitute a solution of the given equations is verified by substituting them for x and y in those equations. 166 ELEMENTARY ALGEBRA [Ch. XI rdx-^y = l, (1) Ex. 2. Solve the equations - x ^ I a; + 2?/ = 9. . (2) Solution. On multiplying Eq. (2) by 2, it becomes 2a: + 42/ = 18, (3) and adding Eq. (3) to Eq. (1) gives 5 a; = 25, whence x= o\ and the corresponding value of y may be found by substituting this value of X in either of the equations which contain both x and y. E.g., by this substitution Eq. (2) becomes 5+2y = 9, whence ^ = 2 ; and it is easily verified as in Ex. 1 that x = 5 and y = 2 is a solution of each of the given equations. r 3 X + 2 ?/ = 26, (1) Ex. 3. Solve the equations -{ ^ 1 5 a: + 9 2/ = 83. (2) Solution. On multiplying both members of Eq. (1) by 5, and of Eq. (2) by 3, they become, respectively, 15 a; + 10 2/ = 130, (3) 15^ + 27 3/ = 249; (4) and subtracting Eq. (3) from Eq. (4) gives 17?/ =119, whence y — 'J. Substituting this value of y in any one of the equations containing both X and y gives _ . it isTfeai and it isTfea^ily verified that x = 4 and ?/ = 7 is a solution of the given system 6i^eqyations. Observl tmxt if Eq. (1) had been multiplied by 9, and Eq. (2) by 2, and if one of the two resulting equations liad been subtracted from the other, then y would have disappeared, and the value of x would have been found before that of y. •( />- 105-106] SIMULTANEOUS SIMPLE EQUATIONS 167 Ex. 4. Solve the equations V-ii=-|, (1) f + ^ = 4i. (2) Solution. Multiplying both members of Eq. (1) by 12, and of Eq. (2) by 6, gives 4a;- 8- 21 =-3 3^, (8) and 3x + 4y=27; (4) and, on transposing and simplifying, Eq. (3) becomes 4:x+'dy = 29. (5) Equations (4) and (5) may now be solved by the method employed in Ex. 3 ; and it is easily verified that their solution, viz., x = 5 and y = 3 is, at the same time, a solution of equations (1) and (2). 106. Elimination. Any process of deducing from two or more simultaneous equations other equations which contain fewer unknown numbers is called elimination. Such a process elimi- nates (i.e., gets rid of) one or more of the unknown numbers, and thus makes the finding of a solution easier. That particular plan of elimination which was followed in the examples given in § 105 is known as elimination by addition and subtraction. It is evident, moreover, that this method is appli- cable to any pair of such equations. The procedure may be formulated thus: (1) Unless each of the ^iven equations is already in the form ax + by = c, wherein a, b, and c are integers, reduce them to this form. (2) Multiply these equations by such numbers as will mahe the coefficient of tlxe letter to be eliminated the same {in absolute value) in both equations. (3) Subtract or add these last two equations (according as the terms to be eliminated have like or unlihe signs), solve the resulting equation for the unhnown number which it contains, and substitute that value in any one of the earlier equations to find the other unhnown number. 168 ELEMENTARY ALGEBRA [Ch. XI (4) Verify that these two numbers really satisfy the two given equations. Note. If the coefficients which are to be made of equal absolute value are prime to each otlier, then each may be used as a multiplier for the other equation ; if, however, these coefficients are not prime, their least common multiple should be divided by each in turn, and these quotients used as the multipliers. EXERCISES Solve each of the following systems of equations, and check the results : (15 X ox 15 a; + 77 2/ = 92, 3. 6 ?/ - 5 X = 18, 12 x - 9 ?/ = 0. 5 a: + 6 2/ = 17, 6 a: +5 3/ = 16. 8. 9. ^-K2/-2)-K^-3)=0, x-\{y-\)-\{x-2) = (i. ^x-ly-m, 3 4 ^ r5;9 + 3(? = 68, • 12;, 6. 7. + 5 ^ = 69. 22 x - 8 ?/ = 50, 26 x + 6 ?/ = 175. 28 a; - 23 ^ = 33, 63 X - 25 2/ = 199. 4.%? Show that this problem includes Prob. 19 as a special case — it is the generalization of Prob. 19 (cf. § 100). 182 ELEMESTARY ALGEBRA [Ch. XI 22. Generalize Prob. 14. Find the solution of the generalized problem, and then show that the answer to the particular problem (14) may be found by merely substituting in the answer to the generalized problem. 23. Generalize Prob. 20, solve, etc., as in Prob. 22. 24. A man rows 15 miles downstream and back in 11 hours. If he can row 8 miles down^^tream in the same time as it takes him to row 3 miles upstream, what is his rate of rowing in still water? and what is the velocity of the current? 25. Divide the number N into two such parts that — of the first 1 *" part, plus - of the second, shall exceed the first part by M. n Specialize this problem, and find the solution of the special problem by substituting in the general solution. 26. Three cities. A, B, and C, are situated at the vertices of a triangle; the distance from A to C by way of B is 50 miles, from A to B by way of C is 70 miles, and from B to C by way of A is 60 miles. How far apart are these cities? Solve this problem by first generalizing it, and then substituting the particular numbers 50, 70, and 60 in the general solution. 27. Two boats which are d miles apart will meet in a hours if they sail toward each other, and the second will overtake the first in b hours if they sail in the same direction. Find the respective rates at which these boats sail. Also discuss fully your solution, i^^ interpret the results when the rate of the second boat is greater than, equal to, and less than, the rate of the first — compare Prob. 3 of § 100. 28. Two men, A and B, had a certain distance to row and alternated in the work ; A rowed at a rate sufficient to cover the entire distance in 10 hours, while B*s rate would require 14. If the journey was completed in 12 boursL how many hours did each row? 29. A mine which is to be emptied of water has two pumps which together can discharge 1250 gallons an hour. The larger pump can do the work alone in 5 hours, but with the help of the smaller pump only 4 hours are needed. How many gallons an hour does each pump discharge ? Solve this problem by first generalizing it, as in Prob. 26 above. 30. Two trains are scheduled to leave the cities A and B, m miles apart, at the same time, and to meet in h hours; but, the train leaving A being a hours late in starting, they met k hours later than the scheduled time. What is the rate at which each train runs ? From the solution of this problem find, by substitution, the solution of the special problem in which m = 800, ft = 10, a = If, and k = ^. 111-112] SIMULTANEOUS SIMPLE EQUATIONS 183 31. Two boys, A and B, run a race of 400 yards, A giving B a start of 20 seconds and winning by 50 yards. On running this race again, A, giving B a start of 125 yards, wins by 5 seconds. What is the speed of each? Generalize this problem. 32. A and B working together can build a wall in 5^ days ; finding it impossible to work at the same time, A works 5 days, and later B takes up the work, finishing it in 6 days. In how many days could each have built this wall alone? Generalize this problem. 33. A railway train, after running 1 hour and 36 minutes, was detained 40 minutes by an accident, after which it proceeded at | of its former rate, and reached its destination 16 minutes late. Under the same cir- cumstances, had the accident occurred 10 miles farther on, the train would have arrived 20 minutes late. At what rate did the train move before the accident, and what was the entire distance traveled? II. THREE OR MORE UNKNOWN NUMBERS 112. Equations containing more than two unknown numbers. It is easy to see that the methods employed in § 105 for solving a system of two simultaneous integral equations, each containing two unknown numbers, may also be employed for solving a system of three or more such equations involving as many unknown num- bers as there are independent equations. (Cf. Exs. 1 and 2 below.) ( x-i-Si/- z = o, (1) Ex. 1. Given | 3 x + 6 ?/ + 2 z = .3, (2) to find the solution of this system of equations. Solution. Adding 2 times Eq. (1) to Eq. (2), member to member, g^^^s 5x+V2y=lS, (4) and subtracting Eq. (3) from 3 times Eq. (1) gives x+12t/=9. (5) Now subtracting Eq. (5) from Eq. (4) gives 4 a: = 4, whence x = 1. (6) On substituting this value of x, Eq. (5) becomes H-12y = 9, whence v = i\ C7\ 184 . ELEMENTARY ALGEBRA [Ch. XI and substituting these values of x and y in Eq. (1) gives whence z = — 2. (8) That these numbers, viz., x = 1, y = ^, and z = — 2, really constitute a solution of the given system of equations is easily verified by substituting them for x, y, and z in these equations. Note. It should be carefully observed that, by principles (i) and (ii) of § 108, Eq. (2) of the given system of equations may be replaced by Eq. (4), —which is derived from Eq. (1) and (2), — and the new system thus formed will be equivalent to the given system, i.e., the system of Eqs. "(1), (3), and (4) is equivalent to the system of Eqs. (1), (2), and (3). So too Eq. (3) may be replaced by Eq. (6), making the system formed of Eqs. (1), (4), and (0) equivalent to the given system; and this last system, being readily solved, furnishes a solution of the given system. The foregoing is another illustration of the fact to which attention has already been called (§ 108), viz., that solving a system of simultaneous equations is accomplished by fix-st replacing the given system by an equivalent system whose solution is more easily obtained. (2x-^y-2z = -l, (1) Ex. 2. Given ' | 3 a: + z = 6, (2) i a: + 2/ + 2 = 3 ; (3) to find the solution of these equations. Solution. Since the second of these equations is already free from the unknown number y^ therefore it is best to combine Eqs. (1) and (3) so as to eliminate y, and thus obtain another equation involving only x and 2. Adding Eq. (1) to 3 times Eq. (3) gives 5 a: + 2 = 8, (4) and subtracting Eq. (2) from Eq. (4) gives 2 a: =2, whence x = \. (5) Substituting this value of x in Eq. (2) gives 2 = 3; and substituting these two values in Eq. (3) gives 2/ = -l. Moreover, it is easily verified that a; = 1, ^ = — 1, and 2=3 constitute a solution of the given equations. Ex. 3. Show that Eqs. (2), (3), and (5), in Ex. 2, form a system which is equivalent to the given system. 112-113] SIMULTANEOUS SIMPLE EQUATIONS 185 113. Formulation of the method of procedure of § 112. The proc- ess of finding a solution of three independent integral equations of the first degree and containing three unknown numbers, which is illustrated in § 112, may be stated thus : Combine any two of the three given equations in such a way as to eliminate some one of the unknown numbers, thus deriving from them an equation containing but two unhnown nujnbers; then combine the remaining equation of the given system with either one of the other two in such a way as to eliminate the same unhnoiun number as before, thus deriving another equation which contains the same two unknown nuinbers as does the first derived equation; next combine these two derived equations so as to elimAnate one of the unknown numbers, thus deriving another equa- tion which contains but one unknown number; from this last equation the value of the unknoiun number ivhich it contains can be found, and then, by successively substituting in earlier equations, the values of the other two unknown numbers can be found. Similarly for the solution of a system of n independent integral equations of the first degree and containing n unknown numbers. When n is greater than 3 the eliminating should be done very systematically, since otherwise the derived equation may not be independent ; the procedure may be stated thus : So combine some one of the given equations {the first, for example) with each of the others, as to eliminate tJie same unknoivn number in each case, thus forming ivhat may be called a first derived system of n — 1 equations, which will be independent, integral, and of the first degree, and which will contain n — 1 unknown numbers ; by proceeding with the first derived system just as with the given sys- tem, a second derived sy stein containing n — 2 equations involving n — 2 unknown numbers is obtained; by continu- ing this process, there is finally obtained a single equation with but one unknown number; from this equation the value of that unknown number is found, and then, by 186 ELEMENTARY ALGEBRA [Ch. XI successive substitutions in earlier equations, the values of all the other unknown numbers are found. Note. It may be remarked that any one of the give)i equations, together with then — 1 equations of the first derived system, constitute a system which is equivalent to the given system ; also that any one of the given system, together with any one of the first derived system, and the n — 2 equations of the second derived system, are equivalent to the given system, and so on; finally, that the system composed of any one of the given equations, any one of the first derived system, any one of the second derived system, and so on including the single equation of the last derived system, is equivalent to the given system. EXERCISES Solve each of the following systems of equations : 1. 2. 3. 4. 2:c + 3?/ + 4z 3a: + 5?/+6^ 20, 26, 31. 4 a; — y — z = 5, 3a;-4?/+16 = 6z, lx + ^y-2z = lQ, 2a; + 5y + 3^ = 39, ^ X — y -\- 5 z = 31. 5x-Qy + 4:Z=l5, 7 X + 4:y- 3z=19, 2x + y+Qz = i6. 2 X + 4 ?/ + 5 2 = 19, Sx-Sy + bz = 2S. 5x + 6y -12z = 6, 2x-2y - 62=- 1, 4x-52/+ 32 = 7^. y + z -8Q = 72 - 5x, 9'^-ix-\y = iy-2z, lx + ly + lz=5S. 10. 12. ^x + ly=12-iz, iy+ Xz = 8+lx, ix + iz = 10. |'2a;-5?/+19 = 0, rdy-4:z-{- 7 = 0, \2z-5x- 2 = 0. 4 3, 4 o 5. '1 + ^ = 6, X V 11. \ 1 1 - + - 10, 1 1 -+ - = Z X 3 2 1, ^ + - + - = 1, X y z V z = 1. 113j SIMULTANEOUS SIMPLE EQUATIONS 187 13. 14. ( X + y -z = a, {x-y = 2b, [x + z = Sa-^h, X -^ y a yz ^1 y + z 6' xz 1 c If 18. 19. i x± xy [x+ z Suggestion = a, i.e., — h- = o y X xy then 15. 16 2 y + 3 a; + y - 2 = 0, 3 3/ ^ 2 X + 2 - 4 <; = 21, 22-3y- ?/+ a: =6, r + 4 a: + 2 ?/ - 3 2 = 12. u + a: + // = 15, X + ?/ + 2 = 18, y + _y + 2 = 17, L I' 4- X + 2 = 16. Suggestion. Adding these equa- tions and dividing the sum by 3 gives v-\-x-\-y + z = 22,. { y + z-Zx = 2a, x + z-3y = 2b, X + y — S z = 2 c, I2x-h2y+v = 0. (Zu + ov-2x + ^z = 2, 2u + 4x-^y-z = Z, u -\- V + z = 2, 6y + iv + u = 2, 5z + 4:x-7v = 0. « 1 1 o - + -+- = 2, X y z 20. J 1 h 2 ^ X y z yz + xz + cxy = 3 xyz. Suggestion. Carefully compare the last equation with either of the other two. 21. 22. abxjjz 4- cxy — ayz = bxz, bcxyz + ayz — bxz = cxy, acxyz + bxz — cxy = ayz. 5xy -\- Q(x+y)= 0, 5yz-2(y + z)=0, Uxz - 3(a:+ z)=0. 17 ( y + z + V — X = 22, 1z + V + X - y = 18, V + X -\- y — z = li, X + y -j- z — V = 10. 23. From the considerations presented in § 113, prove that a system consisting of n independent and consistent equations of the first degree, and containing n unknown numbers, has one and only one solution. (Cf. also § 111.) 24. If there are more unknown numbers than independent equations in any given system, how many solutions has that system? Why ? (Such a system is usually called an indeterminate system.) 25. If there are more consistent equations than unknown numbers in a system, prove that these equations can not all be independent. (Cf. § 111, note.) 188 ELEMENTARY ALGEBRA [Ch. XI 26. Prove that there is no unique solution of the system \ ox + 2y — 2z = 0j [ Sx + ^y - 2 = 2. Is this system indeterminate (cf . Ex. 24) ? Explain. PROBLEMS 1. A grain dealer sold to one customer 5 bushels of wheat, 2 of corn, and 3 of rye, for $6.60; to another, 2 of wheat, 3 of corn, and 5 of rye, for 15.80 ; and to another, 3 of wheat, 5 of corn, and 2 of rye, for $5.60. What was the price per bushel of each of these kinds of grain ? 2. A quantity of water, which is just sufficient to fill three jars of dif- ferent sizes, will fill the smallest jar exactly 4 times; or the largest jar twice, with 4 gallons to spare ; or the second jar 3 times, with 2 gallons to spare. What is the capacity of each of these jars? 3. If A and B can do a certain piece of work in 10 days, A and C in 8 days, and B and C in 12 days, how long will it take each to do the work alone ? 4. Divide 800 into three parts such that the first, plus I of the second, plus f of the third, shall equal the second, plus | of the first, plus I of the third : each of these sums being 400. 5. A merchant having three kinds of tea, sold to one customer 2 lb. of the first kind, 3 of the second, and 4 of the third, for $4.70; and to another he sold 4 lb. of the first kind, 3 of the second, and 2 of the third, for $4.-30. If a pound of the third kind is worth 5 cents more than | lb. of the first kind and I lb. of the second kind taken together, what is the price of each per pound ? 6. Divide 90 into three parts such that I of the first, plus ^ of the second, plus J of the third, shall be 30 ; and that the first part shall equal twice the third part diminished by twice the second part. 7. The sum of the digits of a 3-digit number is 11 ; the double of the second digit exceeds the sura of the first and third by 1 ; and if the first and second digits be interchanged, the number will be diminished by 90. What is the number? 8. The third digit of a 3-digit number is as much larger than the second as the second is larger than the first; if the number be divided by the sum of its digits, the quotient will be 15; and the number will be increased by 396 if the order of its digits be reversed. What is the number ? 113-114] SIMULTANEOUS SIMPLE EQUATIONS 189 9. The sum of the digits of a 4-digit number is 11 ; if the order of the digits be reversed, the number will be increased by 819 ; if 9 be subtracted from the number, the units' and tens' digits will be interchanged ; and the Slim of the units' and tens' digits equals the hundreds' digit. What is the number ? 10. Of three alloys, the first contains 35 parts of silver, to 5 of copper, to 4 of tin ; the second, 28 parts of silver, to 2 of copper, to 3 of tin ; and the third, 25 parts of silver, to 4 of copper, to 4 of tin. How many ounces of each of these alloys melted together will form 600 oz. of an alloy con- sisting of 8 parts of silver, to 1 of copper, to 1 of tin ? 11. If Problem 10 merely demanded that the alloy should contain 8 parts of silver to 1 of copper, how many ounces of each of the given alloys would then be required? Why is this problem indeterminate? 12. A tank whose capacity is 1600 gallons is supplied by two pipes, and has one outlet pipe. If the tank is empty, and all three pipes are opened, it will be filled in 80 hours; if it is | full, and all the pipes are opened for 10 hours, and if the larger supply pipe is then closed, leaving the other two open 10 hours longer, the tank will then be | full; and it can be filled by the larger pipe alone in 26| hours. Find the number of gallons discharged per hour by each of the three pipes, assuming the flow to be uniform. 13. Find an expression of the form ax^ -\- hx + c whose value will be 6, when x = 2, 3 when x = — 1, and 10 when a: = 4. Suggestion. 4a + 2^ + cis the value of ax'^ -i-bx + c when x = 2; therefore. 4a+26 + c = 6, etc. 14. Can such an expression as that in Prob..l3 be found which shall take four prescribed values when four particular values are assigned to a:? Why? What letters represent unknown numbers in Prob. 13? III. GRAPHIC REPRESENTATION OF EQUATIONS* 114. Preliminary remarks. Although an equation in two un- known numbers has an infinitely large number of solutions, and is in that sense entirely indeterminate (§ *101), yet, % a beautiful device, due to a ceiebrated mathematician and philoso- pher named Descartes, a perfectly definite geometric picture of such an equation may be made. The "method by wliich this is done will be explained in this and the next article. * This subject is discussed in detail in a later course in mathematics, — in Analytic Geometry. 190 ELEMENTARY ALGEBRA [Ch. XI Y ,Q P y' M 1 ? y' Let two indefinite straight lines X'X and T^T be drawn at right angles to each other and intersecting in the point — as in the figure. If now it be agreed that distances measured to the right, or upward, be represented by positive numbers, Avhile distances to the left, or downward, are represented by nega- tive numbers, then the position of any point whatever, in the plane of this page, is completely determined by merely giving the distances of that point from the lines X'X and Y'Y. It will be observed that this is similar to locating a place on a map by means of its latitude and longitude. E.g., to locate a point P, whose distances from I'Tand X'X are respectively 3 inches and 2 inches, measure 3 inches to the right from 0, to the point M say, and then measure 2 inches up from M. This point is usually represented by the symbol (3, 2), i.e., by P= (3, 2) ; the numbers 3 and 2 are called the coordinates of the point P, and the lines X'X and r'Fare called the axes of coordinates. Simi- larly, the point Q= (— 3, 4) is located by measuring 3 units toward the left from 0, and then 4 units upward. The point R={—2,, — 3) is also represented in the figure. The student may draw a figure and locate accurately the following points upon it:* (5, -1), (4, 7), (-4, 2), (3i, -4), (-2i -5f), and (8, -6|). 115. Geometric picture, or graph, of an equation. By the geomet- ric picture (or map) of an equation — usually called the locus or graph of the equation — is meant the totality of all those points whose co- ordinates satisfy that equation.. E.g. , since the numbers — 1 and — 5, when substituted for x and ?/, respectively, satisfy the equation 2 a: — ?/=:3, therefore the point Pl= (— 1, — 5) lies on the graph of this equation ; so, too, the points P2=(0, — 3), P3 = (l, — 1), P4=(2, 1), P5=(3, 3), etc., are on the graph of this equation, because each of these pairs of numbers satisfies the equation. If these points are located, by the method of § 114, it is found that they are not scattered * It is recommended that cross-section paper be used for this purpose ; such paper may be obtained from all stationers, 114-116] SIMULTANEOUS SIMPLE EQUATIONS 191 indiscriminately over the page, but that they all lie upon the line AB ; this line is the graph of the given equation.* It is due to this fact that such equations are often called linear equations (cf. § 94). The points F^, P3, P4, ••• were found by assigning the values 0, 1, 2, 3, -■ to z, and then finding the corresponding values of y from the equation; other points between any two of these may be found by assigning intermediate values to X. The above method of finding the graph of any given equation in two unknown numbers may be stated thus: by assigning to ic a succession of values, such as 0, 1, 2, 3, •••, — 1, —2, —3, ••♦, find the corresponding values of y, i.e., find as many solutions of the given equation as may be desired ; locate the points whose coordi- nates are these solutions, and draw a line connecting these points in regular order ; this line will represent the required graph. EXERCISES Draw a j)air of axes, as in §§ 114 and 115, and locate the following points : 1. (5, 4) ; (3, 7) ; (4, - 2) ; (- 3, 1) ; and (- 4, - 6). 2. (3, 0) ; (- 5, 0) ; (0, 8) ; (0, 0) ; and (0, - 2). 3. Where are all points whose second number is 0? Where are those whose first number is 0? Where are all those whose second number is 3|? Draw a fine through this last class of points. 4. Where are those points whose second number is the same as its first number ? Where are those whose second number is the opposite of its first number? Draw a line through each of these two classes of points. 5. What is meant by the graph of an equation ? Find ten pairs of numbers, each of which satisfies the equation 2 x + y = 12. Carefully locate the points determined by these pairs of numbers. 6. How many solutions has such an equation as that given in Ex. 5? Show that its graph may be regarded as a record of all of its solutions. 7. Show that the equation 3 x = 2 (i.e., 3 x + • 2^ = 2) is satisfied by each of the following pairs of numbers: f , 1 ; |, 2; |, 3; f, 4; etc., f, 0; I, — 1 ; f, — 2 ; etc., i.e., by every pair of numbers of which the first is -f . Where do all these points lie (cf. Ex. 3) ? What, then, is the graph of the equation 3 a; = 2 ? Draw it. 8. As in Ex. 7, construct the graph of 2 y = 5. Of x- = - 1. Oi y = ix. Of a:2 = 9. * Students who are acquainted with the theory of similar triangles will find no dilWculty in proving that all these points lie on the same straight line (.-l /?), and 9,lso that the coordinates of every point on AB will satisfy tlie given equation. 192 ELEMENTARY ALGEBRA [Ch. XI Assuming the graph of a first degree equation in two unknown num- bers to be a straight line, construct the graph of each of the following equations by finding two of its points and drawing a straight line through them: 9. 2x + y-4: = 0. 11. iz-y = d. 10. 3y-4a: + 2 = 0. 12. ^ - ^ = -?-. X y xy 116. Intersection of two graphs. Since any two numbers which satisfy an equation are the coordinates of some point on the graph of that equation (§ 115), therefore a pair of numbers which satis- fies each of two given equations must be the coordinates of a point which is on the graph of each of these equations, i.e., these numbers are the coordinates of a point in which these graphs intersect. Hence, to find the coordinates of the point in wiiich the graphs of two equations intersect each other, it is only necessary to solve these equations, regarding them as simultaneous. On the other hand, instead of solving two simultaneous equa- tions in the ordinary way, one may accurately draw the graph of each of these equations, using the same axes for both, and care- fully measure the coordinates of their point of intersection ; these coordinates will constitute an approximate solution of the given equations. EXERCISES 1. Find the coordinates of the point of intersection of the graphs of X + y = 5 and 2 x — y = 4, both by solving these equations and also by measurement, and compare the results. 2. Solve the system of equations 3 a; + 4 ?/ = 7 and 2 x — S y = IQ hy the graphic method, i.e., by measuring the coordinates of the point in which their graphs intersect. Find the coordinates of the point of intersection (as in Ex. 1) of the graphs of each of the following pairs of equations: 3x-§?/ = 3, ^ (2x-Sy=7, 5x-7ly=n. 6. Show that the two equations in Ex. 5 are algebraically inconsistent. How are their graphs related to each other ? Where is their intersection ? 7. In how many points can two straight lines intersect each other? Does this agree with § 111 ? Explain. 3. (4y-^dx=5, ^ r3x-§?/ = 3, ^ j [4x-3y = d. ' [ix-2y = 4. ' [ CHAPTER XII INEQUALITIES 117. Definitions. Expressed in algebraic language, the condi- tions of the problems thus far met with have led to equations; but there are many other problems whose conditions lead only to a statement that one of two expressions is greater or less than the other. A correct analysis of such a statement is often of great importance, and may afford all the desired information concerning the numbers involved in the given problem. The symbols > and < are called the symbols of inequality, and are read "is greater than," and "is less than," respectively. Thus, a>& is read " a is greater than 6," and a< 6 is read "a is less than b." One number is said to be greater than another when the result of subtracting the second from the first is a positive number, and one number is said to be less than another when the result of subtracting the second from the first- is a negative number. Thus, if a — b is positive then a >■ 6, while if a — b is negative, then a 2 ; also, since 2 — (— 6) = 8, therefore 2 > — 6 ; and since 8 — 1.5 = — 7, therefore 8 << 15. The statement that one of two numbers or expressions is greater or less than the other is called an inequality. The number or expression which stands at the left of the symbol of inequality is called the first member of the inequality, while the number or expression which stands at the right of this symbol is called the second member, — the opening of the symbol being toward the greater number. Thus, a > 6 is an inequality of which a is the first member and 6 the second ; it is read, " a is greater than 6." Two inequalities are said to be of the same species (or to subsist in the same sense) if the first member is the greater in each, or if the iirst member is the lesser in each; otherwise they are of opposite species. Thus the inequalities a > 6 and c + dy- c are of the same species, while a-*2 + ?/2>2:2 and m^ 8, and so, also, 10 + 5 > 8 + 5, and 10- 5 > 8 — 5. To prove this principle generally, let the given inequality be n < 6, and let c be any number whatever ; then (a + c) — (6 + c) , which equals a — 6, is negative, since a < 6, and therefore, by definition, a + c <,h -\- c. Similarly, a — c<,h — c. Manifestly the proof would have been just the same if the given inequality had been a>6. From the principle just proved it follows that terms may be transposed in a,n inequality, just as in an equation, viz., by reversing their signs; for subtracting any given term from each member will cause that term to disappear from one member, and to reappear, with its sign reversed, in the other. (ii) // several inequalities of tJw same species be added, member to member, the result will be an inequality of the sajne species. E.g., adding the inequalities 3 < 7, 21 < 30, and — 2 < 1, member to member, we obtain 22 < 38. To prove this principle generally let a > 6, c > d, e >/, •••, /i > A; be any num- ber of given inequalities, all of the same species; then each of the differences a — 6, c — d, e — /, •'•,h — k\s positive, hence their sum is positive, i.e., (a — 6) + (c — cZ) + (e — /) H h (A — k) is positive, hence {a-\-c + e-\ (- A) — (& + d +/H h k) is positive, and therefore, a + c-{-e-\ \-h>h-\-d +/H \-k; which was to be proved. It should be carefully noted that if two or more inequalities which are not of the same species are added, the result may or may not be an inequality. The student may illustrate this statement by means of some numerical examples. 118] INEQUALITIES 195 (iii) // an inequality he subtracted from an equation, or from an inequality of opposite species, member from mem- her, tim result will be an inequality whose species is oppo- site to that of tJie subtrahend. The proof of this principle is similar to that of (ii) above, and is left as an exercise for the student. The student may also illustrate, by appropriate examples, that if one inequality be subtracted from another inequality of the same species, the result may be an inequality of the same or of opposite species, or it may be an equation. (iv) If each mem^ber of an inequality be multiplied or divided by the same positive number, the result will be an inequality of the same species. E.g., 24 > 20, and so, also, 24-^ 4 > 20 -^ 4 ; again, 3 < 5, and so also 3 • 7 < 5 • 7. To prove this principle, let a >> 6 be any inequality, and let c be any positive number whatever; then {a — b)c is positive, since each factor is positive, i.e., ac — be is positive, and hence by definition, ae >■ 6c, which was to be proved. Similarly it is proved that, under the above conditions, ah c c The principle just proved enables one to clear an inequality of fractions, and also to remove any factors that are common to both members. (v) // each member of an inequality be multiplied or divided by the same negative number, the result will be an inequality of opposite species. To prove this principle, let a > 6 be any inequality, and let c be any negative number whatever; then {a — b)c is negative, i.e., ac — be is negative, and hence ac < he, which was to be proved. Similarly it is proved that, under the given conditions, a & c c" (vi) // t?ie signs of all the terms of an inequality be re- versed, then the symbol of inequality must also be reversed. E.g., if2a — 4c' + 3a;>2d + 5y — 7 6, then 4c — 2a-3a;<76-2d — 5y. The proof of this principle follows directly from (v) by putting — 1 for the multiplier c. 196 ^ ELEMENTARY ALGEBRA [Ch. XII (vii) If the first of three numhers is greater than the second, and the second is greater than the third, then the first is greater than the third; and conversely. E.g., 10 > 7 and 7 > 3, and 10 > 3 also. To prove this principle, let a > & and 6 > c be the given inequalities ; then a — 6 is positive, as is also 6 — c, and hence their sum (a — h) + (6 — c), i.e., a — c, is positive, and therefore a > c, which was to be proved. Similarly it is proved that if a < & and 6 < c, then a < c. (viii) If two inequalities which are of the same species, and whose members are all positive, he multiplied togetJier, meinher by member, the result will be an inequality of the same species. E.g., 5 > 3 and 4 > 2, and 5 . 4 > 3 • 2 also. To prove this principle, let a > 6 and c > (^ be two such inequalities ; then by (iv) ac > be, but by (iv) he > hd, whence by (vii) ac > hd, which was to be proved. By proceeding step by step, it is clear that principle (viii) holds for any num- ber of (and not merely for two) such inequalities. The student may modify the above statement and proof so as to apply to the case in which some of the members are negative. EXERCISES 1. When is the first of two numbers said to be greater than the second ? When is it said to be less ? 2. By the definitions of "greater" and "less " given in § 117, show that 5 > 2 ; that - 23 <- 12 ; and that 2 > - 9. 3. li a=^ b, show that a"^ + b^>2 ah. This is a very important rela- tion, and well worth remembering. Suggestion, (a — 6)2 is positive whether a > & or a < 6. 4. If two or more inequalities of the same species are added, what is the species of the resnlting inequality? Prove your answer. Is it neces- sary that the members of these inequalities hQ positive numbers? 5. If an inequality is subtracted from another inequality of the same species, member from member, what is the result? Prove your answer. 6. If two inequalities of the same species are multiplied together, member by member, what is the result? Prove your answer. Is it necessary in this case that the members of these inequalities be positive numbers? 118-119] INEQUALITIES 197 7. What happens if the signs of the terms of each member of an inequality are reversed? Why? 8. May terms be transposed from one member of an inequality to the other ? If so, how and why ? 9. What other operations may be performed with or upon inequali- ties, producing results whose relations are known ? 10. Name and illustrate some operations with inequalities that give results about whose relations there is doubt. E.g., the quotient of two inequalities of the same species, divided member by member, may be an equality or an inequality of the same or of opposite species. 119. Unconditional and conditional inequalities. An unconditional inequality is one which is true for all values of the letters in- volved — e.g., a + 4>a; while a conditional inequality is one which is true only on condition that the values to be assigned to the letters involved shall be somewhat restricted — e.g., a; + 4<3a; — 2 only on condition that the values assigned to x shall be greater than 3.* To solve a conditional inequality means to find those values of its letters for which the inequality is true ; this may be done by means of the principles which were proved in the preceding article = — for illustrations see Exs. 1 and 2 which follow. Ex. 1. Given 3 x — ^^ > ^^ — ar, to find the possible values of x. Solution. On multiplying each member of the given inequality by 3, it becomes ^ 9 a: - 25 > 11 - 3 a:, [§118 (iv) whence 9 a: + 3 a; > 11 + 25, [§ 118 (i) i.e., V2x> 36, whence a:>3;. [§ 118 (iv) therefore, if the given inequality is true, x must be greater than 3. By means of the principles established in § 118 the student may show that each step in the reasoning of Ex. 1 is reversible, and hence that the. converse of that example is also true ; viz., that if x >■ 3, then 3 a; — -^/- >► ^-^- — x. * Let it be observed that conditional and unconditional inequalities are respec- tively analogous to conditional and identical equations; the student may also note the analogy between solving an inequality and solving an equation. 198 ELEMENTARY ALGEBRA [Cri. XII "ig I' ^° ^°^ those values of x and y that will satisfy them both. Solution. On multiplying each member of the inequality by 4, and each member of the equation by 3, they become, respectively, 8x+12?/>20, and 3a:+123/ = 18; whence, subtracting, 5a:>2, [§118 (i) and therefore a:>|. [§ 118 (iv) Now substitute for^a: any number greater than g, in the above equation, and find the corresponding value of y ; these values of x and ?/, taken together, will satisfy both the equation and the inequality. EXERCISES 3. Distinguish between a conditional, and an unconditional inequality. To which of these classes does aP- + h'^ + \>2 ah belong? Why ? 4. Is the expression 6x — 5>3x + 10 true for all values of x ? If not, what is the least value that x may have in this inequality ? To which class does this inequality belong? 5. What is meant by " solving " a conditional inequality ? Describe the procedure. Illustrate what you have said by solving the inequality in Ex. 4. 6. From the inequality in Ex. 4 above it is found that a: > 5, i.e., the range of values that x may have in this inequality is from just above 5 upward ; 5 may here be called the lower limits or minimum, of the possible values of x. Find the minimum value of a: in 3 a; < 5 a; — 9. 7. Show that the range of values of a: in a;^ + 24 < 11 a; is between 3 and 8, i.e., that 3 is the lower limit, or minimum, and that 8 is the upper limit or maximum. Suggestion. In order that {z — 3) (8 — a;) , i.e., Wx — x"^ — 24, may be positive, both factors must be positive or both negative. Find the range of values of x in each of the following inequalities : 8. a:2 > 9. 13. x'^ + o a: > 24. 9. .2 + 24>lla:. , J4.-ll>-, 10. 30>a:+-^>25. ^*- |io_x>5.^ 11. 28>3a: + a;2. j3-4r<7, 12. a:2>9a:-18. ' la:+2<4. 119] INEQUALITIES 199 16. By the definitions of "greater" and "less" given in § 117, show that n + -<2, when n is any positive number,* i.e., show that the sum of n any positive number and its reciprocal is not less than 2. 17. Show that 4 x^ + 9 < 12 x.* 18. Show that26(6a-5 6)X2a + &)(2a-6). If a, h, and c are positive and unequal, prove the correctness of the fol- lowing statements: 19. a2 + 62 ^ c2>> ab + hc ■\- ac. 20. a^-\-b^> a% + abK 21. a^ + 68 + cS > 3 abc, 22. If a2 + 62 = 1, and c"^ + cP = 1, prove that a6 + erf > 1.* 23. If m and n are both positive, which of the expressions ^ ^^ or ^'"'^ is the greater? m '\-n Solve the following systems : r2x-3j/<2, f3a:+2y = 42, fx + y=10, 24. 3 3^. l2x + 52/=:6. |3a:-|>16. l4:c< I 20 15 ^ Find the integral values of x and y in the following systems : l3x-2/<21. |l3a:-^<33. 31. If 16 more than 3 times the number of sheep in a certain flock exceeds 27 plus twice their number, and if 45 less than 4 times their number is less than their number diminished by 6, how many sheep are there in the flock ? 32. Find the smallest integer fulfilling the condition that \ of it decreased by 7 is greater than \ of it increased by 6. 33. Find a simple fraction (in its lowest terms) which, when 2 is added to its numerator and subtracted from its denominator, shall be greater than f, while if 2 is subtracted from its numerator and added to its denominator, it shall be less than \. 34. Three times A*s money and 4 times B's is $ 1 more than 6 times A's ; and if A gives $ 5 to B, then B will have more than 6 times as much as A will have left. Find the range of values of A's money and B's. ♦Compare also Ex. 3, p. 196. The symbol <[ stands for "is not less than." 200 ELEMENTARY ALGEBRA [Cii. XII REVIEW QUESTIONS- CHAPTERS XXII 1. Define and illustrate: conditional equations; equivalent equations; integral equations ; the degree of an equation ; literal equations. 2. Outline the plan for solving a conditional equation in one unknown number, and state the principles upon which this plan rests. 3. How may a fractional equation in one unknown number be solved? 4. Under what circumstances are extraneous roots introduced by- clearing an equation of fractions? How may such roots be detected? 5. By means of the equation \- — — — - = -^^ — -^, illus- trate your answer to Lk. 4. 6. Define and illustrate what is meant by : an indeterminate equa- tion; an indeterminate system of equations; consistent equations; inde- pendent equations ; simultaneous equations. 7. Outline three methods of elimination. 8. Prove that the syetem of equations a^x + b^y = Cj and Og^; + &22/ = Cg has one solution, and only one, if a^b^ ^ «2*i- 9. Outline the procedure for solving a system consisting of n inde- pendent simple equations in n unknown numbers. 10. Find an expression of the form ax^ + bx + - whose value is 16 when X = — 1, 2 when x = 1, and 40 when x = 2. ^ 11. What is meant by the graph of an equation? Illustrate your answer. 12. How may the graph of an equation be constructed ? Construct the graph oi 6y = Sx -\- lO; also of 2 y^ = S x -\- 1. 13. How may a pair of equations, such as that given in Ex. 8, be solved graphically ? Illustrate your answer. 14. Define a conditional inequality, also an unconditional inequality. Illustrate each. 15. How may a conditional inequality be solved? Illustrate your answer by finding the range of values of x in the inequality x — 3 < — . 10 '"' 16. If X - 3 < — , does it follow that a:^ - 3 x < 10 ? X 17. Prove that a positive proper fraction is increased by adding the same positive number to both its numerator and its denominator. CHAPTER XIII INVOLUTION AND EVOLUTION I. INVOLUTION 120. Definitions. If a represents any number * whatever, then it has been agreed that the product «•«•«••• (to n factors), which is called the /ith power of a, shall, for brevity, be represented by the symbol a", which is usually read "a nth." The number a is called the base, and n the exponent, of the power [cf. § 7 (iv)]. The operation of raising a number to any given power is called involution. It consists merely in a succession of multiplications; thus, 43 = 4.4.4 = 64, (-2/ = -32, (a+&)2 = a2 + 2a6 + 6^ etc. Under the above definition the symbol a" has been appropriated only when 71 is a positive integer ; that definition assigns no mean- ing whatever to such expressions as a~^, a°, and a^. In § 44 1 it was shown, however, that in operating with such symbols as a" it is often advantageous to make the further agreement that a"*, where k is any positive integer, shall mean — , and that a" shall mean 1. In Chap. XIV such symbols as a^ will have a meaning assigned to them, and will receive detailed consideration. 121. The exponent laws. Under the above agreements as to the meaning of a", the following laws for exponents are easily estab- lished. (i) First exponent law. If a is any base, and m and n are integers (positive or negative), or zero, then * The word number is here used to include algebraic expression also, t This article should now be reread. J Compare also § 37. 201 202 ELEMENTARY ALGEBRA [Ch. XIII For, if m and n are positive integers, then a"* • a" = (a • a • a ••• to m factors) • (a • a • a ••• to n factors) =z a ' a ' a '• • to (m -\- n) factors [Associative law = a'"+". If either m or n is a negative integer, say n = — Jc, where A; is a positive integer, then a* a* _ g • g ' g • ' • to m factors g . g . g ••• to k factors g* "• according as m > A;, or m < A; ; but (since ri = — fc) g*""* = «"*+'*, and — = g-^*-"*) = g*""* = g^^- : therefore g"* . g" = g'"+", even if one of the exponents is a negative integer. Similarly the student may prove the correctness of this law if both m and n are negative, or if either or both of them are 0. By successive applications of the foregoing law, and with the same limitations upon the exponents, it follows that a"' • a" • a^ • a'' - = fl^«+n+p+'-+-. (ii) Second exponent law. If a is any base, and m and n are integers (positive or negative), or zero, then For, if m and n are positive integers, then (g*")" = (g • g . g ... to m factors)" = g • g • g • • • to mn factors [Associative law 121] INVOLUTION AND EVOLUTION 203 and if either m or n is a negative integer, say m = — k^ where k is a positive integer, then («"•)« = (a-*^)" = /^i-Y=— ' — ' — '" ton factors = -J— = — = a-*^" = a"*". r—k==m If both m and n are negative, or if either or both of them are zero, the proof is similar to that just given; hence, for all these cases, ^^r.y ^ ^«„ (iii) Third exponent law. If a and h are any two bases, and n is a positive or negative integer, or zero, then al" •})'' = {obY. For, if n is a positive integer, then a** • 6" = (a • a • a ••• to n factors) .(p'b-b-" to n factors) = a6 . a5 . a5 . . . to n factors [Commutative and [_ associative laws = (aby; if n is a negative integer, say n = — k, where A; is a positive inte- ger, then a" . 6" = a-* . 6-* = i ^^ = "Fir. = 7^7* = (^^)" = («^)^*' a* b'' a* • &* {aby as before ; and if n = 0, then a" • 5" = 1 = (aby ; [Since a;^' = 1 hence, for all these cases, a" • 5** = (aby. By successive applications of the above law it follows that aj'lf c"^' d^ ••• = {ohcd — )"• (iv) Fourth exponent law. If a is any base and m and n are any integers, or zero, then a"" ^ a""- a^'*. 204 ELEMENTARY ALGEBRA [Ch. XIII The proof of the correctness of this law rests directly upon the first exponent law [(i) above], and the definition of a quotient [§ 3 (iv)], for, since ^m-n , ^^n ^ ^m-n+n ^ ^r.^ [-0 above therefore . a"' ^ a" = a"*-". [§ 3 (iv) EXERCISES 1. Write a carefully worded statement of each of the four exponent laws above, — e.g., the third law may be stated thus : " The product of like powers of any two or more numbers is the like power of the product of those numbers." 2. How is the sign of the power in such a case as (— 6^)5 determined? State, illustrate, and prove a law which shall cover all such cases, bearing in mind that the exponents may be positive or negative integers, or zero (cf. § 18, especially note 2). 3. Tell what the ngn of the result in each of the following expres- sions is, and explain your answer : (-a)3; (-a)^ (a)-^; (- a)-3; (-4)0; (-f^'V'; (-6)^80; (-a:)2"; and (-2)2«-i. ^ ' What is the value of (- 2)3 . 2-2? Of 3-2 • (- 2)3? Of 32 . 2-2? 4. How is a fraction raised to a power ? Why ? Give four illustra- tions of your answer. Read again the second paragraph of § 120. 5. What does a represent in the proofs of § 121? May it represent any polynomial whatever, as well as any number ? What does it repre- sent in Ex. 3 ? 6. By § 62 expand the following expressions : {x + y)^ {x + y)^, and {x + z/)^; then multiply the first two expanded forms together, and thus verify that {x + yY • {x + yY = (x + y^- 7. To what kind of numbers were exponents originally limited ? To what extent has this limitation now been removed ? What is the meaning of such an expression as x~^ ? Of x^ ? Read again § 44, and the third paragraph of § 120. Simplify the following expressions (free them from negative and zero exponents where such occur, etc.) and explain each step of your work fully, always referring to the appropriate exponent laws : 8. a^Wc^\ a-3i-%-3; and (a-2)8. 9. (a2x8^-2)4; (m3^?/-4)2. ^nd (rt2y-3)-2. 10. (rt2:^2)3_^ (_a^2)2. (6a0)2 - (2x0)2; and (- 122a-8j:V)2 -^ (-32a-5x2)3. 121-122] INVOLUTION AND EVOLUTION 205 12. f--^y. 14. l-^^J^!n\\ 16 /_zr!£!^V". 17. State the binomial theorem (§62). 18. How many terms are there in the expansion of (m + n)^? How- many in (a - 6)8? How many in (3 s - 2 0"? 19. What are the signs of the terms in (a - &)8? Compare (a - by with [a + (- b)y, and explain why the alternate terms of the expansion are negative. Write down the expansions of the following expressions, and remove negative exponents where they present themselves : 20. (2a-36)4.t 25. (a-hb-hcy,i.e.,[a+(b + c)f. 21. (x2-i/t)8. 26. (3xY-^fz^y- 22. (x-2 + 3 z/-i)4. 27. (a;3 - 2 ?/-3)5. 23. (3 a + 2 &2)5. 28. (a-2 - ic-i)^. 24. (2 m +3 xy. 29. (2 x8 + 3 a;2 - 5)*. 30. Is (a- b'C- /32 x6?/io. 10. Va^"x-^y^\ 11. f-? 13. v/128 a^^ft-i^y. 8 / 256 yn V« ■ \6561^3%-8 j_^ 3 / 125 a;i2yg ' V 1728 a^z^' 18. .027 a'\r^ ' \ 128a:i4 15. I Hi '^-y) . „ 2na2n2-2^ 19. \ _ 5 I- 32 «5:,40 I, a5xjjx2^dx 12. V- 243 a^^x-^ \ 243 y^s \ 2-^^y^'z' 21. Write a rule for the extraction of such roots as the above, and emphasize particularly the matter of exponents and signs. Does your rule apply to roots of polynomials also ? 208 ELEMENTARY ALGEBRA [Ch. XIII 124. Roots of polynomials extracted by inspection. If a poly- nomial is an exact power of a binomial, a little study will usually reveal the corresponding root ; this is illustrated by the following examples. Ex. 1. Find the square root of m^ + 4 m% + 4 n^. Solution. This expression is easily seen to be (m^ + 2n)2; there- fore Vm*^ + 4 m^n + 4 n^ = i (m^ + 2 n). Ex. 2. Find the cube root of 8 a^ _ 36 a% - 27 b^ -\- 54 ab^ Solution. Since the given polynomial has four terms, two of which, viz., 8 a^ and — 27 h^, are exact cubes, therefore it may be the cube of a binomial (§ 62) ; if it is the cube of a binomial, that binomial must be 2 a — 3 6 (why ?), which, on further examination, proves to be the required cube root. Hence v'S a^ - 36 a% - 27 6^ + 54 ab^ = 2 a -8b. A polynomial which is the square of another polynomial may also sometimes be recognized as such (cf. § 61), and its square root may then be written down by inspection. Ex. 3. Find the square root of a^ + 62 _ 2 aJ - 4 &c + 4 c^ + 4 ac. Solution. Since the given polynomial consists of six terms, three of which are exact squares, and three of which are double products, there- fore (§ 61) it mmj be the square of a trinomial whose terms are the square roots of the square terms ; by a little further examination it is seen that Va2 + 62 _ 2 a6 - 4 &c + 4 c2 + 4 ac =±(a-b + 2c). EXERCISES Extract the following indicated roots by inspection, and verify : 4. ^4 x2 + 12 a: + 9. 6. V(m + n)2 -4:(7n + n)+ 4. 5. V25 2/2 _ 40 y + 16. 7. Vx^ + 2xy + y^ -2xz -2yz + z^. 8. \/8 h^ - 84 h% + 294 hk^ - 343 k\ 9. \/x^ -^xhj + y^-^xy^ + Q x'^y\ 10. a/8 w8 _ 12 u^v - v3 + 6 uv\ 11. v^flS _ 65 _ 5 ^4^, + 5 aj4 + 10 rtS^'i _ 10 an\ 12. Va2 + 9 62 _ 6 a6 + 6 (x - 2 2/) (a - 3 6) + 9 (a:2 - 4 a:?/ + 4 y'^). 13. Vx^ - 6 abx^ + 15 a%^x^ - 20 a%^x^ + 15 a'^b^x'^ - 6 a%^x -f- a%^. 124-125] INVOLUTION AND EVOLUTION 209 125. Square roots of polynomials. Since it is not always easy to lind the square root of a polynomial by the method illustrated in § 124, another method, which is always applicable, will now be given. This method will be better understood by first squaring a polynomial and carefully observing its formation, and then reversing that process. (i) Consider first the binomial A-\-B; its square is, A^ + 2AB-\- B^, therefore the square root of A^ + 2 AB + B'^ is J. + jB ; and the question now to be investi- gated is: given the power A^ + 2 AB-\- B^, how may the root A-\-B he found from it? Since the first term of the power is the square of the first term of the root, therefore the first term of the root is the_square root of the first term of the power ; i.e., the first term of the root is \A^, viz., A.* If the square of the root term just found be subtracted from the given power, then the first term of the remainder, viz., 2AB, will be the double product of the first and second terms of the root, therefore the second term of the root is found by dividing the first term of the remainder by twice the root already found. Twice the root already found at any stage of the work is usually called the trial divisor, and the trial divisor plus the next root term is called the complete divisor. The work of finding the square root just considered may be put into the fol- lowing form : ^2 + 2^5 + ^2 1 ^ + ^ ^2 Trial divisor, 2 A Complete divisor, 2 A 2AB-{-B^ 2AB + B^ =i2A + B)'B Observe that the first and second subtractions are together equivalent to the subtraction of (A + B)^ from the given power. Similarly, to find the square root of 9 m2 — 42 mx^ + 49 x^, the work may be arranged thus (the student should fully explain each step of the process) : 9 m2 — 42 mx^ + 49 x^ \Sm — 7x^ 9m2 Trial divisor, 6 m Complete divisor, 6 m — 7 x^ — 42 ma;3 + 49 x^ — 42 mx^ + 49 a;6 = (6 m - 7 x^) (— 7 x^) (ii) The above plan for extracting the square root of a trinomial power is easily extended so as to apply to polynomial powers of any number of terms. Consider, for example, the expression A + k + B, wherein A stands for the first n terras, k for the next term, and B for all the remaining terms of any poly- * For the consideration of the negative root (viz., — A), see note 2, page 211. 210 ELEMENTARY ALGEBBA [Ch. XIII nomial whatever; and let all of the terms of this polynomial be regarded as already arranged according to the descending powers of some one of its letters. The square of this polynomial is A^ + 2 Ak-\- k^-{-2 AB -i-2 Bk-{- B^, and the question is: c/iven the power A^-{-2 Ak + k^ + 2 AB -{-2 Bk + B'^, how may the root A-{- k + B be found from it ? Let it be assumed that the terms represented by A have already been found,* — by (i) above or any method whatever, — then it is clear that when A^ has been subtracted from the power, the highest term in the remainder is the highest term in 2Ak, hence the next term in the root (viz., k) may be found by dividing the highest term in this remainder by the highest term in 2 A, i.e., by the highest term in the trial divisor. But since A stands for the terms of the root already found, therefore what has just been said shows how to find the next term of the root at any stage of the work, i.e., it shows how to find all the terms of the root. This work may be arranged thus : A^ + 2Ak + k^ + 2AB^2Bk + B2 \A+k A^ Trial divisor, 2 A Complete divisor, 2A-^k 2Ak + k'^-\-2AB + 2Bk+B^ 2Ak + k^ =i2A + k)'k 2AB + 2Bk + B^ Observe that the two subtractions here made are together equivalent to sub- tracting (^ + A;)2from the given power; i.e., by proceeding as above explained, the remainder at any stage of the work is the same as that obtained by subtract- ing the square of the root found at that stage of the work from the given power. Similarly, to find the square root of 9x^ + 6 xhj — 11 xhj'^ — 4 xij^ + 4 y^, the work may be arranged thus (the student should, however, explain each step) : 9 a:4+r,.r3,/_iia;2;/2_4 3^2/3+4 ?/4 [ 3 x^+a;?/— 2 ?/2 9a;4 (> xhj - 1 1 a:2?/2— 4 a;?/3+4 ?/4 6x3//+ x%2 ={Qx'^+xy) 'Xy 1st trial div., 2{Zx'^)=i\x'^ 1st comp. div. , 6 x^+xy 2d trial div., 2(3a;2-f-a-//)=6x2+2x?/| — 12.r2?/2— iajyS-f^^/^ 2d comp. div. , 6 a;2+2 xy-2 r/2 j —12 x^y"^-^ a!?/3+4 ?/4= (fi x'^+2 xy—2 y2) .27/2 The above method for extracting the square root of a polynomial may be stated thus : (1) Arrange the terms of the given polynomial according to the descending powers of soine one of its letters, and write the square root of its first term as the first term of the required root. * The first term at least may always be found as in (i) above. 125] INVOLUTION AND EVOLUTION 211 (2) Subtract the square of the root terrn just found from the given polynofnial, and divide the first term of the remainder by twice tl%e first termj of the root; write the quotient as the next terin of the required root, and also annex it to the trial divisor to form the complete divisor. Q^) Multiply the coTnplete divisor by the last root term, which has just been found, and subtract the product from tixe preceding remainder. (4) Divide the first term of this new remainder by the first term of the new trial divisor; write the quotient as the next term of the required root, and also add it to the trial divisor to form the complete divisor. (5) Repeat the steps (3) and (4) until all the terms of the root are found. Note 1. Observe that if polynomials are arranged according to ascending instead of to descending powers of the letter of arrangement, the above demon- stration still applies; it requires only the verbal change of lowest term for highest term. Note 2. If the negative value, instead of the positive value, of the square root of the first term of the polynomial had been used in the above demonstra- tion, the sign of each term of the result would have been changed, i.e., the result would have been the negative square root of the given polynomial. Note 3. It has been shown above how to find the square root of a polynomial which is an exact square; i.e., if the above process be continued until a zero remainder is reached, then the square of the expression thus found will be the given polynomial. If, however, the same process be applied to a polynomial which is not an exact square, then as many root terms as desired may be found, and the square of this root, at any stage of the work, will equal the result of sub- tracting the corresponding remainder from the given polynomial — such a root is usually called an approximate root, and also the root to n terms. EXERCISES Find the square root of each of the following expressions, and verify the correctness of your result : 1. a;* - 4 a;8 + 8 X + 4. 2. 4 w* - 4 m3 - 3 m2 + 2 Tw + 1. 3. l-6y+52/2+i2y3_^.43,4. . 4. 25 x^ - 40 a%^x^y^ + 16 a^ft*. 5. 4x6 + i7a.2 _ 22 x3 + 13 x4 - 24 X - 4x6 + 16. 212 ELEMENTARY ALGEBRA [Ch. Xlll 6. 4 a4 + 64 64 _ 20 a% + 57 a^"^ - 80 ab^ 7. 6 a:^^ + 2 x^y^ - 28 xy^ -\- 9 x^ + 4: y^ + 45 a;^?/* + 43 x^y^. 8. 3 ^4 - 2 x5 - a:2 + 2 a: + 1 + x^ * 9. 48 a* + 12 a2 + 1 - 4 a - 32 a8 + 64 a« - 64 a^. 10. 46 a:2 + 25 a;4 - 44 a:3 - 40 ar + 4 a:« + 25 - 12 x^. 11. x^-2x'^y + 2 x^z^ - 2 7/^2 + y2 ^ 2*. 12. a:8 - 2 a'^x^ - 3 a^a;* + 4 «6a;2 + 4 «» - 16 a^a;+ 32 aSa;^- 20 a3a;5 + 4 aa;^ 14. — + 16 a^y^ + 8 x'^y^. 15. a;2+2a;- 1 -- + --t X x^ 16. 9a,-2-24a; + 28- — + i. a; a:^ 17. n4 + 4 n3 + - + 2 n + 4 4- 4 w2, 18. x* + 1 + 4 a;8 + - + 6 a:2 + -^ + 5 + 5 a: + -. x^ x^ 4a;2 a; 19. 4 + ^'-^-^ + -^. 62 6 a 4a2 20. (a: - y)2 - 2 (a-^/ + 3:2 - 2/2 _ yz) + (y + 2)2. 21. a;2'-_y2» _ 6 x^^hf^"^ - 30 x^y'^"^ + 10 a:2''-y«+i + 25 a;2'-23/2s+2 _f. 9 ^.y, 22. 1 + a:, to 4 terms. See note 3. 23. a2 + 1, to 3 terms. 24. 1 + a; — a:2, to 4 terms. 25. a;4 + 2 x^y + 2/^ + a:^/^ + a:23^^ to 4 terms. 26. By extracting the square root until a numerical remainder is reached, show that x4 + 4x^+8a:2 + 8a:- 5 equals (^2 + 2 x + 2)2 - 9, and thus find the factors of x* + 4 a;^ + 8 a:2 + 8 x — 5. 27. Similarly, find the factors of x* + 6 x^ + 11 x2 + 6x - 8 and a6 - 6 a* + 10 a8 + 9 a2 _ 30 a + 9. * Check Exs. 8-21 by the method of Ex. 7, § 39. t Show first that this expression is already arranged according to descending powers of «. 125-126] INVOLUTION AND EVOLUTION 213 126. Square roots of arithmetical numbers. Arithmetical num- bers are merely disguised polynomials — e.g., 3862 = 3 (10)* + 8 (10)^ + 6 (10) + 2 — and their square roots are extracted by virtually the same process as that given in the preceding article. Although it is not necessary to do so, yet it is more systematic to find the several digits of these roots in their order from left to right, just as the terms are found in the case of polynomials; to do this the given number is first separated into periods of two figures each, to the right and left of the decimal point. The reason for the separation into periods lies in this : the square of any num- ber of tens ends in two ciphers, and hence the first two digits at the left of the decimal point are useless when finding the tens' digit of the root ; they are there- fore set aside until needed to find the units' digit of the root. So, too, the square of any number of hundreds ends in four ciphers, and hence, for a like reason, two periods are set aside when the hundreds' digit of the root is being found, and so on. Similarly for the periods at the right of the decimal point. The application of the method of § 125 to extracting square roots of arithmetical numbers may be best understood in general by first considering some particular examples. Let it be required, for instance, to find the square root of 1156. Since this number consists of two periods, therefore its square root will consist of two integer places, i.e., of tens and units. Moreover, since 30^ < 1156 < 40^, therefore the required root lies between 30 and 40, i.e., the tens' digit is 3, the square root of the greatest square integer in the left-hand period of the given number. The units' digit may now be found as follows : let Tc represent the part of the root already known (viz. 30), and let u represent the unknown part of the root ; then 1156 = (A; + w)2 = A:2 + 2 ku + w2, and, therefore, 2 ^w + w^ = 1156 - A;2 = 256. [A;2=900 Again, since k represents tens while u represents units, therefore 2 ku is much greater than ifi ; hence the last equation above shows that 2 kit (though somewhat less than 256) is approximately equal to 256, and hence that 256 -^2 A; (though somewhat too great) is approximately equal to u, i.e., 256 -^ 2 Z: will suggest a value for u, which must then be tested by the above equation.* * Since 2.56= (^k-\-u)u, therefore 256^ (2 A;4- w) == w, i.e., the complete divi- sor is 2k-\-u, and 2fc is merely a trial divisor; hence the appropriateness of these names. Since 256-^2 ^ gives too great a quotient, therefore the units' digit in the required square root is either 4 or a smaller number; hence if the units' digit is not 4 (i.e., if it is 3, 2, 1, or 0), then (A: + 4)2 > 1156, i.e., 1156— (A: + 4)2 is negative, and the next smaller number must be tried. This shows that the Jirst one of these numbers (4, 3, •••) w hich leaves a positive remainder in the above subtraction is the units' digit in \/ll56. Similarly in general. 214 ELEMENTARY ALGEBRA [Ch. XIII Finally, since k is already known to be 30, therefore 256 -;- 2 A: = 256 -^ 60 = 4+, hence u is probably equal to 4; substituting this value of u in the equation 266 = 2 A;u + w2^ proves that w = 4, and hence that \/ll56 = 34. The work may be arranged as follows : (A; + w)2=A;2 + 2A:n + w2= 11'56 1 30 + 4 = 34 ^2 = (30)2 = 900 trial divisor is 2 A; = 60 \25Q = 2 ku + u^ complete divisor is 2 & + m = 64 1256= {2k-\-u)-u Again, let it be required to find the square root of 315844. Since this number consists of three periods, therefore its square root will con- sist of three integer places. The work may be arranged as follows (the student should fully explain each step) : 31'58'44 250000 1500 + 60 + 2 = 562 1st trial divisor, 2 • 500 = 1000 Ist complete divisor, 1000 + 60 = 1060 2d trial divisor, 2 • 560 = 1120 2d complete divisor, 1120 + 2 = 1122 &5844 63600 = 1060 . 60 12244 1 2244 = 1122 . 2 NoTB. When some familiarity with the above process has been gained, the work may be abridged by omitting unnecessary ciphers, and annexing to each remainder the two digits which compose the next period in the given number, thus : 31'58'44 25 1562 Ist complete divisor, 106 2d complete divisor, 1122 658 636 12244 12244 Finally, let it be required to extract the square root of 10.5626. 13.25 The work may be arranged thus : 1st complete divisor, 62 2d complete divisor, 646 10.'56'25 9 156 124 13225 13225 The results of the discussion of the present article may be stated thus: (1) Separate the given number into periods of two digits each, beginning at the decimal point and counting both toward the^ right and toward the left, completing the right- hand decimal period by annexing a cipher if necessary. 126] INVOLUTION AND EVOLUTION 215 (2) By inspection find the greatest square integer in the left-hand period, and write its square root as the first digit of the required root. (3) Subtract the square of the root digit already found from the left-hand period of the given number, and bring down the next period as part of the remainder. (4) Divide this remainder, exclusive of its right-lxand digit, by twixie the root digit already found, i.e., by tJie tidal divisor, and annex tlxe quotient digit to the root and also to the trial divisor, thus forming the complete divisor. (5) Multiply the complete divisor by the last digit in the root, subtraxit the product from the former remainder, and bring down the next period of the given nuinber as part of this new remainder. (6) Repeat (4) and (5) above until all the periods of the given number are exhausted. (7). // a negative remainder presents itself in the above worh, it indicates that the corresponding trial root digit is too great, and the one next lower must be tried. (8) For a given number which is not a perfect square as many decimal figures as desired in the root may be found by annexing the necessary number of periods of ciphers to the number (cf. § 125, note 3). EXERCISES Extract the square root of each of the following numbers : 1. 1296. 3. 7396. 5. 667489. 7. 17424. 2. 841. 4. 12.96. 6. 1664.64. 8. 101.0025. 9. How may the square root of a fraction be found? Why? What is the square root of -^-^ ? Why ? 10. Find the square root of f f|. Is — |f also a square root of this fraction? Why? 11. If a number contains 3 decimal places, how many decimal places does the square of this number contain ? Why ? Generalize this relation. 12. Extract the square root of 2 to three decimal places. How many decimal ciphers must be annexed to 2 for this purpose ? Why ? 216 ELEMENTARY ALGEBRA [Ch. XIII Find the square root of each of the following numbers, correct to three decimal places : 13. 13.5. 14. .017. 15. |. 16. 4f. 17. Show by actual trial that, having found the square root of 35.8 correct to 3 decimal places, the next 2 decimal figures of the root may be found by simply dividing the remainder at that stage of the work by the corresponding trial divisor. 18. If the square root of a number is desired, correct to 2 n + 1 figures, prove that when the first n + 1 figures have been found in the usual way, the remaining n figures may be found by ordinary division (cf. Ex. 17). Suggestion. Let N stand for any number whatever, k for the first n+1 figures of its square root (with n ciphers annexed), and r for the remaining n figures of the root. Then N = {k+ r)'^ = k'^ + 2kr+r^, whence — ~ = r-\ , in which -^— is a proper fraction (why ?) ; 2k 2k 2k i.e., merely dividing N— k^ (which is the remainder when the first n + 1 figures have been found) by the trial divisor at that stage of the work (viz., 2 k) gives the next n figures of the root, together with a proper fraction. 19. Find \/84256 to 5 figures, V3.642 to 3 figures, and \/6018274 to 3 decimal places. How many root figures must be found by the usual process, in each of these cases, before the ordinary division may begin ? 127. Cube root of polynomials. The general method for extract- ing the square root of a polynomial, which is given in § 125, may easily be extended so as to apply to cube root also — and indeed to the higher roots as well. The process is in all cases the inverse of that employed in raising a polynomial to a power. The several steps are indicated below.* Since (k + uy=J^ + S k'n + 3 hi' + u^ (1) = k' + (3k'-{-3ku-{- u')u, . (2) therefore : * To avoid needless repetition here the student is referred for fuller statement of reasons to the detailed explanation already given in § 126. 126-127] INVOLUTION AND EVOLUTION 217 (1) Ajrange the terms of the given polynomial according to the descending powers of some one of its letters. (2) The highest term of the required root is the cube root of the highest teriTi of the given power ; i.e., the highest term in the above root is VA;^ viz., k. (3) If the cube of the part of the root already found be subtracted from the given polynoinial, the remainder will be 3 Tihi + 3 kii?' + u^, and the next term of the root may be found by dividing the first term of this remainder by three tiines the square of the first term of the root {which is already known); i.e., t1%e second term of the root is 3 k'^u -J- 3 k^, viz., u. The trial divisor here is 3 • Tc^, i.e., it is three tiines the square of the root already known ; and, from Eq. (2) above, it is clear that the complete divisor is 3 k^ -}-3kii -\- u^, i.e., it is the trial divisor, plus three times the product of the last term of the root by the preceding part of the root, plus the square of the last term of the root. The work may be put in the following form : k^+Zk^u-\-Zku^+u^\k-\-u ;fc8 ' Trial divisor, 3 • k^ Complete divisor, Zk'^-\-Zku-\-u^ 3A;2M + 3ytw2 + tf3 ^k^u-\-Zku'^+u^ = {Zk'^ + Zku + u^) 'U Observe that the two subtractions just performed are together equivalent to the subtraction of {k + w)^ from the given polynomial. (4) By proceeding as in § 125 (ii) it is easy to show that, having found any jiumber of terms of the required root, and having subtracted tl%e cube of this part of the root from the given polynomial, the next root term may be found by dividing the first term of the remainder by the first term of the trial divisor,— the trial divisor being three times the square of the part of the root already found. By continuing this process all the terms of the required root may be found. 218 ELEMENTARY ALGEBRA [Ch. Xlll The work of finding the cube root of x^ - 9 x^ + 30 a;4 _ 45 a;8 + 30 a;2 _ 9 ^ + 1 may be arranged as follows : x6-9x5 + 30x4-45a;8 + 30a;2_9a; + i|;c2_3j;-}-i (X2)8=:x6 9 a;S + ;«x4 -45x8 + 30x2-9x4-1 9x5 + 27x4-27x3 3x4-18 x3 + 30x2_9a; + i 3x4— 18x3 + 30x2- 9x + l 1st trial divisor, 3(X2)2 = 3X4 Ist complete divisor, 3x4-9x3 + 9x2 2d trial divisor, 3(x2-3x)2= 3x4- 18x3 + 27x2 2d complete divisor, 3x4-18x3 + 27x2 3x2-9x + l ' 3x4_i8x8+-30x2-9x + l The student may now solve this example by arranging the terms according to the ascending powers of x and compare his result with the above. EXERCISES Find the cube root of each of the following expressions, and verify the correctness of your results : 1. 8x3-12a;2 + 6x-l. 2. 27 x^ - 189 x^y + Ul xi/ - S^S y». 3. 125n8-150mn2-8m3 + 60 7w2/,. 4. 675 M2y + 1215 wy2 + 125 m8 + 729 y3. 5. a;«-20a;3-6a: + 15x4-6a:5+15a:2+l. 6. 3 a;5 + 9 x^ + x6 + 8 + 12 a: + 13 a:3 + 18 z2. 7. 342 a;2 _ 108 a; - 109 x^ + 216 + 171 x* - 27 x^ + 27 x^ 8. 156 x4 - 144 x^ - 99 x3 + 64 x^ + 39 x^ - 9 x + 1. 48 , 108 12 x2.* 10. 20 15 + 15 c2 + c« + -^ + c-6 + 6 c\ 11. 30 2/-1 + 8 2/-3 + 8 2/8 + 30 y - 12 ?/2 _ 25 - 12 y-^. 12. 6 a^x* - 4 a3x6 - 2 a^x^ + 6 a^x"^ + 3 a^x + a^ + x^ - 3 ax^. 13. 108 3/62 -27 y^- 90 2j^z^ + 8 z^ _ 80 yh^ + 60 y^z^ + 48 yz^ * Compare § 125, Ex. 15. 127-128] INVOLUTION AND EVOLUTION 219 14. ^^i^^8fa/>3^M,2_«6(,,2/,2 + o)rH3fa36+«V4 + ^'_35»c. h^ b \ aJ \ hi a^ a 15. x^ + a%^ - 3 a%^x - 3 ahx^ + 3 a^> ( 1 + ah)x^ + 3 a%'^{l + ah)x'^ - a262(6 4- aJ).r3. 16. a:3y-3 + x-Y + 3 x}j-\{y-^ - 1) + 3 x-hj{x-- - 1) + 3 x-^y-\l + a:-2 + y~'^) - 3 a:''^-^ - 3 a-V - xV + x-^y-^ + 3 xy (^2 j^ y^ - \). 17. 64 y3n_{.117 y3»-3^10 y3H-2_G ^,3n-4_36 y3n-5_144 y3n-l _ g y3n-6, 18. Find the first 3 terms of V\ + x. 19. Find the first 4 terms of v 1 -3x4- x\ 128. Cube root of arithmetical numbers. To extract the cube root of an arithmetical number, proceed as follows : * (1) Separate tJie given number into periods of three digits each, beginning at the deciinal point and counting both toward the right and toward the left, completing the right- l%aihd deciinal period by annexing one or two ciphers if necessary. (2) By inspection {or by trial) find the greatest cube integer in the left-lxaiul period, and write its cube root as the first digit of the required root. (3) Subtract tl%e cube of the root digit just found from the left-hand period of the given number, and bring down tlie next period as part of the remainder. (4) To three times the square of the root digit already found annex two ciphers, thus forming tlxe trial divisor; divide the above remainder by this trial divisor, and annex tl%e first quotient digit to the root. (5) To the trial divisor add three times the product of the last root digit multiplied by the part of the root previ- ously* found with a cipher annexed, and also the square of the last root digit, thus forming the complete divisor. Multiply the complete divisor by the last root digit, and subtract the product from the above remainder, bringing down the next period as part of the new remainder. * The reasoning here is similar to that given in § 126, and should be given by the student. 220 ELEMENTARY ALGEBRA [Ch. XIII (6) Repeat (4) and (5) above until all tl%e periods of the given number are exhausted. Note. As in the case of square root (§ 126), so here, if a negative remainder presents itself in the course of the above worlc, it indicates that the correspond- ing trial root digit is too great, and tlie next lower digit must be tried. As many decimal figures as desired in the root may be obtained by annexing the necessary number of periods of ciphers to a number which is not a perfect cube. The work of finding the cube root of 9800344 may be arranged as follows : 9'800'344 1214 8 1800 [1800-1200 1261 = 1261 • 1 539^ [539344 -f- 132300 539M4= 134836 -4 1st trial divisor, 1200 1st correction, 60 2d correction, 1 1st complete divisor, 1261 2d trial divisor, 132300 5393M [539344^-132300^=4 + 1st correction, 2520 2d correction, 16 2d complete divisor, 134836 Verification of the correctness of the above root : (214)3 — 9800344. Again, let it be required to find the cube root of 43614208. Trial divisor, 1st correction, 2d correction, Complete divisor. 43'614'208 136 27 2700 540 36 3276 16614 19656 [16614 + 2700 = 6+ Since the remainder would be negative, therefore the trial digit 6 is too great, and 5 must be tried. 43'614'208 1 352 27 1st trial divisor, 2700 1st correction, 450 2d correction, 25 1st complete divisor, 3175 2d trial divisor, 367500 1st correction, 2100 2d correction, 4 16614 15875 =3175-5 739208 [739208 + 367500 = 2 + 739208 = 369604 • 2 2d complete divisor, 369604 Verification of the correctness of this root : (352)3 = 43614208. 128-129] INVOLUTION AND EVOLUTION 221 EXERCISES Extract the cube root of each of the following numbers : 1. 1728. 3. 31855.013. 5. 39304. 2. 571787. 4. 148877. 6. 426.957777. 7. 305.909539272. 9. .04, to 3 decimal places. 8. 34.7, to 2 decimal places. 10. 3|, to 2 decimal places. 11. If the cube root of a number consists of 2 n -{■ 2 figures, show- that when n + 2 of these figures have been obtained by the ordinary method, the ramaining n figures may then be found by simple division (cf. Ex. 18, § 126). 12. By the method of Ex. 11, find V.0783259 correct to 6 decimal figures. 129. Higher roots of polynomials and of numbers. The methods for extracting the square and cube roots of polynomials which are given in §§ 125 and 127, respectively, may be easily extended so as to apply to the higher roots. E.g., the identity (k + u)^ = k^ -{- 4: k^u + 6 k'^u'^ + ^ ku^ + n^ shows that the first term of the fourth root is the fourth root of the Jirst term of the power, i.e., of the given polynomial; again, if k and u represent respectively the known and unknown parts of the root at any stage of the work, and if k^ be subtracted from the power, the remainder may be written thus : (4 A;^ + G k'^a + 4 kv!^ + u^) a, which shows that the trial divisor is 4 k^, and that there are three corrections, viz., 6 khi, 4 ku^, and %fi, which must be added to the trial divisor to give the complete divisor. From here on the work proceeds as in the case of cube root. Similarly, in extracting the fifth root the trial divisor is 5k^, and there are four corrections to be added to the trial divisor to form the complete divisor; in the nth root (where n is any positive integer) the trial divisor is nk"--^, and there are n — 1 corrections. The method of extracting any root of a polynomial is easily adapted to the extraction of the corresponding root of an arithmetical number, as has already been illustrated in §§ 12G and 128. Note. If a number be separated into two equal factors, and each of these two factors be further separated into three equal factors, the given number will then really have been separated into 6 (i.e., 3 '2) equal factors; from this it follows that if N repre sents a number which can be separated into 6 equal factors, then 2, therefore the number whose square is 2 must, in absolute value, lie between 1 and 2, and therefore can not be an integer. Moreover, \/2 can not be a fraction such as — because if it were, then — would equal 2, but m ^ ^* rrfi if — is a fraction, it may be supposed to be in its lowest terms, and then — - is n n^ also a fraction in its lowest terms and can not be equal to the integer 2. It is then proved that V2 is neither an integer nor a fraction. Note 2. Although, as has just been shown, such numbers as \/2 can not be exactly represented by integers or by fractions, yet they can be approximately represented, and to any required degree of accuracy, by means of these numbers. E.g., squaring 1, 2, 3, ••• in turn shows that 1 < a/2 < 2, then squaring 1.1, 1.2, 1.3, •.. in turn shows that 1.4 < \/2<1.5, then squaring 1.41, 1.42, 1.43, ••• in turn shows that 1.41 < \/2 < 1.42, etc. Thus it is shown that 1 < \/2 < 2, 1.4/l7 trees in a row show that there is no such number as 1 + VlT? Or does it merely show that the present problem demands what is impossible ? 6. Show how to construct a line which shall be exactly 1 + Vl7 times as long as a given line. 7. Can V— 8 be expressed by means of an integer or a fraction ? Is it then an irrational number? Why not? What kind of number is it? 8. Is the number 21 + VIZ rational or irrational? Why ? What kind of number is 84 V5 - v/ -^ ? Why ? 131. Further definitions. An indicated root of a number is usually called a radical ; if this root is irrational, but the radicand rational, the expression is also called a surd. E.g., V2, v^8, >/5 + \/10, and6v^45 are radicals; and of these \/2and6v^ alone are called surds. The coefficient of a radical is the factor which multiplies it, and the order of the radical is determined by the root index. Two radi- cals which have the same root index are said to be of the same order. E.g., the surds 12 y/5 ax^ and m^V&fi are of the same order, viz., the 7th, and their coeflacients are 12 and m^, respectively. Surds of the second and third orders are usually called quadratic and cubic surds, respectively. If two or more radicals are of the same order, and have their radicands (cf. § 122) exactly alike — or if they can be reduced to such — they are called similar radicals and also like radicals; other- wise the J are dissimilar (unlike). 130-132] IRRATIONAL NUMBERS 227 Expressions which involve radicals, in any way whatever, are called radical expressions ; they are monomial, binomial, etc. (cf. § 27), depending upon the number of their terms. E.g., Vs and 3\/5 a re similar quadratic surds, while 6-\/a2 + 2 6a; + ?/* and {m + 2.n)y/a^ + 2hx + y are similar cubic surds. The four examples just given are monomial surds, while 5 a + 3 V7 and 2^9 + SVx are binomial surds. 132. Principal roots. It has already appeared that a number has tioo square roots {e.g., V9 is + 3 or — 3), and it will be seen later that every number has three cube roots, four fourth roots, Jive fifth roots, etc. E.g., ■\/8 = 2, — l + V— 3, or —1 — ^—3, since t he c ube of each of these numbers is 8 (cf. Ex. 23, § 170) ; and v/I(3 = 2, - 2, 2 V- 1, or - 2\/^. Although, as has just been said, a numbeu has 3 cube roots, 4 fourth roots, etc., some of these roots are imaginary, and when there are two real roots, they are equal in absolute value and of opposite sign.f By the principal root of a number is meant its real root, if there is but one real root, and its real positive root if there are two real roots. E.g., if attention is confined to principal roots, V9=3 (and not —3), v^- 8 = - 2, \^l25 = 5, ^16 = 2, etc. That irrational and imaginary numbers obey the fundamental combinatory laws (commutative, associative, etc.) which have already been established in the case of rational numbers is proved in the appendix ; logically this proof for irrational num- bers should now be read, but it may be deferred until later if the reader will carefully bear in mind that the following discussion assumes that irrational numbers are subject to these laws, and that the results are therefore to be regarded as tentative until this fact is proved. * Such expressions are said to be surd in form even though values may be assigned to the letters involved which make them rational in value. t It should be especially observed that a number can not have two real roots of unequal absolute value. For suppose Va = r^ and also r^, where r^ and r^ are real, and r^^r^i in absolute value; from this it follows that r^^rc^ in abso- lute value, and therefore, if rji = a, then rg** ^ a, i.e., Va ^ r^- 228 ELEMENTARY ALGEBRA [Ch. XIV EXERCISES 1. What is a radical expression? A surd? Give examples to illus- trate your answer. Are all radicals surds? Are all surds radicals ? 2. What is the coefficient of a surd ? Give an example. May this coefficient be a negative number? May it be a fraction? Are there any restrictions upon it ? 3. What is meant by the order of a surd? Illustrate by examples. May the order of a surd as now defined be negative or fractional ? 4. Define similar surds, and illustrate your definition by several examples. May the coefficients differ and the surds still be similar? 5. What factor have any two similar surds necessarily in common? What kind of number, then, is the quotient of two similar surds ? Illus- trate your answer. 6. What is an imaginary number? Give several illustrations. For what vaUies of n is "v^— 5 an imaginary number? Give a reason for calling these numbers " imaginary." 7. Illustrate by examples: monomial and trinomial surds; quadratic and cubic surds ; and the order of a surd. 8. How many values has Vl6? What are they? What is the principal square root of 16? What is the principal fifth root of — 32? Define the principal root of a number. 9. Show that \/'d4:3 is 7. Under what conditions is VK equal to ;?? How, in general, is the correctness of a root tested ? 10. Show that under the definition given in § 132 no number can have more than one principal root of any specified order. 133. Product of two or more radicals of the same order.* Just as V9 . V25 = V^, i.e., V9^^, and ^^^'-y/ 27=^^^216; [Each member of the first of these equations being 15, and of the second, — 6.] SO, too, if X and y are any numbers whatever (cf. footnote, p. 229), and n is any positive integer, •\/x • -y/y = \/xy. * In §§ 133-145 imaginary numbers are excluded, and the proofs are further limited to " principal roots." 132-134] IRRATIONAL NUMBERS . 229 For,, since (-y/x-\/yy = {-y/x^y) • (-Vx-y/y) ••• to n factors = (^xy . (Vyy [§§ 52 and 53 = xy; i.e., since the nth power of -Vx • ^y is xy, therefore (§ 130) Vic • -\/y = ^xy. (1) Similarly, it is easily shown that ■\/x • ^y • -y/z '•• = ^xyz • • •, (2) which may be formulated in words thus : the -product of the nth roots of two or more numhers* is the nth root of the product of those numhers. EXERCISES Express each of the following indicated products as a single radical : 1. V5-\/7. , 4. \/3a.Vl0te. 2. \/3.V7-\/2. 5. y/¥^ • Vb^ • VW^. 3. y/2 • ^6 . y/l • v^. 6. V^T~y • v'^TT^. 7. Verify that vx + y • Vx — y = Vx^ — y^ when x = 5 and ?/ = 4. 8. Is the equation in Ex. 7 true for all values of x and y, or only for, certain particular values, such as a; = 5 and y = 4? Why? ( 9. Is Va • Vh equal to Va6 ? Why ? If Va • v^i were also equal to Vab, how would Vb and Vft compare ? 10. Is V& equal to Vb when & ^t 1 ? Is then Va • V6 equal to Vab or to Vab for all values of a and &? 11. When may the product of two or more radicals be expressed as a single radical? 134. Special cases of § 133. It x = y, then Eq. (1) of § 133, viz., -Vx • -Vy = '\/xy, becomes a/cc • ^x = ^xx, i.e., ('Vxy = -y/x^. * If n is even, these numbers must be positive, since imaginary numbers are excluded from the present discussion. 230 ELEMENTARY ALGEBRA [Ch. XIV Similarly, if x = y = z=-'; then Eq. (2) of § 133, viz., ^/x' ^y ' ^z •" = ^xyz •••, becomes (-v/^)^ = V^, (1) where jp is any positive integer, i.e., the pth poiver of the nth root of a number is equal to the nth root of the pth power of that number. Again, if either ic or ?/ is itself the nt\i power of some number, say X = a", then Eq. (1) of § 133, viz., Vx • Vy = Vxy, becomes Va" • -^y = -y/a^'y, i.e., a^y = Va^'y ; (2) hence, a coefficient of a radical may be inserted (as a fac- tor) under the radical sign by first raising it to a power corresponding in degree to the index of the root ; and (read- ing Eq. (2) from right to left) a factor of the radicand, which is an exact power corresponding in degree with the indi- cated root, may be placed outside of the radical sign (as a coefficient) by merely extracting the indicated root. EXERCISES 1. What is the value of (Vi)^? Of v/i^? How, then, does (\/4)8 compare with \/48 ? Does this agree with Eq. (1) above ? 2. Is (v^)6 equal to -t/75? Why? 3. What is the value of 5 v^ ? Of v/pTs, i.e., of v^lOOO? How, then, does 5\/8 compare with VS^ • 8 ? Does this agree with Eq. (2) above? 4. Is3\/5equalto V32T5? Why? 5. Using the method by which Eq. (2) above was established, prove the correctness of your answer in Ex. 4. In the following expressions insert the coefficients under the radical signs, and explain your work in each case : 134-135] IRRATIONAL NUMBERS 2; 6. 3v/5. 10. fVB. 14. W2i. 7. 2VI0. a 2^. 11. |Vf|. 12. fVSaa: 15. x + i^j 3 x-i ^^ x + i 9. 5^4. 13. ^^/l2 a^a:. 16. ±-y/a^x{x- \). 17. State in words how a coefficient of a radical may be inserted under the radical sign. Write each of the following radicals in a form having the radicand as small as possible : 18. Vi5. Suggestion. \/45 = VsTTB = V32T5, —compare Eq. (2) above. 19. Vl80. 23. y/- 192. 20. vT62. 24. V892a8^. 21. ^^'320, 25. y/imd^¥x^^ ^^ ,^—, 30. V3 x^ -Qxy ^-6 y\ 22. \/- 54. 26. ^-486mV. 31. 12v/-8m4+ 24m8n. 32. Is V^ equal to a;?/ ? Why? 27. V12«3(a: + 2/)5. 28. v/ltja^a:*- 246x6. 29. V18 a - 9. 33. Is Va;2 + ?/2 gq^a] to a; + ?/? Why? 34. Verify your answer to Ex. 33 when x = 3 and y = 4. 35. Is the extraction of roots distributive over a sum ? Over a prod- uct? Compare Exs. 32 and 33. 135. Quotient of two radicals of the same order. Just as . |= = \/|' [Each being I SO, too, if X and y are any numbers whatever (cf . footnote, p. 229), and n is any positive integer. «/7. \ li yy 232 ELEMENTARY ALGEBRA [Ch. XIV To prove this it is only necessary to remember that ('lly^l^.ll.l:!... ton factors \y/yj y/y Vy -Vy -y/g? ♦ -\/x » -y/x » • • to n factors ■\/y . ^y . -s/y ... to n factors [§ 54 (ii) (Vyy y' i.e., the nth power of ^^ is -, and therefore, by the definition ^y y of a root (§ 130), ^^=a/-, — which was to be proved. Vy ^y The student may state in words what has just been proved (cf. § 133). EXERCISES Express each of the following quotients by means of a single radical : 1. V35--V5. 4. \/lQ a^x^ -~ VW^K 2. \/216-^Vl2. 5. v'a;2 - y^ ^ V^+^. 3. \/216-\/l2. 6. \/16 a^b^ - 32 a^x^ - \/4a2. 7. Verify that Va^ - 6^ ~ y/a -h = \/a -\- b when a = 5 and & = 3. 8. Is the equation in Ex. 7 true for all values of a and b, or only for certain particular values of these letters ? Why ? 9. Is y/W^^^Wai equal to ^/^rt^.^ ^j^^. Compare also Ex. 9, § 133. \ Qax 10. If two radicals are of different orders, can their quotient be ex- pressed as a radical of the same order as either one ? 11. iP^'^ : -i/l ^ =? 12 * l^^^y^ ^ 5 /__2rt_ ' \ 35 68 ■ \1b'^xh 3 a \ 35 68 \7 62xV 135-136] IRRATIONAL NUMBERS 233 136. Radicals whose indices are composite numbers. Just as factors i" = I V^J« [Since ( V VS)" = V^ i.e., the n^th power of \^x is ic, and therefore \-\/x=^^- In the same way it may be shown that '^Vx =X^x. This principle is useful in extracting roots whose indices are composite numbers (cf. § 129, note). EXERCISES 1. What is meant by the symbol VN (cf. §§ 122 and 130) ? Point out two places in the above proof where this definition is employed. 2. Using § 136, show that V7 because their respective equivalents, viz., ^625 and \/343, stand in this relation. EXERCISES Reduce the following to equivalent radicals of the same order, and thus compare their values : 1. V5 and y/VL. 3. \/lO, V2, and v^S. 2. \/7 and V3. 4. Vl, v^, and ^5. Reduce the following to equivalent radicals of the same order : 5. V3a6, y/2^\ and y/^a%H^ 7. y/x, V^j, and V^. 6. \/2x% y/ax, and v2m%. 8. Va + 6, Va^ + ft^^ and Va — b. 9. Can the radicals in Ex. 5 be reduced to equivalent radicals of the 6th order? Of the 12th order? Of the 9th order? Give the reasons for your answer in each case. 10. What is the lowest common order to which the radicals in Ex. 6 can be reduced ? Those in Ex. 7 ? Those in Ex. 8 ? 11. Compare the rule, asked for in § 138, with the procedure in solving Exs. 1 to 8, and see whether it meets all the requirements. 12. Which is greater, 3V5 or 2v/IT? Compare §§ 134 and 138. 13. Whichisgreater, 2v/9 or3V3\/2? Why? 14. How may the values of any two numerical radicals (real numbers) whatever be compared ? vn T 236 ELEMENTARY ALGEBRA [Ch. XIV 139. Reduction of radicals to their simplest forms. A radical is said to be in its simplest form when the radicand is integral, when the index of the root is as small as possible, and when no factor of the radicand is a perfect power corresponding in degree with the indicated root. The following examples may serve to illustrate the application of the foregoing principles to the reduction of any given radical to its simplest form. Ex. 1. Reduce v^f to its simplest form. Solution. ^ = y|^ = ^F^ = j ^. [§ 134 Ex. 2. Reduce Vi a^x^y^ to its simplest form. Solution. y/Ia^xY = v^(2a^VF = ^2 axY- [§ 137 Ex. 3. Reduce v8 a^x^y^ to its simplest form. Solution. VsS^ ^ v/4 a^xY • ^2ax [§ 133 = 2 ax^y V2 ax. EXERCISES 4. Is VS ax in its simplest form ? Why ? 5. Is V12ax in its simplest form? Why? 6. Is 5 ViaW in its simplest form ? Why ? 7. Is 12 Vf ax^ in its simplest form ? Reduce it to its simplest form. 8. What is meant by saying that a radical is in its simplest form? Reduce each of the following radicals to its simplest form : 9. Vl2. 19. \/|. 26. 3 \/25 a%^x^ 10. Vl62. 20 A r^ «/ 11. v/16. , "^ y 21. v^. 12. ^250. I—. 28. ^a'+%2.y.-4. 13. \/81. • A/ 9 ^^2* 29. \/a2na;»+5. 14. ^189. ._ ^/TTift^ 3 — 15. VI2-8. "'-^W' 30. V-40.^-B,H. 16. /l2 = V25 • 3 + 3\/4T3'= 5\/3 + 6V3 = ll\/3. Ex. 2. Find the sum of 5 vTS, - VOS, and ^/\. Solution. 5 Vl8 - Vo.5 + Vi- = 5 Vo • 2 - V^ . 2 + V Jg • 2 * = 15 V2 - ^ V2 + i V2 = 14f \^. Ex. 3. Find the sum of V9 ar - 18, 6 V4 x + 8, \/36 x - 72, and - \/25 X + .50. Solution. Vq x - 18 + 6 \/4 x + 8 + \/36 a: - 72 - V25 x + 50 3 Vx^^ + 12Vx + 2 + 6\/x-2-5\/a: + 2 = 9 Vx - 2 + 7 Vx + 2. EXERCISES Find the sum of: 4. VSO, Vl8, and \/98. 6. \/28, V63, and VTOO. 5. \/l2, V75, and V27. 7. \/250, v^, and ^/U. 8. v/500, v^l08, and V^^^. * Since 0.5 = ^ = f , and ^ = i^. 238 ELEMENTARY ALGEBRA [Ch. XIV 9. What is the sum of a, 2 b, and c ? Of 3 x, 4 y, a, 2 a:, and — 5 y ? 10. What is the sum of 3 V2 and Sv'T? Of 3 V2, 5 v^, - 2 V7, and V2? 11. Write out a carefully worded rule for the addition and subtraction of radicals ; provide both for those cases in which the given radicals are similar and for those in which they are dissimilar. Simplify the following expressions as far as possible, and explain your work in each case : 12. v/135 + v^625 - v^320. 18. vl28¥ + \/375^ - v^547. 13. ^+V2-8 + Vl7-5 + ^. ^^ J-l_,JJ_J1, 14. v^375 - Vii - v/192 + V99. ' ^^^ ^^^ ^2' 15. V1+V75-V/12 + 1V3. ^^ ^1^2^, iyq—^ ra 16. Vl47-x4 + iV3 + ^^9. ' ^^V ^ bf ^hf' 17. 6 \^ + 4 v^JI - 8 \^. 21. V(a + b)c - V(^^^h)^. 22. v/192 a;4 - 2 V3 a;4 - v/5 a: + V40 x*. 23. . 5. v^2 + 3 v/2 by Vi. 9. x ~ -Vxyz + yz by V^ + Vyz. 6. 5 + v/4 - 2 v/5 by \/5 + \/6. 10. -Va + Vxby -Va-Vl. Expand the following expressions, and simplify the results : 11. (V2-3v/3)2. 14. (v;^r^ + v;;H^)2. 12. (\/2^-V3^)2. 15. (v/'J^- ^3^2)3. 13. (a + V6 - \/c)2. 16. (v/a + 2\/3)5. 18. (V2a-V6 + V2a + V6)l 143. Division of monomial radicals. By means of §§ 135 and 138 the quotient of any two given monomial radicals (real numbers) may be expressed as a single radical (cf. § 141). Ex. 1. Divide v^4 ax^f by VTcih:. Solution. V±^^ _'^^W^^ V2 a^x V8 a^xs [§138 = :MJ^^ [§135 = a/?^-^' = 1 'V2 a^x^. [§ 139, Ex. 1 142-144] IRRATIONAL NUMBERS 241 EXERCISES 2. What is the quotient of V50 divided by V8 ? 3. What is the quotient of 4V5 divided by VIO? 4. What is the quotient of 1 Vbi divided by 2v^686V 5. How is the quotient of two monomial radicals obtained if these radicals are of the same order ? Express each of the following indicated quotients in its simplest form : 6. 2v^-\/8. » 11. Voi-^V^. 7. 2^6 -^v^. 12. V2^-^^o"^2^. 8. Vl8 - \/500. 13. 2 v/9a2p ^ 3 V3^. 14. a\/^x^y'-^2br yY = -^ ^(x2-?/2)2(x + 3/)8. x+y x+y Verify this equation when x = 64 and ?/ = 0. Is this equation true for all values of x and y, or merely for certain particular values of these letters ? What other name is given to such equations (cf. § 23) ? 144. Division of polynomials containing radicals. If the divisor is a monomial, then, manifestly, the quotient may be obtained by dividing each term of the dividend by the divisor — just as in the case of rational expressions. E.g., 3i^+ivl^lW„3+4Ji-2^| K138 = 3 + 2V6-2v^2. [§139 Instead of dividing directly by a radical, it is usually advan- tageous first to multiply both dividend and divisor by an expres- sion which will make the new divisor rational — indeed, it is frequently necessary to do so. 242 ELEMENTARY ALGEBRA [Ch. XIV E.g., since (3\/2-\/i3) • (3 a/2 + VlS) = (3>/2)2- (Vi3)2 = 5, therefore 5 -^ (3 V2 - Vi3) = 3 V2 + Vi3 , but oue could not easily obtain this quotient by dividing directly. It may be obtained thus: 5(3V2+\/l3) pMultiplying numerator and 3V2— Vl3 (3V2 — Vl3)(3V2 + Vl3) L denominator by 3 V2+\/i3 ^16vl+5Vl3^3^2+Vl3. This method of dividing (usually called division by means of rationalizing the divisor) will often be found very advantageous even when it is not strictly necessary. jEq 3V2 + 4v/3 ^ (3V2+4\/3) . V2 ^ 6 + 4V6 ^ 3 , g^/g V2 (\^)2 2 The factor by which a given radical is multiplied to obtain a rational product is called its rationalizing factor. E.g., of v^4 and ^2 each is a rationalizing factor of the other (why?) ; so also are Vop and \/o"-p (why?), and aVx + hy/y and aVz — hy/.y (why?).* Of two such binomial quadratic surds as a-\/x + &Vy and a^x — h^y, which differ from each other only in the quality sign of one of their terms, each is called the conjugate of the other. EXERCISES 1. Divide Vl5 - V3 by V3. 2. Divide V6 + 2 V3 by \/2. 3. Divide v^ - 4 V5 + 2 v^G by V3. 4. Perform the divisions in Exs. 1-3 by first rationalizing the divisors, and show whether or not there is any advantage here in rationalizing. 5. Show that 2 VB - Vs is a rationalizing factor of 2 VS + \/5. 6. Is \/5 - 2 \/3 a rationalizing factor of 2\/3 + VB? Why? Are these surds conjugate to each other? * The question of finding rationalizing factors for given expressions is further considered in § 161. 144-145] IRRATIONAL NUMBERS 243 Find the simplest rationalizing factor of each of the following surds : 11.5^ 7. V2a. „ ^12ahn 15. 3a -2" ox. 8. V4^2. L ^ _ ^^- oa:-\/2^. 3, ^2. V2-V7. 17^ v^+2V3-6. 9. V4ax-^. ^3^ 2V8+V6. /^^^ 18. ^ii£ + |v^8. 10. Va + 6. 14. 4 + 5 V3. ^ a 19. Divide 31 by 7 + 3 V2. 20. Divide 2 V6 by V5 - VB. 21. Divide 5\/l2 — 2 V6 + 4 by \/4. What is the smallest multiplier that will rationalize v^l ? 22. Divide 3 ■\/2 - 4 V5 by 2 V3 + V7. 23. Divide 4V3 + 5 V2 by 3 V2 - 2 VB. 24. If the result of Ex. 21 were wanted correct to 4 decimal places, say, show in detail that it is far simpler first to rationalize the divisor than to extract roots and divide by the ordinary arithmetical method. 25. What is the product of (2 + VB) - V5 by (2 + V3) + V5 ? Of this result by 2 — 4 \/3 ? What then is a rationalizing factor of 2 + V3 — V5 ? Of 2 + V3 + V5 ? 26. Divide 2 - V3 by 1 + V3 - \/2. Reduce the following to equivalent fractions having rational denomi- nators : a + Vq^ + ar ga ^^ + ^ -y/x - y . 39 E!_ 27. " ^ " "• ^ •". 28. "-^ ^ !f l^ ^. 29. a — V a^ + X Vx + 2/ + Vx — y Va^ + z^ — z 30. Simplify -^ + -^_. 31. Simplify (V2 + 3)(v/5-2), ^_1 ^ + 1 (3-V2)(2 + V5) 32. Find the value of :; -| ^^^ correct to 3 decimal places. 2 - V3 \/2 + 1 145. An important property of quadratic surds. Neither the sum nor the difference of two dissimilar quadratic surds (§ 131) can be a rational number ; for, if possible, let ■y/x-\-Vy = r, (1) Vx and Vy being dissimilar surds, and r rational, and not zero. 244 ELEMENTARY ALGEBRA [Ch. XIV From Eq. (1) Vy==r — Vx, (2) whence, squaring, y = r^ —2 rVx + x, (3) and, solving for Vx, Vx = — -^-- ^, I.e., if Eq. (1) were true, then the surd Vx would equal the rational number — ,^ ~ ^ , which is impossible; hence Eq. (1) can not be true. Similarly, Vx — Vy ^ r. From what has just been shown it at once follows that if x-\- Vy = a + Vb, where x and a are rational, and Vy and Vb are quadratic surds, then x = a and y = b. For, if x-{- Vy = a + Vb, then Vy — Vb = a — x; which, by the above proof, can be true only if each member is zero, i.e., if a = a; and Vy =Vb. In other words, the equation x-\-Vy = a-\-Vb is equivalent to the two equations x = a and y = b. II. IMAGINARY NUMBERS 146. Imaginary numbers. In solving the equations of the next chapter, indicated square roots of negative numb'ers frequently appear; such numbers have already been defined (§ 130) as imaginary numbers ; if they present themselves in the form V—b, where 6 is a positive number, they are called pure imaginary num- bers, while if they present themselves in the mixed binomial form a -{-V—b, where a and b are real and b is positive, they are usually called complex numbers.* * A broader definition of imaginary numbers is given in appendix B, w here it is shown that every such number can be expressed in the form a + bV—l, and where it is proved that these numbers obey the laws already established for real numbers (commutative, associative, etc.). Logically this proof should now be read, but it may be deferred until later if the reader will carefully bear in mind that the following discussion assumes that imaginary numbers are subject to those laws, and is therefore to be regarded as tentative until this fact is proved. The very elementary discussion which is given in the next few pages will suflSce for present needs. 145-147] IMAGINARY NUMBERS 245 E.g., \/—5, 2\/— G, and V— ^ are pure imaginary numbers, while 2 — V— 3 and 7 + 2\/— 5 are complex numbers. Operations with imaginary numbers are greatly simplified by observing that, by the definition of v a, § 130, (V^y^-b, (1) and also (cf. method of § 133, and apply §§52 and 53) that V^b = -Vb'V^. (2) The symbol V— 1 is called the imaginary unit, and is often represented by the letter i. 147. Positive integral powers of V— 1. As a special case of consequently, (V— 1)^ i.e., ( V— 1)^ • V— 1 = — V— 1. Similarly, ( v^^)* = (V^^y . v^^ = - v^T • v=^ = - ( v^' = 1, (V^y = (V^riy . ( v^^ = - 1, ( V^^)^ = ( V^i)^ • ( V^« = - V^^, and so on for the higher powers, i.e., the positive integral powers of V— 1 have only these four values : V— 1, — 1, —V— 1, and 1 ; see also Exs. 5, 6, and 7 below. EXERCISES 1. Define an imaginary number; compare § 130. 2. Which of the following are imaginary numbers : V— 3, v^— 2, ^36, V5, ^/^^^, 3v^^, 4a-J-^ and 2 + i V^s? 3. Is V— a; imaginary when x represents a positive number? When X represents a negative number ? 4. Show that if i = V^^, then i^ = -l, i^ = - i, i* = 1, i^ = i, t^ = — 1, f = — i, i^ = 1, and i^ = i. 246 ELEMENTARY ALGEBRA [Ch. XIV 5. Since any even number may be written in the form 2 n, where n is an integer, and since a^" = (a^)", show that every even power of i is real. 6. As in Ex. 5, show that every odd power of i is either i or — i. 7. Since x"+'^ = x" • x*, and since any positive integer whatever can be represented by one of the following expressions, viz., 4 n + 1, 4 n + 2, 4 n 4 3, or 4 n, show that the positive integer powers of i can have no other values than i, — 1, — i, and + 1, and that these values always recur in this order. 8. Distinguish between pure and complex imaginary numbers, and give three examples of each. 148. Addition and subtraction of imaginary numbers. By first writing the imaginary numbers in the form a + ftV— 1, these numbers may be added and subtracted exactly as though they were real ; this is illustrated below. Ex. 1. Find the sum of V— 4, 4 V— 9, and V— 25. Solution x/i:4 + 4v'^+ V:r25 = 2\^^+4 • 3 V^ + SV"^ [§ 146, Eq. (2) = (2 + 12 + 5) V^ [Footnote, p. 83 = 19^^. Ex. 2. Find the sum of 3 + V^T6, V^^, and 5 - V^T^. Solution. 3 + V- 16 + V^T + 5 - \/^9 = 3 + 5 + V^Te -f v"^^ - v^ITg [§ 50 = (3 + 5) + ( V^iTo + v/i:i - v:r9) [§ 51 = 8 + 3V31. [Ex. 1 Ex. 3. Simplify the expression a:V— 4+V— a:^ — 2x— 1— V— 32. Solution. Since xV^ = 2 xy/^^, V-a:2-2a;-l = V - (x + 1)2 = (x + 1) V^^, and _VZr32 = -V32. \/^l = _4\/2. V^ therefore the given expression becomes {2 a: + (a; + 1) - 4V2} . V^T, i.e., (3 a: + 1 - 4\/2) • \^^. Similarly in general. 147-149] IMAGINARY NUMBERS 247 EXERCISES Simplify each of the following expressions : 4. 3+\/:r36-(l +2\/^25)+3\/- 16. 5. V-49 + 5\/^=lt-(6 + 2\/^^). 6. V^ - 3V~-^ + 6 V^^TS - 2 V^^27 + 8 + V^^^T2. 7. . |_ (9 V-l + 5-3 V-24)+3V 8. V- 16 ah^ + Virrs + 9 V5 18. 30 -V- 9a2x2 + \/-a2x2. 149. Multiplication of imaginary numbers. Multiplication of imaginary numbers is also performed by first writing these numbers in the form a + ftV— 1; this is illustrated below. Ex. 1. Multiply V^^ by V^. Solution. V^ - y/~^ = V2 • V^ • V5 . V^l [ 146, Eq. (2) = ( V2 . V5) ( V^l . V^^l) [§§ 52 and 53 ' . =VT0.(- l) = -\/IO. Similarly in general : V— a • V— 6 = —Vab. Note. The student should carefully observe that (§ 133) the law for the prod- uct of two radicals, i.e., principal roots, does not apply to the product of two imagi nai-y nu mbers; according to that law the product of V—a • V—b would be V{—a) • (~6), i.e., y/ah, and not ~Vab. Errors of this kind are easily avoided by writing an imaginary number in the form a + 6 V— 1 before operating with it. Ex. 2. Multiply 3 + V^5 by 2 - V~^. Solution. Writing these imaginary numbers in terms of the imagi- nary unit, the work may be arranged thus : 3 -h V5 . v^n. 2 - V3 . a/^T 6 + 2V5. v^nr - 3\/.3 • v^i Vi5(v/irT)5 6 + (2\/5- 3V3) . V- 1 +V15. Similarly in general : (a + V^^) • (c + V^) = ac- Vbd H- (a Vd + cV6) V^. 248 ELEMENTARY ALGEBRA [Ch. XIV EXERCISES Find the product of : 3. 3 Vr6 by 5V_ 12. 6. Vr^ + y/ZT^ by VI~6 - VTl. 4. sVITg by 2VI^. 7. 3 + 2Vr^ by 5 - 4VZI;. 5. 2Vi:i by V_ 4 a%3. s. Vrso _ 2Viri2 by V^s - 5V^. 9. Show that the sum and also the product of a + hi and a — 6i (wherein a and 6 are real) is real.* Show that this is also true for Vri _ 3 and - Viri _ 3. 10. Prove that both the sum and also the product of any two conju- gate complex numbers is real. 11. Multiply Vir^ + VITft + V3^ by V- a - V_ ft + VT^. 12. (1 + Vr5)2 ^9 13. (2 - 3 iy = ? 14. (2 a - 3 a;Viri)2 = ? 15. Find the product of aV_ 6 -f &V— a, aV_ a -f ftV— 6, and 16. Show that — J + h^ — 3 and — J — i^^— 3 are conjugates of each other, and also that the square of either is equal to the other. 17. Write a rule for multiplying one pure imaginary number by another, and compare it with the rule for getting the product of two monomial surds of the same order. Wherein do the two rules differ? 18. Reduce — — — ,ZL • + "^-^ — ^ to its simplest form. 150. Division of imaginary numbers. The simpler cases of division of imaginary numbers are illustrated by the following examples : Ex. 1. Divide V^e by V^^. S0.„™.. Q = |l^ = ^ = V| = >^. [§§146,135 Similarly in general : V— a la Va / a -, V— a I a * Of two complex numbers which differ only in the sign of the imaginary term each is called the conjugate of the other (of. § 144). 149-150] IMAGINARY NUMBERS 249 Ex. 2. Divide 12 +V_ 25 by 3 -VZl. Solution. Such divisions are easily performed by rationalizing the divisor (cf. § 144), thus: 12 + V- 25 ^ 12 + 5Vin^ ^ (12 + 5V^rT)(3+2\/^^) 3-V^^ 3-2V^:i (3_2\/^I)(3+2V^n) ^ 36 + 39 V^^ + 10(V^n)2 9_4(V-ri)2 ^ 26_-|-_39>/^ 9+4 = 2+ 3v/^T=2 + \/39. Similarly in general: a + & V£l ^ (a + 6 V;=l)(c - dV^) c + dV - 1 (c + d V- l)(c - dV- 1) _ CTC + ^c? + (&c — ad) V— 1 ~ c^ + d^ EXERCISES 3. Verify the correctness of the result in Ex. 2 above by multiplying the quotient by the divisor. 4. Divide V- 6 + 2V- -8by V^ ~2, 5. Divide 4 by 1 + i. 6. Divide 2 by i* + i\ Simplify the following : 7 2-V^S g V2^ - 3 ai 3 + V-2 V2lc + 2 6t 8. 5 + V-4. 10. ^«-^'^^ 5 - 2 i iV6 + Va 11. Write a rule for dividing one pure imaginary number by another, and compare it with the rule for finding the quotient of two monomial surds of the same order. 12. Divide 3 - V^+ 2 i by 2 + V^ ~ V^ (cf . § 144, Ex. 25). 250 ELEMENTARY ALGEBRA [Ch. XIV 151. Important property of imaginary numbers. Neither the Slim nor the difference of two different pure imaginary numbers can be a real number (cf. also § 145) ; for, if possible, let V^=^- V^=^ = r;* (1) then, transposing, V— a = r + V— &, and squaring, — a = i'^ + 2 r^—b — 6, whence V— 6 = ~^^ ~ — ; z r i.e., if Eq. (1) were true, then the imaginary number V— 5 would equal the real number -^^ — ^^ — , which is impossible, and hence Eq. (1) can not be true. Similarly it may be shown that V— a + V— 6 ^ r. , From what has just been shown it follows that if a; + V— 2/ = a + V— &, wherein a and x are real and V— ?/ and V— 6 pure imaginary numbers, then x = a and y = h. , For, if x + ^—y = a-\- V— 6, then, transposing, ^—y — V—b = a — Xj which, by the above proof, can be true only if each member is zero, i.e., if y = b and x = a, which was to be proved. In other words, the equation x +V— y — a +V— b is equiva- lent to the two equations x = a and y = b. * The expressions V— a and V— 6 represent different pure imaginary num- bers, and r is real, and not zero. . . 151-152] IMAGINARY NUMBEBS 251 152. Complex factors. Solving equations by factoring. Since (a + bi) {a — bi) = r/ + b^, wherein a and b may be any real num- bers whatever, therefore tlie sum of any two real positive numbers may be separated into two imaginary factors. E.g.,x'^-\-^ = {x + 2i)'{x-2i); aH- 3 =(a + iV3)(a- iVi) ; a;2 + 2a: + 5 = (a; + l)24-4 = (a; + l + 2z)(a; + l-2?); x^-x'^ + \ = x^-2x^-\-l + x^ = (a;2- 1)2 + a;2 = (a.2_ 1 + a; . i) (a:2- 1 - X . 0- Note. Observe that the most important step in the above factoring is first to write the given expression as the sum of two squares; the plan for doiug this is precisely that which is followed in § 70, The following examples will illustrate the use of imaginary factors in solving certain kinds of equations ; this method will be more fully treated, however, in Chapter XV. Ex. 1. Solve the equation a;^ + 2 a: + 5 = 0. Solution. Since this equation may be written in the following forms : 22 + 2 a: + 1 + 4 = 0, (x + 1)2 + 4 = 0, (x + l+20(x + 1-20=0, therefore it is clear (§ 72) that the only values of x that satisfy it are those that make a: + 1 + 2 i = or x + 1 - 2 i = ; i.e., the given equation is satisfied if, and only if, x = -l —2i or x = - 1 + 2 I ; i.e., the roots of that equation are — 1 — 2 z and — 1 + 2 t. Ex. 2. Solve the equation x^ = ^x — 22. Solution. This equation may be written in the following forms : a;2 _ 4 ^ + 22 = 0, (a: _ 2)2 + 18 = 0, {x-2 + i vT8)(x - 2 - i VT8) = ; hence its roots are 2-ivl8 and 2 + iVl8, i.e., 2-3V^ and 2 + 3a/^. 262 ELEMENTARY ALGEBRA [Ch. XIV EXERCISES 3. By actual substitution verify the correctness of the roots found in Exs. 1 and 2 on page 251. 4. What must be added to x^ — 8 x in order that the sum shall be the square of a binomial ? 5. Write a:^ — 8 a; + 25 as the sum of two squares. Solve the following equations and verify the correctness of your results : 6. a:2 4- 25 = 8 a;. 8. a;2 - x + 1 = 0. 10. 3x2 -5 a: +21 = 0. 1. x^ + x-\-l=0. 9. 4 x2 + 9 = 0. 11. x-* + a2x2 + a* = 0. 12. Write an equation whose roots are 1, i, and — i (see § 72, note). 13. Write an equation whose roots are 1, —\+li V3, and — \ — \ i V3. 14. If s = - ^ + i i V3, show by substitution that s^ + s + 1 = 0. What other root has this equation? III. FRACTIONAL EXPONENTS 153. Fractional exponents.* In § 137 it is shown that the exponent of the radicand and the index of the root may both be multiplied by any integer, or both be divided by any factor which they may have in common, without changing the value of the expression. This property at once suggests that these numbers may bear to each other relations similar to those of the numerator and denominator of a fraction. For this and other reasons, some of which will presently appear, it is customary to employ, when it is desired to indicate that roots are to be extracted, not only the radical sign, the use of Avhich has already been explained, but also what is known as a fractional exponent. This new symbol may perhaps be best defined by the identity p _ p i.e., the symbol A*" means the pth power of the rth root of A, and r must therefore necessarily represent a positive integer, while p may be positive or negative. E.g., 9^ = (V9)5 = 35 = 243, and 8^3" = ( v/8)-4 = 2-4 = ^ = -1. 24 16 •For a similar treatment of fractional exponents see Tannery's Arithme'tique. 152-153] FRACTIONAL EXPONENTS 253 p The expression A'', whatever the value of A, is usually spoken of as a fractional power of A, just as A^ is called a positive integral power, and A~^ a negative integral power. In the next few articles it is shown how to use this new symbol in the various algebraic operations j these uses will further justify its adoption. For the sake of simplicity, here, as in §§133-145, only the principal roots (§ 132) are considered, and for these roots it has already been shown that (VZ)^ = V^ [§ 134, Eq. (1)] ; hence, in p the following proofs, either (^/Ay or -y/A^ may be used for A"". p p Note. Although '•, in the expression A>' , is called a fractional exponent, and is written in the form of a fraction, and although it will presently appear that such exponents may often he dealt with as though they were really fractions, yet it must he carefully remembered that they are not fractions at a,\\; this fractional-exponent notation is merely another loay of indicating that roots are to be extracted. EXERCISES 1. What is meant by the symbol - ? Has it the same meaning when used as an exponent? 2. Is the exponent — , in the symbol x^, really a fraction ? What is the precise meaning of a." • 3. Is it correct to say that the symbol x« is merely a convenient way of indicating the mth power of the nth root of a;? Is this the same as the nth root of the rnth power of x, when only the principal roots are under consideration? Express each of the following radicals by means of the fractional- exponent notation: 4. W. 6. V^. 8. V{a+2xy. 10. = 0, and thus prove that 1 -f- a" = a «. Com- pare this result with § 44. 10. By means of Ex, 9, show that a' factor may be transferred from numerator to denominator, or vice versa, by merely reversing the sign of its exponent, even when the exponent is fractional (cf. Exs. 22-26, § 93). 157. Product of like powers of different numbers. From § 153 it follows directly that 8^ . 27^ = (8 . 27)^, i.e., 2161 [Each being 36 258 ELEMENTARY ALGEBRA [Ch. XIV So, too, in general, if A and B are any two numbers what- 'er (cf. footnote, p. 254), c which r is positive, then P ever (cf. footnote, p. 254), and if -- is any simple fraction in A^ . B = {ABY. For, since J- = V^ and R = V^, p p [§153 therefore A'- . R="»• e"* • ^"^ ••• = {abed •••)'", wherein m may be positive or negative, integral or fractional, or zero. 20. Make up 4 examples to illustrate the application of law (3) with the various kinds of exponents. 21. Translate law (4) into a rule, and illustrate its application. 22. What is the product of ^'»-" by yl**? What, then, is the quotient of A'^ divided by A'^ [cf. definition of division, § 3 (iv)] ? 23. By means of the suggestion contained in Ex. 22, prove law (4) from law (1*) and the definition of division, — independent of § 156. 160. Operations with polynomials involving fractional exponents. Since the operations with polynomials are merely combinations of the corresponding operations with monomials, therefore the prin- ciples already demonstrated (§§ 155-159) for monomials suffice for operations with polynomials also. Moreover, since fractional exponents obey the familiar laws formerly established for integral exponents, and since any radical expression may be written in the fractional-exponent notation, therefore operations with radicals (real numbers) are usually greatly siraplilied by using fractional exponents ; * this is illus- trated below. Ex. 1. Find the product of 3 Va — 5 v^ by 2-\/a-\-Vy. Solution. Since 3Va — 5\/^ = 3a^ — 5^/^, and 2\/a -}- \/y = 2a2 + ^^> therefore this product becomes 3 a^ - 5 2^i 6 ai+^ - 10 a^y^ + 3 o^y^ - 5 yi+^ 6 a — 7 a^y^ — 5 y^. If it is desired, this product may, of course, be written in either of the following forms :6a — 7 Va v^ — by/y^ or Q a — 7Va^y^ — oVy^. * Although the radical notation and the fractional-exponent notation are each equivalent to the other, and either may therefore replace the other, yet each is frequently met with, and it is desirable that the student should understand how to operate with each form without first converting it into the other. 262 ELEMENTARY ALGEBRA [Ch. XIV Ex. 2. Divide x^ — ij^ by \^x + y/y. Solution. Since Vx + Vy = x^ + y-, this solution may be put into the follow ino- form x^ + x'^y"^ 1 1 x^ + y^ 5 4 1 2 3 x3 — a^3y2 ^ xy — xay^ + 5 1 4. — x^ys^ — x"Sy xsy — y^ x^i -^W' xy^ x^y^ xsy^ - 2/3 x'^t/^ + x^y^ -y' The above quotient may also be written thus : Vx^ — y/x"^ Vy -{- xy — Vx'^ Vy^ + v^a: • y^ — Vp. Note. To appreciate one of the advantages of fractional exponents the student has only to perform the division in Ex. 2, using the radical notation, and compare his work with the above solution. Ex. 3. Extract the square root of v^ — 2 y/x^ + 5 v^x^ — 4 aXx + 4. SoLUTiox. This expression written in the equivalent fractional-expo- nent form is X* — 2 z^ -1- 5 a;5 — 4 x"^ + 4, and in this form its square root may be extracted just as though it were a rational expression (cf. § 125) ; thus: 4 s 9. 1 9. ^ x^ - 2 x"5 -I- 5 x^ - 4 x^ -f 4 [x? - x^-f 2 4 X^ 2 x3 - x^ - 2 x3 + 5 xt -2x^ + xi 2 x^ - 2 x3 + 2 4^!-4x* + 4 4x^-4x3 + 4 hence the square root of the given expression is xs — x3 -f 2, i.e., ^^-Vx-\- 2. 160] FRACTIONAL EXPONENTS 263 EXERCISES Perform the following multiplications : 4. a^ + 62 by a2 - h^ (cf. § 58). 5. a;3 — x'^y'^ + ?/3 by :c3 -|- yz, 6. 771^ — w"5n5 + nS by //?? + n^. 7. m^ — m^y~^ + n~a by w?^ + n~o. 8. i a:t - Jj x?/^ + ,V ^^^ - 2V 2/^ by ^ a:i + 1 yi. 9. 81 ^7^-27 ^^3^^+9^2^^_3^^^^^^ by 3 ^ + ^y. 10. Va - 4 VlK + 6 \/^- 4 v'^ -4- Vx by v^a - 2 v^ + Vx. 11. \/x^ + 2 Vp - \/2^ - ^ Vy + 2 V^ v^s - ^^ Vz by y/x — 2 Vy + Vs. 12. w^ + m"t - 2 m^ + 4 m~^ by 1 + 2 'm"^ - -j^- 13. j9"t + ^-1-6 - p-'!5q-^ by />- -75 + 9-5. 14. 14 n^x a/x + 2 n Vn + 1 a:^-^ + 6 n v^ by Vn — 3 x* + 7— -x^. 15. 5 a-H-i + 3 a-26»x-i - h^-^x^ by x-^ - 3 ir~h-^ + ai Perform the following divisions : 16. a + x2 by a^ + x^. 17. m5' — n3 by m^ — ns. 18. x-i + 3 y~^ - 10 x?/-! by x"! Vy - 2. 19. a* + 2 v^H + ^ by v^ + ft'i 20. x^ + x'^y/y — xy/xy^ — xy + -\/x y^ + y^ 3 by Vx + v^y. Simplify the following expressions : 21. ( ^^ + '^1 '^ _ ^-Vy ^ 23 _^^ 9^ ]_< 1 \Vx-y/'y' \^x + Vy y/a-1 y/a + 1 «i-l a^ + 1 22 3:"» + ?/" _ x" - y"^ 24. ^ ^ 1 'x— + 2/-« x-^-2/— ' y+Vy-^l' yl-l 25 ^ ~ y _ Va:^ — y'^ Vx — Vv ^ — y 264 ELEMENTARY ALGEBRA [Ch. XIV Extract the square root of each of the following expressions : 26. a;2 + 2 a:t + 3 a; + 4 a:^ + 3 + 2 a;"i + x'K* 27. tt^ - 4 as 4- 4 « + 2 a^ - 4 a^ + ai 28. ns — 2 nT^n'^' + 2 rri^n^ + m~^n ^ — 2 m^n^ + m^. Write down, by inspection if possible, the square root of each of the following expressions : 29. 1-2 m3 + wi 31. jo^ - 4 + 4;>~i 30. x^ 4- 4 xt + 4. 32. axt + 2 a^x^ + atx. 33. m + n+jo — 2 ni^n^ + 2 n^jo^ - 2 m^jo^. Extract the cube root of each of the following expressions ; write the results first with all the exponents positive, and then replace all fractional-exponent forms by radical signs : 34. 8 + 12 xt + 6 art + x\ 35. 8 x-i - 12 x~iy + 6 x'^nf - if. 36. r^ - 6 ri + 15 ri - 20 + 15 f^ - 6 f + it. 37. 8 asrt + 9 a&* + 13 at + 3 a^6 + 18 a%-^ + &t + 12 ah-K 161. Rationalizing factors of binomial surds. Another advantage of the fractional-exponent notation is that it furnishes an easy method for finding a rationalizing factor of any binomial surd whatever, — only quadratic binomial surds have thus far been rationalized (§ 144). To illustrate this method, let it be required to rationalize the binomial surd cc^ _j_ y^^ Since (xi)"— (?/^)" is exactly divisible by xs + y'2 whenever n is an even posi- tive integer [§ 68 (ii)], therefore, if n be given such an even integral value as will make both (xs)** and (r/i)** rational, — e.g., G, 12, 18, — , — then the quotient of (xi)** — (?/^)'* divided by x'S -fy^ will be a rationalizing factor of a;3 + y^, because the product of x'S + ?/^ by this quotient will be (a;:?)" — (,?/2)", which is rational for all such values of n. * Observe that this expression is arranged according to descending powers of x. 160-161] FRACTIONAL EXPONENTS 265 In the present case, G is the smallest admissible value of n, and the required rationalizing factor is (a;3)6 — (j/i)6 a;2 _ yS 5 41, 231 5 «3 +,?/2 K^ + y^ Again, a rationalizing factor of x's + y'5 is the quotient \.{x^)^^ -\- {y^Y^'\-^ (a;7 + y'S), i.e., {x^ + y^) -^ (k^ + y's) ; and a rationalizing factor of a^ — 6* is the quotient [(at)i2- (6l)i2] ^ («! _ 5!)^ i.e., (aS- 69) -^ (al- 6^). The student may now, from the above examples, formulate a rule for finding a rationalizing factor for any binomial surd; he should distinguish three cases, viz., (1) when the binomial is a difference; (2) when it is a sum and the L. C. M. of the denomi- nators of its fractional exponents is odd; and (3) when it is a sum and this L. C. M. is even. EXERCISES Find the simplest rationalizing factor for each of the following expressions : 1. a? - ri 2. mk + ni 3. 2 a:l - 3 ^i 4. ahi + 3 v\ 5. x"^ + 2 yl. CHAPTER XV QUADRATIC EQUATIONS I. EQUATIONS CONTAINING BUT ONE UNKNOWN NUMBER 162. Introductory remarks. It has already been shown that the first step in solving an algebraic problem is to translate its condi- tions into algebraic language, and also that this translation leads to equations which contain one or more unknown numbers ; the values of these unknown numbers are then found by solving the equations (§ 26). Although nearly all of the problems thus far met with are such that their conditions give rise to equations of the first degree in the letters representing the unknown numbers,* yet there are many other problems which lead to equations of the second degree in those letters; the solution of equations of this kind will be investigated in the present chapter. Note. It may be recalled, however, that some easy equations of the second degree have already been solved by means of factoring (§ 72) ; it will presently appear that all such equations may be solved by the same method. 163. Definitions. An integral algebraic equation which involves the second but no higher degree of a number, is called a quadratic equation in that number (cf. § 94). E.g., a;2 -f- 5 = 0, 'ix'^ — ^ = lx, and ax^ + 6a; + c = are quadratic equations in the number represented by a;; 4c2 + 2c = 9 and a (c + 4)2 — 3 c + 8 = are quad- ratic equations in c ; and a (?/ — 3)2 -f- 6 (?/ — 3) — 6 = is a quadratic equation in y — 3, and also in y. Unless the contrary is expressly stated, a quadratic equation is understood to mean a quadratic equation in the unknown number. Every quadratic equation in one unknown number, say x, may evidently, by transposing and simplifying, be reduced to the standard form 2.7, a ax^ -+- 6a; + c = 0, * For the solution of first degree equations see Chapters X and XI. 266 162-163] QUADRATIC EQUATIONS 267 wherein a, b, and c represent known numbers and are usually called the coefficients of the equation ; the term free from x, viz., c, is also called the absolute term. Although 6 or c may be zero, a can not be zero, for if a = 0' the equation becomes bx-\- c = 0, which is not quadratic. If neither b nor c is zero, the equation is called a complete quad- ratic equation, while if either 6 or c is zero, it is called an incomplete quadratic equation. If 6 = 0, the equation is also often called a pure quadratic equation, otherwise it is called an affected quadratic equation. E.g., the equation 2x^ + 5 — 3a; = 7a; — 8 becomes, by transposing and uniting terms, 2 a;2 _ lo ^ -{- 13 = o, which is in the above standard form, — the coefficients a, b, and c of the general equation being for this particular case 2, — 10, and 13, respectively; it is a complete, and also an affected, quadratic equation. Again, the equation 8x2 + 4 — 3 a; = ^^— — '- — x + S becomes, by clearing of fractions, transposing and uniting terms, 16x^ — 3 = 0, which is in the standard form, a, b, and c being 16, 0, and — 3, respectively; it is an incomplete, and also a pure, quadratic equation. In the same way evenj quadratic equation in one unknown number may be reduced to the standard form. EXEFiCISES 1. What are the important steps in the solution of an algebraic problem (of. § 26) ? What is meant by the " equation of a problem "? 2. If the conditions of a problem, when translated into algebraic language, lead to a quadratic equation (such as 5 a;^ — 8 x + 10 = 0), can that problem be solved by the methods given in Chapter III or Chap- ter X? 3. What is a numerical equation? a literal equation? a simple equation? a general equation? a particular equation? a root of an equation ? 4. Is 3 a:2 — 2 a: = a complete or an incomplete quadratic equation ? Why ? Is it pure or affected ? Why ? 5. Reduce 5x^ + 2 — 8x = 4(8 — x) to the " standard form." What is its absolute term? Is this equation pure or affected? complete or incomplete ? Why ? 6. Clear the equation 2a: — 3+-=a: + 2of fractions, then reduce it to the standard form, and classify it (pure, complete, etc.) ; also solve it by the method of § 72. 268 ELEMENTARY ALGEBRA [Ch. XV 7. Is the equation in Ex. 6 a quadratic or a simple equation ? Why ? 8. If X andy stand for unknown numbers, tell which of the following equations are simple, which quadratic, and which of a still higher degree : a4ar2 + a^x -\- a = 0', ^^=-^ = -; 5 x - 7 ?/ = 11 ; 5 x + ^'^ - 7 ^ = 11 ; 2 X '2(x2- x)+6 = 2x^; ^ - 4: = ') x + -^ ; 3 ar + 4 a^ _ o ax = 7. y y + - 9. What particular equation is obtained by substituting the values 2, — 7, and 5 for the coefficients in the general equation ax^ + hx + c = 01 10. By assigning different sets of values to the letters a, 6, and c, how many particular quadratic equations can be formed from the general equation ax^ + ftx + c = ? Why is this last equation called a "general" equation, and one in which the coefficients are numerals a " particular " equation ? 164. Solution of quadratic equations. Although the roots of any- quadratic equation whatever may be found by the method of fac- toring (§§ 72 and 165), yet there are various other methods. for solving these equations, and one of these, which will doubtless be more easily followed by the student, will now be explained. Ex. 1. Find the roots of the equation 2x2-3-5a: = 7a:+ll. Solution. By transposing and uniting terms, the given equation becomes 2x^-12x=,U, (1) whence, dividing by 2, x^ — 6 x = 7 ; (2) if now 9 be added to each member of Eq. (2), it becomes x^-Qx + 9 = lQ, (3) i.e. (see " remark " below), (z - 3)2 = 16, (4) whence, taking square roots, x — 3 = ± 4, (5) i-e.y a: - 3 = + 4, or X - 3 = - 4, (6) hence, transposing, x = 7, or x = — 1, and, on substituting these values of x in the given equation, it is found that they each satisfy that equation ; they are, therefore, the roots of the given equation. That this equation has no other roots is shown in Ex. 38 below. 163-164] QUADRATIC EQUATIONS 269 Eemark. Since (x ± k)- = oi^. ± 2 kx + Jc'y therefore the expres- sion x^ ± 2kx, whatever the value of A:,. lacks only the term k'^ of being the square of x ±k, i.e., if the square of half the coeffi- cient of the first power of x he added to an expression of the form x^ + hx, the result will he an exact squared E.g., if ( -^ ) be added to a;2 — (j x, the expression becomes {x — 3)2, as in Eq. (3) above ; if ( - j be added to y'^ + 5 ?/, it becomes (?/+-) ; and if ( 7 ) be added to z'^+hx, it becomes lx-\--\ • (Og l^y, \ 2 " \ , i.e., 72, be added to 4 k'^x'^ + 28 kx, it becomes 2V4/fc2j;2y (2 kx + 7)2 ; this may also be seen by first writing 4 ^2^2 -f 28 kx in the form (2A:x)2^-14(2^•x). Ex. 2. Solve the equation a;2+llx + l = 8a;. Solution. On transposing, the given equation becomes x^ + Sx = -l, (1) whence, adding (|)2, x^ -{- Sx +(1^ = - 1 + (|)2, (2) i-e., (X + 1)2 = I, (3) and hence a: + f = ±A/|=±^ \/5, (4) .^-l±lvE = :^lf^, (5) and each of these values of x, viz., ~ ' "^ — '- and — — ^ — '-, is found, on substitution, to satisfy the given equation ; they are, therefore, the roots of that equation. Ex. 3, Solve the equation ax^ -^ bx + c = 0. Solution. On transposing and dividing by a, this equation becomes x'^ + ^x = -^', (1) a a whence x^ + -x + { — = — — = — , (2) a \2a/ 4a2 a ^a^ ^ ^ I.e., I b \2_ ft2_4ac ,3. * Making this addition to the given expression is usually spoken of as com- pleting the square. 270 ELEMENTARY ALGEBRA [Ch. XV therefore , ^ ^ = ± J^l^^ = ±2^EZ±££, (4) I.e., a; = - — - + = , (5) and as before, each of these values of x, viz., '^^—— — and 2 a — — — — ^—^ — —, is a root of the given equation. 2a Note. Having now shown how to find the roots of any quadratic equation whatever, the method of § 67 may be employed to find the factors of any quadratic expression of the form ax^ -\-hx-\- c (cf. also § 165). E.g., since 7 is a root of the equation x^ — 6 a; — 7 = (see Ex. 1 above) , there- fore X — 7 is a factor of the expression x^—iSx — 1 (cf . § 67) . Similarly, from Ex. 2, the factors of x^ + Zx-\-\ are x — ~^'^ — and z — — — ~ ; and x — ~ "*" ~ — — is a factor of the expression ax^+bx+c. 2 2 d EXERCISES 4. In Ex. 1 above, how was Eq. (1) obtained from the given equation? State also how Eq. (2) was obtained from Eq. (1) ; Eq. (3) from Eq. (2) ; Eq. (5) from Eq. (3). How many equations are expressed in (5) ? How were the roots of the given equation finally found from Eq. (5) ? 5. Show that the essential steps in the solution of Ex. 2, and of Ex. 3, are the same as those in Ex. 1, viz., (1) Transposing and uniting terms, and dividing each member of the new equation by the coefficient of the second power of the unknown number, thus re- ducing the given equation to the form x^ + mx = n ; (2) adding ( — ) to each member, thus making the first member an exact square ; (3) extracting the square root of each member {giving the double sign to the second member^, and solving the two resulting simple equations. By the above method find the roots of the following equations, and verify the correctness of each : 6. 2 a;2 - 27 = 9 X - a;2 + 3. 12. 5 a: = x2 - 14. 7. 2;2+5.r = 21 + a:. 13. 19 a: + 5a:2 = 15 - 5a:2. 8. 2/2-52^-24 = 0. 14. 2y^-by=^y-\-2U. 9. 2 a:2 - a: = 3. 15. 22 f + 3 /2 = 4 ^2 _ 43. 10. 2.^2 - 10^/ = ?/2 + lOy - 51. 16. /2 _ 3 = 10 ^ - 3<2. 11. z^^ z- 1.50 = 4 - 2 e. 17. 9 - 5 0,2 = 12 x. 164] qUADRATIC EQUATIONS 271 18. Write a carefully worded rule for solving such equations as those given above ; also show that by this rule any quadratic equation what- ever, which contains but one unknown number, may be solved. 19. Show that the rule asked for in Ex. 18 will serve to solve such equations as x^ + 6 x = 0. What are the two roots of this equation ? Verify your answer. 20. Show that while such equations as that given in Ex. 19 may be solved by the above method, they may be much more easily solved by the method given in § 72. Prove that if an equation has no absolute term, one of its roots is neces- sarily zero. 21. Does the rule asked for in Ex. 18 apply to such equations as 4 ^2 — 9 = ? What are the roots of this equation ? Verify your answer. Solve the following equations, and verify your results : 22. 5 a;2 = 8 z. 25. ax^ + bx = cx^. 23. lSx'\-2x^=5x + 4x^. 26. ax^ + b = 0. 24. 3f-8y = 2y(y-i)-\-Q. 27. {m + n)x'^ + n^ m 28. What must be added to a;^ + 8 a: to " complete the square " ? 29. What must be added to P^ — 5 P to complete the square ? 30. What must be added to (x + yy — 4:(x -h y) to complete the square ? 31. What must be added to 4: M^ -\- 8 M to complete the square ? 32. What must be added to 9 a'^x^ + 12 ax'^ to complete the square? 33. Show that the answer to each of the exercises 28-32 conforms to what is said in the "remark" under Ex. 1. 34. How many different equations are expressed by P = ± Q ? What are they ? Write them separately. 35. How many different equations are expressed by ± P = ± Q ? What are they? Write them separately. Do the equations + P = + Q and — P = — Q express the same or different relations between P and Q? 36. Show that the equation P = ±Q expresses all the relations between P and Q that are expressed by the equation ±P = ± Q ; and hence show that the double sign (±) need be employed in only one member of an equa- tion which is obtained by extracting the square root of each member of a given equation. Illustrate this in the solutions of Exs. 1 and 2 above. 272 ELEMENTARY ALGEBRA [Ch. XV 37. Prove that the two equations P = ± Q are together equivalent (§ 95) to the equation P^ = Q^ Proof. The equation P^ = Q^ is equivalent to pa _ q2 = o, [§ 95 (1) i.e., to (P-Q)(i'+Q) = 0, and, manifestly, this last equation is satisfied when, and only when, P - Q = or P + Q = 0, i.e., when P = ± Q ; hence the equations P2 = Q2 and P = :kQ are equivalent. 38. In the solution of Ex. 1 above, show that the given equation and Eqs. (1), (2), (3), and (4) are all equivalent to each other, and that each is equivalent to the two equations (5), i.e., to the two in (6). Hence show that the given equation has two roots, and only two. 39. By the method of Ex. 38, show that the equation given in Ex. 2, above, has two roots, and only two. 40. Show that Ex. 3 has two solutions, and only two, and thus prove that every quadratic equation in one unknown number has two roots, and only two (cf. § 97). Solve the following equations, and verify your results : 41. 3 a;2 + 5 x - 4 = x2 - 2 a: + 3. 45. 2y'^ + Z = l y. 42. x2-|x-2 = 0. *6. 3x2-10 = 7x. 47. 6 + 5 « = 6 f2. 43. (2-x){x+\)+^ = x-^. ^ 48 Ix — '^-^ 4- '^ = 44. (2 2/-3)2zz:6(?/ + l) -5. ^ 4 49. What are the roots of x^ — Sar — 2 = 0? Are these roots rational or irrational numbers? Define rational and irrational numbers. Are the above roots real or imaginary ? 50. What are the roots of a;^ — 3a: + 4 = 0? Verify the correctness of your answer. Are these roots real or imaginary ? 51. Solve the equation 3a;2-8a:+10 = 0. Suggestion. The method already explained for solving such equations gives rise to fractions ; these fractions can be avoided by proceeding thus : On multiplying the given equation by 3 (the coefficient of a;2), and transposing, it ^^°°"^«« 9x2_24cc=-30; completing the square, 9 a;2 — 24 a; + 16 = — 30 + 16 = — 14, i.e., (3 X -4)2 =—14, hence 3 « — 4 = =t V^^HI, and a; = i(4 + \/^=n4) or i(4 - V^^14). 164] QUADRATIC EQUATIONS 273 52. Solve the equation 3a;2-5a;-2 = 0. Suggestion. Multiply this equation by 4 • 3 and then proceed as in Ex. 51. 53. Solve the equation ax^ ■}- bx -{■ c = 0. Multiply by 4 a and then proceed as in Ex. 51. 54. Solve the equation mx'^ -\- 2 nx -^ k = 0. Multiply by m and proceed as in Ex. 51. 55. By studying ^xs. 51-54, especially 53 and 54, point out when it is necessary to multiply by 4 times the coefficient of the second degree term in order to avoid fractions in the solution of a quadratic equation ; and also when multiplying by that coefficient alone will suffice. Solve the following equations, avoiding fractions in completing the square : 56. 3 a;2 + 2 X = 7. 60. 2 t^ + 7 t = - Q. 57. 5 x2 + 6 a; = 8. 61. 3 a.-2 - 5 x = 2. 58. 3y^ + 4y = 95. 62. 5 z^ - x - 3 = 0. 59. 2 !/2 + 3 3/ rr 27. 63. 15 y-^ -7 y - 2 = 0. 64. Is 8 a root of a:^ - 5 a: - 24 = ? Why ? What is the correspond- ing factor of a:2 — 5x — 24 (cf. Ex. 3, note) ? What is the other factor of this quadratic expression? What root of the given equation corre- sponds to this other factor ? 65. Since x'^ — 7 x -\- I0~(x — 2)(x — o), what are the roots of the equation a:2 - 7 x + 10 = ? Why (cf . § 72) ? 66. Since 2 and 7 are the roots of x^ — 9 x + 14 = 0, what are the factors of x2 - 9 X + 14 ? Why (cf . § 67) ? 67. Since ^ and f are the roots of 6 x2 — 7 x + 2 = 0, what are the factors of 6x2 — 7x + 2? ^^.g these the only factors, or is there also a numerical factor? 68. By first finding the roots of the equation 15 x2 — 4 x — 3 = 0, find all the factors of the expression 15 x2 — 4 x — 3. 69. Based upon the note under Ex. 3, and upon Exs. 64-68, write a carefully worded rule for factoring quadratic expressions. Apply the rule asked for in Ex. 69 in finding all the factors of the following expressions, and verify their correctness : 70. 5 x2 + 12 X - 9. 73. (x + 1)(2 - x) + 9 - x. 71. 8.2_io,_3. ^^ (2, - 3)2- 6(2, + 1)+ 8. „2 3x2 ^ '^ ~4"~2~ 75. ax2 + 6x + c. 274 ELEMENTARY ALGEBBA [Ch. XV 76. Are the expressions in Exs. 70-75 equal to ? What justification have we then for writing them so ? 77. Write an equation whose roots are 3 and 8 (cf. § 72). 78. Write an equation whose roots are — | and 12; 7 and — 1; f and V- ; 1 + V3 and 1 - \/3 ; i audi; 2 + 3 i and 2 - 3 i. 79. By first finding the factors of x^ — 2 x — 10, prove that the roots of 7(x^ — 3 X — 10) = are also roots of a;^ — 3 a; — 10 = 0, and vice versa. Prove this also from § 95 (2). 80. Is there any number which is a root ofa;^ — 3a: — 10 = and also oi S x^ + X — 10 = O'j i.e., have these equations a root in common ? Suggestion. Solve either of these equations and substitute its roots in the other equation. Also solve by means of § 76. 81. Find the common roots, if any, of 2 x^ — S3 x^ — 5 x -\- Q = and 6a:8 + 7x2 + 4a; + l = 0. 82. Find all the roots of the equations in Ex. 81. 165. Solution of quadratic equations by factoring. In § 72 it was shown how factoring may be employed to solve algebraic equations; it will now be shown that any quadratic equation whatever may be solved by this method. Ex. 1. Solve the equation a:^ + 6 x + 8 = 0. Solution. The expression x^ + 6 a: + 8 = x2 + 6 a; + (1)2 _ (1)2 + 8 [cf . §§ 70 and 164 = a:2+6a; + 9-9 + 8 = (a: + 3)2 - 1 = {(a: + 3) + 1} . {(.r + 3) - 1} = (a; + 4) (a: + 2) ; hence the given equation is equivalent to (x + 4) (a: + 2) = 0, which, in turn, is equivalent to the two equations a: + 4 = and a; + 2 = 0, whose roots are — 4 and — 2, respectively ; therefore, the roots of the given equation are — 4 and — 2. Note. Observe that the plan of the above solution is first to transform the expression a;2 + 6a; + 8 into the difference of two squares, one of which shall con- tain all the terms involving x, and then to factor the resulting expression by the formula A^-B^={A — B) (A + B). 164-165] QUADRATIC EQUATIONS 276 Ex. 2. Solve the equation a;^ — 3 a; + 1 = 0. ^ Solution, x^ - '6 x + 1 = x^ - Sx + 1-]^ -I^Y+ 1 =(^-|-^)-(^-|-f) hence the roots of the given equation are the same as the roots of i.e., they are — — and '—^ — ^• Ex. 3. Solve the equation ax^ + bx -{- c = 0. Solution. The expression aa:^ + &x + c, whatever the values of a, b, and c, may be factored as follows : ax"^ + bx + c = alx^ + - X + -\ a-l X + — 4:ac J_ _ VW^Jac} („ . 6 . Vb^ - 4 ac 2a 2a M-f« a< X -^ y ' i X -\ — >• i 2a \ I 2a ) hence the roots of the given equation are the same as those of , ft-V62_ ■4ac> ' 2a -b + Vb^- -4rtc \ 2a /\ '2 a J z>., they are -^-rv. -:.„. ^^^ _ ft _ V//^ _ 4 «c . ' "^ 2 a 2« Since every quadratic equation is reducible to the standard form ax^ + bx-\-c=:0, therefore the solution of Ex. 3 shows not only how to factor any expression of the form ax^ + bx-j- c, but also that every quadratic equation has two roots, and only two ; compare also § 164, Ex. 40. 276 ELEMENTARY ALGEBRA [Ch. XV EXERCISES 4. By first finding the factors of the expression x^ — 9 x -\- li, solve the equation x^ - 9 x -{• H = 0. 5. By first finding the factors of 15 a:" — 4 x — 3, find the roots of the equation 15 a;^ — 4 a; — 3 =0. 6. Factor 3y'^-2y-20, and thus solve the equation Sy^-2y-20 = 0. Factor the following expressions, both by the method of § 164 and also by that of § 165 ; also point out which method is simpler, and why : 7. 8 /2 _ 10 ^ - 3. 10. 5 m2 + 6 m + 2. 8. (a; - 1) (2 - a;) + 9 - x. 11. x^ + (m + n)x + mn. 9. 3 y2 + 4 2/ _ 1. 12. x^+px-\- q. 166. Solution of quadratic equations by means of a formula. Since every quadratic equation in one unknown number may be reduced to an equivalent equation of the form ax^ + te + c = (§ 163), and since the roots of this equation are — — ^— — — — —^ whatever tJie numbers represented by a, b, and c (§ 165, Ex. 3, and § 164, Ex. 3), therefore the roots of any particular quadratic equa- tion may be found by merely substituting for a, b, and c, in the expressions for the roots of the above general equation, those values which these coefficients have in the particular equation under consideration. E.g., since the roots of ax^ + bx-{-c = are — ^^vft^^ — 4qc ^ therefore the 2a roots of 3 x2 + 10 K — 8 = (in which a = 3, 6 = 10, and c = — 8) are _10j,VlO._4.3.(-8) ^ .^^^ -lOj.14, .^^^ 2 ^^^ __^ 2*3 6 3 So, too, the roots of 6 ?/2 + 19 y — 7 = are 19J:Vl92-4.(^(-7) , 1 ^^^ _7. 2-6 3 2 And the roots of a;2_3a; + 5 = are ( '^)^^{ ^)^ 4.1-5 ^ 2 Note. While the student should, of course, be able to solve quadratic equa- tions without the use of the formula (by the method of § 164, or of § 165), he is advised to commit this formula carefully to memory, and henceforth to employ it freely as in the illustrative examples above ; he will find this well worth his while, because roots of quadratic equations are so very frequently required in mathe- matical investigations. 165-167] QUADRATIC EQUATIONS 277 EXERCISES 1. Write down the formula for the roots of ax^ -{-bx + c = 0. How many values has this expression ? Write two expressions which are together equivalent to this formula. 2. Do these two expressions represent the roots of ax^ + hx + c = for all values of the coefficients a, b, and c, or only for particular values of these letters ? By means of the above formula, write down the roots of each of the following equations, verify their correctness in each case, and point out which are real, which imaginary, which rational, and which irrational : 3. a:2-5a: + 6 = 0. 4. 3w2_4^_10 = 0. 6. (3y + l)(2-r) ^ y(3 - -V). 7. mx^ + nx + jo = 0. 8 -t2——t — ~' a n 2n 9. If the numbers represented by p and q are such that p^>4:q, are the roots oi x^ + px + q = real or imaginary? What are they if jt?2< 4 q'i 10. W^hat are the roots of 36 m^x^ + 36 m^nx - n^ = m^(l -9n^)l Show that each of these roots is real whatever integers or fractions (positive or negative) may be represented by m and n. 167. Character of the roots. It has already been shown (§ 165) that the roots of the equation ax^ -{- bx -\- c = are -6 + V62-4ac -, _6-V62-4ac —-r and ; 2a 2a ' hence, if a, b, and c represent real and rational numbers, these roots can be imaginary or irrational only if V&^ — 4 ac is imagi- nary or irrational. E.g., both roots are imaginary if 6^ — 4 ac is negative. The conditions for discriminating the character of the roots may be summarized thus : if &^ — 4 ac > 0, the roots are real, and unequal, if &^ — 4 ac = 0, the roots are real, and equal, if 6^ — 4ac<0, both roots are imaginary, and the roots are rational only when ft- — 4 ac is an exact * The expression b^ — ^ao is, for this reason, usually called the discriminant of the quadratic equation. 278 ELEMENTARY ALGEBRA [Ch. XV The character of the roots of any particular quadratic equation may, therefore, be determined by merely. finding the value of the expression &^ — 4 ac for that equation. E.g., the roots of 3 x^ — 5a; — 1 = are real, irrational, andunequal, because here h^ — 4tac = '61 (since a = 3, 6 = — 5, and c = — 1), and y/'dl is real and irra- tional ; The roots of 3a;2 — 5ccj— 2 = are real, rational, and unequal, because in this equation y/b^ — 4 ac = \/4y = ± 7, i.e., it is rational ; The roots of 2a;2 + 5x — 8 = 4x — 11 are imaginary, because in this equation V62 — 4ac = V— 23; And the roots of 4 x^ — 12 k + 9 = are real, rational, and equal, because in this equation h^ — ^ac = 0. EXERCISES 1. If 62 = 4 ac, what is the value of &2 _ 4 ac ? of y/h'^ -4:ac ? of -bjW¥Z±^9 of -b-Vb-'-^ac , How, then, do the two roots 2a 2a of ax^ + 6a: + c = compare when 6^ = 4 ac ? 2. State verbally the condition that must hold among the coefficients of a quadratic equation in order that the roots of that equation shall be equal, — instead of "a" say "the coefficient of the second power of the unknown number," etc. 3. For what value of h will the roots of 3 a;^ — 10 a; -f 2 ^ = be equal ? Suggestion. The roots are equal if (— 10)2 = 4 . 3 • 2 A;. Why ? 4. Find the value of m for which mx^ — 6 a; + 3 = has equal roots. 5. Find the values of k for which the roots of 3 x^ — 4 ^a; + 2 = are equal. 6. For what values of a are the roots of ax^ — 5 aa: + 11 = a equal? 7. For what values of m are the roots of a:^ — 3 a; — m{x + 2 a;^ 4- 4) = 5a;2 + 3 equal? Without first solving, tell whether the roots of the following equations are real, imaginary, rational, equal, etc., and explain your answers : 8. a:2-5a: + 6 = 0. 11. ^t^ + lit + 11 = 0. 9. a:2-6a: + 9 = 0. 12. 3:^2 + 2 _ 1 ^ £-j, 7 3 6 10. 3 f2 _ 11 f _ 17 = 0. 13. 7 w2 + 4 w + 1 = 0. 14. Are the roots of the equation in Ex. 13 related in any way (cf. Ex. 9, §149)? 167] qUADRATIC EQUATIONS 279 15. Show that if either root of a quadratic equation is imaginary, then the other root is also imaginary, and that each is the conjugate of the other. 16. For what values of k are the roots of 36 x^ — 24 /:a: + 15 ^ = — 4 imaginary ? Solution. The roots of this equation (§ 166) are 24 fe + V(-24 A:)-^-4 • 36 (15 k+^) ^^^ 2ik- \/(-24 A:)2-4 • 36 (15 k+i) ^ 2-36 2 • 36 . 2k+V4:k'^ — 15k-4: , 2k~V4ck^-15k-4: . 6 6 ' and these roots are imaginary for those values of k for which the expression under the radical, viz., 4 k'^ — 15k — 4, is negative, and for those values only. Now 4 A;2 — 15 ^ — 4j which equals (4 A; + 1) (k — 4) (§ 165) , is negative for those values of k for which one of these factors is positive and the other negative, and for no others ; hence the roots of the given equation are imaginary when k lies between — ^ and 4. 17. From the solution of Ex. 16 point out those values of k for which the roots of the given equation are real, and explain your answer. 18. If k =— \, are the roots of the equation in Ex. 16 real or imagi- nary ? How do they compare in value when k = — ^'i when A: = 4 ? " 19. "Without actually solving the equation, find the values of m for which the roots of 4 m^x"^ + 12 m^x + 10 — m = are equal. 20. Without actually solving the equation, find the values of m for which the roots in Ex. 19 are real, and those for which these roots are imaginary. 21. Find the sum of the two roots of ax^ -\- hx -{■ c = ', also the sum of the roots oi x^ ■}■ px + q = 0. 22. By means of the results of Ex. 21, and without first solving the equation, determine the sum of the roots of x^ + Sx — 2 = 0; also the sum of the roots of 4 x^ — 6 a; = 3. Verify your answers by actually adding the roots. 23. Find the product of the roots oi x^ + px + q = 0', also the product of the roots of ax^ -\- bx + c = 0. 24. By means of the results of Ex. 23, determine the product of the roots of a;2 - 10 X 4- 16 = ; also of 4 a;2 _ 30 a; + 25 = 0. 25. State verbally the relation between the sum of the roots of a quadratic equation and the coefficients of that equation ; also make a similar statement concerning the product of the roots, — compare Exs. 21 and 23. 280 ELEMENTARY ALGEBRA [Ch. XV 168. Sum and product of the roots. If r and r' be employed to represent the roots of the equation ax^ + 6a; 4- c = 0, i.e., if r = !— — and r = , 2 a 2a ' then by adding, and by multiplying, it follows that "cf. Exs. 21 and 23, § 167 b c r -\-7'' = and r - r' = - a a The student should perform these operations in detail, and should also express the results in verbal language. Compare Ex. 25, § 167. Note. Rationalizing the numerators in the above expressions for the roots of az^+bz + c = 0, shows that -& + v/62_ -4ac -2c 2a 6+V62- -2c 4:ac -b-Vb^- -4 ac 2a 6-V62- 4ttc and Since r-r' =-, therefore if c is very small as compared with a, i.e., if - is very a a small, then at least one of the roots (r or r') must be very small ; to see which one this is, and also to see how large the other root is, it is only necessary to examine the above expressions for r and r'. Thus as c = 0,* 4 ac =^ 0, and b^ — 4tac^ b^, i.e., VP^^^^Toc = b, and the first expression for r shows that ?• = 0, — since — = 0. 2 a Similarly it may be shown, from the first expression for r', that when c = 0, then r' = , — observe that the second expression for r' becomes indeterminate a r. when c = 0, i.e., it becomes — What has just been shown is usually expressed by saying "if the absolute term of a quadratic equation is zero, then one root of that equation is also zero*' (cf. Ex. 20, § 164). Again, if a = 0, then the above expressions show that r' becomes — oo (cf . note to Ex. 15, § 55), and that r becomes — -, — the first expression for r becomes -, c ^ which is indeterminate, but the second shows its value to be — -• b What has just been shown may be expressed by saying " a = is the condition that one root of az^ + bz + c = Ois infinitely large.'* EXERCISES 1. Without solving the equations, write down the sum and also the product of the roots of each of the equations in Exs. 6-11 of § 164, and explain your answer in each case. * The symbol = is here used to mean " approaches indefinitely near to." 168] QUADRATIC EQUATIONS 281 2. Give the sum and also the product of the roots of each equation in Exs. 22-27 of § 164, and verify your work. 3. If one root of the equation x^ + 5 a; — 24 = is known to be 3, how may the other root be found from the absolute term? from the coefficient of the first power of x ? Do the results agree ? 4. If one root of any given quadratic equation whatever be known, how may the other root be most easily found? 5. What is the sum of the roots of 3 nfiz^ + (8 m - l)a: + 5 = ? For what value of m is this sum 3 ? 6. For whatr values of A; will one of the roots of 2 (k ■]- lyx^ — ^(2k-i- I) (k + l)x+9k = 0he tlie reciprocal of the other ? Suggestion. Equate one of the roots to the reciprocal of the other, and solve. 7. For what value of k will one root of the equation in Ex. 6 be zero ? With this value of k, what will be the value of the other root? 8. For what value of k will one root of the equation in Ex. 6 be infinite (cf . note, § 168) ? 9. For what values of n will one of the roots of (n — 3)y^ — (2 n + 1) y = 2 — 5 n be double the other ? 10. Prove that one of the roots of ax"^ -\- bx + c = 0, whatever the values of a, b, and c, will be double the other if 2b'^ = 9 ac. 11. If r and r' are the roots of ax^ -\- bx -\- c = 0, find the value of - + — expressed in terms of a, b, and c. r r' 12. It has already been shown that if r and r' are roots of the equation ax^ + bx -\- c = 0, then ax^ + bx -\- c = a(x - r) (x - r') ; from this fact prove that if r" is not equal to r or to r', then r" can not be a root of ax^-{-bx + c = (cf. Ex. 40, § 164). 13. Show that the roots of ax^ + 2bx -\- c = are -b+^b^-ac and b-Vh^ a How do these expressions compare with the expressions a for r and / above ? 14. Apply the formulas of Ex. 13 to write down the roots of 3 a;^ — 8 a; — 3 = 0; also of 2 x^ + 10 a; = 7. Compare these results with those obtained by the formulas of § 166 ; which of these formulas gives the simpler result when the coefficient of the first power of the unknown number is even? 15. Show that when a and c represent numbers having like signs, the roots of ax^ + bx + c = may be real, or may be imaginary, depending upon the relative values of a, 6, and c ; but that these roots are necessarily real when a and c represent numbers having unlike signs. 282 ELEMENTARY ALGEBRA [Ch. XV 16. What relation exists between the roots of ax^ + hx -{- c = when a = c'i when a = — c'i 17. If r and r' represent the roots of ax^ + 6a; + c = 0, form an equation whose roots are — r and — r'. Solution. The equation whose roots are — r and — r' is (§ 72} {x-\-r){x + r')=Q, i.e., a:2+ (r + r')a; + rr' = 0; 6 c but r-\-r' = and rr' = - (§ 168) , hence the required equation is a;2 — -a;+- = 0, i.e., ax^—hx-\-c = 0. a a X 18. Find r^ + r'2 from ax"^ -{-hx + c = 0. Also find the sum of the reciprocals of the roots ofx^ — 5a: + 2 = (cf. Ex. 11). 19. If r and r' are the roots of ax'^ -{■ hx -\- c = 0, form the equation whose roots are r^ and r'^ ; also one wliose roots are - and — • r r 20. What do the roots of nx^ + hx-[-c = become when c = 0? when c = and 6 = 0? when a = ? when a = and 6 = 0? when 6 = 0? Compare the note on p. 280. 169. Fractional equations which lead to quadratics. The general principles underlying the solution of fractional equations are dis- cussed in § 99; manifestly those principles apply whatever the degree of the integral equation to which the fractional equation leads. The following solutions may illustrate the procedure. Ex. 1. Solve the equation ^ "^ + 1 = 3 x. X + 2 Solution. On clearing the given equation of fractions, it becomes a: + 5 + a; + 2 = 3a;2 + 6a;, which reduces to 3 a;^ -f 4 x ~ 7 = 0, whence 4 j: Vl6 + 84 ^^ 6 ^^ -4±10 6 1 or - and since neither x = \ nor a; = — | reduces to zero the multiplier which was used to clear of fractions, therefore they are the roots of the given equation (cf. § 99). 168-170] QUADRATIC EQUATIONS 283 Ex. 2. Solve the equation -^— + * ^ + '^ - ^ ^^ 1 - X X + 1 x2-l Solution. On clearing the given equation of fractions, it becomes -x^-x-^x-S + ix^ + Sx = 2x^, which reduces to a;^ — 2 x — 3 = 0, whence ^ ^ 2 ^ V4TT2 ^ 2_^^ 2 2 I.e., x='d or — 1 ; but although both 3 and — 1 are roots of the integral equation, yet 3 alone is a root of the given fractional equation. Observe that a: = — 1 reduces the multiplier a:^ — 1 to zero ; compare also § 99. EXERCISES Solve the following fractional equations, being careful to exclude all extraneous roots : 3. 15x + ?=ll. - 2x-2_x-l X .5a:+5ar + l X X x + 2x — 2 -C-^) ^ ^ ^ 8. ^+(x-2)-i 2(a;2-l) 4(a: + l) 8 x -1 9. -1^+ 42_ ^g_^ 6 X + 5 (x + 5) (a: - 2) a: - 2 10 ^Q I ^Q I 7= ^^ 12 2a + a; a- 2 a: ^ 8 ■ a; + 3 a;2+4:a; + 3 a; + l* ' 2 a - a; a + 2 a; 3* 11 2a; + l 5^ a:-8 ^3 Ix ^ ^ a(a: + 2 6) l-2a: 7 2 * ' a - a; a + 6 3^4 2 _ 5a: ^ a: + 29 g • 15. a; -5 3 a; + 2 (3 a; + 2) (a: - 5) X X X — 1 a; + 1 170. Irrational equations. Equations which contain indicated roots of the unknown numbers are usually called irrational equa- tions ; they are also sometimes spoken of as radical equations. E.g., V^-o = 0, VF+l + a; = 8, ^^ + 1 = 0, and 3 + ^=\/^2Zri are irrational equations, but x — -\/3 = 5 k is a rational equation. The solution of irrational equations may be illustrated by the following examples : 284 ELEMENTARY ALGEBRA [Ch. XV Ex. 1. Solve the equation Vx — 5 = 0. Solution. The given equation is (§ 95) equivalent to Vx = 5, whence, squaring, x = 25. On substituting 25 for x, the given equation is satisfied, provided that Vx is understood to mean the positive value of the square root ; and in that case 25 is, therefore, a root of the given equation. Ex. 2. Solve the equation y/x + 1 + a; = 11. Solution. The given equation is (§ 95) equivalent to Vx + 1 = 11 - x, whence, squaring, a: + 1 = 121 — 22 a: + a:^, t.e., a;2 - 23 X + 120 = 0, , 23 ± V232 - 480 whence x = — — — > i.e., X = 15 or 8, and, on substitution, it is found that 15 satisfies the given equation if Vx + 1 means the negative value of this root, while 8 satisfies it if the positive value of this root is intended. Ex. 3. Solve the equation ^^l£ + 1 = 0. Vx Solution. The given equation is equivalent to 6 — X = — Vx, whence, squaring, 36 — 12 x + x^ = x, and therefore x = 9 or 4 ; of which 9 is a root of the given equation if the positive value of the square root is meant, otherwise 4 is a root. The above procedure may be formulated thus : (1) isolate the radical, or one of the radicals, if there are two or more, (2) hy involution rationalize the given equation, (3) solve this rational equation, and (4) test the results hy sub- stituting them in the given equation. Note 1. Observe that a quadratic irrational equation is ambiguous unless it is stated which of the two values of the radical is intended. E.g.t the equation Vx — 5 = really contains in itself two equations, viz., Vx — 5 = 0* and Vx — 5 = 0; and the equation Vx + Vs — x = 3 contains in itself * Let V and V indicate the positive and negative values, respectively, of the roots. 170J QUADRATIC EQUATIONS 285 four equations, viz., \/z -h ^/5 — x = 3, Va: + VS — x = 3, 'y/z + V5 — x = 3, and Vx + Vo — X = 3, Hence, in order to avoid ambiguity, it is always neces- sary to specify in connection with a radical equation which root is intended. Note 2. It sliould also be observed that if both members of any given equation be raised to the same positive integral power, then every root of the given equa- tion will be a root of the new equation thus formed, and the new equation will, in general, have one or more additional roots which were introduced by the involution. To prove this, let the members of the given equation be represented by u and V respectively (where u and v may be expressions containing the unknown number x) ; then the given equation is u = v, and from this it follows by squaring that u^=v^, which is equivalent to u^ — v^ = 0, i.e., to {u — v){u + v)=0; but every root of the given equation makes u = v, i.e., makes u — v^O, and hence satisfies the equation {ii — v) (w + u) = 0, while the new equation is also satisfied by those additional values of x which make w + u = 0; hence the correctness of the above statement. Similarly if the members of the given equation had been raised to a higher power than the second. Hence the roots of any given irrational equation are to be found among the roots of the equation resulting from rationalizing the given equation, and if none of the roots of the rational equation prove to be roots of the irrational equation, then that equation has n o root w hate ver. E.g., the equation V'Sx + i + 2Vx + 5 - Vx = leads to 3 x2 -f 4 x - 64 = 0, whose roots are 4 and — V> neither of which is a root of the given equation, hence that equation has no root whatever. EXERCISES 4. Show that if the signs of the radicals are left unrestricted, then the equation V8 a; -j- 4 + 2 Vx -f 5 — Vx = has two roots. What are these roots? Solve the following equations, and show what restrictions, if any, must be made on the signs of the radicals in order that your results shall be roots of the equations : 5. \/5-a: = a;-5, 11. V4a:-|-1- Va:-f:3: 6. x+V^ = 4x-4Vi. "• V^T^+V. + 6=V2x+« + ft. 13. Va: + 3-l-V4a;-l-l = Vl0x + 4. ^^ V3 a: + 1 + V3^ 8. \/4!/-f 17 + \/z/ + l- 4 = 0. V3F+T - >/3x ^•^ + ^^ = ^«- ^^^ V,37TT+V8^ _, 9. Vx -f 1 -f (a; 4- l)-2 = 2. ^^^ Vx- 2_ Vi-f- 1 Vx + 3 Vx + 21 10. VS + x-l-Vx^A. i6_ J^%6_>'_ \X ^ X ^ X 286 ELEMENTARY ALGEBRA [Ch. XV Find all the roots of the following restricted equations (cf. note 2, above), and verify your results : 17. V^in + V^^^ = 2. 20. V3^^+V^^-2v'^^=0. 18. VxT4:+Vx^^ = -2. 21. V3X-5+ v^^-2Va:-l = 0. 19. vTT5+Vx^^ = 2. 22. Va^^+Vx^-2Vi^=0. 23. By first rationalizing the equation x = VT, and then transposing and factoring, show (§ 72) that this equation has 3 solutions; i.e., show that 1 has 3 distinct cube roots, viz. : 1, i( — 1+ V^) and |( — 1— V — 3). Similarly it may be shown that any number whatever has 3 cube roots (cf. § 132). 171. Problems which lead to quadratic equations. The directions already given for solving problems whose conditions lead to simple equations (§ 26) are also applicable to problems which lead to quadratic and still higher equations ; the three important steps are : (1) Translate the conditions of the problem into equations, (2) Solve these equations, (3) Test and interpret the results. Special emphasis is to be laid upon the testing and interpreting of the results, because a problem often contains restrictions upon its numbers, expressed or implied, which are not translated into the equations, and therefore the solutions of the equations may or may not be solutions of the problem itself, — compare the illustrative problems which follow, and also § 100. Prob. 1. A farmer purchased some sheep for $168 ; later he sold all but 4 of them for the same sum, and found that his profit on each sheep sold was |1. How many sheep did he purchase ? Solution Let X = the number of sheep purchased. 1 ftft Then = the number of dollars each sheep cost, X 1 OQ and = the number of dollars received for each sheep, a? — 4 and hence i^ - 1^ = 1 ; fSi^ee the profit is X — 4: X L$lon each sheep therefore (§ 169) x = 28 or - 24. 170-171] QUADRATIC EQUATIONS 287 The first of these values, viz., 28, is found to be a solution of the prob- lem as well as of the equation, but while the second satisfies the equation it can not satisfy the problem, because the number of sheep purchased is necessarily a positive integer. Prob. 2. At a certain dinner party it is found that 6 times the num- ber of guests exceeds the square of f their number by 8 ; how many guests are there? Solution ' Let X = the number of guests. Then the expressed condition of the problem is t.e., 2 a:2 - 27 X + 36 = 0, whence a: = 12 or f . Here, too, an implied condition of the problem is that the answer must be a positive integer, and hence, although f satisfies the equation, it is not a solution of the problem. Prob. 3. If 4 times the square root of a certain number be subtracted from that number, the result will be 12 ; what is the number? Solution Let X = the required number. Then the problem states that a; — 4\/x = 12, Le., ' a;2 - 40 a; + 144 33 0, whence a: = 36 or 4. If the above square root is understood to be positive, then 36 is the solution, but if the negative root is meant, then 4 is the solution. Prob. 4. If the number of dollars that a certain man has, be multi- plied by that number diminished by 4, the product will be 21. How much money has he ? Solution Let X = the number of dollars he has. Then the problem states that x{x — 4) = 21, whence a; = 7 or — 3. Each of these numbers will satisfy the conditions of the problem, pro- vided, in the case of the second, that a negative possession be regarded as an indebtedness; i.e., the man may either possess $7, or he may owe |3. 288 ELEMENTARY ALGEBRA [Ch. XV Prob. 5. The sum of the ages of a father and his son is 100 years, and one tenth of the product of the number of years in their ages, minus 180, equals the number of years in the father's age ; what is the age of each ? Solution Let X = the number of years in the father's age. Then 100 — x = the number of years in the son's age, and the condition of the problem states that a:(100-.)_^3Q^ 10 whence a; = 60 or 30. Although each of these numbers is a positive integer, yet the second is not a solution of the problem, since it would make the son older than the father. Hence the father is 60, and the son 40 years old. If, in the above problem, "two persons" be substituted for "a father and his son," then both solutions are admissible, and their ages are either 60 and 40, or 30 and 70 years. PROBLEMS 6. Find two numbers whose difference is 11, and whose sum multi- plied by the greater is 513. 7. A man purchased a flock of sheep for $75. If he had paid the same sum for a flock containing 3 more sheep they would have cost him $1.25 less per head. How many did he purchase? Is each solution of the equation of this problem a solution of the prob- lem itself? Why? 8. A clothier having purchased some cloth for $30 found that if he had received 3 yards more for the same money, the cloth would have cost him 50 cents less per yard. How many yards did he purchase? Has this problem more than one solution? 9. Divide 10 into two parts whose product is 22|. 10. Find two numbers whose sum is 10 and whose product is 42. Are there any two real numbers which satisfy these requirements? 11. Find two consecutive integers the sum of whose squares is 61. How many solutions has the equation of this problem? Show that each of these solutions of the equation is also a solution of the problem itself. 171] QUADRATIC EQUATIONS 289 12. Are there two consecutive integers the sum of whose squares is 118? Are there two numbers whose difference is 1, and the sum of whose squares is 118? What are they? How does the second of the above questions differ from the first ? 13. Find three consecutive integers whose sum is equal to the product of the first two. 14. Is it possible to find three consecutive integers whose sum equals the product of the first and last? How is the impossibility of such a set of numbers shown ? 15. If the number of eggs which can be bought for 25 cents is equal to twice the number of cents which 8 eggs cost, what is that number? How many solutions has the equation of this problem ? Is each of these a solution of the problem itself ? Explain. 16. A farmer, having taken some eggs to market, found that their price had risen 2| cents per dozen, and he also discovered that he had broken 6 eggs. He received $2.88 for his eggs, which was exactly what he would have received if he had broken none, and if the price had not risen. How many eggs did he take to the market? Is each solution of the equation of the problem a solution of the prob- lem itseK ? Explain. 17. Find two numbers whose sum is f, and whose difference is equal to their product. How many solutions has this problem ? 18. The product of three consecutive integers is divided by each of them in turn, and the sum of the three quotients is 74. What are these integers? How many solutions has this problem? Explain. 19. If the product of two numbers is 6, and the sum of their recipro- cals is II, what are the numbers? How many solutions has the equation of this problem? How many solutions has the problem itself? Explain. 20. A merchant who had purchased a quantity of flour for |96 found that if he had obtained 8 barrels more for the same money, the price per barrel would have been $2 less. How many barrels did he buy? How many solutions has this problem? Explain. 21. Why is it that the solutions of the equation of a problem are not always solutions of the problem itself? Compare the last paragraph in §171. 22. The area of a rectangle is 55^ sq. in., and the sum of its length and breadth is 15 in. How long is the rectangle ? 290 ELEMENTARY ALGEBRA [Ch. XV 23. Find the length of a rectangle whose area is 464 sq. in., and the sum of whose length and breadth is 16 in. What is the interpretation of the imaginary result in this problem (cf. note 1, § 100) ? Does an imaginary result always show that the con- ditions of the problem are impossible of fulfillment (cf . Prob. 10, above) ? 24. A boating club on returning from a short cruise found that its expenses had been $90, and that the number of dollars each member had to pay was less by 4| than the number of men in the club. How many men were there in the club ? 25. If in Prob. 24 the expense of the cruise had been $145, the other conditions remaining unchanged, how many members would the club contain ? What is the significance of the fractional and negative results in this problem ? Do such results always indicate that the conditions of a prob- lem are impossible of fulfillment ? 26. The cost of an entertainment was $20, and was to have been shared equally by those present. Four of them, however, left without paying, and this made it necessary for each of the others to pay 25 cents extra. How many persons attended the entertainment? 27. The number of miles to a certain city is such that its square root, plus its half, equals 12. What is the distance ? Has this problen\ more than one solution? Explain. 28. When a certain train has traveled 5 hours it is still 60 miles from its destination ; if it is also known that, by traveling 5 miles faster per hour, 1 hour could be saved on the whole trip, what is the entire distance? And what is the actual speed ? 29. The diagonal and the longer side of a rectangle are together five times the shorter side, and the longer side exceeds the shorter by 35 yards. What is the area of the rectangle ? 30. It took a number of men as many days to dig a trench as there were men. If there had been 6 more men, the work would have been done in 8 days. How many men were there ? 31. A crew can row 5J miles downstream and back again In 2 hours and 23 minutes ; if the rate of the current is 3| miles an hour, find the rate at which the crew can row in still water. 32. A crew can row a certain course upstream in 8f minutes, and if there were no current, they could row it in 7 minutes less than it takes them to drift down the stream. How long would it take them to row the course downstream ? 171-172] QUADRATIC EQUATIONS 291 33. The hypotenuse of a right-angled triangle is 10 inches, and one of the sides is 2 inches longer than the other ; required the length of the sides. 34. From a thread whose length is equal to the perimeter of a square, one yard is cut off, and the remainder is equal to the perimeter of another square whose area is | of that of the first. What is the length of the thread at first? 35. If one train by going 15 miles an hour faster than another, requires 12 minutes less than the other to run 36 miles, what is the speed of each train ? 36. A tank can be filled by one of its two feed-pipes in 2 hours less time than by the other, and by both pipes together in IJ hours. How long will it take each pipe separately to fill the tank ? 37. A man who owned a lot 56 rods long and 28 rods wide constructed a street of uniform width along its entire border, and thereby decreased the available area of the lot by 2 acres. What was the width of the street? 38. Of two casks, one contains a certain number of gallons of water, and the other | as many gallons of wine; 6 gallons are drawn from each cask, and then emptied into the other, after which it is found that the percentage of wine is the same in the one cask as in the other. How many gallons of water did the first cask originally contain? 39. A and B together can do a given piece of work in a certain time ; but if they each do one half of this work separately, A would have to work 1 day less, and B 2 days more, than when they work together. In how many days can they do the work together ? 40. In going a mile, the hind wheel of a carriage makes 145 revolu- tions less than the front wheel, but if the circumference of the hind wheel were 16 inches greater, it would then make 200 revolutions less than the front wheel. What is the circumference of the front wheel ? 172. Equations above second degree, but solved like quadratics. Two important classes of equations of higher degree than the second can be solved like quadratics ; they are discussed below, (i) Equations in the quadratic form. Equations which contain only two different powers of the unknown number, the exponent of one being twice that of the other, may all be reduced to equivalent equations of the form ax^"" + &ic" + c = ; such equa- tions are said to be in the quadratic form, and may be solved like quadratics. 4 L y=l or - f, c2 = 1 or - li x = ±l or ± V - 1; 292 ELEMENTARY ALGEBRA [Ch. XV Eac. 1. Solve the equation 2 x^(x^ + 1) = 6 - z^ Solution. The given equation is equivalent to 2 x* -{- 3 x^ — 5 = 0, and on putting y in place of the lower power of x, i.e., putting y = x^, this equation becomes 2y^ + 3y-5 = 0, whence ^ = llliJ^SE, ,;§ igg i.e., and therefore whence i.e., the roots of the given equation are : + 1, — 1, + ^ V— 10, and - ^ V3T0. Ex. 2. Solve the equation x^ + 6 x^ = 3 + a;^ — a;3. Solution. The given equation is equivalent to 2 x^ + 5 a; * — 3 = 0, or, on putting y ior x^,to 2y^ + 5y — 3 = 0-, whence i.e., and therefore whence Ex. 3. Solve the equation \^x^-^5x + 10 = 2 x^ - 10 a: + 14. Solution. Since the rational part of this equation is, so far as the terms containing x are concerned, simply a multiple of the part under the radical, therefore the equation may be easily transformed into the quad- ratic form ; thus, the given equation is equivalent to y = — o± V25 + 24 4 y = h or -3, .1 = h or -3, X = i or -27. Va:2 - 5 a; + 10 = 2 (a;2 - 5 x + 10) - 6 ; and, on letting y stand for Vx^ _ 5 x + 10, the given equation becomes y = 2y'^-Q, whence y = 2 ov — f , i.e., Va;2 _ 5 a: + 10 = 2 or - f , and therefore a;^ — 5 x + 10 = 4 or |, whence a: = 2, 3, ^.A^^Zl or 5^:2^. 2 2 172] QUADRATIC EQUATIONS 293 EXERCISES 4. Show that rationalizing the equation given in Ex. 3, leads to an equation of the 4th degree. Is this rational equation easily reduced to the quadratic form? Of the methods of §§ 170 and 172 which is prefer- able in such equations ? 14_ y2 i, + 1 _ 7 Solve the foil owing equations 5. X*- 8a;2 + 12 = 0. 6. 3d:6 -4 v^ = 10. 7. x^ + 1 _ x^ 8. .- .yl = 6. y + 1 y2 12 [Observe that ^-^ is the reciprocal of -^.1 v + lj v^ - y + 2 2(y2 + 4) _ 51 15. JLJUL^ 9. a;2-7a: + Va;2-7a;-l-18=24. 3/2^.4' ^^2 5 10. (z2 +1)2 + 4 (x2 + 1) = 45. 16. a:4 + 4 a;8 - 8 x + 3 = 0.' 11. x2-5a: + 2Vx2-ox-2 = 10. 17. 3/* + 2 ?/3 + 53/2 + 4 ^ ^ 60. 12. a;-t + 5 a^'s +4 = 0. 18. 16 a;* - 8 a:^ - 31 a;2 + 8 a: 13. (12_i\V8(12_iU33. +15 = 0- V w / \u J 19. a;8 + 2 a:2 - 9 a: = 18. (ii) Reciprocal equations. An equation which remains un- changed when, for the unknown number, its reciprocal is sub- stituted, and the new equation is cleared of fractions, is called a reciprocal equation. Reciprocal equations of the fifth and lower degrees are readily solved like quadratics, as is shown in the following examples : Ex. 1. Solve the equation ax^ + &a:2 + 6a; + a = 0. Solution. This equation is equivalent to a (a;^ -f- 1) -f bx (x + 1) =0, i.e., to (^ + 1) • {« (a:2 - X + 1) + hx} = 0, which is equivalent to the two equations, X + 1 = and ax^ — ax -{• bx + a = 0, from which the values of x are easily found. * By extracting the square root of the first member, show that this equation may be written in the form (a;2 + 2 a; — 2)2 = 1, from which the complete solution readily follows. 294 ELEMENTARY ALGEBRA [Ch. XV Ex. 2. Solve the equation ax* + bx^ + cx^ ■}■ bx -\- a = 0. Solution. This equation is equivalent to ax^ -\-bx + c-\ h — = 0, i.e., to a(x^ + -\ +b(x + -]-{- c = 0; 1 f 1\^ and, remembering that x^ + — = i x -\- - ) — 2, this equation becomes a(x+-] +blx+-\ + c — 2a=:0. Now, on putting y for x -{--, this last equation becomes X aif + by + (c - 2 a) = 0, whence y = - ^>± ^^'-4a(c-2a) ^ ^^ ^^^^ ^^^ -j^^ ^^ ^^^ . then af + - = k., and x H — = ^9, X X " i.e., a;2 - ^^a: + 1 = 0, and x'^ - k.p: + 1 = 0, whence the four values of x are easily found when a, 6, and c are known. EXERCISES 3. Prove (from the definition) that if ax^ + bx'^ + cx^ + tZx^ + ea;+/=0 is a reciprocal equation, then a = f, b = e, and c = (/, or a = — /, J = — e, 1 and c = — d. Also generalize this result. ' 4. Show from Ex. 3, by grouping terms as in Ex. 1, that a reciprocal equation of odd degree contains the factor a: + 1 or a: — 1. 5. By comparing Ex. 3, show that every reciprocal equation of even degree may have its terms grouped as in Ex. 2. Solve the following equations : 6. 2x8 + 3^2+ 3a; + 2 = 0. 8. ^^ - 3 ?/» + 4?/2 = 3?/ - 1. 7. a:4 + a;8-4a;2 + a;+ 1 =0. 9. 3a;H6a;4-2 a;8-2a;2 + 6a: + 3 = 0. 173. Maximum and minimum values of quadratic expressions. I Evidently such an expression as 3 + 5 ic — a?^ will, in general, have ^ different values when different values are assigned to x ; and it is often important to determine the greatest or the least value (i.e., the maximum * or the minimum value) that such an expression may have, for real values of the letter or letters involved in the expression. * While this definition is somewhat narrow, it serves present purposes best. 172-173] QUADRATIC EQUATIONS 295 Ex. 1. Find the maximum value of the expression 3 + 5 x — z^, for real values of x. Solution. Let m stand for the numerical value of the given expression, i.e., let 3 -{- 5 X — x^ = m. Then x^ - 5 x + m - 3 = 0, , 5 ± V25 - 4(7/1 - 8) 5±V87-4/n .. ,«« whence x = —^ ^ 1 = -^ [§166 From this last expression it is clear (§ 167) that x will be real only so long as 4 my>S7,i.e., so long as m>>^V 5 hence the greatest value that the given ex- 54- v37— -4w pression may have, while x is real, is ^^. Moreover, since x= '-^ — , therefore, x — | when m = V ; «-e-) f is the value of x which gives the above expression its maximum value. Ex. 2. Find the least positive value of x + - , for real values of x. 1 ^ Solution. Let x +-= m. [Wherein m is positive X Then x^ - mx + 1 = 0, 1 m ± Vm^ — 4 whence x = — =^= — In order that x may be real, m^ — 4 < 0, i.e., m < 2; hence the least positive value of m is 2 ; and the corresponding value of x is 1. Note. This exercise may also be solved thus: for any real value of x, (x — 1)2<0, i.e., x2 — 2a; + l<0, whence cc2 4- 1 ^ 2 x-, whence x + -<2 — since the problem requires that x be positive (why?) — i.e., 2 is then the least value of x + - ; and the expression takes this value when x = l. X Ex. 3. Find the range of values of the fraction ' " , for real values of x. Solution. Let Then a:2_6a: + 2_ x+1 ;2 _ - (6 + m)x + 2 - m = whence x = ^ + ^ ±^(^ + ^0' - ^("- ^) = 6 + m j: Vm2+ 16 >» + 28 Hence, in order that x may be real, m2 + 16 m + 28 < 0, i.e., (w + 14) • (m + 2)<0, and, therefore, the factors m + 14 and m + 2 must both be positive or both be negative (in order that their product shall be positive) ; hence m 8 " 296 ELEMENTARY ALGEBRA [Ch. XV may have any value whatever from -co to — 14, and from — 2 to + oo, but it can not have a value between — 14 and — 2. In other words, for real values of x the given fraction has no value between — 14 and — 2. Ex. 4. A window consisting of a rectangle surmounted by a semi- circle, is to have a perimeter of 18 ft. ; what shall be the dimensions of the rectangle in order that the window shall admit the maximum amount of light? And what will be the window's area? Solution. Let x stand for the number of feet in the width of the win- dow; * then - is the radius of the semicircular part, and tt- is the semi- circle's length. And since the entire perimeter is 18 ft., therefore the height of the rectangular part must be | ( 18 — x — tt- j , i.e., 9 — 2!_Jl_ x. From these dimensions it follows at once that the area of the window is hence, if a represents the area, 9 a:- "^^-t^^a = a, 8 whence (tt + 4)a:2 - 72 x + 8 a = 0. Solving this equation gives ^^ 36±V(36)2-8a(,r + 4) 7r + 4 and hence, in order that x be real, (36)'-^ - 8 a(7r + 4)< 0, i.e., a> — ^^^^, which is 22.68 (nearly) ; 8(7r + 4) hence the maximum area of the window is nearly 22.68 sq. ft. ; and the width and height are, therefore, (nearly) 5.04 ft. and 2.52 ft., respectively. EXERCISES For real values of x, find the maximum, or the minimum, value of each of the following expressions ; also the corresponding value of x : 5. x^-8x+ 10. 6. 9 - 2a;2 + 16a:. 7. 12 + x^ -2ax. 8. Find the range of values of ^^ + ^ ^ - ^ . 9. Find the dimensions of the largest rectangular field that can be inclosed by 160 rods of fence. How many acres does this field contain ? * The student should draw a figure to represent the window ; it will make the solution easier to understand. 173-175] QUADRATIC EQUATIONS 297 10. Solve Ex. 9 if a be substituted for 160. 11. Divide 20 into two parts such that the sum of their squares shall be a minimum. 12. A man who can row 4 miles per hour, and can walk 5 miles per hour, is in a boat 3 miles from the nearest point on a straight beach, and wishes to reach in the shortest time a place on the shore 5 miles from this point. Where must he land? II. QUADRATIC EQUATIONS IN TWO OR MORE UNKNOWN NUMBERS 174. Introductory remarks. The really essential thing in solv- ing any system of simultaneous equations, is first to combine the given equations so as to eliminate all but one of the unknown num- bers, and then to solve the resulting equation containing that unknown number. When each equation of the given system is of the first degree, this elimination, as well as the solution of the resulting equation, is easily effected (§ 112) ; but these operations become much more difficult if one or more of the given equations is quadratic, or of a still higher degree. The next few articles are devoted to a study of the procedure in cases where the given system consists of two equations one or both of which are quadratic. 175. One equation simple and the other quadratic. In this case elimination by substitution (cf. § 107) is usually advisable. Ex. 1. Solve the following system of simultaneous equations : 3:^-2^=3, 1 (1) ,.l X^ + 4:f=ld.} (2) Solution. From Eq. (1), x = i±-^, (3) o whence, by substituting this value of x, Eq. (2) becomes + 4 2/2 = 13, (4) (H^J and, on expanding and simplifying, Eq. (4) becomes lOy^+Sy-27 = 0, (5) whence (§ 164) y = I or - |. (6) 298 ELEMENTARY ALGEBRA [Ch. XV But Eq. (3) — also Eq. (1) — shows that to every value of y corresponds one, and only one, value of x ; and that when ?/ = f then x = 2, and when y = — f then x = — ^. It is, moreover, easily verified that each of these pairs of numbers is a solution of the given system of equations. Manifestly the above method is applicable whenever one equa- tion of the given system is simple and the other quadratic. EXERCISES Solve the following systems of equations and verify the correctness of your results : ^ (4x-\-Sy = 9, [2x^ + 5x!j = 3. ^ (x^ + xy- [x-y = '2. . r(^ + 3)(.y-7)=48, *• [x + y=lS. . ( 2s + St = 10, 12 = 0, uv — V = 10 u, + 2 = v. ^ (2x^-\-y^ = 3xy-^U, ' [2x-y = 7. (1Q + 4:v + 2u^ = 5uv, [llv-5u = 4.. 9. 10. x^ -\- 2 X + y _ 4 ^2 _ 5 X + 3 ~ 9 xy 2 1 +H=i+4 = 7.* 11. Write a rule for solving a pair of simultaneous equations one of which is simple and the other quadratic, and which contain two unknown numbers. Could two such equations containing three unknown numbers be solved? Compare § 111 note, and explain. 12. How many solutions has each of the above systems of equations (Exs. 2-10) ? Has every such system two solutions, and only two ? Why (see also § 176, Exs. 1 and 2) ? 176. Principles involved in § 175. The success of the method of solution employed in § 175 depends upon the fact that, if X, Y, * Solve first for - and 1- X y 175-177] QUADRATIC EQUATIONS 299 and Z represent any expressions whatever which contain either X or y, or both, then the system of equations I Y'Z=0,} is equivalent to the two systems r.::;) - If:::! To prove this equivalence, it need only be observed that every solution of either of the last two systems is evidently a solution of the first system; and every solution of the first system is found among the solutions of the last two systems, for it must make X=0 and also either F= or Z = 0.* EXERCISES 1. By means of the proof just given show that Ex. 1, § 175, has two solutions, and only two. Suggestion. The given system of equations is equivalent to Eqs. (1) and (5) (Why ?), and Eq. (5) may be written in the form (2 ?/ — 3) (5 ?/ + 9) = 0. Compare also § 108 (iii) and § 111. 2. By means of the suggestion just given show that every system con- sisting of two equations, one of which is simple and the other quadratic, and containing two unknown numbers, has two solutions, and only two. 3. Show that the solutions mentioned in Ex. 2 may be imaginary (cf. Ex. 6, § 175), and also that one or both of these solutions may be infinite (cf. note, §168). 4. In the solution of Ex. 1, § 175, are Eqs. (2) and (6) equivalent to the given system ? May then the values of y from Eq. (6) be substituted in Eq. (2) to find the corresponding values of x? In which two equations may they be substituted? Why? Does your "rule" (Ex. 11, § 175) pro- vide for this ? 177. Both equations quadratic, — one homogeneoust. If both of the equations of a given system are quadratic, then elimination by substitution, as in § 176, leads to an equation of the 4th degree ( W • X = ) * Similarly it may be shown that the system J „ 7 _ n' [ ^^ equivalent to the four systems { j.^^|, j^^^j. j j. ^ ^^ j , and j ^ ^ J . t An equation is said to be homogeneous if all of its terras are of the same degree in the unknown numbers (cf. § 41). 300 ELEMENTARY ALGEBRA [Ch. XV » in one of the unknown numbers,* and this equation can not, in general, be solved by the methods already studied. If, however, one of the given equations is homogeneous, then the solution of the system may always be made to depend upon the solution of a quadratic equation in one unknown number ; this is illustrated below. Ex. 1. Solve the following system of equations : r6x2^-5xy-6^/2 = 0,| (1) \ 2x^-y^+5x = 9.l (2) Solution. On dividing Eq. (1) by y% it becomes 6(^y+5(.^)-6 = 0, (3) whence (§ 164) ^ = |, or ^ = - |, (4) y d y 2 i.e., x = ly,orx = -^y. (5) On substituting the Jirst of these two values of x, viz., |y, in Eq. (2), that equation becomes 2(|y)2-3/2 + 5(fy)=9, (6) i.e., y'-30y + 81 = 0, (7) whence (§ 164) y = 27 or y = 3, (8) and, since x = ^y, the corresponding values of x are 18 and 2. By substituting these pairs of numbers, viz., x=18, y = 27, and x = 2, y = 3, in the given system of equations, it is easily verified that each pair is a solution of that system. Similarly, if the second of the two values of x in Eq. (5), viz., — f y, be substituted in Eq. (2), two other solutions of the given system of equa- tions will be found; these are : x = — ^, y = S, and x = f , y = — f . It is, moreover, evident that every such system of equations may be solved by this method. Note 1. The success of the method of solution here employed is due to the fact that the two systems of equations from which the values of x and y were finally found, are together equivalent to the given system. * For example, given the system x^ — Sx + Sy = 4: and Sx^ — Ifi y^ + 20 y = 9. Solving the second of these equations for y gives y = i (5 ± \/l 2a:^— 11), and on suhstituting this value of y, Eq. (1) becomes a;2 _ 3 a- -j- 5 -j- Vl2 a;2 — 11 = 4, which, when rationalized, is x* — 6 a;8 — a;2 — 6 a; + 12 = 0. 177-178] QUADRATIC EQUATIONS 301 This equivalence may be seen by writing the given system thus : and recalling that, by § 176, this system is equivalent to the two systems [2a;2 — 2/2 + 5x = 9, J \2x^ — y^ + 5x = 9, ) from which the above solutions were obtained. Moreover, since each of these systems has two solutions, and only two (§ 176), therefore the given system has four solutions, and only four. Note 2. In practice the above method may be somewhat simplified by putting a single letter, say v, in place of the fraction - in Eq. (3), i.e., by putting x = vy y in the homogeneous equation. Thus, on substituting vy for x in Eq. (1), it becomes 6 «2j/^+ 5 ?;|/2 _ 6 2/2 = 0, and hence, dividing by ?/2, 6 ?;2 -|- 5 ^ — 6 = 0, whence (§ l&i) w = f or w =— f ; and, since x = vy, therefore a; = | y and x=~^y. From here on the work is the same as that already given. EXERCISES Solve the following systems of equations and verify the correctness of your results : 5x'^ + ^xij = y% ( 2{x^ -}- f) = 5 xy, a;2 + 3x = 5+ I/. ' \x^-y^=^75. ^ fx^^xij-U = y-x, ^ (x^-2xy-Sy^ = 0, [2x^-3y^ = xy. ' \ y(x -\- y) = ^. 6. Show that every such system of equations as those above has four solutions (real or imaginary, finite or infinite), and only four. 178. Both equations homogeneous in the terms containing the unknown numbers. The solution of a system consisting of two quad- ratic equations, each of which is homogeneous m the terms which contain the unknown numbers, is easily made to depend upon § 177. Ex. 1. Solve the following system of equations : I 3^2 + 3x^ + 22/2 = 8, (1) I a;2 _ X2/ - 4 2/2 = 2. (2) Solution. On subtracting Eq. (1) from 4 times Eq. (2), the result is a;2 _ 7 ^y _ 18 ^2 ^ 0, (3) and the given system of equations is equivalent to the system consisting of Eq. (3) together with either Eq. (1) or Eq. (2) ; but of this last system Eq. (3) is homogeneous, and hence the system can be solved by the method of § 177. 302 ELEMENTARY ALGEBRA [Ch. XV r x2 - 7 i.e., solve the equations \ „ EXERCISES 2. By the method of § 177 complete the solution of Ex. 1 above, xy -42/2 = and verify the correctness of your results - Solve the following systems of equati( and verify your results : 3 (4:X^-xy-S2f = 2, ^ ^ ( y^ -\- 15 = 2 xy, [x^+Qxy-y'' = ~Q, ^^, \x^ + y^ = 21 + xy. (2x^-xy = 28, g I ta:2 + 2 2/2=18. ' [ 2x^-xy = 28, ^ {x'^+6xy = 3-6y% x^-2o = 2y(y -\-2x). 7. Substitute vy for x in each of the equations of Ex. 6 ; then solve each of the resulting equations for y^ in terms of v ; from the first equation 3 25 you will find y^ = — — , and from the second, y^ = — — -; now equate these two values of y% solve the resulting equation in v, and from its values find the values of y, and thence the corresponding values of x. 8. Solve Exs. 4 and 5 above, by the method outlined in Ex. 7. 9. Is the method of Ex. 7 easier or more difficult than that outlined in Ex. 1 ? In what respect ? 10. Is the method of Ex. 7 applicable to all such exercises as those given above? ( Sx^- 5xy-4:y^ = 3x, 11. Solve the system ^ •^ [9x^-\- xy-2y^ = Qx. Suggestion. Subtract the second of these equations from twice the first, and then proceed as in Exs. 1 and 2 above. 12. By the method of Ex. 11, solve the following system of equations, and verify your results : (4:X^ + Qxy-y^ = ly, \Qx^ -Qxy -\-2y^ = 2y. 13. Show that the method suggested in Ex. 11 may be successfully applied to any system of equations whatever of the form ax^ + hxy + cy^ = dx, a'x^ + b'xy + c'y^ = d'x. 14. Could the method suggested in Ex. 7 be employed in such systems of equations as those given in Exs. 11, 12, and 13 ? Explain. 178-179J QUADRATIC EQUATIONS ^^^ . 303 Solve the following systems of equations, and verify your results : c^ — XT/ — y^ = 2 :2 - 3 x?/ + 2 3/2 r2 - xy - ?/2 = 20, (u^ + 3uv + v^= 61, ^^' ^ - - - - ' I m2 _ y2 = 31 _ 2 My. 2 3 17. i I 4_ 4y2 _ y2 + 2 xy [3 !;-l~2(l-a;)' 179. Special devices. . kinds of systems of equations speci- fied in §§ 175, 177, and 1 >ccur frequently, and, although they present themselves in a grt. u variety of forms, they may always be solved by the methods there given. It is worth remarking, however, that special devices of elimina- tion sometimes give simpler and more elegant solutions, not only for the systems already considered, but also for many others which need not now be classified. Some of these special devices are illustrated in the following examples, where it is also shown that they apply to some exercises in which equations above the second degree are involved. Facility in the use of these special devices can be acquired only by practice, but a little study of any particular problem will often suggest a suitable method for attacking it. x-y = 6, (1) xy = -Q. (2) Solution. From Eq. (1), x'' - 2 xy + y^ = 25, (3) fromEq. (2), 4 2:3/ = -24, (4) adding Eq. (4) to Eq. (3), x^ + 2 xy + y"^ = 1, (5) whence x + y = ±1; (6) and from Eq. (1) and Eq. (6), a; = 3 or 2. The corresponding values of y are ?/ = — 2 or — 3. Observe that this exercise belongs to the class of § 175, and could have been solved by the method tlere given. + 3a;2/ = 54, (1) a:?/ + 4 2/2 = 115. (2) Solution. On adding Eqs. (1) and (2), we obtain a:2 + 4 a:?/ + 4 ?/2 = 169, i.e., ix + 2yy=m, (3) whence x + 2 y = ± 13. (4) Ex. 1. Solve the equations 1 ^ ^ ^o> f x^ Ex. 2, Solve the equations \ 304 ELEMENTARY ALGEBRA [Ch. XV From the first of the two equations in (4), and either Eq. (1) or Eq. (2), by § 175, it is found that x = S, y = 5 and x = 3Q, y = — 11^ are solutions. Similarly, by using the second equation in (4), it is found that x'= — 36, y = 11 J and x = — 3, y = — 5 are also solutions of the given system of equations. Observe that this exercise belongs to the class of § 178, and could have been solved by the method there given. r a:2 + 2/2 = 6, (1) Ex. 3. Solve the equations \ ^ , ^ ^ ^ [xy = 2(x-\-y)-5. (2) Solution. On adding 2 times Eq. (2) to Eq. (1), we obtain x^ + 2xy + y^ = ^x + y)-^y -■ ^ (3) i.e., (x + 3/)2 - 4(a: + 2/) + 4 = ; W^\^ ) ^ ^J (4) whence x -\- y = 2. ^- (5) Substituting this value oi x -{■ y in Eq. (2) gives xy = ^-5 = -l; (6) and 2 times Eq. (6) subtracted from Eq. (1) gives x^-2xy + y^ = S, ' (7) whence x - y =±2\/2, (8) From Eq. (5) and Eq. (8), it follows that x = 1 + a/2, y = 1 - V2, and X = 1 — V2, y = 1 + a/2 are solutions of the given equations. Equations like those in Ex. 3, which are not changed by inter- changing X and y, are usually said to be symmetric with regard to those letters. If the equations of a given system are symmetric, or symmetric except for the signs of one or more terms, their solution is often facilitated by substituting u+v for one of the letters and u—v for the other ; this method of solution is illustrated in Exs. 4-6 below. f x^ + w2 =r 6, (1) Ex. 4. Solve the equations ■! ' ' .„. \xy = 2(x + y)-5. (2) Solution. On putting x = u -{- v and y = u — v, the given equations become, respectively, 2 m2 + 2 y2 = 6, and m2 _ „2 = 4 ^^ _ 5 . (3) therefore, eliminating i?2 and simplifying, u2 - 2 w + 1 = 0, whence - w = 1. 179] QUADRATIC EQUATIONS 306 Substituting this value of u iu either one of Eqs. (3), gives v=±V2, whence (since x = u + v, and y = u — v) X = 1 ± \^, and 3/ = 1 T V2, which agrees with the result found in Ex. 3 above. Ex. 5. Solve the equations -i ' \x-y = 5. (2) Solution. On putting x = u + v, and y = u — v^ the given equations become, respectively, ^2 _ „2 ^ _ g, and 2 . = 5. (3) From the second of these, w = |, and substituting this in the first gives whence a: = 3 or 2, and y = — 2 or — 3 (cf . Ex. 1, above) . „ ^ o 1 , . [ x^ + y^ = xy — h, Ex. 6. Solve the equations \ 1 X + ?/ + 1 = 0. Solution. On putting x = u + v and y = u — v, the given equations become, respectively, 2 w3 _f. 6 My2 _ ^2 + y2 4. 5 ^ 0, and 2 u + 1 = 0. From the second of these equations, u =- i, and substituting this value in the first gives y =± f, whence x = 1 or — 2, and ?/ = — 2 or 1. x^ + y^= 17, (1) Ex. 7. Solve the equations , „ ' x + y = 3. (2) Solution. This example may be solved like Exs. 4, 5, and 6 ; another solution is as follows : On raising each member of Eq, (2) to the 4th power, we obtain x^ -\- i x^y -\- 6 xhf + 4 a:2/3 + y4 ^ 81, (3) whence, by subtracting Eq. (1) from Eq. (3) and simplifying, xy (2 x2 + 3 a:?/ + 2 y"^) = 32 ; (4) from Eq. (2), 2 x2 + 3 xy + 2 y^ = 18 - xy, (5) whence, on substituting from Eq. (5), Eq. (4) becomes x^/(18-xy)=32, (6) Le., (xyy - 18 (xy) + 32 = 0, (7) whence (§ 164) xy = 2 or 16. r (8) , x8-8= (X2-/)V, (1) Ex. 9. Solve the equations ' 306 ELEMENTARY ALGEBRA [Ch. XV By combining Eq. (8) with Eq. (2) it is now easy to show that X = 1, 2, or ^ , and the corresponding values of y are O ^ y/ KK y = 2, 1, and ^ , respectively. If one of two equations is exactly divisible by the other, mem- ber by member, their solution may often be greatly simplified, as is shown below. ra;2-3/2=3, (1) Ex. 8. Solve the equations \ ^ ^^. [ x-y=l. (2) Solution. On dividing Eq. (1) by Eq. (2), member by member, we obtain 3, ^ 2/ = 3, (3) whence, from Eqs. (2) and (3), X = 2, and y = 1. r x8 - 8 = [ x + y = 2. (2) Solution. By transposing, Eq. (2) becomes x-2=-y, (3) and, dividing Eq. (1) by Eq. (3), member by member, we obtain a;2 + 2 a: + 4 = - a;2 + ?/2, (4) whence, from Eqs. (2) and (4), by § 175, a; = or — 6, and ^ = 2 or 8. Note. That this method of division must be applied with some caution is, however, evident from Ex. 9, for, while it is easily verified that the two pairs of numbers there found are solutions of the given system of equations, that system has another solution, viz., x = 2, and y = 0, which the above process has failed to reveal. This last solution is found by equating each member of Eq. (3) sepa- rately to zero.* * The general theory for such cases may be stated thus : if P, Q, E, and S represent any expressions whatever, which contain either a; or ?/ or both, then ( P'Q=R'S,] the system of equations < ^ Ms equivalent to the two systems I -P = "S, J Q = R, \ r P = 0, 1 n „ t S'lid ] „ y because every solution of either of the last two sys- P = S, j [ 5 = ; J tems is evidently a solution of the first system, and every solution of the first system is found among the solutions of the last two systems. In Ex. 9 above, P=x — 2, S = -7j, Q= x^ + 2x + 4:, and R = — x^ + y^. 179] QUADRATIC EQUATIONS 307 Ex. 10. Solve the equations 1 + i 13. Solution. These equations being fractional, the first step toward their solution would ordinarily be to clear them of fractions ; in cases like this it is, however, easier to regard - and - as the unknown numbers, and to X y eliminate without first clearing of fractions. If, for brevity, u and v be substituted for - and -, respectively, the given equations become, respectively, " and whence (§ 175) and therefore 3 M - 2 y = 3, m2 _|. 4 „2 = 13^ M = 2 or — I, and u = | or — |, a; = J or — 5, and 2/ = | or — |. EXERCISES Solve the following systems of equations x'^-\-y^ = 13, xy ■■I 6. 12. ^^^ + ^^ = 1- 25 a:y 4- 12 = 0. a;2 + 7/2 + a: = ?/ + 26, xy = 12. V.2 j^ „2 13. 14. 15. 16. 17. r x2 + ?/2 = a, yx-]-y = h. r m2 4. j,2 := 61, \u + v = 11. 1 + 1 = 0. xy l^ 1 1 a:2 ^- 1 1 l + i = 74, .2 ^ „2 ' = 2. 18. 19. 20. 21. 22. ] y X [x -y ■- r a:3 + f [x + y -- {r^ -p [r-p = 1+i: X^ yS ( x^ + y^ 1 x^ + ?/^ 16 15' z2. = 26, --2. = 91, 7. 91, 23. \''^'^' [x + y = 7. = 2, = 26. a, 24. 2/ a:^+ ?/* X + y = 97, • 1. 308 ELEMENTARY ALGEBRA [Ch. XV , rri^n^ =96-4 n,n, ' ( x + y . x -y _ 10 25. «« + — 26. J rri'n^ = 96 - \m + n = 6. (x^-\-xy-\-y^ = 8^, - ^ \x-Vry-,y = ^. 33. ( ^(-^ + ^^) [ X-^ + y-l : 32. \ X - y X + y 6 + y^ = 45. : 5xy, 1.5. 27. !„,,„ ,,_,o o. f(2 + :r)(2/ + l) = 4 .s8 - ^8 = 37, s« (s - f) = 12. 34. (2 + x)^-(i/ + l)i = i. 28. Va; + Vy = 7. 35 x'-^ — 3 a;?/ + ?/2 = 5, a;4 + V* = 2. 36. I 2 Va; + 2/ = 2Vx -y + d. 36. 37. 5 a;-2 _ (a;2,)-i + 2 2/-2 z= 3. 30. a; + y + 2V x + y = 24, f 3 a:^ + 3 a;?/"! = 5^ — y + 3va; — y = 10. ■I 3 a:.y + 3 x-^y = 2.5. 31. ,^^^ + ^^+6V^^T?=55, 38.,_&_!.^^_,.«_5. |x2- ,^ = 7. y X 180. Systems containing three or more unknown numbers. Al- though the solution of a system consisting of three or more simultaneous quadratic equations (involving as many unknown numbers as there are equations in the system) can not in general be made to depend upon the solution of a quadratic equation in one unknown number, yet some solutions of special cases of such systems may be found in this way. (x^+xy-hxz = 2, (1) Ex. 1. Solve the equations \ xy -\- y^ + yz = — 2, (2) [xz +yz + z^ =4:. (3) Solution. Since these equations may be written in the form {x(x + y + z)=2, (4) y(x + y + z) = -2, (5) z(x + y-^z) = 4, (6) therefore, dividing Eqs. (5) and (6) by Eq. (4), member by member, we obtain ^ = -1, and- = 2, (7) X X Le.j y = — X, and z = 2x\ (8) 179-180] QUADRATIC EQUATIONS 309 substituting these values of y and z, in terms of x, Eq. (1) becomes x^ = l, whence x = ±l; and, substituting these values of x in Eq. (8), we obtain x — 1, y = — l, 2 = 2, and also x= — 1, y z=l, z = — 2, as solutions of the given system of equations. i^xy-Sx-2y = 0, 2xz—Sx — 6z = 0, 5 3/2 + 3 y - 4 z = 0. (1) (2) (3) Solution. On dividing these equations by xy, xz, and yz, respectively, they become y X 2-?-? Z X 5 + ?_! = 0. z V These last equations, being of the first degree in the fractions -, -, and 1 1 "" y ~, may be readily solved for -, etc., and hence the values of x, y, and z z X themselves be found. { 2x-\-2v — z = ^ Ex. 3. Solve the equations -I x-Qy-\-z = 2, [a;2-8?/2 + 3?/2=l6. Solution. From Eqs. (1) and (2), y ^^"^ (1) (2) (3) and z 7 a,- -11 4 2 ' substituting these expressions for y and z in Eq. (3), and reducing, it ^^^°'^^« 5x2-12:r-9=:0, whence a: = 3 or — | , and the corresponding values of y and z are readily found. 4. 5. xy = 30, yz = 60, xz = 50. a:2 + ?/2 = 13, 2/ + z^ = 34, a-2 + 22 = 29. EXERCISES 6. x + i xyz y + z xyz z + x = 1.2, = 1.5, 310 ELEMENTARY ALGEBRA [Ch. XV 7. (z + x)(z + y)=Q. x^ + 2/2 + z^ = 29, xy -\- yz + zx = - 10, X + y + 5 = z. 9. X3/2 ^2 + z2 5 3' a;.y2 2-^ + X^ xyz 13 6' 181. Square roots of binomial quadratic surds. Having now learned how to solve simultaneous quadratic equations, it is pos- sible to deal with an interesting problem which was necessarily postponed from Chapter XIII ; this problem is the extraction of the square root of a binomial quadratic surd. Ex. 1. Find the square root of 8 + V60. Solution. Let . Vx + V^ = V 8 + V60. Then, by squaring, a: + 2 Vary + ?/ == 8 + V60, i.e., x + y + 2y/xy=^ + y/m, whence (§ 145) x + y=^ and 2Vxy = VOO ; combining these last two equations — after squaring the second — easily leads (§ 175) to the solution x = 3, ?/ = 5 ; therefore Vs + \/60 = V3 + V5, as is easily verified by squaring each member of this last equation. Ex. 2. Find the square root of a — V&. Solution. Let ■\/x — -\/y = V a — Vb. Then, as before, x + y = a and 4:xy = b, whence (§ 175) x = ^(a+Va^-ft) and y = i(a - Va^ - &), and, therefore, ^^^~7b= yjl+S^^H _ ^/«Z^^, as is easily verified. Note. The above solution shows that although an expression can always be found whose square is a — \/b, yet, unless a^ — b happens to be a perfect square, the expression so found is more complicated than v a — \/&; in other words, the procedure of Exs. 1 and 2 is of advantage only when «2 — 6 is a perfect square. 180-182] QUADRATIC EQUATIONS 311 EXERCISES 3. In Ex. 1 above, why is x + y equal to 8, and 2Vxy equal to V60? Find the square root of each of the following expressions : 4. 25 + 10 V6. 5. 11 + 6\/2. 6. 47 - 12vTl. 7. 18 - 6V5. 8. If the numerical value of v 21 -f SVS is required, is it easier to find first the binomial whose square is 21 + 8V5, or to begin by extract- ing the square root of 5 ? Explain. Also answer this question if 12 — 6 VZ be substituted for 21 + 8V5. 182. Square roots of complex numbers. The square root of a complex number may be found by a process similar to that used in § 181. E.g., to find the square root of 5 + 12 V^, let Vx+ V^\/-l= Vs + l-iV-l. Then, by squaring, x-{-2 yjxy V— 1 — ?/ = 5 + 12 V— 1, whence (§ 151) x — 7/ = 5 and lyjxy = 12, and therefore (§ 175) a; = 9 and ?/ = 4, whence V5 + 12 y/^1 = 3+2 V^, as is easily.verified. Similarly in general. Note. By means of extracting square roots of complex numbers every imagi- nary number may be reduced to the form a + 6 v'— 1, wherein a and b are real, and b^O. E.g., ^V^l=^e^=,J/3i [ B, "where A and B are any two quan- tities of the same kind, and n is a number, then the quantity A is said to have the ratio n to the quantity B. E.g., since a line 10 inches long equals 2 times a line 5 inches long, therefore the ratio of a 10-inch line to a 5-inch line is 2, i.e., it is the same as the ratio of the numbers 10 : 5. Similarly the ratio of S 6 to $ 9 is the same as 6 : 9, i.e., as 2 : 3. Since, by the above definition, the ratio of any two like quanti- ties is the same as that of the numbers which represent these quantities, therefore it is sufficient for present purposes to study the ratios of numbers only. If the ratio of two numbers (or quantities) is a rational num- ber (§ 130), then the given numbers (or quantities) are said to be commensurable * with each other, but if this ratio is an irrational number, then they are said to be incommensurable with each other. * In this case the uumbers have a common measure, hence the name. 820 ELEMENTARY ALGEBRA [Ch. XVI E.g., since VB : 3 is an irrational number, tlierefore y/b and 3 are incom- mensurable with each other; the diagonal and a side of a square are incommen- surable with each other, their ratio being V2 (§ 130) ; but the two irrational numbers 3 ■\/2 and 6\/2 are commensurable with each other, since their ratio is 3 : 5. Note. An irrational number is also often called an incommensurable number, since it is incommensurable with the unit 1. EXERCISES 1. Show that the following ratios are all equal : 8 bu. oats : 6 bu. oats ; 4 tons of coal : 3 tons of coal ; 1 12 : $ 9 ; 10 qt. of milk : 7^ qt. of milk ; 4:3; and ^ : i^. 2. Find the value of each of the following ratios : 8:6; 32 lb. : 4 lb. ; 4V3 in. : 3V2 in. ; 2.7:9; 9:2.7; 4v^:V2; 4\/2 : 2 ; 8.46 cm. : 2.35 cm. ; and ^ 5.80 : 29 cents. 3. Which of the pairs of numbers (or quantities) in Ex. 2 are com- mensurable with each other? Which are incommensurable? Why? 4. Which of the individual terms in Ex. 2 are irrational? II. PROPORTION 188. Definitions. An expression of the equality of two or more ratios is called a proportion. E.g., if a: 6 equals c: d, then the equation a : 6 = c : d is a proportion, and the numbers a, 6, c, and d are said to he proportional {a\^o in proportion) ; thus, since 6 : 3 = 10 : 5, therefore the numbers 6, 3, 10, and 5 are in proportion. The proportion a: b = c:d is sometimes written in the form a:b : :c: d, which is read " a is to & as c is to d." E.g., the proportion 6 : 3 : : 10 : 5 is read " 6 is to 3 as 10 is to 5 " ; its meaning is the same as 6 : 3 = 10 : 5, i.e., the same as S = ¥• The first and fourth terms of a proportion are called the ex- tremes, while the second and third terms are called the means, and the fourth term is called the fourth proportional to the other three. The antecedents and consequents of a proportion are the antecedents and consequents of its two ratios. E.g., in the proportion a:b = c:d, the extremes are a and d ; the means, 6 and c; the antecedents, a and c ; the consequents, 6 and d ; and the fourth proportional to a, b, and c is d. If the first of three numbers is to the second as the second is to the third, then the second is said to be a mean proportional between 187-189] RATIO, PROPORTION, AND VARIATION 321 the other two, and the third is called the third proportional to the other two. E.g., in the proportion a:b=^b:c the number 6 is a mean proportional between a and c, and c is the third proportional to a and b. A succession of equal ratios in which the consequent of each is also the antecedent of the next, is called a continued proportion. E.g., it a:b = b: c = c:d= d:e= '•-, then this expression is a continued pro- portion. EXERCISES 1. Is it true that 8: 12:: 10: 15? Why? How is this proportion read ? What does it mean ? 2. Is it true that 8 : 10 : : 12 : 15 ? What are the means, and what the extremes, of this proportion ? What is the fourth proportional to 8, 10, and 12 ? What are the antecedents? What are the consequents ? 3. How does the proportion in Ex. 1 compare with that in Ex. 2? If any four numbers are in proportion, will they be in proportion after the means have been interchanged? Try several numerical cases, and also compare § 189, Prin. 5. 4. Show that the numbers 3, 4, 6, and 8 are proportional in the order in which they now stand. Arrange these numbers in three other ways in each of which they will be proportional. 5. Show that 6 is a mean proportional between 4 and 9 ; also between 2 and 18. Is — 6 also a mean proportional between these numbers? What are the third proportionals in these cases? 189. Important principles of proportion. Since a proportion is merely an equation whose members are fractio7is, the principles of proportion may be easily derived (as is shown below) from those already demonstrated for equations and fractions. » Principle 1. If four numbers are in proportion, then the product of the means equals the product of the extremes* * Before reading the proofs of these principles the student is urged to make several numerical illustrations of each, and also to try to make a general proof for himself, which he may then compare with that given in the text. Verbal statements of these principles should be committed to memory. If the terms of a proportion are quantities, they may first be replaced by their representative numbers (cf . § 187) , after which the above principle may be applied ; the product of two quantities is meaningless. 322 ELEMENTARY ALGEBRA [Ch. XVI For, let a, b, c, and d be any four numbers which are in propor- tion, then ^ . 5 ^ c : d ; a_c b d whence ad = be, [Multiplying by bd which was to be proved. Principle 2. If the product of two nurrbbers equals the product of two others, then these four numbers form a pro- portion of which the two factors of either product may be made the means, and those of the other product the extremes.* For, if ad = be, then ^ = ^ , [Dividing by bd i.e., a:b = c:d. In the same way it may be shown that, if ad = be, then b : a = d : c, e: a = d:b, etc. ; hence the correctness of Principle 2. Remark. From the proof just given it follows that the correct- ness of a proportion is established when it is shown that the product of the means equals the product of the extremes; this test is very useful. Principle 3. Tlie products of the corresponding terms of two {or more) proportions are proportional. For, if a:b = e:d and e \f=g : h, then (multiplying) «.! = £. |,,>.,p = ^^, hence ae : bf= eg : dh, which was to be proved. Principle 2 is the converse of Principle 1. 189] RATIO, PROPORTION, AND VARIATION 323 Principle 4. Tlie quotients of the corresponding terms of two proportions are proportional. For, if a:h=^c:d and e : /= g : h, then ad = be and eh =fg, whence adfg = bceh ; [§24 (2) on dividing each member of this last equation by ehfg, it becomes ad _ be . a d _b c eh fg' ' '' e h f g' and from this last equation, by Principle 2, a . b_ c , d e'f~g'h' which was to be proved. Principle 5. If a:b = c:d, then (1) b:a = d:e', (2) a:c =b:d', (3) {a + b): a(or b) = (c-\-d): c(or d)-, (4) (a — b): a{or b) = (c — d): c{or d) ; and (5) {a-\-b) : (a-b) = (c + d) : (c-d). The correctness of these proportions [(1) to (5)] easily follows from the remark at the end of Principle 2; the detailed proofs are left as an exercise for the student. Eemark. Proportion (1), above, is usually said to be formed from the given proportion by inversion ; (2) by alternation ; (3) by composition ; (4) by division (or by separation) ; and (6) by compo- sition and division. The student should translate each part of the above principle into verbal language, and commit it to memory ; e.g., (3) thus translated is : If four numbers are in proportion, then they are also in propor- tion when taken by composition; i.e., the sum of the first and second is to the first (or the second) as the sum of the third and fourth is to the third (or the fourth). 324 ELEMENTARY ALGEBRA [Ch. XVI Principle 6. In a series of equal ratios the sum of the antecedents is to the sum of the consequents as any ante- cedent is to its own consequent. Thus, if a:b = c:d = e:f=g:h= "' = .^:y, then (a + c + e + 9'+ ••• +»): (6 + d +/+/i + •-. -\-y)=:e:f. To prove this theorem, let each of the given equal ratios be represented by a single letter, say r ; then l=r, ^=r, ^ = r, f = r, ..., and ?= r, b d f h y hence a = hr, c = dr, e = fr, g = hr, • • •, and x — yr, and, adding these equations, member to member, a + c + e-l-gr+ ... ^x=(b-\-d+f+h+ "- + y)r, and therefore ^ + c +6 + ^ + - + o^ ^^e which proves the principle. Note. As in the proof just given, so it will often be found advantageous to represent a ratio by a single letter. Principle 7. Lihe powers of proportional numbers are proportional; so also are like roots; i.e., if a:b = c: d, then a"* : 6** = c'* : c?". For, if ^ = ^, then f^X = f^X, i.e., ^- = ^; ' b d' \bj \dj ' b^ d"' hence, if a:b = c:d, then a~ : 6** = c" : c^",* which was to be proved. EXERCISES 1. Find the fourth term of the proportion of which the first three terms are 5, 12, and 15. Suggestion. Let x represent the fourth term, and apply Principle 1. 2. Find a mean proportional between 4 and 25. How many answers has this problem ? 3. Find the third proportional to 25 and 40. * According as n is an integer or its reciprocal, a»» is a power or a root of a. 189] EATIO, PROPORTION, AND VARIATION 326 4. If a line 18 inches long is divided into two parts whose ratio is 4 : 5, how long is each part ? 5. If x:15=(a;-l):12,finda:. 6. If 32 : x^ = iL : (a; + 2), find x. 7. Find the mean proportionals between am^ and a^m ; also between a + b and a — b. 8. li a : b = c : d, show that am :bn = cm: dn, wherein m and n are any numbers whatever ; also translate this principle into verbal language. 9. Show that the product of the means of a proportion, divided by either extreme, equals the other extreme. 10. Show that the mean proportional between any two numbers is the square root of the product of these numbers. 11. Prove Principle 6 by means of the remark under Principle 2. 12. Prove Principle 4 by using a single letter to represent a ratio (compare proof of Principle 6). 13. Add 1 to each member of the equation a:h = c :d, write the result in the form of a proportion, and thus prove (3) of Principle 5. 14. It a:b = c : d, and if a is not equal to b nor to c, show that no num- ber whatever can be added to each term of the proportion and leave the results in proportion. If p : q = r : s, prove that : 15. r : .9 = - : - • 17. pr : as = r^ : s^. q p 16. 5p:dr=5q:Ss. 18. (p -]- q) : (r + s) = Vp^ + (/^ : Vr^ + s^. xy. v.iven ^^y^2x):(rj-2x) = (12x + 6y~Sy.iQy-12x-l)^^ find x and y. 20. Given x :27 = y : 9 = 2 : (x - j/); find x and y. 21. li a : b = c : d - e :f = g : h = •", and I, m, n, p, ••• are any numbers whatever, prove that (jna + Ic — ne + pg + •••) : (mb-j- Id — nf + ph -\- "•) = a:h. 22. If a : X = b : y = c : z = d : w = ••• , show that (a« + 6" + c» + ...) : (x~ +?/'» + 2« ...) = a" . x\ 23. If (p + q -h r -h s) (p - q - r + s) = (p - q + r - s) (p -\- q - r - s), show that p : q = r :s. 326 ELEMENTARY ALGEBRA [Ch. XVI 24. It a:b= c :d = e :f, show that c:d = Va2 + 4 ac + 5 c2 : Vft2 + 4 6c? + 5 c?2 25. If (x — y) : (y — 2) : (2 — x) = / : m : n*, and x ^y =^z, show that / + w + n = 0. By the principles of proportion, solve the following equations : 2g^ Vx + 7 + Vx ^ 4:+ Vx Vx + 7 — Vx 4 — Vx Suggestion. Apply Principle 5 (6). 27 x+ VT^n: ^ IS a; _ V.r - 1 7 28. (a - V2 ax - a;^) : (« _ 6) = (a + V2 ax - x^) : (a + 6). Suggestion. First apply Principle 5 (2). 29. If qx±^^ay±_cz^azj^^^ ^^^^ ^^^^ ^^^^ ^^ ^j^^^^ ^^^.^^ &y + c?2 62 + c?x 6x + z _ yz X- y whence — - = ^7— , from which the conclusion is evident. X y z 17. The area of a triangle varies as its altitude if its base is constant, and as its base if its altitude is constant. If the area of a triangle whose base and altitude are, respectively, 6 and 5 in., is 15 sq. in., what is the area when the base and altitude are 13 and 10 in. respectively? 18. If the volume of a pyramid varies jointly as its base and altitude, and if the volume is 20 cu. in. when the base is 12 sq. in. and the altitude is 5 in., what is the altitude of the pyramid whose base is 48 sq, in. and whose volume is 76 cu. in. ? 19. The distance (in feet) fallen by a body from a position of rest varies as the square of the time (in seconds) during which it faUs. If a body falls 257| ft. in 4 sec, how far will it fall in 5 sec. ? how far during the 5th second? how far during the 7th second? 20. If the intensity of light varies inversely as the square of the dis- tance from its source, how much farther from a lamp must a book, which is now 2 ft. away, be removed so as to receive just one third as much light? 330 ELEMENTARY ALGEBUA [Ch. XVI 21. A rectangle moves with its center on a given straight line and its plane perpendicular to that line. If one of its sides varies as the dis- tance, and an adjacent side as the square of the distance, of the rectangle from a certain point on this line, and if at the distance 3 ft. the rectangle becomes a square 2 ft. on a side, what is its area when the distance is 5ft.? 22. In order that two weights attached to a rod should balance each other when the support on which the rod rests is between them, their distances from the point of support should vary inversely as the weights. Find the point of support for a 12-foot plank on which two boys weigh- ing 75 and 90 lb., respectively, wish to play see-saw. 23. The number of oscillations made by a pendulum in a given time varies inversely as the square root of its length. If a pendulum 39.1 inches long oscillates once a second, what is the length of a pendulum that oscillates twice a second ? 24. The volume of a sphere varies as the cube of its radius, and the volume of a sphere whose radius is 1 ft. is 4.19 cu. ft. Find the volume of a sphere whose radius is 3 ft. 25. Three metal spheres whose radii are 3, 4, and 5 in. respectively, are melted and formed into a single sphere. Find the radius of this new sphere. Suggestion. If S^ and S^ are the volumes of two spheres whose radii are rj and r^, and if -S is a sphere of radius r and equivalent to S^ -i- S^, then Si = kr^, and S = Si -1- Sa = A {r^ + r^^) = kr». CHAPTER XVII SERIES - THE PROGRESSIONS 191. Definitions. A series is a succession of related numbers which conform to some definite law. The numbers which con- stitute the series are called its terms. The law of a series may prescribe the way each of the terms, after a given term, is formed from those which precede it, or it may state how each term is related to the number of the place it occupies in the series. E.g., in the series 1, 2, 3, 5, 8, 13, ••• eacli term, after the second, is the sum of the two preceding terms. In the series 2, 6, 18, 54, ••• each term, after the first, is 3 times the preceding term; and 3, 7, 11, 15, 19, ••• is a series of which each term, after the first, is formed by adding 4 to the preceding term. On the other hand, in the series 1, 4, 9, 16, 25, ••• each term is the square of the number of its place in the series; and the law of the series §, |, f, f, tt» •" is expressed by , where n is the number of the term's place in the series. 1 ~T" ^ M If the number of terms of a series is unlimited, it is called an infinite series, otherwise it is a finite series. E.g., in each of the five examples given above the series is infinite, but the series 1, 2, 3, 5, 8, 13, ••• 89 is finite, consisting of 10 terms. Only the simplest kinds of series — the so-called "progres- sions " — will be studied in the present chapter. I. ARITHMETICAL PROGRESSION 192. Definitions and notation. A series in which the difiierence found by subtracting any term from the next following term is the same throughout the series is an arithmetical series ; it is also often called an arithmetical progression, and is designated by "A. P." This constant difference, which may be either positive or negative, is called the common difference of the series. 331 332 ELEMENTARY ALGEBRA [Ch. XVII E.g., the numbers 2, 5, 8, 11, 14, ••• form an A. P. because 5 — 2 = 8 — 5 = 11 — 8 = 14 — 11 = ••• ; the common difference of this series is 3. So, too, 18, 11, 4, — 3, — 10, ••• is an A. P. whose common difference is — 7. In any given A. P. it is customary to represent the first term, the last term, the common difference, the number of terms, and the sum of all the terms, by the letters a, I, d, n, and s, respec- tively ; and these are called the elements of the series. E.g., in the series 2, 5, 8, ••• 32, the elements are : a = 2, 1 = 32, d = 3, n = 11, and s = 187. EXERCISES 1. Define a series. If a row of numbers be written down quite at random, will they constitute a series ? Explain. 2. Define an arithmetical series. Is 1, 7, 13, 19, 25, an A. P. ? What are its elements? 3. If the series 7, 11, 15, 19 be continued toward the right, what is the next term? Why? Extend this series by writing the next four terms at the right, and also the next three at the left. 4. Do the numbers 7, 11, and 15 belong to the same A. P. as 27, 31, and 35? What is d for each of these two series? Write the series which includes both of these sets of numbers. 5. If the first, third, and fifth terms of an A. P. are 18, 24, and 30, respectively, find d and write 8 consecutive terms of this series. Also write 10 consecutive terms of the series of which 19, 9, and 4 are the first, fifth, and seventh terms, respectively. What is d for this last series? 6. Are the numbers 5, 5 + 3, 5 + 6, 5 + 9, and 5 + 12 an A. P.? What are the values of a, d, I, n, and s for this series? How may the second term of this series be fonned from the first? the third from the second? any terra from the one preceding ? 7. Are the numbers x, x -{■ y, x -{■ 2 y, x -\- Sy, a; + 4y, ••• an A. P.? Why? What is d in this series? How may the second term be formed from the first? the third from the first? the fourth from the first? the tenth from the first? the fifteenth from the first? How may any term whatever (say the nth) be formed from the first? 8. Show from the definition of an A. P. that such a series may be written in the form a, a -\- d, a -{- 2 d, a-j-Zd, •■- I - 2 d, I - d, I, wherein a, d, and / represent, respectively, the first term, common differ- ence, and last term . 192-193] SERIES — THE PROGRESSIONS 333 193. Formulas. The elements of an A. P. are connected by two fundamental equations (formulas), which will now be derived. Since each term of an A. P. may be derived by adding d to the preceding term (cf. Exs. 6-8, § 192), therefore, if I stands for the ^*^*^^^' l = a+(7i-l)d. (1) Again, since the sum of the terms of an A. P. may be written in each of the two following forms, s = a + {a-{-d) -{- {a -{-2 d) + '" + {1-2 d) -\- (I- d) + 1, and s = l-j-{l-d)-{-{l-2d)-\ f- (a + 2 d) + (« + ^) 4- «, therefore, by adding these equations, term by term, 2s = (a + Z) + (a + ^)4-(a+/)+-H-(a4-0 + (« + + (« + 0; i.e., 2s = n(a-\-l), [n terms whence g = M«ilil. ' (2) Note. If any three of the five elements of an A. P. are given, the other two can always be found from formulas (1) and (2) above, because, in that case, the remaining two unknown elements will be connected by two independent equations (cf. Ex. 17, p. 334). EXERCISES AND PROBLEMS 1. Verify formulas (1) and (2) above, in the case of the arithmetical series 7, 10, 13, 16, 19, 22, 25. What is the value of a in this series? of d? of n? of Z? 2. Verify formulas (1) and (2) above, for the arithmetical series 26, 19, 12, 5, - 2, - 9, - 16, - 23, - 30 ; also for the series - 8, - 5|, - 3i - 1, H, 3i 6, 8i lOf, 13. 3. By means of formula (1) find the 17th term of 7, 11, 15, ••• ; then, using formula (2), and without writing all the terms, find the sum of the first 17 terms of this series. 4. Using formulas (1) and (2) find the 8th term, and also the sum of the first 8 terms of 1, 3.5, 6, 8.5, •-. 5. Find the 26th term, and also the sum of the first 18 terms of the series 1, 5, 9, — . 6. Find the sum of 10 terms of 4, 11, 18, •••. 7. Find the sum of 30 terms of - 2, - 0.5, 1, 2.5, •... 334 ELEMENTARY ALGEBRA [Ch. XVII 8. Find the sum of 19 terms of 2, 5, 8, ••• ; also find the sum of k terms of this series. 9. Find the sum of n terms of the series 5, 5 + /:, 5 + 2 ^, 5 + 3 ^, •••. 10. Find the sum of t terms of the series h, 2h,^h, •••. What is this sum if ^ = 2 and f = 50 ? 11. Find the sum of the even numbers from 2 to 100 inclusive. Compare your result with that found in Ex. 10. 12. How many strokes does a clock make during the 24 hours of a day? 13. Suppose that 50 eggs were placed in a row, each 2 yds. from the next, and a basket 2 yds. beyond the last eg^, how far would a boy, starting at the basket, walk in picking up these eggs and carrying them, one at a time, to the basket ? 14. If a body falls 16.1 feet during the first second, 3 times as far during the next second, 5 times as far during the third second, etc., how far will it fall during the 8th second? how far during the first 8 seconds? 15. If the 6th and 11th terms of an A. P. are, respectively, 17 and 32, find the common difference, and also the sum of the first 11 terms. Suggestion. Since the 6th term is 17, therefore 17=a + 5d. Similarly, 32 = a + 10 d. From these two equations find a and d, and then find s. 16. By means of formula (1) find the number of the terms in the series 2, 6, 10, •••, 66. Also find the sum of the series. 17. How many terms are there in the series — 1, 2, 5, ••• if the sum of this series is 221 ? Suggestion. Since in this series a = — 1, d? = 3, and s = 221, therefore formulas (1) and (2) of § 193 become, respectively, Z =— 1 + (n - 1) 3 and 221 = - (— 1 + ; and from these equations it is easy to determine n and I. 18. Determine the unknown elements in the series •••, 10, 13, 16, ••• if s = 112 and n = 7. 19. If s, 71, and d are given, find a and I, i.e., find a and I in terms of s, n, and d (cf. Ex. 18). 20. Find a and n in terms of c?, Z, and s. Make up and solve eight other examples of this kind. 21. Show that an A. P. is fully determined when any three of its elements are given. 22. Prove that the products obtained by multiplying each term of an A. P. by any given number are themselves in arithmetical progression. If each term of an A. P. be divided by any given number, or be in- creased or diminished by any given number, will the results be in arith- metical progression? Explain. 193-194] SERIES — THE PROGRESSIONS 335 194. Arithmetical means. The two end terms of an arithmetical series are called the extremes of the series, while all the other terms are called the arithmetical means between these two. E.g., in the series 5, 9, 13, 17, 21, the extremes are 5 and 21, and 9, 13, and 17 are arithmetical means between 5 and 21. Ex. 1. Insert 5 arithmetical means between 3 and 27. Solution. Since there are to be 5 means between 3 and 27, therefore the complete series will consist of 7 terms, and therefore, for this series, a = 3, Z = 27, and n = 7 -, whence, from formula (1) of § 193, d = i, and the series is : 3, 7, 11, 15, 19, 23, 27. EXERCISES AND PROBLEMS 2. Insert 4 arithmetical means between 12 and 27. 3. Insert 15 arithmetical means between 19 and 131. 4. Insert 20 arithmetical means between 16 and — 40. 5. If m arithmetical means are inserted between two given numbers, such as a and b, show that the common difference for the series thus formed is d = (h — a)-^ (m + 1). 6. If X is the (one) arithmetical mean between a and b, show, directly from the definition of an A. P., that x =(a+ b)-^2. Does this agree with the statement in Ex. 5? Explain. 7. Without actually finding the means asked for in Ex. 2, find the sum of the series formed by inserting them. 8. Find 3 numbers in A. P. whose sura is 15 and the sum of whose squares is 107. Suggestion. Let x — y,x, and x + y represent the required numbers. 9. The sum of 7 terms of an A. P. is 105, and the sum of its third and fifth terms is 10 times its first term. Find the series. 10. The product of the extremes of an A. P. of 3 terms is 4 less than the square of the mean, and the sum of the series is 24. Find the series. 11. The sum of 4 numbers in A. P. is 14, and the product of the means is 12. What are the numbers ? Suggestion. Let x — Sy, x — y, x + y, and x + 3y represent the series. 12. The sum of an A. P. of 5 terms is 15, and the product of the ex- tremes is 3 less than that of the second and fourth terms. Find the series. 13. How many arithmetical means must be inserted between 4 and 25 so that the sum of the series may be 116? 336 ELEMENTARY ALGEBRA [Ch. XVII 14. A number consists of 3 digits which are in A. P. ; and the sum of the digits multiplied by 30.4 equals the number, but if 9 be added to the number, the units' and tens' digits will be interchanged. What is the number ? 15. In the series 1, 3, 5, ••• what is the nth term? Prove that the sum of the first n odd numbers, beginning with 1, is n^. II. GEOMETRIC PROGRESSION 195. Definitions and notation. A series in which the quotient of any term (after the first) divided by the next preceding term is the same throughout the series is a geometric series ; it is also often called a geometric progression, and is designated by " G. P." This constant quotient is called the common ratio, or simply the ratio, of the series. E.g., the numbers 2, 6, 18,54, ••• form a geometric series, whose ratio is 3; while §, — 1, ^, — I, ¥-, ••• is a G. P. whose ratio is — f . It is customary to represent the elements of a G. P., i.e., the first term, the last term, the number of terms, the ratio, and the sum of all the terms, by the letters a, I, n, r, and s, respectively. E.g., in the G. P. 2, -(5, 18, -54, 162, -486, 1458, a = 2,^=1458, n=7, r=— 3, and s=1094. EXERCISES 1. Is 7, 21, 63, 189, 567 a geometric series? Why? What are its elements ? 2. Is 2, 8, 32, 96, 288 a geometric series? If not, why not? 3. Is - 6, 12, - 24, 48, - 96, 192, - 384, 768 a G. P.? What are its elements ? How may the second term be obtained from the first ? the third from the second ? the sixth from the fifth ? 4. If the series in Ex. 3 be continued toward the right, what is the next term? the next after that? Extend this series 5 terms toward the left also. 5. If a represents the first term of a G. P., and r the ratio, what is the second term? the third? the fourth ? the fifth? the fourteenth? the twenty-third? the nth ? Explain. 6. Show that x, xy, xy^, xy% xy'^, ... is a G. P. What are a and r in this series ? ^ _ Answer these questions with regard to ~, p^, p^q\ p(i\ q^, — also. 194-190] SERIES— THE PROGUESSIONS 337 7. What is r in the series 2, |, f, ••• ? in the series 21, 7, |, ••• ? Are these two series merely parts of the same series? Explain. 8. If the first, third, and sixth terms of a G. P. are 12, 3, and f, respectively, find r, and then write down the first 8 terms of this series. 196. Formulas. The elements of a G. P. are connected by two fundamental equations which will now be derived (cf. § 193). Since each term of a G. P. may be obtained by multiplying the preceding term by r (cf. Exs. 5 and 6, § 195), therefore, if I repre- sents the nth term of such a series, then / = ar^'-K (1) Again, if s represents the sum of a G. P. of n terms, then s = a + a/' + ar^ -I- ar^ H h ar"~^ + ar"~^, whence sr = ar + cn^ + a?*^ + • • • + ar"^"^ + ar", [multiplying by r and therefore, by subtracting the second of these equations from the first, member from member, s — sr = a — ar"", hence s = (2) 1 — ?' EXERCISES AND PROBLEMS 1. By means of formula (1) above, write down the fifth term of the G.P. 7, 21, 63, .... 2. By formula (1) write down the seventh term of 3, 6, 12, •••, and then find the sum of the first 7 terms of this series by means of formula (2). Verify your answers by actually writing the first 7 terms of the given series. 3. Find the G. P. whose third term is 18 and whose eighth term is 4374. Suggestion. Since the third term is 18, therefore, by formula (1), 18 = ar^; similarly, 4374 = ar^; therefore, by dividing the second of these equations by the first, 243 = r6, i.e., r = 3; etc. 4. Find the G. P. whose fifth term is f and whose ninth term is ^|f . Also find the sum of this series. 5. Find the sum of the first 10 terms of the series 1, 2, 4, •••. 6. Find the sum of the first 6 terms of 1, 1.5, 2.25, •.-. 338 ELEMENTARY ALGEBRA [Ch. XVll 7. Find the sum of the first 7 terms of 2, - |, |, ■••. 8. Find the sum of the first 7 terms of 1, — 2 a:, 4 x% •••. 9. Find the sum of the first k terms of — 5, — 2, — .8, .... 10. Find the sum of the first 9 terms of the series whose first term is 13.5 and whose fourth term is 4. 11. By actually dividing a - ar'\ i.e., a (1 - r"), by 1 - r, verify the correctness of formula (2) of § 196 [cf. § 68 (1)]. 12. Show that the sum of n terms of a G. P. may be expressed in each of the following forms: a — li rl — a nr" — « « ^ « «^'* 1 — r'r— 1 r — 1 1— r 1 r 13. If r, n, and I are given, find a and s ; i.e., find a and s in terms of r, n, and I (cf. Ex. 19, § 193). 14. By means of formulas (1) and (2), § 196, show that a G. P. is fully determined when any three of its elements are given (cf. Ex. 21, §193). 15. If r = 3, do the terms of the series a, ar, ar% ar% ■- ar'^~'^ increase or decrease in going toward the right? Can you name a number so large that it will exceed the nth term of this series for all values of n, however large ? 16. If r > 1 (numerically), show that the terms of the series a, ar, ar% ar% ••• grow larger and larger in passing toward the right, and that, by taking n sufficiently large, the nth term, i.e., ar"'~^, may be made to exceed any given finite number however large. 17. If r< 1 (numerically), show that the terms of the series a, ar, ar^, ar^, ..• grow smaller and smaller in passing toward the right, and that, by taking n sufl&ciently large, ar^-'^ may be made to differ from zero by less than any given number however small.* ♦ Suggestion on Exs. 16 and 17. Let h be any positive number, then since (1 + hy- (1 + hy-l = (1 + hy--^{{l+h)-l} ^h{l+hy-\ and since hO--\-hy-^> h, when 5 — 1 is positive, therefore {1 + h)^— (l + h) >h, (l-j-h)^— (l-\- h)^>h, (l + A)4-(l + /i)3>/i, (i + /i)5_(i + /i)4>/i, ...and (l + /i)«- (1 + /i)'»-i> A. Now adding these inequalities, and the equation l-{-h = l-{-h, member to mem- ber, we have {1 + h)n ';> 1 -{■ nh ; but 1 + «^ > Q (where Q is any given number however large) when n>(Q — l)-^A, hence, for this or larger values of n, (l + 7i)">Q; and therefore, by taking n large enough, the nth power of any number greater than 1 can be made to exceed any number however large. Again, letp1, and therefore q^^i.e., l-^p", may be made larger than any given number however large, hence p" may be made smaller than any given number however small. 196-197] SERIES — TEE PROGRESSIONS 339 18. Three numbers whose product is 216 form a G. P., and the sum of their squares is 189'. What are the numbers? Suggestion. Let - , a, and ar represent the requh'ed numbers. r 19. If the population of the United States was 76,000,000 in 1900, and if it doubles itself every 25 years, what will it be in the year 2000 ? 20. Thinking $1 per bushel too high a price to pay for wheat, a man bought 10 bu., paying 3 cents for the first bushel, 6 cents for the second, 12 cents for the third, and so on. AVhat did the tenth bushel cost him, and what was the average price per bushel ? 21. A gentleman loaned a friend $250 at the beginning of each year for 4 years. If money is worth 5 % compound interest, how much should be paid back to him at the end of the fourth year to discharge the obligation ? 22. Divide 38 into three parts which are iii G. P., and such that when 1, 2, and 1 are added to these parts, respectively, the result shall be in A. P. 197. Infinite decreasing geometric series. If r 1 (numerically), it is an increasing series. Formula (2) of § 196, which gives the sum of the first n terms of the series a, ar, ar^, ai^, ••• may evidently be written in the form a ar'' 1 — r 1 — ?• Now, for a decreasing series the value of becomes smaller 1 — r and smaller, and approaches zero as a limit when n becomes in- finite (cf. Ex. 17, p. 338) ; therefore the sum of the first n terms of an infinite decreasing G. P. may, by taking n sufficiently large, be made to differ from by less than any given number how- ever small. This is usually expressed by saying that the sum to infinity of a decreasing G. P. is ^ ; and if s^ stands for "limit of «„ when 7i becomes infinite," it may be written thus : s - ^ 1 — r 340 ELEMENTARY ALGEBRA [Ch. XVII EXERCISES AND PROBLEMS 1. From a line one foot long cut off one half, then one half of the re- mainder, then one half the next remainder, and so on ; if this process were continued without end, show that, when expressed in inches, the parts cut off form the G. P. : 6, 3, I, I, f, ^, ^2, ii, .-. 2. By means of formula (2), § 196, find s^ for the series in Ex. 1. Also find Sg, Sq, *iq, and s„. 3. Based upon the manner in which the series in Ex. 1 was formed, show that 5„< 12, however large n may be. How near to 12 will s„ ap- proach as n is made larger and larger? Explain. Also find Soo by § 197. 4. Find s„ for the series 0.6, 0.06, 0.006, ••., and thus show that 0.6, i.e., that 0.666 •••, equals |. Find s^ for each of the following series : 5. 1, - h h •••• 9. 0.3. 6. 1, h h •••• 10. 0.i2. 7. i - f, A, -. 11. 1.362. 8. V2, 1, \/o:5, •••• 12. 4.7523. .5. If, in a G. P., r is positive and le 13. l,k,k^ (wherein k <1). 14. ^,-,K X x^ (wherein x >1)' the series is greater than all the terms that follow it. 16. If a point moves from a given position, and along a straight line, with such a velocity that during any given second it moves 75 % as far as it did during the preceding second, and if it moved 24 feet during the first second, how far will it move before it comes to rest? 17. If a sled runs 80 feet during the first second after reaching the bottom of a hill, and if its distance decreases 20% during each second thereafter, how far will it run on the level before coming to rest ? 18. If a ball, on being dropped from a tower window 100 feet above the pavement rebounds 40 feet, then falls and rebounds 16 feet, and so on, how far will it move before coming to rest? 19. The president of a woman's charity organization starts a " letter chain" by writing 3 letters, each numbered 1, requesting each recipient to remit 10 cents to the society, and also to send out 3 other letters, each numbered 2, with a similar request, the chain to close with the letters 197-198] SERIES — THE PROGRESSIONS 341 numbered 20. If evefy one addressed complies with the requests, how much money will be realized for the society ? 20. Although Sao for the series ^, \, I, ••• is 1, show that for the series hhhh •••> ^n grows larger beyond all bounds, by sufficiently increasing n. Suggestion. Write the series tlius: Sn=^+(J+i) + (i+B+7+J)H , putting 8 terms in the next group, 16 in the next, and so on, and show that each group is greater than 5- 198. Geometric means. The two end terms of a finite G. P. are called its extremes, while all the other terms are called the geo- metric means between these two. E.g., in the series |, ^, |, |, and ^^ the extremes are f and ^, and I, I, and | are geometric means between them. Ex. 1. Insert 4 geometric means between f and — ^. Solution. Since 4 means are to be inserted, therefore the complete series will consist of 6 terms, and therefore, for this series, a = ^, I = — ^-, and n = 6; hence, by formula (1) of § 196, - ^= I • r^ therefore r^ = - »-^^, i.e., r = - f , and the _series is : I, — I, 1, — I, f, and — ^. EXERCISES 2. Insert 4 geometric means between 3 and 96. 3. Insert 3 geometric means between 2 and ^\ (two answers). 4. Insert 5 geometric means between x^ and y^ (two answers). 5. If m geometric means are inserted between any two given num- bers, such as a and b, show that the common ratio for the series thus formed is ""^y/b -^ a. 6. If X is the (one) geometric mean between a and b, show directly from the definition of a G. P. that x = Vab. Does this agree with the statement in Ex. 5? Explain. 7. Insert a geometric mean between 12 and 3. Give two solutions, and compare Ex. 6. 8. Insert a geometric mean between 0.5 and 3.5 ; also between (a + 6)2 and (a — b)^; and between dm^x^ and 75m-^x. 9. If the difference between two numbers is 24, and if their arithmeti- cal mean exceeds their geometric mean by 6, what are the numbers ? 342 ELEMENTARY ALGEBRA [Ch. XVII 199. Arithmetico-geometric series. A series formed by multi- plying corresponding pairs of terms of an A. P. and a G. P. is usually called an arithmetico-geometric series. The sum of n terms of such a series may be found by the method of § 196.* Ex. 1. Find the sum of the series 1, 2 r, 3 r^, 4 r^, 5 r*, .••• nr''-\ Solution. Let s = 1 + 2 r + 3 r2 + 4 r3 + ... . + n/-"-i, then rs — r + 2r2+ 3r3 + .- • + (n - l)r"-i + nr^, whence s - -rs — 1 + r + r^+r^+. ... + 7- i.e., .(1- -r) = 1 - 1 - r [§ 196, ] form and therefore s = 1 - (1- - r)2 1 - r EXERCISES 2. By the method of Ex. 1 find the sum of the n terms of the series obtained by multiplying the corresponding terms of the two series a, a + rf, a -f 2 c?, ••• a + (n - l)d and 1, r, r\ ••• r"-i. 3. Find the sum of the series whose (n + l)th term is (a + nh)x''\ i.e., find a +(a + &)a; + (a + 2 6)a;2+ ... + (a + nh)x'^.* III. HARMONIC PROGRESSION 200. Harmonic series. A series of numbers whose reciprocals form an A. P. is called an harmonic series ; it is also often called an harmonical progression, and is designated by " H. P." E.g., the series 1, i, \, iV, ••• is an H. P. because the reciprocals of its terms are 1, 4, 7, 10, ••., and these form an A. P. It follows immediately from the above definition that questions concerning harmonic series, which admit of solution,! ^ay be solved by treating the reciprocals of the terms of the given series as an A. P. E.g., to extend the H.P. ?, \, i\, ^ps three terms further at each end it is only- necessary to take the reciprocals of these numbers, which form the A, P. 5, 4, V. '/> in which d = §, and extend it three terms at each end, and write the reciprocals of its terms. Thus, the given series extended is — §, — 1, f , ?, \, ^, ^^, i, a'l. * For an extension of this subject see Chrystal's Algebra, Part I, p. 489. t There is no general formula for the sum of n terms of an H. P. 199-200] SERIES — THE PROGRESSIONS 343 EXERCISES 1. If X is the harmonic mean between a and b, show, as above, that 1 _ 1 =. 1 _ 1, and hence that x = ■^^' X a X a-^ b 2. Insert 5 harmonic means between 2 and — 3. 3. The arithmetical mean between two numbers is 5, and their har- monic mean is 3.2. What are the numbers ? 4. The difference between two numbers is 2, and their arithmetical mean exceeds their harmonic mean by |. Find the numbers. 5. Given (b — a) : (c — b)= a:x, prove that x equals a, b, or c, accord- ing as a, b, and c form an A. P., a G. P., or an H. P. 6. If the sixth term of an H. P. is ^, and the seventeenth term is ^j, find the thirty-seventh term. 7. If a and b are any two unequal positive numbers, show that their arithmetical mean is greater than their geometric mean, and that this, in turn, is greater than their harmonic mean ; also that the geometric mean is a mean proportional between their arithmetical and harmonic CHAPTER XVIII MATHEMATICAL INDUCTION — BINOMIAL THEOREM 201. Proof by induction. An elegant and powerful form of proof, and one that finds extensive application in almost every branch of mathematics, is what is known as "proof by induc- tion." Suppose it to have been found, by trial or otherwise, that x — y is a factor of a? — y^, o? — y^, and x^ — y^, and that one wishes to know whether it is a factor of x^ — if, x^ — y^, ••• also. Actual trial with any one of these, say x'—'if', would show that it is exactly divisible by x — y, but, besides being somewhat tedious, this division gives no information as to whether ic — ?/ is or is not a factor of x^ — y^, ••• also ; each successful trial increases the proh- dbility of the success of the next, but it really proves nothing beyond the single case on trial. That x — y is a factor of a?" — ?/", for every positive integral value of n, may be proved as follows : Since ic" — y^ = x (a?""^ — 2/""^) + y^~^ (x — ?/), therefore x — y is sl factor of .t" — ?/", if it is a factor ofx'^''^ — ^/""K In other words : if x — y is a, factor of the difference of two like integral powers of x and y, then it is a factor of the difference of the next higher powers also. But since, by actual trial, x — y is already known to be a factor of x'^ — y*, therefore, by what has just been proved, it is also a factor of a^ — y^', again, since it is now known to be a factor of a^ — y^, therefore it is a factor of x^ — y^ -, and so on without end : i.e., x — y is a factor of a;** — ?/" for every positive integral value of n [cf. § 68 (i)]. The proof just given is an example of what is known as a proof by mathematical induction ; it consists essentially of two steps, viz. : 344 201] MATHEMATICAL INDUCTION 345 (a) Showing, by trial or otherwise, the correctness of a given proposition when applied to one or more particular cases, and (h) Proving that if the proposition is true for any given case, then it is true for the next higher case also. From (a) and (6) it then follows that the proposition under consideration is true for all like cases.* EXERCISES 1. Prove that the sum of the first n odd integers is n^. Solution, (a) By trial it is found that 1 + 3 = 22 and 1 + 3 + 5 = 32. (6) Moreover, t/ 1 + 3 + 5 H \-{2k-l) = k2, (1) then, by adding the next odd integer to each member of Eq. (1), we have l + 3 + 5 + ... + (2A;-l) + (2A; + l) = A;2+(2A; + l) = (A: + l)2; i.e., if the law in question is true for the first k odd integers, then it is true for the first k + 1 odd integers also. But, by actual trial, this law is known to be true for the first 3 odd integers, hence it is true for the first 4; and, since it is 7iow known to be true for the first 4, therefore it is true for the first 5, and so on without end ; hence the sum of any number of consecutive odd integers beginning with 1 equals the square of that number. By matliematical induction prove that : 2. 1 + 2 + 3 + ... + n = -1 n (n + 1). 3. 2 + 4 + 6 + ••• + 2w = n(n + 1). 4. 12 + 22 + 32 + ... + 7i2 = 1 n (n + 1) (2 n + 1). 5. 13 + 28+33+ ... +n8 = in2(n + 1)2^(1 + 2 + 3 + ... + n)2. 6. A: + ;r^+:r^ + .-.+ ^ 1-2 2.3 3.4 n(n+l) n + 1 7. 1.2 + 2.3 + 3.4 + ... + n(n+l) = in(n + l) (n+ 2). 8. Having established (a) and (b) in the inductive proof of any prop- osition, show the generality of the proposition by showing that there can be no first exception, and therefore no exception whatever. * The student should carefully distinguish between mathematical induction , as here defined, and what is known as inductive reasoning in the natural sciences; a proof by mathematical induction is, from its very nature, ahsolutely conclusive. On the other hand, the inductive method in physics, chemistry, etc., consists in formulating a statement of a law which will fit the particular cases that are known, and regarding it as a law only so long as it is not contradicted by other facts, not previously taken into account. From the nature of the case step (&) above can not be applied in physics, etc. 346 ELEMENTARY ALGEBRA [Ch. XVIII 202. The binomial theorem. The method of induction (§ 201) furnishes a convenient proof of what is known as the binomial theorem; this theorem, which was presented without formal proof in § 62, may be symbolically stated thus : (a; + yy = a;" + nx^-'y + ^ ^^ 7^^) x^^-y wherein x-\-y represents any binomial whatever, and n is any positive integer. To prove this theorem by mathematical induction, observe first that it is correct when n = 2, for it then becomes 2 • 1 (x -i- yy = 3(^ -\- 2 xy i- — — xy, i.e., (x + yy = x^-{-2xy + y^, which agrees with the result of actual multiplication. Again, if Eq. (1) is true for any particular value of n, say for n = k, i.e., if (x-{-yy=x^-^7c^-'y-^ ^^^ x'^-y+ ^^^~^^%~^^ ^"V + • • •, (2) then, on multiplying each member of Eq. (2) hj x-\-y, it becomes (x+yy^'=x'^^^+kx'y-^^^^^^a^-y+^^^ 1 • ^ i.e., (x+yy+'=x'-^' + (k+l)x'y+ ^\~^'^^^ x'-y + (±i^l|fclla--y+.., (3) * The student should now re-read § 62, and observe that the second member of this identity conforms in every detail to the statement there given. 202-203] BINOMIAL THEOREM 347 which is of precisely the same form as Eq. (2),* merely having A: + 1 wherever Eq. (2) has k. Moreover, Eq. (3) is obtained from Eq. (2) by actual multiplication, and is therefore true if Eq. (2) is true ; hence, if the theorem is true when the exponent has any particular value (say k), then it is also true ivhen the exponent has the next higher value* But, by actual multiplication, the theorem is known to be true when n = 2, hence, by what has just been proved, it is true when 71 = 3 ; again, since it is now known to be true when n = 3, there- fore it is true when w = 4 ; * and so on without end : hence the theorem is true for every positive integral exponent,* which was to be proved. EXERCISES 1. In the expansion of (x + yy* what is the exponent of y in the 2d term ? in the 3d term ? in the 4th term ? in the 12th term ? in the rth term? What is the sum of the exponents of x and y in each term ? 2. In the expansion of (x + ?/)" what is the highest factor in the denominator of the 3d term ? of the 4th term ? of the 10th term ? of the rth term? How does this factor compare with the exponent of y in any given term ? 3. What is subtracted from n in the last factor of the numerator^ in the 3d term of the expansion of (x + yYl in the 4th term? in the 5th term? in the 9th term? in the rth term? 4. Based upon your answers to Exs. 1-3, write down the 6th term of (x + 2/)". Also write the 10th term ; the 17th term ; and the rth term. 203. Binomial theorem continued. Strictly speaking, all that was really proved in § 202 is that, for every positive integral value of the exponent, the first four terms of the expansion follow the law expressed by Eq. (1) ; that all the terms follow this law will now be shown. In multiplying Eq. (2) of § 202 by x + y the 2d term of the product (3) is x times the 2d term plus y times the 1st term of (2) ; so, too, the 10th term of (3) would be found by adding x times the 10th term to y times the 9th term of (2), and the rth * Only the first four terms are given in Eqs. (2) and (3) ; see § 203 for com- plete proof. 348 ELEMENTARY ALGEBRA [Ch. XVIII term of (3) by adding x times the rth term to y times the (r— l)th term of (2). But the (r — l)th and the rth terms of (2) are, respectively, 1.2-3- .••(r-2) ^ and A;(A; - l)(k - 2) .•> (fc - r + 3)(fe - r + 2) .^^.,_, ^"""^ 1.2.3....(r-2)(r-l) ^ "V , ■ therefore the rth term of (3) is fe(fc-l)(fe-2)..-(fe-r + 3) 1.2.3. •.. (r- 2) I fc(A:-l)(fe-2)...(A;-r + 3)(fc-r + 2) \ ^-.+2^.-1 ■^ 1.2.3....(r-2)(r-l) j ^ ' (fe+l)fe(A;-l) ... (fe-r + 3) ._,^, , * *' 1.2.3. ...(r-1) ^ ' which conforms to the law for the rth term expressed by (1) of § 202. Hence the rth term (i.e., every term) in (3) conforms to the law expressed by (1), which was to be proved. EXERCISES 1. Write down the expansion of (a + by-, also of {p — qY- Explain why the alternate terms in the expansion of (p — qY are negative. 2. Write down the first 3 terms of {x + y)^^ ; also the 8th term. 3. Write down the 4th and 7th terms of (a — xy^. 4. How many terms are there in the expansion of {x -f yY^l Write down the first three, and also the last three terms of this expansion, and compare their coefficients. 5. Write down the coefficient of the term containing a^y^ in (a — yy^. 6. Expand (3 a2 _ 2 xy^y\ compare Ex. 2, p. 93. 7. Write down the 4th term of (f a: - | yy^ ; also the 9th term. 8. How many terms are there in ( a: — ] ? Write down the 10th term. Also write the 5th term of (y^- +'V~)*" 9. Write down the term of (3 x* - 2 a;2)7, u., of {x'^y {^ x^ - 2y , which contains x^. 203-204] BINOMIAL THEOREM Md 10. Write down the term of ( a^ — ^ ) which contains a^^. 11. Expand (a^ -{■ ^ a^x-^y, and write the result with positive ex- ponents. 12. Expand (1 — a: + x^y by means of the binomial theorem (cf. Ex. 25, p. 205). 13. By applying the law expressed in Eq. (1) of § 202, show that the coefficient of the (n + l)th term of {x + iy)« is 1 ; also show that the coefficient of every term thereafter contains a zero factor, and hence that (x + yy contains only n + 1 terms. 14. Since (a + !))"■ — {h + a)", show that the coefficients equally dis- tant from the ends of (a + &)" are equal ; show this also by comparing the coefficient of the rth term from the beginning with that of the rth term from the end [i.e., with the (n — r + 2)th term from the beginning]. 15. Show that the sum of the binomial coefficients, i.e., of 1, n, n(n-\) n(n-l)(n-2) • «„ 2 ' 1-2.3 '••''" ^ • Suggestion. Let x = ?/ = 1, after expanding {x + y)^. 16. Show that the sum of the even coefficients {i.e., the 2d, 4th, •••) in Ex. 15 equals the sum of the odd coefficients, and that each sum is 2**-^ Suggestion. Let x = 1 and y =—lin {x + yy. 17. Show that the coefficient of the rth term in {x -f yy may be ob- tained by multiplying that of the (r — l)th term by ^~ ^"^ , and thus r — 1 show that the binomial coefficients increase numerically in going from term to term toward the center (cf. also Ex. 14). 18. Show that the coefficient of the rth term is numerically greater than that of the (r — l)th term so long as r< \(n + 3) ; and thus write down the term whose coefficient is greatest in the expansion of {x + ?/) ^M and also in {x + yY^. 204. Binomial theorem extended. It may be remarked in passing that the binomial theorem (§ 202), which has thus far been re- stricted to the case where the exponent is a positive integer, is greatly extended in Higher Algebra, where it is shown that, under certain restrictions, it admits negative and fractional exponents also. Although the proof of this fact is beyond the limits of this book, its correctness may be assumed in the following exercises. 350 ELEMENTARY ALGEBRA [Ch, XVIII EXERCISES 1. By means of the binomial theorem wi^te the first four terms of (1 + a:)^ ; the first five terms of (a + b)-^; the 5th term of (1 - 3 a;)i 2. Show that the application of the binomial theorem to such cases as the above gives rise to an infinite series (cf. Ex. 13, § 203). 3. Expand (1 — x)-^ to 8 terms by the binomial theorem and compare the result with the first 8 terms of the quotient 1 -f- (1 — x). 4. Show that (25 + 1)^ = 5 + ^V— nks + z when expanded by the binomial theorem and simplified ; compare this result with V'26 as found by the usual method. 5. By means of the expansion of (9 — 2)* show how to get an approximate value of the square root of 7. 205. The square of a polynomial. In § 61 it was pointed out that, by actual multiplication, the square of a polynomial consist- ing of 3, 4, or 5 terms, equals the sum of the squares of all the terms of the polynomial, plus twice the product of each term by all those that follow it. It will now be shown that if this theorem is true for polynomials of n terms, then it is also true for those of n + 1 terms, and from this it will follow, as in § 201, that it is true for polynomials of any finite number of terms whatever, since it is already known to be true for polynomials of five terms. Let a-\-h-^c-\ \-p + qhe a. polynomial of n terms, and let (a + b + c-\ [- p -{- qf = a^ + b^ -\ ^q'~-\-2ab + 2ac-\ \-2aq + 2bG-\ \-^bq-{ \-2pq. In this identity replace a everywhere hj x-\-y; then the number of terms in the polynomial in the first member will become n 4- 1, and the second member will still consist of the sum of the squares of all the terms of the polynomial, plus twice the product of each term by all those that follow it (the student should work this out in detail) ; therefore, if the theorem is true for polynomials of n terms, then it is also true for those of n + 1 terms, which was to be proved. EXERCISES Expand : 1. (a + b-3x + 2ah- 1)2. « / . 2 „ .2 2. (2-3a? + 4ma:2-3ma;+3a;- 3a2a:)2. \ X ml , APPENDIX A IRRATIONAL NUMBERS [Supplementary to § 132] 206. Irrational numbers are defined and illustrated in Chapter XIV, and it is there tentatively assumed, not only that the earlier definitions of sum, product, etc., apply to these numbers, but also that they are subject to the combinatory laws previously established for rational numbers. These definitions will now be restated from a somewhat broader point of view, and one from which the proofs of the combinatory laws are easily established. As in § 130, note 2, two infinite series may be found such that the square of each term of the first series is less than 2, while the square of each term of the second series is greater than 2. These series may be conveniently written in the form 1, 1.4, 1.41, 1.414, 1.4142, .•• *3 • *'3' ••• ^H • *'n, — , (5) and flj -4- h\, a^ h- 6'2» as "^ ^V •- «n ^ 6'«, -" „ - a'„, .-. (6) Note 1. Observe that if k = k' , as defined above, then these two irrational numbers have the same decimal expressions , however far they may be carried out. For suppose that some decimal figure, say the 14th, in k is greater than the corresponding figure in k', then the corresponding* a woulci be equal to, or greater than, the corresponding 6', and k would not equSil k' under t^e above definition. . ' ' - Note 2. In applying the above definitions, say that of the sum, it may happen that ai + a'l = 03 + a'2 = — = an + a'n = — = &n + b'n = •.• ; in this case k + k' = an-{-a'n= bn + b'n, i.e., this sum is a, rational number. To illustrate this fact numerically, let k = y/2 and k' — 5~ \/2. Note 3. The above definitions [inequalities (3)-(6)] apply also when negative irrational numbers are involved : those of sum and difference apply directly, and those of product and quotient apply by regarding the numbers as positive and attaching the proper sign to the result. 208. Comparisons and operations between rational and irrational num- bers. A given rational number r is said to be less than k (see § 207) if, and only if, some of the a's are greater than r, otherwise it is greater than k. The sum of a rational and an irrational number, say A: + r, is defined by the series a^ + r, a^-\-r, flg + r, .- a„+r < ^+r < ft^ + r, h^-^r, 63+r, — 5„ + r, ...; and the difference, product, and quotient of a rational and an irrational number are defined in a similar manner. 209. Combinatory laws of irrational numbers. That the irrational numbers are subject to the same combinatory laws as are the rational numbers follows easily from the definitions given in §§ 207 and 208. Thus, by (3) of § 207, aj + a'i, a2+«'2' «3+«V •*• < k-\-k' "e!-5lCM - LD 2t-100w-7.';?nM'V>. ^ /3V9/9 THE UNIVERSITY OF CALIFORNIA UBRARY