-NRLF [iij t iiiji !l^^ i' ||||> ^0 IN MEMORIAM FLORIAN CAJORl Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/firstcourseinalgOOwellrich A FIRST COURSE IN ALGEBRA BY WEBSTER WELLS, S.B. PROFESSOR OF MATHEMATICS IN THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY ^v-t ^W CL. ^ v^ 5i-w^~~ BOSTON, U.S.A. D. C. HEATH & CO., PUBLISHERS 1908 COPYRIGHT, 1908, BY WEBSTER WELLS. All rights reserved. PREFACE In the preparation of this text the author acknowledges joint-authorship with Robert L. Short. This book meets the demand that the pupil be given an elementary algebra containing no more than can be accom- plished in the time allotted to the subject. It is not intended for a complete course, but gives the student a good working knowledge of the subject through simultaneous quadratics. It should be followed by a second course by those intending to pursue the study of higher mathematical subjects. This book is sufficient preparation for geometry, and the frequent introduction of geometric ideas and geometric problems not only prepares for geometry but also makes that subject at- tractive to the learner. This text is as brief as the algebra of years ago, and yet contains all that is good in modern mathematical thought. Attention is called to the introduction of graphical methods through simple horizontal and vertical measurements (Exer- cise 4, Exercise 41, problems 28-30). This procedure makes the transition to Cartesian coordinates a natural one. Teach- ers will find that the color scheme recommended in graphs will greatly aid the student in connecting related data. Peda- gogical advantage is gained through the combining of related and reverse processes. (Chapters III, VII, X, XII, XIII.) The use of the fractional exponent in operations involving surds is recommended, thereby avoiding confusion, since the four fundamental laws and the exponential laws of Multipli- cation, Division, Involution, and Evolution, are the only ones involved. The complete index will be found helpful to both pupil and teacher. No attempt is made toward technical iv PREFACE definition. Definitions for the beginner must be explanatory and descriptive. The lists of queries will aid in fixing both definitions and principles. The authors thank the many teachers of mathematics who have made this book better and have brought it close to actual class-room conditions by their timely criticism and suggestion. Webster Wells. CONTENTS II. III. IV. V. VI. VII. VIII. IX. X. XL XII. XIII. XIV. XV. XVI. XVII. Definitions and Notation. Axioms. Equations Algebraic Expressions Positive and Negative Numbers Addition and Subtraction. Parentheses Multiplication of Algebraic Expressions Division of Algebraic Expressions Integral Linear Equations Products and Factors .... Solution of Equations by Factoring Highest Common Factor Lowest Common Multiple . Fractions Fractional Equations Ratio and Proportion . . . . Simultaneous Linear Equations Graphs Involution and Evolution Theory of Exponents .... Irrational Numbers .... Imaginary Numbers .... Quadratic Equations. Graphs Equations in Quadratic Form Factoring of Quadratic Expressions Simultaneous Quadratic Equations . Graphs Binomial Theorem Hints on Checking .... ALGEBRA I. DEFINITIONS AND NOTATION SYMBOIiS REPRESENTING NUMBERS 1 . In Algebra the symbols usually employed to represent numbers are the Arabic numerals and the letters of the alphabet. The numerals represent known or determinate numbers. The letters represent numbers which may have any values whatever, or numbers whose values are to be found. EQUATIONS 2. The Sign of Equality, =, is read ''equals,'' Thus, a = 6 signifies that the number a equals the number b. 3. An Equation is an expression of equality. The Jirst member of an equation is the number to the left of the sign of equality, and the second meinber is the num- ber to the right of that sign ; thus, in the equation 2x — Z = b, the first member is 2x — ^, and the second member 5. AXIOMS 4. An Axiom is a statement which is assumed as self- evident. Algebraic operations of finite numbers are based in part on the following axioms : 1. Any number equals itself. 2. Any number equals the sxmi of all its parts. 3. Any number is greater than any of its parts. 4. Two numbers which are equal to the same number, or to equal numbers, are equal. 2 ALGEBRA 5. If the same number, or equal numbers, be added to equal numbers, the resulting numbers will be equal. 6. If the same number, or equal numbers, be sub- tracted from equal numbers, the resulting numbers will be equal. 7. If equal numbers be multiplied by the same number, or equal numbers, the resulting numbers will be equal. 8. If equal numbers be divided by the same number, or equal numbers, the resulting numbers will be equal. Numbers cannot be divided by the number 0. SOLUTION OF PROBLEMS BY ALGEBRAIC METHODS 5. The following examples illustrate some uses of alge- braic symbols ; I. The sum of two numbers is 30, and the greater exceeds the less by 4 ; what are the numbers ? We will represent the less number by x. Then the greater will be represented by x+4. By the conditions of the problem, the sum of the less nuftiber and the greater is 30; this is stated in algebraic language as follows: a;4-x + 4 = 30. (1) x-\-x = 2x. Therefore, 2a: + 4 = 30. The members of this equation, 2 .T-f4 and 30, are equal numbers; if from each of them we subtract the number 4, the resulting numbers will be equal (Ax. 6, § 4). Therefore, 2 a: = 30 - 4, or 2 a: = 26. Dividing the equal numbers 2 x and 26 by 2 (Ax. 8, § 4), we have a;=13. Hence, the less number is 13, and the greater is 13-1-4, or 17. The written work will stand as follows : I^et X = the less number. Then, a:-f-4 = the greater numbtM-. By the conditions, a; -f x -f- 4 = 30, or 2 x -H 4 = 30. Whence, 2x = 26. Dividing by 2, a? = 13, the less number. Whence, ar f4 = 17, the greater number. DEFINITIONS AND NOTATION 3 2. The sum of the ages of A and B is 109 years, and A is 13 years younger than B ; find their ages. Let n represent the number of years in B's age. Then, n— 13 will represent the number of years in A's age. By the conditions of the problem, the sum of the ages of A and B is 109 years. Whence, n - 13 + n = 109, or 2 n - 13 = 109. Adding 13 to both members (Ax. 5, § 4), 2n=122. Dividing by 2, n = 61, the number of years in B's age. And, n — 13 = 48, the number of years in A's age. The written work will stand as follows: Let n = the number of years in B's age. Then, n — 13 = the number of years in A's age. By the conditions, n-13 + n = 109, or 2n-13 = 109. Whence, 2n = 122. Dividing by 2, n = 61 , the number of years in B's age. Therefore, n — 13 = 48, the number of years in A's age. In Ex. 2, we do not say " let n represent B's age/' but " let n represent the number of years in B's age." 3. A, B, and C together earn $66. A's share is one-half as much as B's, and C's is 3 times as much as A's. How much has each? Let X = the number of dollars A has. Then, 2 a: = the number of dollars B has. and 3 a: = the number of dollars C has. By the conditions, x4-2x + 3rc = 66. But the sum of x, twice x, and 3 times x Is 6 times x, or 6 x. Whence, 6x = 66. Dividing by 6, a: = 11, the number of dollars A has. Whence, 2 a: =22, the number of dollars B has, and 3 X =33, the number of dollars C has. (By letting x represent the number of dollars A has, in Ex. 3, we avoid fractions.) 4 ALGEBRA EXERCISE 1 Write the following in algebraic symbols : 1. One number is 4 more than another. What is their sum ? (Hint : Let or = the smaller number.) 2. There are three numbers such that the second is twice the first, and the third thrice the first. What is their sum ? 3. The sum of two numbers is 20 and one of the numbers is X, What is the other number ? 4. If one number is 4 times another, what is their differ- ence? 5. Write: the sum of 5 times a certain number and 3 times the number, divided by 3. 6. The sum of two numbers is a and one of the numbers is 6. What is the other ? 7. The greater of two numbers is 8 times the less, and exceeds it by 49 ; find the numbers. 8. The sum of the ages of A and B is 119 years, and A is 17 years older than B ; find their ages. Q. Divide $74 between A and B so that A may receive 148 more than B. 10. Divide $108 between A and B so that A may receive 5 times as much as B. n. Divide 91 into two parts such that the smaller shall be one-sixth of the greater. 12. A man travels 112 miles by train and steamer; he goes by train 54 miles farther than by steamer. How many miles does he travel in each way ? 13. The sura of three numbers is 69 ; the first is 14 greater than the second, and 28 greater than the third. Find the numbers. 14. The area of a trapezoid is equal to the product of one- half the sum of the parallel sides and the altitude. In the trapezoid ABCD, AD is 8 more than BC, EB is 6, and the DEFINITIONS AND NOTATION area of the trapezoid is 54. Find the length of BC and of AD. Is the drawing cor- rect ? '^ E '^ 15. The sum of two angles a and b is 180°. The angle b is three times as great as the angle a. Find the number of degrees in each angle. 16. Divide $6.75 between A and B so that A may receive one-fourth as much as B. 17. A man has $2, After losing a certain sum, he finds that he has left 20 cents more than 3 times the sum which he lost. How much did he lose ? 18. A, B, and C in partnership gain f 140 ; A is to have 4 times as much as B, and C as much as A and B together. Find the share of each. 19. One side of a rectangle is thrice the side adjacent to it. The opposite sides are equal, and the 1 — sum of the sides is 24 inches ; find the w length and breadth. 20. At an election two candidates, A and B, had together 653 votes, and A was beaten by 395 votes. How many did each receive ? 21. A field is 7 times as long as it is wide, and the dis- tance around it is 240 feet. Find its dimensions. 22. My horse, carriage, and harness are worth together f 325. The horse is worth 6 times as much as the harness, and the carriage is worth $65 more than the horse. How much is each worth ? 23. The sum of three numbers is 87 ; the third number is one-eighth of the first, and the second number 15 less than the first. Find the numbers. 24. The sum of the three angles of a triangle is always 180°. In a triangle ABC, angle B is 30° larger than angle A, and a^ angle C is 30° larger than angle B. Find the number of degrees in each angle. 6 ALGEBRA 25. The sum of the ages of A, B, and C is 110 years ; B*s age exceeds twice C's by 12 years, and A is 9 years younger than B. Find their ages. 26. A pole 77 feet long is painted red, white, and black; the red is one-fifth of the white, and the black 21 feet more than the red. How many feet are there of each color? 27. Divide 70 into three parts such that the third part shall be one-fifth of the first, and one-fourth of the second. 28. In an algebra class of 27 pupils there are twice as many girls as boys. How many girls in the class ? 29. A, B, and C have together $22.50 ; B has $1.50 more than A, and C has $8 less than twice the amount that A has. How much has each ? 30. In a triangle, ABC, angle C is 90°, angle B is twice angle A, The sum of the three angles is 180°. Find the angles A and B, 31. The sum of the three sides of a triangle, ABC, is 35 ^ feet. Side AB h 4: feet more than side BC and side AC is 7 feet more than side ic BC, Find the length of AB and AC. 32. Three straight lines are drawn from a point O forming the angles a, 6, and c. 6 is 30° larger than a ; c is 30° larger than b. The sum of the three angles ^.^a is 360°. Find the number of degrees in each angle. DEFINITIONS 6. The continued product of a number by itself any num- ber of times is called a Power of that number. An Exponent is a number written at the right of, and above another number called the Base, to indicate what power of the latter is to be taken ; thus, DEFINITIONS AND NOTATION 7 o^ read "a square ^'^ ov "a second powevy' denotes aXa; a^ read "a euhe^' or "a ^Airc? power,'''' denotes aXaXa; aS read "a fourth^'' " a fourth power y''' or " a exponent 4," denotes aXaXaXa, etc. The meaning of exponent will be extended in Chap. XIII. If no exponent is expressed, the^rs^ power is understood. Thus, a is the same as a^ 7. Symbols of Aggregation. The parentheses ( ), the brackets [ ], the braces {}, and the vinculum , indicate that the numbers enclosed by them are to be taken collectively ; thus, ia-{-b)Xc, [a+b\Xc, {a+b}Xc, and a-\-bXc all indicate that the result obtained by adding 6 to a is to be multiplied by c. If an expression involves parentheses^ the operations indi- cated within the parentheses must be performed first. EXERCISE 2 Write the following in symbols : 1. The result of subtracting 6 times n from 5 times m, 2. Three times the product of the eighth power of m and the ninth power of n. 3. The quotient of the sum of a and b divided by the sum of c and d. What operations are signified by the following ? 4. 2xY' 8- 3-(i/+z). f^+^y. 5. m(x-y). 9. (m-ny. ' ^^'^^ 12. (2a-h6)(4c-5d). Ttih a c 7. 3+{y-z). \x yj 13- 2// 8 ALGEBRA Write the following in symbols : 14. The product of 3^-\-y and z\ 15. The result of subtracting 7/ — 2 from x. 16. The product oia — h and c — d. 17. The result of adding the quotient of m by ii, and the quotient of x by y. 1 8. The square of m + n. 19. The cube of a — 6 -f- c. 20. Translate into English ; -^ 21. In the above example is a a number ? What value has it? If a were 5 and b were 3, what would be the value of the fraction ? 22. Translate into English — ^• x-y 23. In example 22, if x is 7 and y is 5, find the fraction. ALGEBRAIC EXPRESSIONS 8. An Algebraic Expression, or simply an Expression, is a number expressed in algebraic symbols ; as, 2, a, or 2x^-3ab+5, 9. The Numerical Value of an expression is the result obtained by substituting particular numerical values for the letters involved in it, and - performing the operations indi- cated. 1. Find the numerical value of the expression when a=4, 6=3, c=5, and d=2. Wehave, 4a+^^-ci3=4x4+5^-23 = 16+10-8=18. b 3 2, Find the numerical value, when a = 9, 6 = 7, and c == 4, of a+h (a-b){b+c)- b-c DEFINITIOJSS AND NOTATION First perform the operations indicated in parentheses. We have, a-b =2, b + c =11, a-f-6 =16, and b-c =3. Then the numerical value of the expression is 2X11- 16 3 =22- 16. 3 ' 50 3 ' EXERCISE 3 Find the numerical values of the following when a = 6, 6=3, c=4, d=5, m=3, and n=2; I. a^b—cd^, 2. 2 abed. 4. oT^b'', a' -{-If, 9- I 5. 6. 7. 8. a b 3. 3 a6+4 6c— 5cd. c a be ad 15c"» 1 d a beH a^ ,2^52 11 c d 2 rf2 28 rZ'* Find the numerical values of the following when a=5, 6=2, c=3, and d=4: ^2a±d\\ 15. 5 a2(a-6) -2 6^(0+^). .2 6+cy * (a'^b'^d^y. ^^' 8(a-6)H3(c-fd)^ 17. (a-6)2+(2a-3 6)2-(6+c)^ 18. (2a~6-c+rf)(2a+6+c-d). 8a4-3 6 — 6 c __ a—b , a — c , a — rf 13. 14. 19. -6 , a — c , a- 20. -H \ -• a4-6 a+e a+d 9a-46-3c Find the numerical values of the following when a=f, 6=|, c=^, and x=4: a4-c_a— c ^^ 8a+66 — 15c a—c a+c 23. a:H(2a+3 6)a;2- 16a4-10 6H-9c (5 a — 4 c)x+^ abc. ,2,32.^ 5,4 13 x^ \-—x^-] — H — X — • a b abc abc 10 ALGEBRA II. POSITIVE AND NEGATIVE NUMBERS 10. In financial transactions, we may have assets or lia- bilities^ and gains or losses ; we may have motion along a straight line in a certain direction, or in the opposite direc- tion ; etc. Taken in pairs, these ideas have opposite mean- ings or opposite sense. In each of these cases, the effect of combining with a mag- nitude of a certain kind another of the opposite kind is to diminish the former, destroy it, or reverse its state. Thus, if to a certain amount of asset we add a certain amount of liability, the asset is diminished, destroyed, or changed into liability. 1 1 . The signs + and — , besides denoting addition and subtraction, are also used, in Algebra, to distinguish between the opposite states of magnitudes like those of § 10. Thus, we may indicate assets by the sign +, and liabilities by the sign — ; for example, the statement that a man's assets are —$100 means that he has liabilities to the amount of $100. EXEBCISE 4 1. If a man has assets of $400, and liabilities of fOOO, how much is he worth ? 2. If gains be taken as positive, and losses as negative, what does a gain of —$100 mean ? 3. In what position is a man who is — 50 feet east of a certain point P? See figure. r —^o p. - t ^ ° West East^ 4. In what position is a man who is — 3 miles north of a certain place ? Draw the figure showing this. 5. How many miles north of a certain place is a man who goes 5 miles north, and then 9 miles south ? Draw figure. 12. Positive and Negative Numbers. If the positive and negative states of any concrete magni- tude be expressed without reference to the unit^ the results are called positive and negative numhers^ respectively. POSITIVE AND NEGATIVE NUMBERS 11 Thus, in 4- $5 and — $3, + 5 is a positive number, and — 3 is a negative number. For this reason the sign + is called the positive sign, and the sign — the negative sign. If no sign is expressed, the number is understood to be positive ; thus, 5 is the same as + 5. The negative sign must never be omitted before a nega- tive number. 13. The Absolute Value of a number is the number taken independently of the sign affecting it. Thus, the absolute value of — 3 is 3. ^ ADDITION OF POSITIVE AND NEGATIVE NUMBERS 14. We shall give to addition in Algebra its arithmetical meaning, so long as the numbers to be added are positive in- tegers or positive fractions. We may then attach any meaning we please to addition involving other forms of numbers, provided the new meanings are not inconsistent with principles previously established. 15. In adding a positive number and a negative, or two negative numbers, our methods must be in accordance with the principles of § 10. If a man has assets of $5, and then incurs liabilities of $3, he will be worth S2. If he has assets of $3, and then incurs liabilities of $5, he will be in debt to the amount of $2. If he has liabilities of $5, and then incurs liabilities of $3, he will be in debt to the amount of $8. Now with the notation of § 11, incurring liabilities of $3 may be regarded as adding — $3 to his property. Whence, the sum of +$5 and -$3 is +$2; the sum of -$5 and +$3 is -$2; and the sum of — $5 and ~$3 is — $8. Or, omitting reference to the unit, 12 ALGEBRA ( + 5)+(-3)= + 2; (-5) + (+3) = -2; (-5) + (~3)=-8. To indicate the addition of +5 and —3, they must be enclosed in parentheses (§7). We then have the following rules : To add a positive and a nega,tive number, subtract the less absolute value (§13) from the greater, and prefix to the result the sign of the number having the greater absolute value. To add two negative numbers, add their absolute values, and prefix a negative sign to the result. 16. Examples. 1. Find the sum of + 10 and —3. Subtracting 3 from 10, the result is 7. Whence, ( + 10) + (-3)= +7. 2. Find the sum of — 12 and +6. Subtracting 6 from 12, the result is 6. Whence, (-12) + ( + 6)= -6. 3. Add -9 and -5. The sum of 9 and 5 is 14. Whence, (-9) + (-5)= -14. EXERCISE 6 Find the values of the following : 2. (+8) +(-3). V 9/ V ey 3. (-9) + (+5). ,7+?W-?Y 4. (+4)+(-ll). \ 8j \ 7) 5. (-13)+(-18). 10. (-15j) + (+12^). 6. (-42) + (+57). II. (+171-) + (-10t^). 7. (-34) + (+82). 12. (-14|)+(-21^;f). POSITIVE AND NEGATIVE NUMBERS 13 MULTIPLICATION OF POSITIVE AND NEGATIVE NUMBERS 17. If one algebraic expression is multiplied by another, the first is called the Multiplicand, and the second the Mul- tiplier. 18. We shall retain for multiplication, in Algebra, its arithmetical meaning, so long as the multiplier is a positive integer or a positive fraction. That is, to multiply a num- ber by a positive integer is to add the multiplicand as many times as there are units in the multiplier. For example, to multiply —4 by 3, we add —4 three times. Thus, (-4)X( + 3) = (-4) + (-4)-f(-4) = -12. 19. In Arithmetic, the product of two numbers is the same in whichever order they are multiplied, that is, which- ever is taken as the multiplier. Thus, 3X4 and 4X3 are each equal to 12. If we could assume this law to hold for the product of a positive number by a negative, we should have (+3)X(-4)= (-4)X(+3) = -12 (§ 18)= -(3X4). Then, if the above law is to hold, we must give the follow- ing meaning to multiplication by a negative number : To multiply a number by a negative number is to mul- tiply it by the absolute value (§13) of the multiplier, and change the sign of the result. Thus, to multiply +4 by —3, we multiply +4 by +3, giving +12, and change the sign of the result. Thatis, (+4)X(-3) = -12. Again, to multiply —4 by —3, we multiply —4 by +3, giving —12 (§ 18), and change the sign of the result. Thatis. (-4)X(-3) = +12. 20. From §§18 and 19 we derive the following rule : To multiply one mmaber by another, multiply together their absolute values. Make the product plus when the multiplicand and multiplier are of like sign, and minus when they are of unlike sign. 14 ALGEBRA 21. Examples. !• Multiply +8 by -5. By the rule, ( + 8)X(-5)= -(8X5) = -40. 2. Multiply -7 by -9. By the rule, (-7) X(-9)= + (7X9) =+63. 3. Find the numerical value when d=4 and fe= —7, of (a+by. We have, (a + 6)3=(4-7)(4-7)(4-7) = (~3)(-3)(-3)='27. BXEBCISE 6 Find the values of the following : 1. (+5)x(-4). 6. (-24)x(-5). 2. (-ll)x( + 3). 7. (-14)x(+15). 3. (-8)x(-7). 8. (+27)X(-19). 4. (4-9)x(-6). 9. r-ivf-!> 5. (-12)x(+9). \ SJ \ 5J III. ADDITION AND SUBTRACTION OF ALGEBRAIC EXPRESSIONS. PARENTHESES 22. A Monomial, or Term, is an expression (§ 8) whose parts are not separated by the sign + or — ; as 2 a;^, — 3 aft, or 6. 2x^y ~ 3 aft, and + 5 are called the terms of the expression 2x2~3aft + 5. A Positive Term is one preceded by a + sign ; as + 5 a. If no sign is expressed, the term is understood to be positive. A Negative Term is one preceded by a — sign ; as — 3 aft. The — sign must never be omitted before a negative term. 23. If two or more numbers are multiplied together, each of them, or the product of any number of them, is called a Factor of the product. Thus, a, ft, c, aft, ac, and be are factors of the product aba. ADDITION AND SUBTRACTION 15 J4. Any factor of a product is called the Coefficient of the product of the remaining factors. Thus, in 2 a6, 2 is the coefficient of a6, 2 a of 6, a of 2 6, etc. 25. If one factor of a product is expressed in Arabic numerals^ and the other in letters^ the former is called the numerical coefficient of the latter. Thus, in 2a6, 2 is the numerical coefficient of ab. If no numerical coefficient is expressed, the coefficient 1 is understood ; thus, a is the same as 1 a. 26. By § 20, (-3)Xa= - (3Xa)= -3a. That is, —3a is the product of — 3 and a. Then, — 3 is the numerical coefficient of a in — 3 a. Thus, in a negative term as in a positive^ the numerical coefficient includes the sign. 27. Similar or Like Terms are those which either do not differ at all, or differ only in their numerical coefficients ; as 2 x^y and — 7 x^y. Dissimilar or Unlike Terms are those which are not sim- ilar ; as 3 x^y and 3 xy^. ADDITION OF MONOMIALS 28. The result of addition is called the Sum. 29. The adding of 6 to a is expressed a + 6. The sum is (a + &). But where no ambiguity is to be feared, parentheses may be omitted. 30. The addition of monomials is effected by uniting them with their respective signs. Thus, the sum of a, —b, c, —d, and — e is ^■b a — b + c — d — e. 3 1 . We assume that the terms can be united in any order ^ provided each has its proper sign. Hence, the result of § 30 can also be expressed c+a — e — d — 6, — d — 6-f c — e-fa, etc. 16 ALGEBRA 32. To multiply 5 + 3 by 4, we multiply 5 by 4, and then 3 by 4, and add the second result to the first. Thus, (5 + 3)4=5X4 + 3X4. We then assume that to multiply a + 6 by c, we multiply a by c, and b by c, and add the second result to the first. Thus, (a + b)c=ac + bc. 33. Addition of Similar Terms (§ 27). 1. Required the sum of 5 a and 3 a. We have, 5 a + 3 a= (5 + 3)a (§ 32) = 8 a. That is, we do not add the a's but the coefficients of the a's. 2. Required the sum of — 5 a and — 3 a. We have, (-5a) + (-3 a)=(-5)Xa+(-3)Xa (§ 26) =[(-5) + (-3)]Xa (§32) = (-8)Xa (§15) = -8a. (§26) 3. Required the sum of 5 a and —3 a. We.have, 5 a+(-3)a =[5 + (-3)]Xa (§32) = 2 a. (§15) 4. Required the sum of — 5 a and 3 a. We have, (-5)a + 3a=[(-5) + 3]Xa (§32) = (-2)Xa(§ 15)= -2a. Therefore, to add two similar terms, find the sum of their numerical coefficients (§§ 15, 25, 26), and aflBlx to the result the common letters. 5. Find the sum of 2 a, —a, 3 a, -- 12 a, and 6 a. Since the additions may be performed in any order, we may add the positive terms first, and then the negative terms, and finally combine these two results. The sum of 2 a, 3 a, and 6 a is 11a. The sum of —a and —12 a is —13 a. Hence, the required sum is 11 a+ ( — 13 a), or —2 a. 6. Add 3(a-6), -2(a-6), 6(a-6), and ~4(a-fe). I ADDITION AND SUBTRACTION 17 "The sum of 3(a-6) and 6(a-6) is 9(a-6). The sum of -2{a-b) and -4(a-6) is -6(a-6). Then, the result is [9 + (-6)](a-6), or 3(a-6), If the terms are not all similar, we may combine the simi- lar terms, and unite the others with their respective signs (§ 30). 7. Required the sum of 12 a, —5 a?, —3 y^, —5 a, S x, and -3x, The sum of 12 a and —5 a is 7 a. The sum of —5x,8x, and —3 x is 0. Then, the required sum is 7 a — Sy^. ^ EXERCISE 7 Add the following : 1. 8 m and 4 m. 8. 31 c^d^ and -31 c^d^ 2. 12 a and —5 a. g* — 6(c+rf) and — 4(c4-6?). 3. 12 a and —16 a. 10. —5(x^+y^) and — 9(a?^+i/^). 4. — 12 a and —5 a. 11. 7 a:, 4 a:, and —Sx. 5. — 8 y^ and — 20 2/^. 12. 16 a, —5 a, —3 a, and a. 6. —15 cd and 13 cd, 13. 2 a, —5 a, and —11 a. 7. 24 a^6 and —23 a^b. 14. 3 a;?/2;, 6 ari/z, and —9xyz, 15. 8(a;+i/), — 14(a;+2/), and 3 (a? +2/). 16. 8n^ — ?^^ 14 n^ — 4?i2, and 7n^ 17. 3a^62^ -5a^62, a'b\ -'9a'b', and 20 a^ft^. 18. 3 ax, —4bx, 5 ax, and —2 6a;. 19. 3(a+6), 4(a-6), -2(a+6), and 6(a-6). 20. 4 /?, 3 i, —5 a, 2 A:, — /?., and 2 a. ADDITION OP POLYNOMIALS 34. A Polynomial is an algebraic expression consisting of more than one term ; as a + 6, or 2 a;^ — a;?/— 3 y^. A polynomial is also called a multinomiaL A Binomial is a polynomial of two terms ; as a + 6. A Trinomial is a polynomial of three terms ; as a + 6 — ^. 18 ALGEBRA 35. A polynomial is said to be arranged according to the descending powers of any letter, when the term containing the highest power of that letter is placed first, that having the next lower immediately after, and so on. Thus, a?^-h3 x^y-2 x^-^-^ xy^-4 y* is arranged according to the descending powers of x. The term —4 y*, which does not involve x at all, is regarded as con- taining the lowest power of x in the above expression. A polynomial is said to be arranged according to the ascending powers of any letter, when the term containing the lowest power of that letter is placed first, that having the next higher immediately after, and so on. Thus, x^+3 x^y-2 xY+S xy^-4y* is arranged according to the ascending powers of y, 36. Addition of Polynomials. Let it be required to add 6 + c to a. Since 6 + c is the sum of b and c (§ 29), we may add 6+c to a by adding b and c separately to a. Then, a+(b+c)=a+b+c, (To indicate the addition oib + c, we write it in parenthesis.) The above assumes that, to add the sum of a set of terms, we add the terms separately. 37. From § 36 we have the following rule : To add a polynomial to a quantity, add its terms with their signs unchanged. I. Add 6a-7a;2, Sx^-2a+Sy\ and 2x^-a-mn. We set the expressions down one underneath the otlier, similar terms being in the same vertical column. We then find the sum of the terms in each column, and write the results with their respective signs ; thus, Qa-7x^ -2a-\-Sx^ + 3y^ — a + 2 x^ —mn Sa-2x^-\-3y^-mn ADDITION AND SUBTRACTION 19 2. Add 4a:-3a;2-ll+5ir^ 12 a;^- 7-8 x^- 15 a;, and 14+6a;3+10ar-9a;2. It is convenient to arrange each expression in descending powers of x (§35); thus, 5^_ 3^,^ 4^_jj -8 0:3 + 12x2 -15 a;- 7 Qa^- 9x2 + 10x4-14 3x3 _ a;- 4 3. Add 9(a+6)-8(6+c), -3(6+c)-7(c+a), and 4(c+a)-5(a+6). 9(a + 6)- 8(6 + c) - 3(6 + c)-7(c+a) -5(a + 6) +4(c+a) 4(a + 6)-ll(6 + c)-3(c+a) Add fa-f |6- Jc and ^a-f 6+f c. ia+ §6- ic ja- ib+ ^c «a-i|6 + Ac EXSHCISE 8 Add the following : (Results may be checked as in Chap. XVII.) I. 2. 3. 2a-5 6 - 4 0^2+3 2/2 -Sxy+2st -7a+6 6 x^-iy"" -{-2 xy-7 st 9 a- b -Ux^'+Sy^ -Sxy-\-5st 4. 7 d-4 r-e n and 3 d+9 r+2 n. 5. 5a^-iab+b\ 4a^+4ab+5b\ and -QaHefc^. 6. 2 m--3 x+/, m+x—f, and m-f/. 7. 3 6i/+2 pk-qt, 5 6i*-7 pA:+2 ^^ 8. &-2+3 62-8 b\ 6+6- 6H7 b\ and 6 + 2 6^-4 6^ 9. 3(a+6)-7(6+c), 5(a+6)+5(6+c), and -2(a+6)-3(6+c). 10. J a-^ 6+f c and - 1 a+| 6-^ c. 11. 4<+3w-5c, -2^-a+3c, 2a-9c+2i^, and 5<+3a-4i^. "• ife ^+1^+1^2 and -^x-^^Z-i 2. 20 ALGEBRA 13. Add these equations (Ax. 5, § 4) , then find the value of a:: .x-2/ = 7. After X is known can you find y ? 14. Find y by adding these equations : hx\2y^ 16, -5x + 32/=~l. What value has xl 15. Find X and y in these equations: r2a:+3 2/=ll, 1 x-^y=\. Add the following : 16. U{x-\-y)-\l{y-\-z), 4(2/+z)-9(2+a:), and -3(a;+2/>-7(2+a;). 17. 6 c+2 a- 3 6, 4 d- 7 c+12 a, 8 6-5 cZ+c, and -10a-116 + 9rf. 18. ^7(a-6)2+8(a-6) + 2, 4(a-6)2-5(a-6), and 3(a-6)2-9. EXERCISE 9 1st No. 2ndNc ). Sum. 1st No. 2nd No . Sum. I. -f 8 + ? = 5 d. X + 9 = ^ + 2/ 2. -10 -f ? = -7 7- a 4- ? = a — 6 3- -f 10 + ? = -7 8. a -f ? = 6 4. + 6 4- = 11 9. -6 + ? = a 5. - 3 + ? = —9 10. c + ? = b 38. In the above examples we have given the sum of two numbers and one of the numbers to find the other number. SUBTRACTION OP MONOMIALS 39. Subtraction is the process of finding one of two num- bers when their sum and one of them is given. The Minuend is the sum of the numbers. The Subtrahend is the given number. The Difference is the required number. ADDITION AND SUBTRACTION 21 40. Therefore, to subtract one number from another is to find a number, which added to the subtrahend will produce the minuend. For example, to subtract 3 from 10, we find the number, which, added to 3, will produce 10. By remembering the result in addition such num- ber is seen to be 7. Thus 7 is our difference. To subtract —4 from 9, find the number which, added to —4, will produce 9. By inspection this number is evidently 13. Subtract —6 from —8. — 6 plus —2 gives —8, hence our difference is —2. Subtract +3 from —9. 3 + (-12)=-9, hence — 12 is our difference. EXERCISE 10 Subtract the following : 1. 7 from 2. 4. -3 from 8. 7. 6 from 13. 2. 3 from -8. 5. -6 from - 11. 8. -9 from 3. 3. -11 from -10. 6. 36 from 12.. 9. 10. II. 12. 9a; 4a -4a 13 < 3 X —5 a —7 a -- t 41. Notice that in each of the above examples the result is the same as if we had changed the sign of the subtrahend and proceeded as if adding the subtrahend to the minuend. 42. Similarly, from § 4X this rule follows : To subtract one number from another, change the sign of the subtrahend and proceed as in addition. (The sign of the subtrahend must be changed mentally,^ EXEKCISE 11 Subtract the following : (The accuracy of all results may be checked by adding the difference to the subtrahend.) 1. 5 ax from ax. 3. 14 aW from 11 a'^lP. 2. 3 afec from — 9a6c. 4. 15(a— 6) from 19(a— 6). 22 ALGEBRA 5. i my from ^ my. 8. From 8 a take 3 b. 6. -- 11 c^s from —Qc^s. 9. From 7 a; take — 2 y. 7. —21 cy from 13 cy. 10. From —3a take 4 6^. SUBTRACTION OP POLYNOMLAXS 43. Since a polynomial may be regarded as the sum of its separate terms (§ 30), we have the following rule: To subtract a polynomial, change the sign of each of its terms, and add the result to the minuend. 1 . Subtract 7 afe^ - 9 a% + Sb^ from 5 a^ - 2 a'6 + 4 ab\ It is convenient to place the subtrahend under the minuend, so that similar terms shall be in the same vertical column. We then mentally change the sign of each term of the subtrahend, and add the result to the minuend; thus, . 5 0^-2 0^6 + 4 06^ -9a^6 + 7a6'-f8 6^ 5a^ + 7a^b-Sah^-8b^ 2. Subtract the sum of 9 x^ — Sx + x^ and 5 — x^ + a; from Gx^-7x-4:, We change the sign of each expression which is to be subtracted, and add the results. 6x3 -.7aj-4 - a^-dx^ + Sz + x^— re— 5 5a;3_8a;2 -9 EXERCISE 12 Subtract the following: I. 2. 3. a;^+13a:— 11 — 2m^— 4mnH- 9n^ ab-^bc-hca — Sx^+ 6 a:— 5 8 m^— 7 mnH-14 n^ ab—bc^-ca 4. From 9 a4-4 h—b k take 9 a— 4 A+5 k, 5. From 6 x^-b a:H4 a;- 3 take x^-Z x^-2 a:-f 1. 6. From 11 a- 9 6+2 z subtract -3 2-f 2 a- 14 6. ADDITION AND SUBTRACTION 23 7. Take -S(h+k)+S{h-k) from (h+k)-^h-k), 8. Subtract 74 z2_47 2;A;+30 fc^ from 24 i^-SO zfc+lO z^. 9. From 7^-8 8-^-75 tHvike -16 v+19s. 10. What must be added to 4 g+lS z^—x to give ? 11. By how much does 8 x^—7x^+b x—1 exceed x^+Ux''-3x+7? 12. From a:^— 11 x+4: subtract 8 x^—3 x— 1. 13. From aH2 ab+b^ take a^-2ab+b\ 14. Find the sum of a^+2 ab-{-¥ and a^-2ab+b^, 15. From the sum of a?^+4 icH4 x and 2 ir2+8 o^+S take 6a;H12a;. 16. From the sum of a^—2a^b+ab^ and —0^6+206^—6' take the sum of a^+2 a^b+ab^ and a^b+2 a¥-\-b^. 17. From I 5— I a+f 6 take | s+i a— ^ 6. 18. From f gt^-{-v-^t take gr^^^l v+Q t 19. Take a^-6 a^- 15 0^-8 a+4 f rom 7 aH 3 a^- 5 a^- 1 1 a- 9. 20. From h m— J n+f p take f m— f n-f J p, 21. From n*— lOn^a;— nV+8na?^+3 X* take 5 n*+4 n^a;— 9 n^a;H2 na;^— 12 x*. 22. Take 18ar^-8a;+6a;H12-8a;3 I from -10a;3^2-15a;2+lla;^-4a?. 23. Take a'- 10 0^6^+13 0^6^-7 ab'-5 ¥ from 9 aH3 a*6+6 0^62-0^6^- 16 6^ 24. From the sum of 2 x^—6 x]/—y^ and 7a:2— 3 a:y+9 y^ subtract 4 a?^— 6 a;?/ +8 y^. 25. From subtract the sura of 4 a^ and 3 a— 5 a^— 1. Add the following pairs of equations to find ar, subtract them (Ax. 6, § 4) to find y. Verify results by substituting the values of x and y in the given equations : ,6. |^+2'=5' 27. (2^+5 2/= 16, ^g |5y+x=9. I ra:+2/=5, ^^ (2x + 52/=--4. (5^—07 = 1 24 ALGEBRA PARENTHESES 44. Removal of Parentheses. By § 30, a + {b-c)=a + b-c, HcDce, Parentheses preceded by a + sign may be removed without changing the signs of the terms enclosed. Again, by § 43, a- {b-c)=^a-b + c. Hence, Parentheses preceded by a — sign may be removed if the sign of each term enclosed be changed. The above rules apply equally to the removal of the brackets^ braces, or vinculum (§ 7). It should be noticed in the case of the latter that the sign apparently prefixed to the first term underneath is in reality prefixed to the vin- culum; thus, +a — b means the same as + (a — 6), and —a — b the same as —(a — 6). 45. I. Remove the parentheses from 2 a-3 6- (5 a-4 6) + (4 a-b). By the rules of § 44, the expression becomes 2a-3 6-5a + 4 6 + 4a-6=a. Parentheses sometimes enclose others ; in this case they may be removed in succession by the rules of § 44. Beginners should remove one at a time, commencing with the innermost pair ; after a little practice, they should be able to remove several signs of aggregation at one operation, in which case they should commence with the outermost pair. 2. Simplity 4x—{3x+{-2x-x-a)}, We remove the vinculum first, then the parentheses, and finally the braces. Thus, 4,x-\3x+(-2x-x-a)\ ==4:X-{Sx+i-2x-x+a)\ =^x—{3x—2x—x+a\ = 4 X — 3 x+2 x+x — a=4: x — a. EXEBCISE 13 1. What is the sign of 2 a; in 3 x^-4:c- (2 x-|- 1)? 2. What is the sign of a in 4 a' — a — 4 c^ -f 9 ar^? What is the coefficient of a after the vinculum is removed? ADDITION AND SUBTRACTION 25 Simplify by removing the signs of aggregation and then uniting similar terms : 3. 11 a-(-6m-f5c)~(3a+4c). 4. iX'-Sy-[7y-d] + {'-4iX-3y}. 5. x'+[-3 x'-{2 y^-2 x'')+2 y^], 6. 7t+u-{Qt-u+7-S}. 7. (a^+2 ab + b'')-(a^'-2 ab+b'). Compare Ex. 13, Exercise 12. 8. x^- (-3 x^'y-S xy^) -^-y^- (x^-[3 x^'y-'S xy"" +y^]). Compare Ex. 16, Exercise 12. 9. 7 x-{-Sy-10x-ny}. 10. a^- ( - 6 a^- 12 a + 8) - (a^+ 12 a). Compare Ex. 16, Exercise 12. 46. Insertion of Parentheses. — To write terms in paren- thesis, we take the converse of the rules of § 44. Any number of terms may be written in parenthesis preceded by a -|- sign, without changing their signs. Any number of terms mdty be written in parenthesis preceded by a — sign, if the sign of each term be changed. JEx. Write the last three terms of a — b-\-c — d + e in pa- renthesis preceded by a — sign. Result, a — b — { — c+d — e), EXERCISE 14 In each of the following expressions, write the last three terms in parenthesis preceded by a — sign : 1. a-\-b+e~d. 5. Sx^—y^—y-hz, 2. m^+3m-2-j-h. 6. a^^b^+c^-d^ 3. x^—3x^+3x'-l. 7. x^—2xy—y^—2yz—z^. 4. 4a^-3a3-2a2-a. g. 2 a^-lO a^-8 aH5 a^-G a-}-9. 9. In each of the above results, write the last two terms in parenthesis in brackets preceded by a — sign. 26 ALGEBRA 47. Addition and Subtraction of Terms having Literal Coefficients. — To add two or more terms involving the same power of a certain letter, with literal, or numerical and literal, coefficients, it is convenient to put the coefficient of this letter in parenthesis. 1. Add ax and 2x, By § 32, ax + 2x = {a + 2)x. 2. Add {2m-{'n)y and (m— 3n)2/. (2m + n)2/+(m-3 n)y — [{2 m4-n) + (m-3 n)]y = (2 m + n+m — S n)y = {S m — 2 n)y. (The pupil should endeavor to put down the result in one operation.) 3. Subtract {h—c)x^ from ax'^. By §§ 32, 42, 44, ax" - (6 - c)x^ = [a - (6 - c)\a? = (a — 6 + c)x^. EXERCISE 15 Add the following: 1. ax and bx, 5. ah, he and —m^h. 2. 3 a6^ and —4 6^. 6. cV and {a—3d)x^. 3. — a^6 and 5a^2/- 7* (7 a+4 fe)a;^ and (3 m-f n)a;'. 4. 3 a^hc and —8 erf. 8. (4 x—y)z* and (3 a;-fc)2*. Subtract the following : 9. 3 ex from dx. ii. — ca;?/ from —dxy. 10. —4 mz/ from 3cy. 12. (c-hrf)a: from ax. 13. (3 c-4 d)x^ from (6 c+9 d:)x\ QUEBIES 1 . What is the difference between 4 x and 3 2/ ? 2. Express the sum of three times a certain number and four times the same number. 3. Do you make any distinction between a factor and a coefficient ? 4. Regarding 3 mxy and 5 cdx as the coefficients of the expressions 3 amxy and 5 acdx, find the sum. Regard ax as the common part and find the sum. 5. Does the sum of a and 6, (a + 6), have the same significance to you as the sum of 5 and 2, (5 + 2)? MULTIPLICATION 27 6. Express algebraically : the sum of 6 times the sum of a and 6, and 9 times the sum of the same two numbers. 7. What must you add to a polynomial to produce 0? Give an example. 8. Did you work problem 7 by addition or subtraction? Is there any difference between the two processes as here used? 9. Have you noticed which forms of the signs of aggregation are used most? Does the vinculum appear often? 10. Subtract 3 m{a + b) from 6 m{a — h). 11 If your difference is x^ — ix + l and your subtrahend is 3 a;^ + 4a; — 9, what is your minuend? IV. MULTIPLICATION OF ALGEBRAIC EXPRESSIONS 48. The Rule of Signs. If a and b are any two positive numbers, we have by § 20, (+a)x(+6) = +a6, (+a)x(--6) = ~a6, (~a)x(-f6) = --a6, (--a)x(~6)= +afe. From these results we may state what is called the Rule of Signs in multiplication, as follows : The product of two terms of like sign is positive ; the product of two terms of unlike sign is negative. 49. We have by § 48, (-a) X (-6) X (-c) = (ab) X (-c) ^-abc; (1) (^a)x(~6)x(-c)x(-d) = (~a6c)x(-d), by (1), =abcd; etc. That is, the product o£ three negative terms is negative ; the product of four negative terms is positive ; and so on. In general, the product of any number of terms is posi- tive or negative according as the number of negative terms is even or odd. 50. The Law of Exponents. Let it be required to multiply a^ by a\ 28 ALGEBRA By §6, a^=aXaXa, and a^-=aXa. Whence, a^Xa^=aXaXaXaXa The general case. — Let it be required to multiply a"^ by a'*, where m and n are any positive integers. We have a^=aXaX'*' to m factors, and a^==aXaX''' to n factors. Then, a'^Xa^'^aXaX"- to m+n factors = a''*"*'''. (The Sign of Continuatiortf •••, is read " and so on") Hence, the exponent of a letter in the product is equal to its exponent in the multiplicand plus its exponent in the multiplier. This is called the Law of Exponents for Multiplication. A similar result holds for the product of three or more powers of the same letter. Thus, a^Xa^Xa''==a^''^-^^=^a^\ MUIiTIPIilCATIGN OF MONOMIALS 51 . I. Let it be required to multiply 7 a by —2 6. By §26, -2 6 = (-2)X6. Then, 7oX(-2 6)=7aX(-2)X6 = 7X(-2)XaX6=-14afe. (§48) In the above solution, we assume that the factors of a product can be written in any order. 2. Required the product of —2 a^b^, 6 a6^ and —7 a*c. (-2 a^b^)X6 ab'X{-7 a*c) =-{-2)a%^XQ.ab^X(-7)a*c ^i-2)X6X{-7)Xa^XaXa*Xb^Xb^Xc = 84 a'b^ by §§ 49 and 50. We then have the following rule for the product of any number of monomials : To the product of the numerical coeflacients (§§ 25, 26, 49, 50) annex the letters; giving to each an ex- ponent equal to the sum of its exponents in the factors. MULTIPLICATION 29 3. Multiply -5 a^ft by -8 ab\ ( - 5 a'b) X ( - 8 ab^) = 40 a^+'b'+^ =40 a*b*, 4. Find the product of 4n^ —3 n^, and 2 n^ 4 n2x(-3 n^) X2 n*= -24 ri.2+5+4 ^ _24 ^n. 5. Multiply -x'^ by 7 x^. (-x"»)X7x»=-7a;'«+\ 6. Multiply 6(m+ii)* by 7{m+ny, 6 (w + n)*X7 (m + n)3=42 {m+ny. EXERCISE 16 Multiply the following : 1. 4 a^ by 7 a^. 5. — 12 x^y by 9 xy^. 2. 6 m^d by 9 md^. 6. 4 x'^yh by —11 x^y'^z^, 3. 11 ax by -8 aft. 7. 3 {x^-yf by 16 (x+2/)^ 4. -lex by -10 r^. 8. -4 (a-6) by 6 {a-hf. Find the product of : 9. {x—yY, Mx—yY, and — 7(a:— y). 10. 3m^/is^, — 5 m^i^ and — Gmns*. 11. aV, — 2a'cS — 5ac^ and Q a^bc. 12. a^fty, 4 a^fe, and -11 a^^ft'^^. MULTIPUCATION OF POLYNOMIALS BY MONOMIALS 52. In § 32, we assumed that the product of a + ft by c was ac + 6c. We then have the following rule for the product of a polynomial by a monomial : Multiply each term of the multiplicand by the multi- plier, and add the partial products. Ex. Multiply 2 x^-5 x+7 by -8 x\ = (2a:2)x(-8r») + (-5x)X(-8r') + (7)X(-8x»)- 2x^- 5x4-7 The written work should stand as follows: — 8 x^ -16xH40x*-56x3. 30 ALGEBRA EXERCISE 17 Multiply the following : 1. 12a-3 by 5 a. 2. Sa'b+7ab^hy -9aW. 3. x^—S x^y^+S xy^ by —x^y^, 4. 4 m*— 3 71^-4 by 7 m^ 5. 5z2 by 82^-822 + 13. 6. ^9cd^ by 5 c^- 10 c''d+2 d\ 7. 8a^-4a«+9a^ by ~8a^ 8. 8a;'*-3x2'»by 15 ^c^n 9. ~15a2&+7 62--4a^by -8a^62 10. 4 a;^!/^ by x^+6 x^i/— 6 a^y^+S y^ 11. 12a26-6a62~8 6Ha3 by -8a6. 12. 6 a3-8 a'c+l ac^- 11 c^ by 14 aV. 13. a3-9 a26+27 a62-27 6^ by -3 h, 14. ia^-Jaft + ifc^b^ _j5 ^5- 1% c^- 1 h^c-^ cH by 5 cd. MULTIPLICATION OP POLYNOMIALS BY POLYNOMIALS 53. Let it be required to multiply a-^-hhy c-\-d. As in § 32, we multiply a-\-h by c, and then a + & by r/, and add the second result to the first ; that is, (a+6)(c+cZ) = (a+6)c+(a+6)cZ =^ac-\-hc-\-ad-{-hd. Rule : — Multiply each term of the multiplicand by each term of the multiplier, and add the partial products. 54. I. Multiply 3 a~4 6 by 2 a- 5 ft. In accordance with the rule, we multiply 3 a — 4 6 by 2 a, and then by — 5 6, and add the partial products. A convenient arrangement of the work is shown below, similar terms of the partial products being in the same vertical column. MULTIPLICATION 31 3a -46 2a -56 4 6 o^ — 8 a6 -15a64-20 6 ^ 6a2-23a6 + 20 62 The work may be checked by solving the example with the multipli- cand and multiplier interchanged. 2. Multiply 4 ax^-ha^—S x^—2 a^x by 2x+a. It is convenient to arrange the multiplicand and multiplier in the same order of powers of some common letter (§ 35), and write the par- tial products in the same order. Arranging the expressions according to the descending powers of a, we ^^^® a^-2 a'x+^ax'-S a^ a +2x a*-2a3a;+4aV-8aar» 2a^x-4a^x^-hSax^-lQx* a* -16 X* EXERCISE 18 Multiply the following : 1. 3a;-4 by 8a:+5. 5. a^-a-f-l by a + 1. 2. 4a+6by 4a+6. 6. F- A:- 12 by i- 7. 3. 7t-4u by 11 tSu. 7. 2a;+3 by 2xS. 4. 6 a6+62 by 3 ab-5 h\ 8. 3 a:+7 by 3 x-T. 9. \c-\-\d by lc-\d, 10. \c-\-\d by ^c-\-\d. 11. Jc~id by \c-\d. (Note carefully in what way examples 9 to 11, and 12 to 14 differ.) 12. ic+3 by a;+5. 13. a:~3 by a;+5. 14- ^+3 by x— 5. 15. a^+a + l by a^-a + l. 16. 6(m-f^)^-5(m+n)-f 1 by 7(m+n)-2. 17. 3(a~6)*-2(a-6) by 4(a-6)H(a-6). 18. 7<2+8^-l by 7<2_8f+l. 19. 6a6H-aH9by 3a2~4 + 2a6. 32 ALGEBRA 20. 2 h^- 10 h+b by fe^-f 5 /i- 10. 21. <^— f^w+^w^—i^^ by <+w. 22. h^-3hk+9 ¥ by A4-3 k, 23. x^+a?2/"^2/^ '^y ^""^z- Note the similarity in examples 21-23. 24. 3 a^+Tfoy by 3 a^~7 6^. 25. 5(a:+2/)+7(a:-2/) by ^(x^y)-2{x^y). 26. 6^+^by e^-^y. 27. a^-\-2 a^b-\-2 ab^+b^ by a2-2 ab-\-b\ 2S, ^ar-^^b^-Sa'b'hy ar+''b'-2abo'-K 29. 5 x^-6 x^-4: a?2+2 a;~3 by 3 X'-2, 30. a3-3 a2x+3 ax^-ar^ by a^+3 a^x+3 ax^-\-x\ 55. If the product has more than one term involving the same power of a certain letter, with literal, or numerical and literal, coefficients, we may put the coefficient of this letter in parentheses, as in § 47. Ex. Multiply a:^—aa: — bx+ab hy x— a. 7? —ax —hx -{-ah X —a a^—ax^ —hx^ +abx —ax^ +a^x-habx — a^b a^-{2a+b)x^+ia'-{-2ab)x-a'b As in § 47, —2ax'^-bx^ is equivalent to — (2 a + 6)x^ and a^x-\-2ahx to (a^ + 2 ab)x. EXEBCISE 19 Multiply the following : 1. x^+ax+bx-]-ab by x-^c. 2. x^—mx+nx—mn by x—p. 3. x^—bx — cx + bc by x—a. 4. x^+ax—bx^Sab by x+b. 5. x^-\-ax + 2bx-{'2abhy x — c. 6. x^-^px—5qx'-5pq by x—r. .7. x^—S ax—bx-\-3 ab by x+2 a. 8. x^—4imx-]'nx—4:mnhyx-\-3 7h MULTIPLICATION 33 9. x^+S ax— 2 bx + Q ah by a;— 4 c. 10. (a— 6)a;— 3a6 by 2x—{a—h), 11. a;2'*-5aa:^4-4 6a:'*-2a6by af^+c. 12. (2a-l)a:2 + (a + 2)a;~(a + 3) by (a~2)a:-a. 56. Ex. Simplify (a-2a;)2-2(3aH-a:)(a-a:). To simplify the expression, we first multiply a—2xhy itself (§ 6) ; we then find the product of 2, 3 a + o:, and a — x, and subtract the second result from the first. a -2x 3a +x a —2x a —X a^-2ax 3a^+ ax -2ax+4x^ -Zax- x^ a^-4:ax+4:x^ 3a^-2ax- x^ 2 6a^-^ax-2x^ Subtracting the second result from the first, we have a^-^ax + 4:X^-6a^ + 4:ax+2x^=:-5a^-{'QxK EXEBCISE 20 Simplify the following: 1. (3a+5)(2a-8) + (4a~7)(a+6). 2. (3a:+2)(4:r + 3)-(3ar-2)(4a:~3). 3. (a-2x)(b+3y) + (a+2x)(b''3y), 4. (3m + l)2(3m^l)2. 5. (x-y){x''-y^)'-(x+y){x^+y^), 6. (2a + 3by-^a-b)(a+5b). 7. [Sx-(5y-\-2z)][Sx-{5y'-2z)l 8. [m-\-2n-{2m-n)][2m + n-{m-2n)l 9. {a+b+cy-(a-b-cy, 10. (a+2)(a + 3)(a-4) + (a-2)(a-3)(a + 4). 12. [2 ^24.(3 ^_ i)(4 a: 4-5)] [5 x^- (4 :r+3)(a;-2)]. 13. (a+2b'-c-3dy. 34 ALGEBRA 14. (a-2)(a-f3)-(a-3)(a+4)-(a-4)(a+5). 15. (ir+2)(2 X- 1)(3 a?-4)- (.T-2)(2 x + l)(3 a:+4). DEFINITIONS 57. A monomial is said to be rational and integral when it is either a number expressed in Arabic numerals, or a sin- gle letter with unity for its exponent, or the product of two or more such numbers or letters. Thus, 3 a^6^, being equivalent to 3 • a • a • j[; • 6 • 6, is ra- tional and integral. A polynomial is said to be rational and integral when each term is rational and integral ; as 2 a:^ — f at + c^. 58. If a term has a literal portion which consists of a sin- gle letter with unity for its exponent, the term is said to be of the first degree. Thus, 2 a is of the first degree. The degree of any rational and integral monomial (§ 57) is the number of terms of the first degree which are multi- plied together to form its literal portion. Thus, bah is of the seco/icZ degree ; 3 a^6^, being equivalent to 3 • a • a • 6 • & • 6, is of the fifth degree ; etc. The degree of a rational and integral monomial equals the sum of the exponents of the letters involved in it. Thus, a¥c^ is of the eighth degree. The degree of a rational and integral polynomial is the degree of its term of highest degree. Thus, 2 a^b — 3c + d^ is of the third degree. Frequently the degree of a term or polynomial with respect to some particular letter is required. Tims 3 a^x^~4hxy^ + 2 c* is of the third degree in x. 59. Homogeneity. — Homogeneous terms are terms of the same degree. Thus, a*, 3 b\ and — 5 x^y^ are homogeneous terms. A polynomial is said to be homogeneous when its terms are homogeneous ; as a^ + 3 fo^c — 4 xyz. DIVISION 36 V. DIVISION OF ALGEBRAIC EXPRESSIONS 60. Division, in Algebra, is the process of finding one of two numbers, when their product and the other number are given. The Dividend is the product of the numbers. The Divisor is the given number. The Quotient is the required number. 61. The Rule of Signs. — Since the dividend is the pro- duct of the divisor and quotient, the equations of § 48 may be written as follows : + ab , , —ab , , -^ab ^ , +afe , — — = +6, = +0, — — = -6, and = -6. + a —a +a —a We may state the Rule of Signs in division as follows : The quotient of two terms of like sign is positive ; the quotient of two terms of unlike sign is negative. 62. Let ^=x, (1) b Then, since the dividend is the product of the divisor and quotient, we have ^ 7 _ a=ox. Multiply each of these equals by c (Ax. 7, § 4), ac=bcx. Regarding ac as the dividend, be as the divisor, and x as the quotient, this may be written ^=x. (2) be From (1) and (2), ^=^' (Ax. 4, § 4) be b That is, a factor common to the dividend and divisor can be removed, or cancelled. 63. The Law of Exponents for Division. — Let it be re- quired to divide a^ by a^. By § 6, — = — — . a' aXa 36 , ALGEBRA Cancelling the common factor aXa (§ 62), we have Consider the general case : Let it be required to divide al^ by a% where m and n are any positive integers such that m is greater than n. We have a^_ a X a X a X • • • to m factors a^ aXaXaX'" to n factors Cancelling the common factor aXaXaX*-'ton factors, — =aXaXaX-" to m — n factors =a^~". Hence, the exponent of a letter in the quotient is equal to its exponent in the dividend, minus its exponent in the divisor. This is called the Law of Exponents for Division, DIVISION OF MONOMIALS 64. I. Let it be required to divide — 14a^6 by 7a^ BvS51 -14a26 ^ (-2)X7Xa^Xb ^ la' 7Xd' Cancelling the common factors 7 and aM§ 62), we have :i?l^=(-2)X6=-2 6. Then to find the quotient of two monomials : To the quotient of the numerical coefficients annex the letters, giving to each an exponent equal to its expo- nent in the dividend minus its exponent in the divisor, and omitting any letter having the same exponent in the dividend and divisor. Tlie work may be checked by multiplying the divisor by the quotient. 2. Divide 54 a^bV by —9a*b^, -9a*¥ 3. Divide -2x'^'^ifz' by -x^ifz^. — x''^^Z^ DIVISION 37 4. Divide 35(a-6)7 by 7(a-6)^ 7(a-6)* ^ EXERCISE 21 Divide the following : 1. 30 by -5. 4. -64 by 8. 7. ~Hbyr5- 2. -42 by 6. 5. -135 by -9. 8. 21ai'^by3a^ 3. -48 by -4. 6. 176 by -11. 9. -63mVby7mV. 10. 5 x^y'^ by —x^y. 15. 75 x^y^ by — 15 x^i/*. 11. 12(c-h d)' by 3(c+rf)^ 16. 81 ab'^c by 27 i^c. 12. a^fe^p by — a6c. 17. \x'^y by |^ arz/. 13. 60(a-6)« by 12(a-6). 18. -f a^feV by -bab\ 14. 8(2m-h3) by 4(2 m+3). 19. 6(a+6)5 by 3(a+6)2. Find the numerical value when a = 2, 6 = — 4, 6=^^^ and rf=-3of: 7a + 146-12c 2 a-h a+46 20. ! • 21. ■ • 13a-96 + 17c c-bd 3cH-c? DIVISION OF POLYNOMIAIiS BY MONOMIAIiS 65. We have, from § 32, {a-\-h)c=ac-\-hc. Since the dividend is the product of the divisor and quo- tient (§ 60), we may regard ac-\-hc as the dividend, c as the divisor, and a + 6 as the quotient. Whence, =a + 6. c Hence, to divide a polynomial by a monomial, we di- vide each term of the dividend by the divisor, and add the results. Ex. Divide 9 a?h'' - 6 a'c + 12 a?hc^ by - 3 a\ -3a2 38 ALGEBRA EXERCISE 22 Divide the following : 1. 30 a^-25 a^+20 a^ by 5 a. 2. —33 am^+22 a^m by 11 am. 3. 18 sHh- 24: sHV- 12 stV by -3 sth, 4. 72 /i^PiT^-eO /iPa;^ by - 12 hxK 5. la'b^-^a^b'hjiab, 6. 9(a+6)2-6(a+6) by 3(a+6). 7. a;^+2^2 :i;2^+i--3 ^r^^^^ ][jy ^m+i^ 8. 36 aiH28 a^2_4 a^-20 a« by 4 a«. 9. 45(a-by-40{a-by+25(a-by by 5(a~6)2. 10. 8 m%~24 ^2^3 + 12 m^-31 m^n^ by 6 m\ 11. (a;+2/)'-9(a?+2/)2+27(a?+y) by (x+y). 12. ^ m*— 2 m^ + l m^ by f m^ 13. 15 a^-4 aH8 a^-5 a-2 a*+3 a^ by -2 a. 14. a2(2 m+3)H2 a6(2 m+3)2+6'(2 w+3) by (2 m +3). DIVISION OF POIiYNOMIAIiS BY POLYNOMIALS 66. Let it be required to divide 12 + 10 o?^ — 1 1 a; — 21 a:^ by 2ar^-4-3aT. Arranging each expression according to the descending powers of X (§ 35), we are to find an expression which, when multiplied by the divisor, 2 rc^— 3 a; — 4, will produce the dividend. 10 a:' — 21 x^— 11 x+ 12. It is evident that the term containing the highest power of x in the product is the product of the terms containing the highest powers of x in the multiplicand and multiplier. Therefore, 10 ic* is the product of 2 x^ and the term containing the highest power of x in the quotient. Whence, the term containing the highest power of x in the quotient is 10 x^ divided by 2 x^, or 5 x. Multiplying the divisor by 5 x, we have the product 10 a;^ — 15 x^ — 20 a; ; DIVISION , 39 which, when subtracted from the dividend, leaves the remainder -6xH9a;+12. This remainder must be the product of the divisor by the rest of the quotient; therefore, to obtain the next term of the quotient, we regard — 6 a;^4-9 a;+ 12 as a new dividend. Dividing the term containing the highest power of x, — 6 a:^, by the term containing the highest power of x in the divisor, 2 x^, we obtain — 3 as the second term of the quotient. Multiplying the divisor by —3, we have the product — 6 x^ + Q ar+ 12; which, when subtracted from the second dividend, leaves no remainder. Hence, 5 a; — 3 is the required quotient. 10^3-21 x2-lla:+12 I 2x^-Sx-4, Divisor. 10x3-15x^-2Qa; | 5x -3, Quotient. - 6x2+ 9a;+i2 ~ 6x^-h 9x+12 The example might have been solved by arranging the dividend and divisor according to the ascending powers of x. From the above example, we derive the following rule : Arrange the dividend and divisor in the same order of powers of some common letter. Divide the first term of the dividend by the first term of the divisor, and write the result as the first term of the quotient. Multiply the whole divisor by the first term of the quotient, and subtract the product from the dividend. If there be a remainder, regard it as a new dividend, and proceed as before ; arranging the remainder in the same order of powers as the dividend and divisor. I. Divide 9 aft^+a^- 9 6^-5 a^fo by 3 b^+a^'-2 ab. Arranging according to the descending powers of o, a^-5a^b + 9ab^-9¥ I a^-2ab-}-3¥ a^-2a^b + 3ab^ | a -3 6 -Sa'b + Qab^ -Sa'b + 6ab^-9b^ In the above example, the last term of the second dividend is omitted, as it is merely a repetition of the term directly above. The work may be verified by multiplying the divisor by the quotient, which should of course give the dividend. 40 ^ ALGEBRA 2. Divide 4 + 9 a;*-28 a;2 by -3 ;r2+2-f4 a?. Arranging according to the ascending powers of a;, 2 + ^x-Sx^ 4-28 xH 9x' 4+ Sx - 6x2 2-4:X-3x^ - 8x -22^2+ 9x* - Sx -16x2+12 a;« - 6x2-12x3.^9^4 - 6x2-12x3 + 9x^ EXERCISE 23 Divide the following : 1. Idx^'-llx-U by 3a:+2. 2. 12a2-32a+21 by 6a-7. 3. 32^2^28 5^-1552 by 4^+5^. 4. c3-8c2-5c+84byc~7. 5. a2-2 afo + 62 ^y ^_{, .7, ^2^4 ^^4 ^^^ ^_|.2, 6. a2+2 a6 + 62 by a+6. 8. x^-Q x+9 by a;-3. Note the form of examples 5 to 8, also the results obtained. Have you similar forms in Exercise 18? 9. P~8 A: + 15 by A:"-3 11. a^-l by a-l. 10. h^'-h-12 by h-^. 12. a^^S b^ by a-2 6. Have you had multiplication problems similar to examples 9 to 12? 13. 64 2^+27^3 by 4 z+3d. 14. S(b-{-xy-y^ by 2{b+x)'-y, 15. x^—5x^y + 9 xy^ — 9 y^ by x^—2 xy + 3 y^. 16. 71^-16 by n+2. 17. aH243 by a + 3. Do the quotients in examples 11, 12, 13, 14, 16, 17 seem to have similarity of form? 18. 16(a-6)2~9by 4(a~6)4-3. 19- fa^-Ja-f by |a + |. 2^- ^ 9'-i\ 9'^-^lE 9fc'-ek ^' by ^g-lk. DIVISION 41 21. e'-Sl by ^•^-3^2_^9^-27. 22. a*— 256 6^ by a — 4 6. Compare example 16. 23. 13 x^ + 6 0^^-70 a;H71 a;-20 by 4+3 x^^-l x, 24. 42(c4-rf)'-47(c+c/)2 + 17(c+^)--2by 7(c+d)-2. 25. m^-~18m^-3mH24m2 + 52m-21 by m+m2-7. 26. i^x'+l^ by t:r+|. 27. 9 h'-b2 h^P+64: k' by 3 h'-2 hk-S k\ 28. 6xH5a?^--57ir3-x'2 + 67a;+28by -4 + 3ir2~5a;. 29. a2x_52«^2 6V-c2^ by a'^-VV-c'. 30. 4 a^^+^t^^ii a"*+«6^+^ + 6 a^ft^^-^ by a^+25~2 afe^-^ 67. The operation of division is often facilitated by the use of parentheses. Ex. Divide x^+{a-{-h — c)x'^-{- {ah — he — ca)x — ahc by x + a. a^+{a + b — c)x^-{-(ah — hc — ca)x — ahc I x +a x^+ a^ I x'^+(h — c)x — bc (J)-c)x^ (b — c)x^ + (ah — ca)x — hex — hex — ahc EXERCISE 24 Divide the following : 1. a:^ + (a— 6—c)x^ + (--a6 + 6c— ca)a;H-a6c by x'^-{-{a—h)x—ab, 2. x^ -\- {a-\-h— c)x'^ -{• {ah—hc— ca)x — ahc by x—c. 3. x^—{a^-h-\-c)x^-\-{ah-\-hc-\-ca)x'~ahc by x'^—(b-\-Q)x-\-hc. 4. a:^- (a-2 6-3 c)x2 + (-2 a?> + 6 6c~3 ca)a!-6 ahc by x^— (a— 3 c)x—Z ac, 5. a;H(3 a+6+2 c)x^-\-(S ah + 2 hc+Q ca)x-\-6 ahc by a:H-3 a. 6. m(m+n)a?^— (7?i^+n^)x+n(m— n) by mx—n. 42 ALGEBRA 7. x^— (m--2 n)x—2 m^ + 11 mn—15 n^ by x+m—3 n. 8. (2m2 + 10m7i)a;2 + (8m2-9mn-15n2)x-(12mn-9n^) by 2 ma?— 3 n, 9. a;3~(3 a + 2 6-4 c)a;2 + (6 a6-8 6c+12 ca)a;-24 afec by x-2 6. 10. a(a—b)x^ + (—ab+b^-\-bc)x—c{b+c) by (a— 6)a;+c. QUEBIES 1. Is the continued product of six numbers, one half of wliich are positive and one half negative, a positive or a negative number? Why? 2. What three numbers are involved in these two problems: {a + b){a + b); {a^ + 2 ah + b^)-^(a+b)? 3. Make a rule governing the result of (a +6) (a + 6). 4. Will the rule in 3 govern the result of {x+7){x-{-7)? 5. Apply your rule to example 10, Exercise 18. 6. Find the product of (a + b) and (a ~ b) and make a general rule for such a product. 7. Can you solve example 8, Exercise 18, by this rule? 8. One of two factors of 30 is 6, what is the other? How did you find it? 9. One of two factors of a:^ — a;— 12 is x — 4, what is the other? Does this correspond to your definition of division? 10. Does your rule in 3 aid you in solving such problems as example 6, Exercise 23? Such forms are of frequent occurrence. 11. What is the quotient of a^-\-12 a+36 divided by a + 6? 12. Given the multiplicand = m, and the product = p, what is the multiplier? 13. The product is 2* + 2^4-1, the multiplier is z^ — z+1; find the multiplicand. VI. INTEGRAL LINEAR EQUATIONS 68. Any term of either member of an equation (§3) is called a term of the equation. 69. A Numerical Equation is one in which all the known numbers are represented by Arabic numerals ; as, 2x-7=x + Q. INTEGRAL LINEAR EQUATIONS 43 An Integral Equation is one each of whose members is a rational and integral expression (§ 57) ; as, 4x— 5=§j/+l. 70. An Identical Equation, or Identity, is an equation which is always true for specified values of the letters which enter; as, (^a+b)(a-b)^a^-b\ The sign = , read *' is identically equal to"is frequently used in place of the sign of equality in an identity. 71. An equation is said to be satisfied by a set of values of certain letters involved in it when, on substituting the value of each letter in place of the letter wherever it occurs, the equation becomes identical. Thus, the equation x — y = 5 is satisfied by the set of values ic = 8, y = S; for, on substituting 8 for a:, and 3 for t/, the equation becomes 8 — 3 = 5, or 5 = 5; which is identical. 72. An Equation of Condition is an equation involving one or more letters, called Unknown Numbers, which is sat- isfied only by particular values of these letters. Thus, the equation a:; + 2 = 5 is not satisfied by every value of ir, but only by the particular value x = S. An equation of condition is usually called an equation. Any letter in an equation of condition may represent an unknown number. 73. If an equation contains but one unknown number, any value of the unknown number which satisfies the equation is called a Root of the equation. Thus, 3 is a root of the equation a: + 2 = 5. To solve an equation is to find its roots. 74. If a rational and integral monomial (§ 57) involves a certain letter, its degree with respect to it is denoted by its exponent (§ 58). 44 ALGEBRA If it involves two letters, its degree with respect to them is denoted by the sum of their exponents ; etc. Thus, 2 ah^x^y^ is of the second degree with respect to x^ and of th^ffth with respect to x and y, 75. If an integral equation (§ 69) contains one or more unknown numbers, the degree of the equation is the degree of its term of highest degree. Thus, if X and y represent unknown numbers, ax— by = c is an equation of the first degree ; a:^+4 a;= — 2, an equation of the second degree ; 2 x^—3 xy^==5, an equation of the third degree ; etc. A Linear, or Simple, Equation is an equation of the first degree. PRINCIPLES USED IN SOLVING INTEGRAL EQUATIONS 76. Since the members of an equation are equal numbers, we may write the last four axioms of § 4 as follows : 1. The same number, or equal numbers, may be added to both members of an equation without destroying the equality. 2. The same number, or equal numbers, may be sub- tracted from both members of an equation without de- stroying the equality. 3. Both members of an equation may be multiplied by the same number, or equal numbers, without destroying the equality. 4. Both members of an equation may be divided by the same number, or equal numbers, without destroying the equality. 77. Transposing Terms. — Consider the equation x + a — b = c. Adding —a and +6 to both members (§76, 1), we have x=C'-a + h. INTEGRAL LINEAR EQUATIONS 45 In this case, the terms +a and — 6 are said to be transposed from the first member to the second. Similarly, any term may be transposed from one mem- ber of an equation to the other by changing its sign. 78. It follows from § 77 that If the same term occurs in both members of an equa- tion affected with the same sign, it may be cancelled. 79. Consider the equation a—x = b'-c. (1) Multiplying each term by — 1 (§ 76), we have x—a = c—b; which is the same as equation (1) with the sign of every term changed. * Similarly, the signs of all the terms of an equation may be changed, without destroying the equality. 80. Clearing of Fractions. — Consider the equation 2 5 5 9 - X = - X 3 4 6 8 Multiplying each term by 24, the lowest common multiple of the denominators (Ax. 7, § 4), we have 16 a;-30=20 0^-27, where the denominators have been removed. Removing the fractions from an equation by multiplication is called clearing the equation of fractions, SOLUTION OF INTEGRAL LINEAR EQUATIONS 8 1 . To solve an equation involving one unknown number, we put it into a succession of forms, which finally leads to the value of the root. This process is called transforming the equation. 46 ALGEBRA Every transformation is effected by means of the principles of §§ 76 to 80, inclusive. 82. Examples. 1. Solve the equation 5 a:— 7=3 x+1. Transposing 3 x to the first member, and —7 to the second (§77), we ^^^^ 5a:-3a: = 7+l. Uniting similar terms, 2 x=8. Dividing both members by 2 (§ 76, 4), x=4. To verify the result, put a; =4 in the given equation. Thus, 20 - 7 = 12 + 1 ; which is identical. 2. Solve the equation 7 5_3^_1 6 3 5 4' Clearing of fractions by multiplying each term by 60, the L. C. M. of 6, 3, 5, and 4, we have 70^-100=36^-15. Transposing 36 t to the first member, and — 100 to the second, 70^-36^=100-15. Uniting terms, 34 ^=85. Dividing by 34, ^==1=1* 5 Verify this result by substituting t = ~ia. the given equation. 3. Solve the equation (5-3 x)(3+4 x) =62- (7-3 a?)(l-4 x). Expanding, 15+11 x- 12 3:^ = 62- (7-31 x+ 12 ar^). Or, 15+11 a;-12 0:2=62-7 + 31 a;-12x2. Cancelling the - 12 a:^ terms (§78), and transposing, llx-31x=62-7-15. Uniting terms, — 20 a; = 40. Dividing by - 20, a; = - 2. Verify the result by substituting a;= -2 in the given equation. INTEGRAL LINEAR EQUATIONS 47 To expand an algebraic expression is to perform the operations indicated. From these examples, we have the following rule for solv- ing an integral linear equation with one unknown number : Clear the equation of fractions, if any, by multiplying each term by the L. C. M. of the denominators of the fractional terms. Remove the parentheses, if any, by performing all the operations indicated. Transpose the unknown terms to the first member, and the known to the second; cancelling any term which has the same coefficient in both members. Unite similar terms, and divide both members by the coefficient of the unknown number. The pupil should verify every result, SXEBCISE 25 Solve the following equations, in each case verifying the result : 1. 5a;+13=28. 8. 24 ^-28 = 14 <~48. 2. 7 21=4 2-33. 9. 26-4a:=31-2a;. 3. 11 71+71=6 71+76. 10. 16i?-47=8/J-43. 4. 8d--2 = 5d-26. II. 17 + 14 a; = lla:+16. 5. 15a; + 19 = lla;-5. 12. 43 A:- 27= 37- 149 fe. 6. 13-21 ik=34- 14 fc. 13. 12i/+15 = 15y + 17. 7. 25 a;- 3 =4 + 18 a:. 14. 98-16 a; =23- 41 a:. 15. 29aj-8 + 17 = 32a;-14a:-24. 16. 35s-41 = -81+63 2-58z. 17. 0=31<-14f+3^+30. ol.loSl 2 15,1 18. -m+-=2 m, 19. - '2^— - =- ^+;;* 3 2 62 3663 48 ALGEBRA 20. - k-}-- k-\-- k= — • 22. -(7-)--== — q-\ a. 6 6 3 3 7^4 14^ 28^ 5 7 3 13 ^ 2 38 8 4 21. ~s=-S S • 23. -X = -X X. 648 48 5393 2 4 7 1 23 24. -V q) = - q)^ — q; . 3 5 8 20 24 25. 5(ir+3)-7 = 6(2a:-3)4-40. 26. 12A:-(4A;^7)=3A;-(9A:~28). 27. 75-8(7 2/+5) = 6i/-(4w+52). 28. 5/i-3(2-8ii) = 9iJ-4(l-4/i). 29. (4-3 2)(5+4 z) = (8+2 z)(l-6 z)-82. 30. l(4x + l) + l(Qx-2)^h5x-\-S)==2. 3 5 6 PBOBIiEMS LEADING TO INTEGRAIj LINEAR EQUATIONS WITH ONE UNKNOWN NUMBER 83. For the solution of a problem by algebraic methods, the following suggestions will be found of service : 1. Represent the unknown • number, or one of the un- known numbers if there are several, by some letter, as x, 2. Every problem contains, explicitly or implicitly, at least as many distinct statements as there are unknown numbers involved. Use all but one of these to express the other unknown numbers in terms of x, 3. Use the remaining statement to form an equation. 84. Problems. I. Divide 45 into two parts such that the less part shall be one-fourth the greater. Here there are two unknown numbers; the greater part and the less. In accordance with the first suggestion of § 83, we represent the greater part by x. The first statement of the problem is, implicitly: The sum of the greater part and the less is 45. The second statement is : The less part is one-fourth the greater. INTEGRAL LINEAR EQUATIONS 49 In accordance with the second suggestion of §83, we use the first statement to express the less part in terms of x. Thus, the less part is represented by 45— a;. We now, in accordance with the third suggestion, use the second statement to form an equation. 1 Thus, 45 — a;=- x. Clearing of fractions, ]80-4a;=x. Transposing, -4x-x=-180, or-5a:=-180, Dividing by —5, a; =36, the greater part. Then, 45 — re =9, the less part. Verify by substituting aj=36 in the given equation. 2. A had twice as much money as B ; but after giving B $28, he has | as much as B. How much had each at first ? Let X represent the number of dollars B had at first. Then, 2 x will represent the number A had at first. Now after giving B $28, A has 2 a: — 28 dollars, and B, x+28 dollars; we then have the equation 2x-28 = ^(x + 28). Clearing of fractions, 6 a: - 84 = 2(x + 28) . Expanding, 6 x - 84 = 2 a; + 56. Transposing, 4 x = 140. Dividing by 4, x = 35, the number of dollars B had at first ; and 2 x = 70, the number of dollars A had at first. Verify the result. 3. A is 3 times as old as B, and 8 years ago he was 7 times as old as B. Required their ages at present. Let n-- = the number of years in B's age. Then, 3n = =the number of years in A^s age. Also, n — 8 = =the number of years in B's age 8 years ago, and 3n-8 = = the number of years in A's age 8 years ago. But A's age 8 years ago was 7 times B's age 8 years ago. Whence, 3n-8 = 7(n-8). Expanding, 3n-8 = 7n-56. Transposing, -4n=-48. Dividing by — 4, n = 12, the number of years in B's age. Whence, 3 n=36, the number of years in A's age. Verify the result. 50 ALGEBRA 4. A sum of money amounting to f 4.32 consists of 108 coins, all dimes and cents ; how many are there of each kind? Let x = the number of dimes. Then, 108 — a: = the number of cents. Also, the X dimes are worth 10 x cents. But the entire sum amounts to 432 cents. Whence, 10 a; +108 -a; =432. Transposing, 9 a: = 324. Whence, a; =36, the number of dimes; and 108 — a; = 72, the number of cents. Verify the result. EXEBCISE 26 1. The difference of two numbers is 12, and 7 times the smaller exceeds the greater by 30. Find the numbers. 2. The sum of two sides, AB and jBC, of the triangle ABC is 23, and the lesser side exceeds their dif- ference by 7. Find the sides AB and BC. Can more than one such triangle be drawn ? p,y^ \c 3. Find two numbers whose sum is |, and difference J. 4. The sum of two numbers is 35, and their difference is three-fifths the larger number. Find the numbers. 5. A is 5 years older than B, and the sum of their ages is 39 years. How old is each ? 6. A rectangle is 4 feet longer than it is wide. w If 4 feet were added to the length the area would I w-M- be increased 40 square feet. Find the length of the sides. 7. A rectangle is 6 feet longer than it is wide. If 3 feet be added to its width and 4 feet be subtracted from its length its area will not be changed. Find the length and breadth. 8. A man counting the coins he has in his hand finds that he has three times as many quarters as half dollars, five INTEGRAL LINEAR EQUATIONS 61 more dimes than quarters, and twice as many five-cent pieces as dimes. The entire sum of money is $2.85. How many coins of each kind ? 9. A is 25 years of age, B is 9 years of age. In how many years will A be twice as old as B ? 10. The length of a rectangle is 5 feet more than the width. If 4 feet be taken from the length and 4 feet from the width the area of the rectangle will be diminished 124 square feet. Find the length and breadth of the rectangle. 11. A certain number of two digits is equal to 9 times the sum of the digits and the digit in ten's place is 7 greater than the digit in unit's place. Find the number. 12. Divide f300 among A, B, and C so that ^oi B's share plus $20 may equal A's share, and C and B may have equal amounts. 13. A man has $4.10, all five-cent and fifty-cent pieces; and he has 5 more five-cent pieces than fifty-cent pieces. How many has he of each ? 14. The difference between | and ^ of a certain number exceeds ^ of it by 44. What is the number? 15. A has $5.50 and B $3.50; how much money must A give B in order that B may have | as much as A ? 16. A room is | as long as it is wide ; if the length were diminished 3 feet and the width increased by the same amount, the room would be square. Find its dimensions. Note: Oranges come packed in boxes, a box containing 86, 90, 110, 126, 150, 175 oranges. A box marked 90's indicates that there are 90 oranges in that box. 17. A merchant buys oranges, 150's, a certain number of boxes at $3.25, twice as many at $3.00 and six boxes at $3.50, paying $39.50 for the entire lot. Find the average cost per dozen oranges. 52 ALGEBRA 1 8. A merchant buys oranges, 90's, some at $2.00 per box, I as many at $2.20 per box, paying $28.80 for the entire lot. Can he make a profit retailing them at 290 per dozen, no allowance being made for expense of handling ? Note : Banana dealers estimate the value of a hunch of bananas by the number of hands on a bunch. A hand is a cluster of bananas grouped together and contains 12 to 16 bananas. 1 9. A merchant bought three lots of bananas ; some, 8 hands, at 850 ; three times as many 12 hands at $1.15, and 5 bunches, 10 hands, at $1.05, paying $18.15 for all. Find the approximate average cost per dozen bananas, averaging 15 bananas to the hand. 20/ A given square has 39 square feet more area than a given rectangle. The length of the rectangle is 3 feet more than a side of the square, and the breadth of the rectangle is 5 feet less than a side of the square. Find the dimensions of each figure. 21. Divide $480 among A, B, C, and D so that B shall have twice as much as A, B shall have $6 more than C, and C and D together as much as A and B together. 22. Find two numbers whose difference is 17, such that the square of the greater exceeds the square of the less by 1037. 23. A room is | as long as it is wide, and 60 feet of pic- ture molding are required to go around it. Find the number of square feet in the floor. 24. A starts to walk from Boston to Rockland, 19 miles, at the same time, B starts to walk from Rockland to Boston. A walks \ mile an hour faster than B. They meet in 3i hours. Find the rate of each. (Let 72= number of miles per hour A walks.) 25. The sum of $900 is invested, part at 4%, and the rest at 5%, per annum, and the total annual income is $42. How much is invested in each way? INTEGRAL LINEAR EQUATIONS 53 26. In 9 years B will be | as old as A ; and 12 years ago he was f as old. What are their ages ? (Let n represent the number of years in A^s age 12 years ago.) 27. A man buys irrigated farm land, some at $17 per acre, and three times as much less 160 acres at $15 per acre, pay- ing $17,440 for the entire farm. He also pays $2.50 an acre for a water right. He sells the land for $21 per acre. What is his profit? 28. A has ^ of a certain sum of money, B has |, C $5 less than |, D the balance which is $44. Find C's share. 29. Find three consecutive numbers such that the square of the greatest exceeds the product of the other two by 70. 30. Find three consecutive numbers such that if the square of the least number be subtracted from the product of the other two the remainder will be 47. 3 1. A number consists of two digits, and the ten's digit is 5 greater than the unit's digit. The difference between the squares of the digits is 65. What is the number? 32. A is 10 years older than B ; 4 years ago B was f as old as A will be in 5 years. Find the age of each. 33. There are two heaps of coins, the first containing 5-cent pieces, and the second 10-cent pieces. The second heap is worth 20 cents more than the first, and has 8 fewer coins. Find the number in each heap. 34- A certain number is composed of two digits ; the num- ber is six more than six times the sum of the digits, and the digit in unit's place is f the digit in ten's place. Find the number. 35. Find four consecutive odd numbers such that the pro- duct of the first and third shall be less than the product of the second and fourth by 64. 54 ALGEBRA VII. PRODUCTS AND FACTORS 85. A Power of a Power. — Required the value of (0^)3. By § 6, (ay=a^ Xa^ Xa''=a\ The general case : — Required the value of (a'^)'*, where m and 71 are any positive integers. We have, (a'")'* =a'" Xa'^X — to n factors __^m4-mH — to n terms __ ^mn 86. A Power of a Product. — Required the value of (a6)^. By § 6, (aby=abxabxab=a%\ The general case : — Required the value of (a6)'*, where n is any positive integer. We have, (aby=ab Xab X««* to n factors=a'*6\ In like manner, (abc'"y=a%V''', whatever the number of factors in abc*". 87. A Power of a Monomial. 1. Find the value of (— 5a*)^» By §26, (-5a*)« = [(-5)XaT = (-5)»X(a*)M§86) = -125a»2(§85). 2. Find the value of (^—2m^ny, We have, {-2m^n)* = {-2yx{myXn* = 16m^^n*. 88. From §§ 85 and 86 and the examples of § 87, we have the following rule for raising a rational and integral mono- mial (§ 57) to any power whose exponent is a positive integer. Raise the absolute value of the numerical coeflacient to the required power, and multiply the exponent of each letter by the exponent of the required power. Give to every power of a positive term, and to every even power of a negative term, the positive sign ; and to every odd power of a negative term the negative sign. PRODUCTS AND FACTORS 55 EXERCISE 27 Expand the following : 1. {xyz')\ 5. (7a'"62n)3^ p^ (a^'b^cy. 2. (m'^n^pyK 6. (-nVyy, lo. (x^'^y^^z^pyK 3. (-a6V«y. 7. (2mV)«. ii. (-3mVa;«)^ 4. (~lla;y)2. 8. (~4a;V')'- ^2. (-2amV)«. Find the factors of the following : 13. 25a^b\ 17. Sa\ 21. 343 a V. 14. 32 m^ 18. a2«. 22. 243 mV. 15. 48a^62^. 19. a"*-^'. 23. 165 aV. 16. 21a^ 20. a2«+^ 24. -282 a^^c^^ 89. Type I. Product of the Sum and Difference of Two Numbers. — Let it be required to multiply a+6 by a— 6. a+b a—b a^+ab ^ab-¥ Whence, («+&)(«-&) = «'' -^^• That is, the product of the sum and difference of two numbers equals the difference of their squares. 1. Multiply 6 a+ 5 &^ by 6a-56^ By the rule, (6 a + 5 6«)(6 a-5 6^) = (6 a)^- (5 6^)2=36 0^-25 h\ 2. Multiply — ar^+4by — a;^— 4. (-a;2 + 4)(-a:2-4)=[(-a:2)+4][-(_^2)_4j EXERCISE 28 Expand the following : 1. (4a+3 6)(4a~3 6). 3. (3 c+8)(3 c-8). 2. {2x-\-4ty){2x'-^y). 4. (8 fi + l)(8 fi-1). 56 ALGEBRA 5. (6ci-f-5 0(6t/-5 0. 8. (15a + 13 6)(15a-13 6). 6. {nk'-]-sP)(nk*-^P). 9. (-x+7)ix+7), y. (9 a+2)(9 0—2). (Prove this last result by actual multiplication.) 10. (12x+y)(12x-y). 11. (-l9c^-\-4:d')(-19c''-4d*). 12. From what factors do you obtain x'^—9 ? 13. From what do you obtain 4 a 2— 25? 14. Find the factors of 9 c2-49 d\ By reversing the product rule in § 89, this rule follows: To factor the difference of two squares, extract the square root of the first square, and of the second square ; add the results for one factor, and subtract the second result from the first for the other factor. Note : It is not always possible to factor an expression ; there are, how- ever, certain forms which can always be factored; these will be con- sidered in the present work. Factor the following : (Check: If results are correct, the product of the factors will equal the given expression.) 15. 9a^-U\ 19. 49-4^2 23. (m+ny-z\ 16. 36m2-25P. 20. 36a;^-121^«. 24. 49a«-1446«. 17. a^-9c\ 21. 16-25a«. 25. (x-{-yy-(a+by. 18. 25e^'-Slh\ 22. 100mV2-1692/^^ 26. {x+yy-(a-hy. 27. (2a+xy-{a-2xy. Expand the following : 28. (5a2 + 12 63c)(5a2-12 6V). 30. (6m + 4b^)(6m-4b^), 29. (a^-fe^)(a^-e»'). 31. (c^'-\-d^^){c^'-d^^). Sometimes the factors of an expression admit of further factoring : PRODUCTS AND FACTORS 57 32. o;^— 81 = (a;2 + 9)(a:^ — 9) (The second factor can be factored) = (x^ + 9){x + S)(x-3). 33. Factor 16m^-625y\ 34. Factor a«-6^ 35. Factor 81 c«~ 16 cZ^ 36. Factor (/2+S+3)2-(/J-S-4)2. 37. Does 36 0^—2 belong to this type? 38. Can you factor a;^ + 9 by this type? 90. By division : (Compare Exercise 23, Ex. 18.) 1. Divide 25 yh'-Q by 5 yz^-S. By § 88, 252/^2* is the square of 5yz^; then by (2), 2. Divide ^2 — (2/— 2)2 by x+(y — z). EXEBCISE 29 Find, without actual division, the values of the following : a+2 * x^-9 x-S' "' SnHa:' "' 7xh+S 91. Type II. Square of a Binomial. — Let it be required to square a+b, a+b a+b 25n*~l 5. 1-lU a'^'b^ 5n^-l 1-12 a^b' Mn^-x^" 6. 49xV-64 a^-\- ab Whence, (a-^by = a^+2ab+b\ (1) 58 ALGEBRA That is, the square of the sum of two numbers equals the square of the first, plus twice the product of the first by the second, plus the square of the second. I. Square Sa-\-2b. We have, (3 a + 2 by = (S ay + 2(3 o)(2 6) + (2 by =9a^+12ab + 4h\ Let it be required to square a—h, a—b a—b o^— ab - ab^b' Whence, (a~6)^ = a^-2a6^-6^ (2) That is, the square of the difference of two numbers equals the square of the first, minus twice the product of the first by the second, plus the square of the second. In the remainder of the work we shall use the expression " the differ- ence between a and 6 " to denote the remainder obtained by subtracting b from a. The result (2) may also be derived by substituting —6 for 6, in equa- tion (1). 2. Square 4 x'^—b. We have, (4 a;^- 5) 2 = (4 x^) 2- 2(4 a;^) (5) + 5' = 16 x*- 40^2 +25. If the first term of the binomial is negative, it should be written, negative sign and all, in parenthesis, before applying the rules. 3. Square — 2a^ + 9. We have, {-2 a^ + 9y = [{-2a^)+9Y = (-2a3)2-f2(-2a3)(9)+9' = 4a«-36a3 + 81. EXEHCISE 30 The following 18 examples are for mental drill : 1. (a:+3)2. 5. (4 2/-6z)2. 9. {h-liy. 2. (a~4)2. 6. (3ac-4 6)2. 10. {v-UwY 3. (c4-9)2. 7. (ir+4)2. II. (4a + 13 6)2. 4. {2x-\-l)\ 8. (-4A:+3d)^ 12. (15 a;- 1)^ PRODUCTS AND FACTORS 59 Note that in each of these trinomial squares, the first and third terms are perfect squares and positive, and the middle term is twice the pro- duct of the square roots of the first and third terms. What sign does the middle term have? In each of the following expressions supply the missing term which will form a perfect trinomial square : 13. x^+4:Xi. 15. c^H-lG. 17. 6^—4 6. 14. a'-f9. 16. x^' + Ux. 18. 5^4. Can you substitute other numbers than those you used and still form a perfect square? 19. a;2+10 a;+25 is the square of what ? 20. x^ — 6x+9 is composed of what factors? (Compare example 1.) 21. Factor x^+2 xy+y\ 22. From what two factors do you obtain 16 a^H- 8 a -h 1 ? By reversing the product rule in § 91, this rule follows: To factor a trinomial square, extract the square roots of the first and third terms, and connect the results by the sign of the second term. This gives one of the equal factors. 23. Factor 4 a:2+ 12 0:2/4-9 2/2. 26. Factov25 P-\- 60 hk+ 3d h\ 24. Factor 9 2/2+6 2/4-1. 27. Expand (3 a: +2 2/) 2. 25. Factor c^'+S c + 16. 28. Expand (8 x^ + 9 x'^y. 29. From what do you obtain a^^z/^-f 14 xy-}-49? Sometimes the factors of an expression admit of further factoring : 30. x'-Sx^ + 16==(x^-4)(x''-i) = {x+2){x-2){x + 2){x-2) [by §89]. This result may be written ix-{-2y(x — 2y. 31. Factor a^-18a2 4-81. 32. Factor 49 <2 4-l68 tu+lU u^. 33. Factor 25(a4-6)24-40(a4-6)c4-16 c^. 60 ALGEBRA 34. Factor 16 m'-72 mV+81 v*. 35. Expand {x+y+z){x-y-\'z), (x+y + z)iz-'y + z)=[{x + z)-\-y][ixi-2)-y] = {x + zy-y' =x^ + 2xz+z^-y\ 36. Expand (a-|-6— c)(a— 6-fc). By §46, (a4-6-c)(a-64-c)=[a+(6-c)][a-(6-c)] = o^ — (6 — c) ^ by the rule, 37. (X^+X + 1)(X^+X'-1), 38. (a2 + H-3a)(a2 + l-3a). 39. (x-\-y+S)(x'-y-S). 40. (a2 4-5a^4)(a2~5a-f4). 41. Factor a^+2 ab+b^-c\ = (a+6)2-c2 = (a+6-f-c)(a4-6-c). 42. Factor aH6 a + 9~4 c^. 43. Factor 9-a^+2 ah-b^ (§ 46). 44. Factor a^ + 2 ab+P-c^-2 cd~d\ 45. Factor a2-4aa:+4x2-6H6 6i/-9 2/2. 46. Factor a;^— 1/2— 22/2— z^. 47- Is a;2-8a;+25 a perfect square? Why? 48. Square both members of the equation (fiC) = (5Z))-(CZ)). (See figure.) I 1 f 92. Type III. Product of Two Binomials having the Same First Term. — Let it be required to multiply x-\-a by x-Vb. x-\-a x-\-b .T^-f ax + bx-\- ab Whence, (x + a)(ic+b) = x^ + (a-^b)x^ab. PRODUCTS AND FACTORS 61 That is, the product of two binomials having the same jlrst term equals the square of the first term, plus the algebraic sum of the second terms multiplied by the first term, plus the product of the second terms. 1. Multiply x—5 by x-\-S, The coefficient of x is the sum of —5 and +3, or —2. The last term is the product of —5 and -}-3, or — 15. Whence, {x-5)ix-\-S) = x^-2 x-15. 2. Multiply x— 5 by x—S. The coefficient of x is the sum of —5 and —3, or —8. The last term is the product of —5 and —3, or 15. Whence, (x-5)(x-3) = aj^-S x-f 15. 3. Multiply a&-4 by ab+7. The coefficient of ab is the sum of —4 and 7, or 3. The last term is the product of —4 and 7, or —28. Whence, (ah - 4) (ah + 7)= a^h^ + Sab -28. 4. Multiply ^2+6 y^ by x^i-S y\ The coefficient of x^ is the sum of 6 y^ and 8 y^, or 14 y^. The last term is the product of 6 y^ and 8 2/^ or 48 2/". Whence, (x^ + 6 y^) (x^ + 8 2/') = x* + 14 xY + 48 ?/». EXERCISE 31 Expand the following by inspection : 1. (a;+2)(a;+3). 8. (a^+3)(a^-f 9). 2. (a;-3)(a:+7). 9. (i?+2 C)(/? + 9 C). . 3. (x-12)(a:-l). ID. {e-Sy){e-^y), 4. (a:~9)(a:+2). 11. {a'^-\-2){a^-b), 5. (z2 + 13)(z2+2). 12. (a:^-l)(x« + 7). 6. (a3-l)(a3-h27). 13. {x^-2^){x^+A), 7. (c5~4)(c^+6) 14. (6^4-3)(^^-ll). 15. From what factors do you obtain ir'-f8rr-fl5? (Compare example 9, Exercise 23.) 16. What are the factors of xH7 x + 12 ? 62 ALGEBRA 17. Factor x^ +4 X— 12. By the rule in § 92, the product takes the form To factor a trinomial of the form x^+ax+b, reverse this process. Hence, to obtain the second terms of the binomials re- verse the rule for products and find two numbers whose algebraic sum is the coefficient of x, and whose product is the last term of the trinomial. The numbers may be found by inspection. 18. Factor x^ + 14 x+45. We find two numbers whose sum is 14 and product 45. By inspection, we determine that these numbers are 9 and 5. •Whence, x^+14:X + 45 = {x + 9){x + 5). Factor the following : 19. x^-^-Sx-lO. 28. m2 + 6m-16. 20. a;2-12a;+ll. 29. l+2a-99a2. 21. x^-5x-U. 30. a2 + 18a+56. 22. aH16a2 + 15. 31. c^-lOc-TS. 23. m2+5m-24. 32. k''-Qk-72. 24. C2-C-72. 33. m^+27m+72. 25. dH37d+36. 34. a2 + 17a + 72. 26. k*-h5k^-U, 35. (ar~y)2-9(x-2/)-h20. 27. R^-lSR^+22. 36. (a + 6)' + (a-|-6)-56. 37. (c+dy-i{c+d)-m. Expand by inspection : 38. (a2-8)(a2 + 12). 40. (A4-3)(fe4-3). 39. (c+7)(c4-7). 41. [{x+y)+2][{x+y)^U]. 42. [(m+iJ)^8] [(m4-i?)+6]. Find numbers which will make the following factorable : 43. x24-(?)x4-36. 44. a^ ( )a-~72. 45. c^ ( )c-48. PRODUCTS AND FACTORS 63 EXERCISE 32 Select the type to which each of the following belongs and then factor : 1. a:^-4x2-32. 7. S6x''-9y\ 2. a^+Sa + lG. 8. a''-lQ+2 ab+b\ 3. a2 + 17a + 16. 9. a^-625. 4. a^-flOa + lG. 10. 22-2^132. 5. P-12A: + 36. II. m2-50m+49. 6. 0^2 + 2 07 + 1. 12. m2-14m4-49. 13. Can you factor x^+x+1 by any type you have had ? The accuracy of your factors can always be proved by finding the product of your factors. 14. Factor (a;+j/)2-ll(ii;+y)H-30. 15. Factor x^-\-(2 m-f 3 k)x + Q mk, 93. Type IV. Product of Two Binomials of the Form tnx+n and px+q. — We find by multiplication : mx+n X px+q mpx^+ npx + mqx+nq mpx^ + (np+mq)x+nq The first term of this result, mpx^, is the product of the first terms of the binomial factors, and the last term, nq, the product of the second terms. The middle term, (np-\-7nq)x^ is the sum of the products of the terms, in the binomial factors, connected by cross lines. Ex. Multiply 3 a;+4 by 2 a:-5. The first term is the product of 3 rr and 2 x, or 6 x^. The coefficient of x is the sum of 4X2 and 3X(-5); that is, 8-15, or —7. The last term is the product of 4 and —5, or —20. Whence, (3 x + 4)(2 x-b) =6 x^-? x-20. 64 ALGEBRA SXERCISE 33 £xpand the following by inspection : I. (x+2)(4x+3). 8. (2c^~l)(5rf-f2). 2. (3a:-2)(2a: + l). 9. (3m+4a;)(2m-3a;). 3. (2a:-7)(5x+3). 10. (2a^ + 3y)(3a^+5 2/). 4. (8:r-l)(7a:+2). II. (6a2+a;2)(8o2-5a:2). 5. (a^6)(3a-4). 12. (5iJ-4iJ)(3iR-fllH) 6. (2A; + 15)(4)fc-ll). 13. (m + 116)(llm4-6). 7. (6 6-5)(4e-3). 14. (6A:~-5Z)(5^'+6/). 94. Note that the product of two factors of the above form is a trinomial of the form (Type IV.) 005^ + 6a5 + c To factor a trinomial of the form reverse the above process. Hence, To resolve a trinomial of the form ax^'\-bx + c into two binomial factors, the first terms of the binomials must be such that their product is ax^; the second terms must be such that their product is c ; the sum of the cross products must be hx. I. Factor Sx^ + Sx + 4, The first terms of the binomial factors must be such that their product is 3 a;^; they can be only 3 x and x. The second terms must be such that their product is 4. The numbers whose product is 4 are 4 and 1,-4 and —1,2 and 2, and — 2 and —2; the possible cases are represented below: x+4 x+1 x-4 X X X 3x+l 3a;4-4 3a;-l 13 X 7x -13x x-l x+2 x-2 XXX 3a;-4 3x-f2 3.T-2 -7 X Sx -Sx PRODUCTS AND FACTORS 66 The corresponding middle term of the trinomial, obtained by cross- multipHcation, as in §93, is given in each case; and only the factors jc-f 2, 3 x+2 satisfy the condition that the middle term shall be 8 a:. Then, 3 x' + S a:-f 4 = (a; + 2)(3 x + 2). 2. Factor 6a:^ — a:-2. The first'terms of the factors must be 6 a; and x, or 3 x and 2 x. The second terms must be 2 and — 1 , or — 2 and 1 . The possible cases are given below : Qx + 2 Qx-1 6a:-2 Qx + 1 X X X X x-1 x-f-2 x+1 x-2 — 4 X 11 a; 4:x —11 x 3a: + 2 3x-l 3a;-2 3x + l X X X X 2a;-l 2x4-2 2a;+l 2a:-2 X 4i X —X —4 X Only the factors 3 a; — 2 and 2a;+l satisfy the condition that the middle term shall be —x. Then, Qx^-x-2^(S x-2){2 x+1). The following suggestions will be found of service : (a) If the last term of the trinomial is positive, the last terms of the factors will be both + , or both — , ac- cording as the middle term of the trinomial is + or — . Thus, in Ex. 1, we need not have tried the numbers —1 and —4, nor — 2 and — 2 ; this would have left only three cases to consider. (&) If the last term of the trinomial is negative, the last terms of the factors will be one + , the other — . If the x^ term is negative, the entire expression should be enclosed in parentheses preceded by a — sign. If the coefficient of a:^ is a perfect square, and the coeffi- cient of X divisible by the square root of the coefficient of x^, the expression may be readily factored by the method of § 91. 3. Factor 9x^-lSx+6, In this case, 18 is divisible by the square root of 9. We have 9 x^-lS x-f 5 = (3 a;)^-6(3 a;)+5. We find two numbers whose sum is —6, and product 5. The numbers are —5 and —1. Then, 9 a-^-lS a:-f-5 = (3 ar-5)(3 a:-l). 66 > ALGEBRA JiiXEBCISE 34 Factor the following by inspection : I. 3a;2+20x + 12. 9. 10 aV- 3 ax- 18. 2. Ux^+5x-l, 10. 30x^ + 17 dx-2d\ 3. 8x^-Ux-l5. II. S^x^-ldxy-^y^ 4. 20a2~27a + 9. 12. 49a2-42a6+8 62. 5. 16m2 + 16m+3. 13. 54 a^^ + 15 a^y+y^. 6. 15/JH4/J~-4. 14. 48a^-22aV-5a;^ 7. 22a^-19a2+4. 15. 50^2^55 ^^^14 ^2 8. 30cH41cH6. 16. 72 cW -13 abed- 15 a^b\ EXERCISE 35 Select the type to which each of the following belongs and then factor : 1. 9b^-20bc+4c\ 7. A;H14%+49 2/2. 2. 9 62-12 6c+4c2. 8. 15 c^-19 cd-5Qd\ 3. 9¥-4c\ 9. 6^2-7 a;- 20. 4. (235)2- (234)2. 10. a^- 16 ab+M b\ 5. 9 62- 16 6c- 4 c2. II. 36d2a;2-36cima; + 9m2. 6. A;2-13)ki/-48 2/2. 12. 256 a^-800 a262 + 625 6^ 13* x^—y^. 14. Can you factor 3 x^— 2 a: + 12? Solve the following equations and verify each result : 15. (x + 3y + (x+5)(3x-4) = {2x+5y. 16. (3t'{-5){3t-5)-{t-\-7){t-l) = {St+3){t-l). 17. (2m-3)2 + (m+8)(m-8) = (5m-l)(m+3). 95. It is not possible to factor every expression of the form x^-\-ax+bhy the method of § 92. Thus, let it be required to factor .r^-f 18a:+35. We must find two numbers whose sum is 18, and product 35. The only pairs of positive integral factors of 35 are 7 and 5, and 35 and 1 ; and in neither case is the sum 18. PRODUCTS AND FACTORS 67 It is also impossible to factor every expression of the form aod^+bx+c by the method of § 94. Thus, it is impossible to find two binomial factors of the expression 4a^+4a;— 1 by the method of § 94. In § 236 will be given a general method for the factoring of any expression of the form x^+ax-\-by or ax^-\-hx+c, 96. Type V. When the expression is in the form Qc'^ + ax^y^ + y^ Certain trinomials of the above form may be factored by expressing them as the difference of two perfect squares, and then employing § 89. 1. Factor a' +aW+b\ By § 91, a trinomial is a perfect square if Its first and last terms are perfect squares and positive, and its second term plus or minus twice the product of their square roots. The given expression can be made a perfect square by adding a ^6^ to its second term; and this can be done provided we subtract a%^ from the result. = {a^ + by-a^b\ by ^ 91, =-(a' + b^+ah){a^ + b^-ab), by § 89, = {a^ + ab + b'){a'-ab + b'). 2. Factor 9.T^-37a;2+4. The expression will be a perfect square if its second term is — 12 x^. Thus, 9 a;*-37a;2 + 4 = (9x^-12x2 + 4) -25x2 = (3x2-2)2-(5x)2 = (3 x2 + 5 x-2)(3 x'-5 x-2). The expression may also be factored as follows : 9x*-37x2 + 4 = (9x*+12x2 + 4)-49x2 = (3 x2 + 2)2- (7 x)2 = (3 x2 + 7 x + 2)(3 x2-7 x + 2). Several expressions in Exercise 36 may be factored in two different ways. The factoring of trinomials of the form x* + ax^y^ + y*, when the factors involve surds, will be considered in § 237. 68 ALGEBRA EXERCISE 36 Factor the following : 1. x*+5x^ + 9, 5. 9a:* + 6a:y-f-49y^ 2. a^-21 0^62+36 6*. 6. 16a^-81aH16. 3. 4-33x^+4 x'. 7. 64-64 m2+25m^ 4. 25 m^~ 14 mV+n*. 8. 49 a^~ 127 aV+81 x\ Factor each of the following in two different ways (com- pare §§ 92, 94) : 9. a;*- 17 0^2^16. n. 16m*-104 mV+25 x^ 10. 9-148a2 + 64a^ 12. 36 a*- 97 a^mH 36 m^ 97. Type VI. We find by division, a+6 a— o That is. If the sum of the cubes of two numbers be divided by the sum of the nunabers, the quotient is the square of the first number, minus the product of the first by the second, plus the square of the second number. If the difference of the cubes of two numbers be di- vided by the difference of the numbers, the quotient is the square of the first number, plus the product of the first by the second, plus the square of the second num- ber. If an expression can be resolved into three equal factors, it is said to be a perfect cube^ and one of the equal factors is called its cube root. Thus, since 27 aW is equal to 3 a^6x3 a^bxS a% it is a perfect cube, and 3 a^b is its cube root. Similarly to extract the cube root of a positive monomial perfect cube : Extract the cube root of the numerical coefficient, and divide the exponent of each letter by 3. PRODUCTS AND FACTORS 69 Thus, the cube root of 125 a^6V is 5 a^b^c, 1. Divide 1+8 a^ by 1-f 2 a. By § 88, 8 a^ is the cube of 2 a ; then, by the first rule, l±8af^Lt(2a}!=l_2a+(2a)^ = l-2a + 4a^ l+2a l+2a ^ (Compare Exs. 11-14, 26, Exercise 23.) 2. Divide 27 a;«- 64 y^ by 3 a;^- 4 y\ By the second rule, 3 a;* — 4!/^ 3x^ — 4 2/^ =9a;*+12a;V+16 2/'. EXEKCISB 37 Find, without actual division, the values of the following : 1. ! — . 4. ! , 7. H-. x-\-l a^+b^ Sx^-5y l-a' aH125 ^ 343mV+8i)3 2, . 5. . 8, L—, 1— a a+5 7mn+2p n^-27 . 64ir«^-l 64a«63+216c^ n~3 4a;2'^-l 4a26+6c3 Factor the following : 10. aH6'. 13. 8a'H27c^ 16. 64w^~7i3 11. x^-y^, 14. 1-27 n^ 17. a'63-216c^ 12. l+mV. 15. a«-i-l. 18. S m^P -\-21 7i^\ 98. Type VII. We find by actual division, = a^ + a^b + ab^ + b\ ^5-l4- = a4^^3^^^2^2^^^3^.j>4. etc. a — b 70 ALGEBRA In these results, we observe the following laws : I. The exponent of a in the first term of the quotient is less by 1 than its exponent in the dividend, and de- creases by 1 in each succeeding term. II. The exponent of h in the second term of the quo- tient is 1, and increases by 1 in each succeeding term. III. If the divisor is a— &, all the terms of the quo- tient are positive ; if the divisor is a -f &, the terms of the quotient are alternately positive and negative. (Compare Exs. 14, 16, 17, Exercise 23.) 1. Divide a^—V hy a — b. By the above laws, a — b 2. Divide 16 x'-Sl by 2 x+3. We have l^-^^(Al)lz^ 2a;+3 2a: + 3 = (2x)'-(2x)2.3 + 2a;.32-3' =8x3-12x2 + 18a;-27. EXERCISE 38 Find, without actual division, the values of the following: I. h-k 3. x-1 5. 2. 4. l-a« 1+a 6. a'- be' Fa< 3tor the following : 7. x'+y\ II. a'-b\ 15. n'P+S2, 8. a'-l. 12. a«-l. 16. 2^3x^+y\ 9. l~mV. 13. x^i-n^ 17. m^H128 7i^ ID. 1'\-X\ 14. S2a'-¥\ 18. 32a56^^'^-243c^^. PRODUCTS AND FACTORS 71 99. The following statements will be found helpful if n is a positive integer : x + 2/ is a factor of x'^-\-y'^ il n\s odd. x — y is never a factor of x^ + y^. x — yis always a factor of x^ — y^. x + y is a, factor of x^—y^ if nis even. When one factor is x—y all the terms of the other factor are positive, and when one factor is x + y the terms of the other factor are alternately positive and negative. 100. A Common Factor of two or more expressions is an expression which is a factor of each of them. 101. Type VIII. When the terms of the expression have a common factor. 1. Factor Uab*- 35 a^b\ Each term contains the monomial factor 7 ab^. Dividing the expression by 7 ab^, we have 2 6^ — 5 a^. Then, 14:ab*-S5a^b^ = 7 ab\2b^-5a^). 2. Factor (2 m+3)x'^ + (2 m+3)y\ The terms have the common binomial factor 2 m + 3. Dividing the expression by 2 m + 3, we have x^ + y^. 2 m + 3)(2 m + 3)a;2-h (2 m + 3)y^ x^ +y^ Then, {2 m-h3)x^+{2m + 3)y'' = {2 m + 3)ix^-{-y^). (See example 6, Exercise 22.) 3. Factor (a—b)m + {b—a)n. By §46, 6-a=-(a-6). Then, (a — b)m+(b — a)n — {a — b)m — (a — b)n = (a — b)(m—n). We may also solve Ex. 3 as follows : {a—b)m+(jb — a)n = {b — a)n—(b — a)m — (jb — a){n—m). IK 4. Factor 5 a(x—y)--3 a(a:4-2/). Hi 5a{x-y)-3a{x + y)=a[5{x-y)-S{x+y)] B =a(5 x-5 y-S x-3 y) ^^ =a(2 x-8 2/) =2 a(a?-4 y). After a common factor is removed one or both of the factors may admit of further factoring. 72 ALGEBRA 5. Factor a^x^+2 a^xy +a^y^. Dividing by the common factor a^, we have for the factors a^ and The trinomial is factorable by § 91. Whence, a^x^-\-2 a^xy-\-a^y^ — a^{x+y){x-\'y) —a\x+yy. EXEBCISE 39 Factor the following : 1. 36m2-48m2p. 7. {h'-k)a^'-{k-h)^c\ 2. a^-3a^6 + 3a^62-a2R 8. c\c^-2)+4y^{2-c^). 3. 21x^y-33xy^-\-12xy, 9. (ar+2/)H4 ^•(:r-f ^y). 4. 14 z^xc- 28 zV+7z*a:c2. 10. 4d3(df-l)~(l-d!). 5. (a+2y'-(a+2)d^a. 11. 4(3 a?4-2)-h4(2 x+S). 6. (2a;+7)x2 + (2x+7). 12. (a-a;)3~5(a-a!)2. 13. (2m+3)a2-(2m + 3)62. 14. (m-l)a'-'(m-l)b\ 15. (a4-6)a2 + (a+6)2a6 + (a+6)62. 17. (m—dy—2m(m—dy+m^(m—dy. In every expression to be factored first remove the common factor, if any, then factor the remaining part if possible. Sometimes it is necessary to group the terms (§§ 46, 47), to show a com- mon factor, then apply the method of Type VIII. 18. ab—ay + bx—xy. By § 46, ah — ay-\-hx — xy = a{h — y) + x{b — y). The terms now have the common factor h — y. Whence, ab — ay + hx — xy = {h — y) (a + x) . 19. a^+2a2-3a-6. If the third term is negative it is convenient to write the last two terms in parenthesis preceded by a — sign, § 46. Thus, a'4-2 a^-S a~6 = (a» + 2 a") - (3 a + 6) =a2(a + 2)-3(a + 2) = (a4-2)(a2-3). PRODUCTS AND FACTORS 73 20. ac-fod+tc+td. 24. Sxy + 12ay + 10 bx + 15 ah. 21. xy — Sx+2y—G. 25. m^+6 m'— 7m—42. 22. mx+my—nx—ny. 26. 6— 10 a-f-27 a^— 45 a^ 23. ah-a-bh+b, 27. 20 a6~28 arf-5 6c-f 7crf. Be sure that the factors of your final result will not admit of further factoring. 28. x^+2x^y+xy\ 30. x\a+b)-^^ y''{a+b). 29. a+^ab + ^ab\ 31. 108 A:V--36 fc'+S 5\ 32. m2(2m+3)-3m(2m4-3)-10(2m4-3). 33. 9 <2(3 ^+2) +8 1^(3^+2) +4(3^+2). 34. d\d+^c)+21c\d+Zc), 35. 5 aV-10 a^xy+5 aY-20 aV. 36. 48a«-243a26^ Solve the following by inspection : 37. 982 = (100~2)2 =(10000-400+4) =9604. 38.992=? 42.982-22=? 46.762-42=? 39. 1042=? 43. 1022-982=? 47. 972-932=? 40. 352=? 44. 682=? ^8. 1112-112=? 41. 652=? 45. 782=? The examples under Type IV afford a valuable application of the method in Type VIII. 49. Factor 6 a;2— 7 a;— 20. Multiply 20 by 6 (the coefficient of x^). Factor -120 so that the sum of the factors is —7 (the coefficient of x). These factors are — 15, 8. Thenwrite 6 a:2-7x-20=6x^-15x + 8x-20. Group by Type VIII, =3x(2x-5) + 4(2a;-5), whence, 6x^-7 x-20={2x-5)(3x + 4:). 50. Factor examples 1-10, Exercise 34, by this method. 74 ALGEBRA 102. Hints on Factoring. For all expressions : First, try Type VIII. Sometimes the common factor is disguised as in examples 8 and 19, Exercise 39. Second, select the type form to which the expression belongs : Test binomials by means of Types I, VI, VII. Sometimes the binomial form is disguised. See examples 26 and 28, Exercise 40. Test trinomials by means of Types II, III, IV, V. Third, be sure that no factor in the result will admit of further factoring. TYPE FORMS I. a^-62 = (a + &)(a-&). (§89) II. a^ + 2a6 + &^= (« + &)(« + &), o^-2«6+6^ = (a-&)(a-6). ^^^^^ III. x'-^-ax + h. (§92) IV. ax' + hx^tc (§94) V. a;^ + aaJV+l/^ (§96) VI. a^-\-h^:=z{a'\-h){a'--ab^h^), VII. a^-b'^^j an+^n, (§98) VIII. ax-[-ay-^az = a(x + y+z). (§101) MISCELLANEOUS AND REVIEW EXAMPLES EXERCISE 40 Factor the following : 1. 42a^bc-7ab, 3- S a{a-x)+S a{c+d). 2. a;2-5x-36. 4- SQ d^- 72 dR-h 35 R\ PRODUCTS AND FACTORS 75 5. a*— 64. 10. c* — d^. 1 II. I25a'-50a'b + 5a^b\ 273 ' 12. a(6+c)-a(6-c). 7. 8a^-14a6~155^ 13. x\5y-2zyx\2y-{-z). 8. 27a:'-8z^ M- a^~16 aV+64 c^ . , , 1 ,3 15. a«-26a'-27. 2^ i6. a;^^-2x^ + l. 17- ax — ay-\-az — bx-{-by — bz, 18. (a + 6)2 + 14(a-|-fe)+24. 19. (a;— 2/)^— 15(a?— 1/) — 16. ' 20. 4 c\c+d) + 12 cd(c+d) +9 d\c-\-d). 21. 2 c'(2 c+S d)+5 cd{2 c + 3 d)+2d\2 c + 'dd). 22. 18a?2-27a6x-35aW. 23. x^ + {bc+2d)x + l^cd. 24. 7a:2(3a-2 6)-3a;2(2a-3 6). After factors are found always unite any similar terms which occur in parenthesis. 25. {x^-\-x-2y-{x''-x-\'^y, 26. 64a3-fl000. 2^. a'-c^-d''-\-¥-2ab-2cd, 28. mm?-{x-yy + 12 m + l, 29. 3(m+n)2-2(m2-n2). 30. 2 a^x-S a^x^+2 a^x^-S ax\ 31. 2:ri/-2a:y-264a:V. 32. 8a(2-3y+a;)+5c(3 2/-a:-2). 33. h^—k^+h + k. 34. m^4-/M4-a:^ + .t\ Find the factors common (§ 100) to the following ex- pressions : 35. x^+x-6. 4 a:2-ll x+e. 36. a2-9 c2, a2+4 ac-21 c\ a^-21 c\ 37- xy + S cx + 2 cy + 6 c^, y^ — 5 cy^ — 24c^y. 76 ALGEBRA 38. xix''+2x+2)-\-2(x''-\-2x-i-2), x'+i. 39. (2c-y)ic'-(2c-y)4cy-\-(2c-y)y\ (2c-yy 40. 6a2+a-2, 90 a^ -25 a^- 10 a, 4a2+2a-2. 41. 07^+2 x2+2x + l, x^ + 1. 42. Solve, using factoring: A square, 441 feet on a side, has a grass plot within it, 432 feet on a side. The remain- ing part of the square is a concrete walk. Find the cost of the walk at 140 per square foot. Additional work in factoring will be found in §§ 236 and 237. SOLUTION OF EQUATIONS BY FACTORING 103. The solution of equations affords an important and interesting application of factoring. Let it be required to solve the equation (x-3)(2a;+5)=0. It is evident that the equation will be satisfied when x has such a value that one of the factors of the first member is equal to zero ; for if any factor of a product is equal to zero, the product is equal to zero. Hence, the equation will be satisfied when x has such a value that either ^_3^q /i\ or 2ic+5 = 0. (2) 5 Solving (1) and (2), we have a: = 3 or — -• It will be observed that the roots are obtained by placing the factors of the first member separately equal to zero, and solving the resulting equations. 104. Examples. I. Solve the equation ar^ — 5 a: — 24=0. Factoring the first member, (a:-8)(x4-3) =0. (§ 92) Placing the factors separately equal to (§ 103), we have a; — 8=0, whence x = S; and a: + 3=0, whence x= —3. Verify by substituting x = S,x=—S successively in the given equation. PRODUCTS AND FACTORS 77 2. Solve the equation 4 x- — 2 a;=0. Factoring the first member, 2 x{2 x — 1) =0. Placing the factors separately equal to 0, we have 2 x—0, whence x=0; and 2 a; — 1=0, whence a:=-- Verify these results. 3 . Solve the equation o[^ + 4:X^—x-'4:=0. Factoring the first member, we have by §§ 89, 101, (.r + 4)(a;2-l)=0,or (a; + 4)(a: + l)(x-l) =0. Then, x + 4 = 0, whence a; = — 4 ; x + 1 =0, whence a;= — 1; and X— 1=0, whence x = l. Verify these results. 4. Solve the equation a:^-27-(a:2 + 9 a;-36)=0. Factoring the first member, we have by §§ 92 and 97, (x-3)(x2 + 3 x + 9)- (x-3)(x+ 12) =0. Or, {x-S){x^-{-Sx + 9-x-l2)=0. Or, (x-3)(a;2 + 2x-3)=0. Or, . {x-S){x + S){x-l)=0. Placing the factors separately equal to 0, x=3, —3, or 1. Verify. The pupil should endeavor to put down the values of x without actually placing the factors equal to 0, as showTi in Ex. 4. EXERCISE 41 Solve each equation and verify results : 1. a^2_4x-21=0. 5. ^'-/-12=0. 2. x^-4:x = 0. 6. 22_8z + 12 = 0. 3. 6ir^-12x2=0. 7. F + 7A: + 12=0. 4. (2a:-7)(a:2~16)=0. 8. 6 i^^,. 17 ^,^12 = 0. 9. 9 v\2 v-S)-9v(2 7;-3)-4(2 i;-3) = 0. ID. S x^ — kx—4: k^=0. (Solve for a;, then solve for A*. ) 11. I0u^-7u-12 = 0. 14. 4x3+20.r2_9^_45_0. 12. a;z + 2a:-3 2-6=0. 15. 28/2-^-2 = 0. 13. 15^Hv-2=0 16. IS x^^- 27 abx- 35 a'b'=0. 78 ALGEBRA 17. n2-fl4n~32=0. 18. a;24-8a: + 16=0. 19. m^ + 6 7/1^—9 m--54=0. 20. {x-sy-(Sx+2y=o, 21. 10 '^2-39 ^4.14=0. 22. 15a;2+ii;_6=0. 23. (4 :r2-49)(a;2-3 a;- 10)(8 a;2-fl4 x- 15)=0. 24. (x-2)(5a;H8a;-4)-(ar2-4)=0. 25. What number added to its square gives 30 ? 26. What number subtracted from 4 times its square gives 1? 27. If to 4 times the square of a certain number we add three times the number the result is 10. Find the number. 28. A rectangular room is 4 feet longer than it is wide, and its area is 96 square feet. What are its dimensions? Let w= the number of feet in the width, then w+^— the number of feet in the length. w(w + 4:)=9Q, w^ + 4w-9Q=0, {w-S){w + 12)=0. Whence, ti;=8or — 12. Then, W7 + 4 = 12or-8. Since we are finding dimensions of a room, these negative roots have no significance and can be rejected. There is, however, a very interesting geometrical interpretation which may be given. Consider § 10 and Exercise 4. If measurement to the right is positive, then measurement to the left is negative. If distance upward is 4- , then distance downward is — . "7 , "^^ Now draw this rectangle : _8' This gives two rectangles which fulfill the condi- L tions of the problem, if one remembers that —12 is algebraically less than —8. 29. In a right triangle ABC^ the base, ^C, is 3 feet more than the altitude, J5C, and the area is 14 square feet. Find A C and BC, Make a diagram with your results. +12 PRODUCTS AND FACTORS 79 30. The perimeter of a rectangular field is 180 feet, and its area 1800 square feet. Find its dimensions. Make a diagram of your results. Find the equations whose roots (§ 73) are : 31. 2, -f. Subtracting each root from x, we have (x-2), (x-f). By reversing § 103, the product of these expressions equated to zero gives the required equation. Whence, (x-2)(a;+f)=0, or expanding, 3 x2-a;-10==0. 32. 1, 3. 35. 2, -3, 4. 38. 6, -^. 33. f , f . 36. a, 6. 39. J^, 0. 34. -1, 4. 37. ^, ^, a. 40. a-l-2 6, a-2 6. 41. The sides of a rectangle are 8 and 11. Form a problem similar to problem 28. State the equation. QUEBIES 1 . Is 2 a a number? Is it a sum? Is it a product? What are its factors? 2. Is a + 6 a number? Isit the sum of two numbers? Can you factor it? 3. Translate a^ + b^ into EngHsh. Can you factor it? 4. Given two numbers F and S; if their sum is multiplied by their difference, what is the result? 5. Given two numbers F and S ; if their sum be multiplied by itself, what is the result? Express in English. 6. Does the definition of division bear any relation to your idea of the process of factoring f 7. Is 4 a^ + 2 a+1 a perfect square? Why? 8. The following are for mental drill: (30 J)2 = ? (20})^ = ? (29i)'=? 9. Is 3 a root of the equation 3 x=^-4 a; + 7=0? Why? Is x-S a factor of the expression? 10. Is2arootof 2m^— 9m+10=0? Is 7^—2 a factor of the expression? 11. How do you form the equation whose roots are 3 and 7? 12. If one root, 5, of a;^ — 8 a; H- 15=0 is given, can you find the other root without solving the equation? 13. Using your knowledge of § 91, can you make a general statement covering the results of examples 13 and 14, Exercise 12? 80 ALGEBRA VIII. HIGHEST COMMON FACTOR. LOWEST COMMON MULTIPLE (We consider in the present chapter the Highest Common Factor and Lowest Common Multiple of Monomials, or of Polynomials which can be readily factored by inspection. The Highest Common Factor and Lowest Common Multiple of poly- nomials which cannot be readily factored by inspection, will be con- sidered in a more advanced course in algebra.) HIGHEST COMMON FACTOR 105. The Highest Common Factor (H. C. F.) of two or more expressions is their common factor of highest degree (§58). If several common factors are of equally high degree, it is understood that the highest common factor is the one having the numerical coeffi- cient of greatest absolute value in its term of highest degree. For example, if the common factors were 6 x and 2 x, the former would be the H. C. F. 106. Two expressions are said to be prime to each other when unity is their highest common factor. 107. Case I. Highest Common Factor of Monomials. Ex. Eequired the H. C. F. of ^2aW, 7Qa?bc, and 98 a'hH\ By the rule of Arithmetic, the H. C. F. of 42, 70, and 98 is 14. It is evident by inspection that the expression of highest degree which will exactly divide a^6^ a'^hc, and a^W is a'^h. Then, the H. C. F. of the given expressions is 14 a^6. It will be observed, in the above result, that the exponent of each letter is the lowest exponent with which it occurs in any of the given expressions. EXERCISE 42 Find the H. C. F. of the following : 1. 14 x^y, 21 xyK 3- 36 m^b\ 48 mW, 60 m'b. 2. 64 a^h\ 112 ¥c\ 4. 25 ac\ 30 aV, 35 ac. HIGHEST COMMON FACTOR 81 5. 32 a'x\ 128 a%''x\ 192 aVy\ 6. 136 a^V, 51 b^mn\ llOc^mV. 7. 60(x-y)^ 84(a;-z/)^ 108. By §48, (+a) x(+6) = +a6, (+a) x(-6) = ~a6, (-a) x(+6) = ~a6, (-a) x(-6)= +a6. Hence, in the indicated product of two factors, the signs of both factors may be changed without altering the product ; but if the sign of either one be changed, the sign of the product will be changed. If either factor is a polynomial, care must be taken, on changing its sign, to change the sign of each of its terms. Thus, ib — a)(n — m) may be written in the form — {b — a){m — n), or —{a — b)(n — m). In like manner, in the indicated product of more than two factors, the signs of any even number of them may be changed without altering the product ; but if the signs of any odd number of them be changed, the sign of the product will be changed (§ 49). Thus, {a — h){c — d){e — f) may be written in the forms (a-6)W-c)(/-e), {h-a){c-d){f-e), -{b-a){d-c){f-e), etc. 109. Case II. Highest Common Factor of Polynomials which can be readily factored by Inspection. I. Required the H. C. F. of 5 x^y—^5 x^y and 10 a;^^/^— 40 x^y^-'210 xy^. By §§ 101, 89, and 92, 5 x'y-45 xhj==5 x^y{x^-9) = 5x'y{x + S){x-S); (1) and 10 a^V-40 0^^-210 xy^ = 10 xy^{x^-4: a; -21) = 10xy'(x-7){x + S). (2) The H. C. F. of the numerical coefficients 5 and 10 is 5. It is evident by inspection that the H. C. F. of the literal portions of the expressions (1) and (2) is xy(x + S). Then, the H. C. F. of the given expressions is 5 xy{x-\-3). 82 ALGEBRA It is sometimes necessary to change the form of the factors in finding the H. C. F. of expressions. 2. Find the H. C. F. of 0^+2 a - 3 and 1 - a\ By §92. «2_^2o-3 = (a-l)(a + 3). By § 97, 1 -a' = (1 -a)(l +a-f a^). By § 108, the factors of the first expression can be put in the form -(l-a)(3 + a). Hence, the H. C. F. is 1 -a. EXERCISE 43 Find the H. C. F. of the following : 1. 10a;y-40ary, 25xy^-15xy\ 2. c^-25b\ c2-10 6c+25R 3. a2~5a~36, a^-4a-32. 4. tz+5z-7 tS5, th+Stz + 15z. 5. 2a^-ab-Sb\ Sa^+ab-'2b\ 6. 9a2-25 62, 9 a"" - SO ab +25 b\ 7. SuHl, 4n2-2n4-l. 8. /i2-3n~40, n2+4n-5, 2n2+6n-20. 9. t^-\-2t^+t+2, t*+3P+2. 10. v^—v^—v + 1, v^—2v^-\'\, 11. 6a2-fa-2, l2a?+ba-2, 6 aH4 a- 15 az- 10 2. 12. 25P-16, 25 4^-40 Jfc + 16, 30P^A:-20. 13. a^-32, a2+9a-22, a^-S. 14. I~lla + 18a2, Sa^-l, 18a2-5a-2. 15. x^ + 3.t2-40, a:*-25, a^+a^-Sa-S. 16. a2-(6+c)2, {b-ay-c\ b'-(a-cy. 17. (x2+.r-f2)(a?2~a?-2), a:2^5a:-6, x''-Sx-9. 18. 2 a2(2 a+3 0+5 at{2 a + S t) -f 3 i\2 a+3 /) and 4 a'ii+a) + 12 a/(/-f a) +9 /^(/-f a). LOWEST COMMON MULTIPLE 83 LOWEST COMMON MULTIPIjE 1 10. A CommQn Multiple of two or more expressions is an expression which is exactly divisible by each of them. 111. The Lowest Common Multiple (L. C. M.) of two or more expressions is their common multiple of lowest degree. If several common multiples are of equally low degree, it is understood that the lowest common multiple is the one having the numerical coeffi- cient of least absolute value in its term of highest degree. For example, if the common multiples were 4 x — 2 and 6.t — 3, the former would be the L. C. M. 1 12. Case I. Lowest Conmion Multiple of Monomials. Ex. Required the L. C. M. of 36 a^x, 60 aY, and 84 c:x^. By the rule of Arithmetic, the L. C. M. of 36, 60, and 84 is 1260. It is evident by inspection that the expression of lowest degree which is exactly divisible by a^x, d^y"^, and cx^ is ahx^y^. Then, the L. C. M. of the given expressions is 1260 a^cx^y^. It will be observed, in the above result, that the exponent of each letter is the highest exponent with which it occurs in any of the given expressions. EXERCISE 44 • Find the L. C. M. of the following : 1. 5 xy, 6 xy. 5. 105 a% 70 b% 63 c^a. 2. 18 a% 45 b^c. 6. 50 xy, 24 xy, 40 xy, 3. 28 x\ 36 y'. 7- 2*1 ab\ 35 bV, 91 a^cK 4. 42 m'n\ 98 ny. 8. 56 a^b\ 84 bx\ 48 xy, 9. 60 a'bc\ 75 a'b% 90 a'c'd'. ID. 99m*nx^, Q^mVy^, lQ5n^xy. 113. Case II. Lowest Common Multiple of Polynomials which can be readily factored by Inspection. I. Required the L. C. M. of x^— 5 x-\- 6, x^— 4 x+ 4, and x^— 9 x. By §92, x'-5 x + 6 = {x-S){x~2). By §91, a;2-4a; + 4 = (x-2)2. By § 89, x^-9 x=x{x + 3){x-3). 84 ALGEBRA It is evident by inspection that the L. C. M. of these expressions is x{x-2y{x+S)ix-S), It is sometimes necessary to change the form of the factors. 2. Find the L. C. M. of ac-bc-ad+bd and b^-aK By §101, ac-bc-ad + bd={a-b){c-d). By § 89, b'-a^ = (b + a)ib-a). By § 108, the factors of the first expression can be written {b-a){d-c). Hence, the L. C. M. is (b + a)(b-a)(d-c), or {b^-a^){d-c). EXERCISE 45 Find the L. C. M. of the following: 1. x2-7a!+10, a;2-8x+15. 2. A:2-4, A:2-7A;+10, P-5F+4A:-20. 3. 2a^-a-h 2a2+^a+2, 2a2+7a+3. 4. R^-SR+2, R'-5R+G, R'-4R+3. 5. a2-8a-3, a^-3a+2, a^-l. 6. m-2, m2-2m+4, m^-6m^ + 12 m-S. 7. x^+x^ + l, x^+x+1, x^-x + 1, 8. k+l, k+Sl, l-k, k-SL 9. (X-2XX-3), {x-S)(x-i), (4-x)(2-'x). 10. a2~9, a^-27, a-3, a^+3a + 9. Find the H. C. F. and the L. C. M. of the following : 11. m2+3m+2, m^+5m+6, m''+4x+3. 12. a^+iab+4b\ a^-4b\ a^ + 2ab. 13. 2k'+7k-4, 3P + 13A:+4. 14. 2x''-3ax+a\ 2x''-5ax+2a\ x'''-3ax+2a\ 15. 9t^-25v\ 6tx + l0vx, 12tx + 20vx. Find the L. C. M. of the following : 16. dx^-Ux^'-hix, 18ax' + 12ax^+Sax\ and 27.r^-8. 17. {x-{-zy-y\ {x+ijy~z\ x^-iy+zy. 18. (c- 1)2 + 3 c, c^-l, c-1. 19. (e-\-yy—4ey, e^ + 2 e-y + ey^, e*+ey^. 20. utHS, 4ir2-(.T2 + 4)2.* FRACTIONS 85 IX. FRACTIONS 1 14. The quotient of a divided by b is written ?. b The expression - is called a Fraction; the dividend a is called the numerator^ and the divisor b the denominator. The numerator and denominator are called the terms of the fraction. 115. It follows from § 62, (3), that If the terms of a fraction be both multiplied, or both divided, by' the same expression, the value of the frac- tion is not changed. 1 16. By the Rule of Signs in Division (§ 61), +«_— a_ +«_ —a + b~ -b~ -b~~ +b That is, if the signs of both terms of a fraction be changed, the sign before the fraction is not changed; but if the sign of either one be changed, the sign before the fraction is changed. If either term is a polynomial, care must be taken, on changing its sign, to change the sign of each of its terms. Thus, the fraction ^^^, by changing the signs of both numerator and c — d T_ denominator, can be written -; (§ 44). d—c 117. It follows from §§ 108 and 116 that if either term of a fraction is the indicated product of two or more factors, the signs of any even number of them may be changed without changing the sign before the fraction : but if the signs of any odd number of them be changed, the sign before the fraction is changed. Note : To change the sign of a factor is to change the sign of every term of the factor. I Thus, the fraction - — ^^— may be written {c-d){e-f) a — b b—a b—a ,^ , etc. (^d-c){f-c) {d-c)(e-f) (d-c){f-c) 86 ALGEBRA EXERCISE 46 Write each of the following in three other ways without changing its value : a ^ n+3 ^ 8 ^ 2x-7 _ 6a:-5 2 7 2-x x^-2 (a:-3)(t/+4) 6, Write ^ — in four other ways without changing its value. REDUCTION OF FRACTIONS 1 18. Reduction of a Fraction to Lower Terms. A fraction is said to be in its lowest terms when its nu- merator and denominator are prime to each other (§ 106). (We consider in this text those cases only in which the numerator and denominator can be readily factored by inspection. The cases in which the numerator and denominator cannot be readily factored by inspection are considered in the second course.) 119. By § 115, dividing both terms of a fraction by the same expression, or cancelling common factors in the numera- tor and denominator, does not alter the value of the fraction. We then have the following rule : Resolve both numerator and denominator into their factors, and cancel all that are common to both. 1. Beduce — — - to its lowest terms. 40 a'h^cH^ We have 24 a^h^cx ^ 2^X3 Xa^6^cx ^3a^ 40a26Vd3 23x5Xa26Vd3 5 ^^3 ' by cancelling the common factor 2^Xa^6^c. * ' x^—21 2. Reduce — to its lowest terms. x^-2x-^ By §§97 and 92, ^3-27 ^(x-3)UH3 x + 9)_x^4-3 x-f9, •^ "^^ ' x2-2x-3 (a;-3)(a:+l) x-hl 3. Reduce ,^^ j^ ^ to its lowest terms. FRACTIONS 87 By §§89 and 101, ax-6x-ay + by ^ (a-6)(x-t/) . 0^ — a^ (o + a)(6 — a) By § 117, the signs of the terms of the factors of the numerator can be changed without altering the value of the fraction; and in this way the first factor of the numerator becomes the same as the second factor of the denominator. Thpn ax-bx-ay+by ^ (b-a)(y-x) _y-x ' b'-a' (6+a)(6-o) b + a If aU the factors of the numerator are cancelled, 1 remains to form a numerator; if all the factors of the denominator are cancelled, it is a case of exact division. EXERCISE 47 Reduce each of the following to its lowest terms : 5 xyz\ ^ 54mn\ 126aW^ 90 a^m''n\ Sxyh^' '^* 99 mV ^' 14 aV ' ^* 36amV' 12 a%\ 63 xYz\ 6 26 m^nY g 88 x'yV 42 bV '^* 84 xy/ * ISO m'ny* ' 66 x^yz^' 120a76V<^ 15x*y + 10xY a;2~9a?4-18 Q. • ID. ^ ^« II. 1 . 75abV 6a;32/H4a:y a?2-fx-12 12. ^'-5m-84 4a^-16ad+l5d\ a'm^-a'm-m a} 4 a?- 12 ad-\-^ d? dx^-7 xz-20z\ a^^a-12 4 0^2-25 z2 3a2-13a + 12 16. (a?^--49)(a:^-16a?+63) 17. (x^-U x-^i9)(x^-2 X-6S) 12 a^m^+48 a'- 10 m^b-40 b 36a«-60a^6+25 62 o 36a2 + 97ac+36c2 27 6^-8 a^ 15. • 20. 9a2 + 13ac+4c2 16 a2-32 a6 + 12 6^ 18a2-3ac-10c2 165t^-{-2t-l 19. • 21. 36a2-25c2 15 t'+Ut^-t^ 88 ALGEBRA ^^' (4a-2by-{3c-dy ''^' {x-2vy-(u'-yy 4 c^(2 c-3 d)-6 cd(2 c-S d) + 9 d\2 c-3 d) '''** 10c'+cd-24d' 120. Reduction of a Fraction to an Integral or Mixed Expression. A Mixed Expression is a polynomial consisting of a ra- tional and integral expression (§ 57), with one or more frac- tions. Thus, a + - , and - H — ^^^ are mixed expressions. c 3 x—y 121. A fraction may be reduced to an integral or mixed expression by the operation of division, if the degree (§ 58) of the numerator is equal to, or greater than, that of the denominator. 1. Keduce to a mixed expression. O X By§65, §-^!±lA£zi2^6x^ + 15^_A = 2:r + 5-^. Sx 3a;3.r3a; 3x i^^A ^ Ux^-Sx^+^x-d. . . 2, Keduce to a mixed expression. 4 ^2 + 3)12 x^-S x2 + 4 x-5(3 a;-2 12 x^ +9x -8x^-5x-5 -Sx^ -6 -5x + l Since the dividend is equal to the product of the divisor and quotient, plus the remainder, we have 12 x'-S a:2 + 4 a:-5=(4 a;2 + 3)(3 a:-2) -f (-5 x-f 1). Dividing both members by 4 x^ + 3, we have 12a;^-8a;^ + 4a;-5 _3^ o. -!)X-\-\ 4x^ + 3 4x^ + 3 ' FRACTIONS 89 Thus, a remainder of lower degree than the divisor may be written over the divisor in the form of a fraction, and the result added to the quotient. If the first term of the numerator is negative, as in Ex. 2, it is usual to change the sign of each term of the numerator^ changing the sign before the fraction (§ 116). Thus, Ux'^-Sx^+Ax-b ^3^_2-^ 5x-l I 4x^ + 3 4x^+3 EXERCISE 48 Reduce each of the following to a mixed expression : 25a2-10a+ll ^ a^+32 1. • o. • 5 a a— 2 16m^ + 12m3+8m2^9 c^+d' 2. • O, • 3m2 "^ c+d ^' 2a:2 + l* '^* 4ir2~2ir+5 ^' a;2+3a; + 9 "* x^-^2x-7' a:Hv' . 3a;3 + 7a;2 x — y 3a;^+a;— 9 ^ 8a3-27c^ 8a2-22a6-2162 0. . 13. . 2a + 3c 2a-7& Sa'+12a^b^-15b\ ^ ^dx"" -96xy+27y\ 2a' + Sh' ' ''^' rx^'+xy-lSy^ 122. It is evident from § 121 that a mixed expression may be result of division. Since the dividend is equal to the pro- duct of the divisor and quotient plus the remainder, to reduce a mixed expression to a fraction, Multiply the integral part by the denominator of the fraction, add the numerator to this result and write the denominator under this sum. Note : If a minus sign precedes the fraction, change each sign in the numerator. 90 ALGEBRA Ex. Reduce 2 a;— 3— to a fractional form. x-\-\ 2x 3 ^^-5 - (2a;-3)(a;+l)-(4a;-5) x+l x-hl _ 2x^-x-S-4:X-\-5 2x^-bx + 2 -3+^'' 5a-l- 3. 2/t + ll + 2c 6a'-2 5 a 3 x+l x+1 EXERCISE 49 8. x-4:a- -4a2 a; 9. a; 4- 5 a— 20 a x—b a 10. 2t-^u- 8/^27^'' II. 2a-56- 4a2-2562 2a-56 12. a:^4-2 xu ..M l^.y^ 6n+2 3a;— 4i/ aH2a6+6^ ^ 4 a6 ~ '^ ' "^^ ' ^ ' x—2y {x-yY 4a-2b ^ (Sc-Sdy , 10a2^29ac + 10c2 ,^ 9 0^—64^2 3 a— c 123. Reduction of Fractions to their Lowest Common De- nominator. — To reduce fractions to their Lowest Common Denominator (L. C. D.) is to express them as equivalent fractions, each having for a denominator the L. C. M. of the given denominators. Let it be required to reduce — ^ . — —, and -^^ to their 1 J. J • ^ 3 a^b^ 2 ab^ 4 a^b lowest common denominator. The L. C. M. of 3 a^b\ 2 ab\ and 4 a^b is 12 a^b' (§ 112). By § 115, if the terms of a fraction be both multiplied by the same expression, the value of the fraction is not changed. Multiplying both terms of -^^ by 4 a, both terms of -^-^ by 6 a% 3 a^b^ 2 ab^ and both terms of — ?- by 3 6^, we have FRACTIONS 91 16 acd 18 a^bm , 15 6^n 12 a'b^' 12 a'b^' 12 a'b^' It will be seen that the terms of each fraction are multi- plied by an expression, which is obtained by dividing the L. C. D. by the denominator of this fraction. Whence the following rule : Find the L. 0. M. of the given denominators. Multiply both terms of each fraction by the quotient obtained by dividing the L. C. D. by the denominator of this fraction. Before applying the rule, each fraction should be reduced to its lowest terms. 124. JEx, Reduce— — -and— to their lowest com- . . . a2-4 a2-5a+6 mon denominator. We have a2-4 = (a + 2)(a~2), and a2-5a + 6 = (a-2)(a-3). Then, the L. C. D. is (a + 2) (a -2) (a -3). (§ 113) Dividing the L. C. D. by (a + 2) (a — 2), the quotient is a — 3; dividing it by (o — 2)(a — 3), the quotient is a + 2. Then, by the rule, the required fractions are 4a(a-3) ^^^ 3a(a + 2) (a + 2)(a-2)(a-3) (a + 2) (a -2) (a -3) EXERCISE 50 Reduce the following to their lowest common denominator: 7ab Sbc 2ca ^ 4 a^ 2 I- -7r-> -TTT* -7^- 5 2. 6 ' 10 ' 15 * 4a2-9' Qa^'-da 1 3 mn 2 m^n^ 2 7n?n 5m^n^' 7 mn^ m—n' 2(m— n)^' 3(m— n)' 3 a;+4 z 6 x—5 y 3 n 5 22 xy'' ' 33 2/2^ ' n^-S' n2-4n+4* llc^p 9a^m 8 hhi g 2 3a Ua^h' \4tb'c 21 c^a ' a^-hS a^-f 2 a-f 6' aH27' 92 ALGEBRA ADDITION AND SUBTRACTION OF FRACTIONS 125. By§65, ^ + £_^ = *±£r:i. a a a a We then have the following rule : To add or subtract fractions, reduce them, if necessary, to equivalent fractions having the lowest common de- nominator. Add or subtract the numerator of each resulting frac- tion, according as the sign before the fraction is -h or — , and write the result over the lowest common denomi- nator. The final result should be reduced to its lowest terms. 126. Examples. I. Simplify -J-ZI-+ -^-rr-' The L. C. D. is 12 a^b^; multiplying the terms of the first fraction by 3 6', and the terms of the second by 2 a, we have 4q + 3 l-6b^ _ 12a6^ + 9b^ 2 0-12 ab' 4 0^6 6ah^ 12 a^b^ 12 a^b^ ^ 12 ab^ + 9 b^-\-2 a-12 ab^ ^ 9 bH2 a 12a^¥ 12 a^b^ ' If a fraction whose numerator is a polynomial is preceded by a — sign, it is convenient to write the numerator in parenthesis preceded by a — sign, as shown in the last term of the numerator in equation (A), of Ex. 2. If this is not done, care must be taken to change the sign of each term of the numerator before combining it with the other numerators. 5 a? --4 y _ 7 x—2y 6 The L. C. D. is 42; whence, z. Simplify ^ ^^ FRACTIONS 93 5x — 4:y 7 x — 2y _ 35x — 28y 2ix — 6y 6 14 42 42 3. Simplify _ 35a;-28y-(21 x-6y) ,.. 42 ^^^ ^ S5x-28y-21 x-{-6 y ^ U x-22 y ^ 7 x - 11 i/ 42 42 21 1 1 x^+x x^—x We have, x^ + x=x{x+l)j and x^ — x=x{x — l). Then, the L. C. D. is x{x + l)(x-l), or x{x^-i). Multiplying tlie terms of the first fraction by x — l, and the terms of the second by x + 1 , we have 1 1 ^ x-1 x+1 x^-\-x x'^ — x x{x^ — l) xix"^—!) ^ a;-l-(a;4-l) ^ a:-l-a;-l ^ -2 x{x^-l) x{x^-l) x{x^-l)' By changing the sign of the numerator, at the same time changing the sign before the fraction (§ 116), we may write the answer -— = x{x^ — l) Or, by changing the sign of the numerator, and of the factor x^ — 1 of 2 the denominator (§ 117), we may write it x{l-x^) t o 4. Simplify a^-3a+2 a^-4a+3 a'^-Ba+Q We have, a^-S a + 2 = (a-l)(a-2), a2-4 a + 3 = (a-l)(rt-3), and a2-5a + 6 = (a-2)(a-3). Then, the L. C. D. is (a-l)(a-2)(a-3). Whence, — -^^ — ? + - ^ a2-3a + 2 a2-4a + 3 a^-5a-hQ a-S 2(a-2) (a-l)(o-2)(a-3) (a-l)(a-2)(a-3) (a-l)(a-2)(a-3) _ a-3-2(a-2)4-a-l _ a-3-2a + 4 + a-l (a-l)(a-2)(a-3) (a-l)(a-2)(a-3) ^ -0. (a-l)(a-2)(a-3) 94 ALGEBRA EXERCISE 51 Simplify the following : 2a;H-9,3x-5 5R'\-2t 3R+St I, — — r — 77: — • 4 16. 12 6RH^ 9Rt^ 8 ^ 2c-7(/3 5c+2d 4aV 7a^c IS d^ 26 c 2a-3 6 Sa-Sb ^ 4a + 3b c+2b 5a-c 10 15 ' ' 2ab Sbc iac' 2(6 n+5) 3(n + 6) 4(5 n-4) 11 22 44 ' ^ 2a-f3f Sa+2t . 5 a-7 t a. — _- -_ y. 9. 14 21 28 3a^-4 _ 4aH2 _ 6a^-2 10 a^ .5 a' 25a^~' 5a;-4 3iy+2 2 2+5 Sx 12 y 6z ' bx-l 9ar-8 12x--ll 2x + 9 5 15 20 10 7^-4 3^-8 7^+7 . 6<-5 12. • 4 5 8 10 13. |(3a+4 6)-A(2a-56) + ^(a+26). 3,2 5c , c2+8c-9 3a;-l 2a; + l 5c-3 c2-|-4c-2 jg _l 2_^ ^^ _^ X 2 a? - 6 ' m + 3 m-5' ^' 2x-3 2a:+3 4x2-9 5 m ^^ 3a^+2 _ 9a;2+4 m-f5 m-3* ^^* Sx-2 9x^-4* ^ a;— 3 X— 4 6a— 5 . 2a— 1 17. — . 23. x-4 x-5 a2-2a-15 2a2-5a-25 g 3 X y 6 a^— 4 a a — 5 ' Sx-i-y Sx-y ^^' a'+27 6^^ + l7a-S 19. . 25. — '-^— — --^ — 2 u. 4a-12 lOa+15 3x-2y ^ Note: We may regard an integer as a fraction whose denominator is 1. FRACTIONS 96 ^ 2c-l , . 2c-f 1 26. h4 — c • 3 c-hl 27. 1 ; 2. m— 1 m + l Tnr—l 28. 30-1-1 222_i0z + 12 5z-7 10 2;2-49z+49 3< 15 ,. 29. h5. ^ + 1 3/2+^-2 c-^d , c^—d? c , d 30. — + TT~:}-~ + - c— d c+c/ c-\-2d In certain cases, the principles of §§ 116 and 117 enable us to change the form of a fraction to one which is more con- venient for the purposes of addition or subtraction. 3 2 6-ha ^''' a-h b'-a'' Changing the signs of the terms in the second denominator, at the same time changing the sign before the fraction (§ 116) (see Exercise 46), we have 3 _ 2b + a a-h a^-b^' The L. C. D. is now 0^-52. rr.. _3 2 6-f-a ^ 3(a + 6)-(2 6 + a) a-n a'-b' a^-b^ Sa+Sb-2b-a 2a+b 33. 02_52 ^2_52 1 11 (x-y)(x-z) (y-x)(y-z) (z-xXz-y) By § 117, we change the sign of the factor y — xin the second denomi- nator, at the same time changing the sign before the fraction; and we change the signs of both factors of the third denominator. The expression then becomes 1 + 1 1 {x-y){x-z) {x-'y){y-z) ix-z)(y-z) The L. C. D. is now {x —y){x — z)(y — z) ; then the result _ {y — z)-\-{x — z) — {x — y) _ y — z-{-x-z — x+y {x-y){x-z){y-z) {x-y){x-z){y'-z) 96 ALGEBRA _ 2y-2z ^ 2{y-z) ^ 2 {x-y)ix-z){y-z) {x-y){x-z){y-z) {x-y){x-z) 2 3 3a:(a-6) a-2b a-h 3a:-12 4-a: x''-h^ x-^h h-x 2 a , 5 n h h^-n2 35. -t-^ + t; 40. a^— 9 3 — a k—a k—h b^—bk . 2e + 3a , 3e-\-4a 2(a+t) , a+t , t-a 30. 1 • 41. — ^^ ^ -| 1 • 2e— 3a 4a— 3^ t a—t t+a ___^--l__ 2-^^ 2 22^+7 3^-5 17^4-2 ^ ' a;2-8a: + 15 a:-5* "^^^ 4-6w 9i*+6 4-9^2' o 'y— 6 t;+6 , 6^— 4a^ m—2 3— m , m— 5 38. ■ f- . 43. 1 • v—2 a v+2a ia^—v^ m—3 4 — m 6— m MULTIPLICATION OF FRACTIONS 127. Required the product of - and - • b a Let f • 3 = ^- (1) a (Multiplication may be indicated by either X or •.) Multiplying both members by h ' d (Ax. 7, § 4), |.£.6. From (1) and (2), ^ • ^ = ^. To multiply fractions, multiply the numerators together for the numerator of the product, and the denominators for its denominator. 128. Since c may be regarded as a fraction having the denominator 1, we have, by § 127, FRACTIONS 97 a a c ac - . c= ~ • -= —• b bib Dividing both numerator and denominator by c (§ 115), a _ ci b '^~bTc' Then, to multiply a fraction by a rational and integral expression, if possible, divide the denominator of the fraction by the expression ; otherwise, multiply the numerator by the expression. 129. Common factors in the numerators and denominators should be cancelled before performing the multiplication. Mixed expressions should be expressed in a fractional form (§ 122) before applying the rules. X. Multiply 1^^ by ^-^. ^^ ^bx"^ ^ 4 ay 10 a^y . 3 6V ^ 2 ' 5 ■ 3 • a^h^x^y ^ bh^x 9 6a;2 '4 aV 32.22. a^hx^y"" 6 y ' The factors cancelled are 2, 3, a^, b, x^, and y. 2. Multiply together i?-±^, 2- 5^, and ^^. a;^+a?— 6 a;— 3 x^— 4 x^-\-2x /o x-A\ ^2-9 (-a) ir^-fx-e \ a;-3/ ^2-4 ^ ^2+2 re , 2a;-6-a; + 4 . ^2-9 rc2+a;-6 x-Z ' x2-4 (aH^)(x-2) ' j^--3 ' (;5-^)C»-^) x-2 The factors cancelled are x + 2, x — 2, x + 3, and x — 3. 3. Multiply ^] by a- 6. Dividing the denominator by a — b, ^^ . • (a — 6) = ^ • a2-6^ a + b 4. Multiply by m+n. Multiplying the numerator by m + n, -^^ . (m + n) = ^^"^^^ . I ALGEBRA EXERCISE 52 Simplify the following : 8 am\ ^ g ^^5 5. 14 6^c ^ 5 c^a ^ 6 a^b 27 6 V 15 a« 12 6^ 7 c 21 a^6^ ^^ 4 c^d° ^ 28 m^ 15 nV 5 a:'^ 8 cd« 35 a^ft^' ' 25 rv'x' ' 14 m^x^ ' 21 m^// ^' Sx^ . 15 y^ . 28z3 g 5c+a: lOj/^ * 72 * 90^2* ' 0^2 + 4 ir-12 35a^6 ^ yi^-6ndH-9f/^ ^' 4n2-36cZ- * 20a62 a2-2a-35 4 0^-9 a 10. • • 2a^-3a^ a-1 {x-2). II* 12. 13. 16 2^-9 2/^ ^ 2 2^H-llziy + 14y^ S z^ +22 zy -21 y^' 4 z'' + 11 zy + Qy""' 4P+4t + l ^ Stc+5c+6td+10d 3t+5 * 4/2 + 10^4-4 a^-Sb^ ^ a^+4ab+4b^ a2-4 62 * 2a^+4:a''b+Sab^' 2ab+b^ \ f 4ab ab )w-2ab+b^ i4.(4-^!±l?^Y-V-^^^,-fl 3 a?-4 y 9 0:^24 a?y + 16 y^ ^ 3a:4-4 2/ 3 0:2—4 xy Note: In problems similar to example 15, indicated multiplication or division must be performed before addition or subtraction is made. 2 X is to be added to the product of the fractions, not to the second fraction. For example, in 13 + 4X3 + 6-5-2-4, 4X3 and 6-5-2 must be per- formed before uniting the terms of the expression. ^ 25m2-40m-fl6 Sm'^-Um^ o . "^ 16. ■ — • 2 m + 4m2-9 25m2-16 2mH-3 FRACTIONS 99 DIVISION OF FRACTIONS C 130. Required the quotient of - divided by - a tr- (1) Then since the dividend is the product of the divisor and quotient (§ 60), we have a c ^ Multiplying both members by - (Ax. 7, § 4), c bed c (2) From (1) and (2), ^-^^==^X-- babe (Ax. 4, § 4) Then, to divide one fraction by another, multiply the dividend by the divisor inverted. If the divisor is an integer, c may be regarded as a frac- tion having the denominator 1. Mixed expressions should be expressed in a fractional form (§ 122) before applying the rules. T^. ., Ga^fe , 9a263 1. Divide — — - by - We have 6a^b . 9 a^fe^ ^ 6 a^b . 10xV ^4y« 5 x^y"^ ' 10 x'y'' 5 x^y^ 9 a^b' 3 b^x 2. Divide 2 - ^-^^ by 3 - ^^'~^^ - /g 2x-3 \ . /g 3a;^-13 \ _ 2a; + 2-2x + 3 . 3 3-^-3-3 3;'^+ 13 x^-\ ' " x^-X ^ 5 . a;^-1 ^ 5(a;+l)(a;-l) _a;-l a;+l * 10 2.5- (a:+l) 2 3. Divide '~ , by m — n. Dividing the numerator by m - n, ^?^^ ' ^ (^ _ n) = ^'^ + ^n + n'^ 100 ALGEBRA 4. Divide - — — by a +6. a—b Multiplying the denominator by a+b, ^— t^ ^ (a+6) = «!±^. a — h a^ — b^ If the numerator and denominator of the divisor are exactl}'^ contained in the numerator and denominator, re- spectively, of the dividend, it follows from § 127 that the numerator of the quotient may be obtained by dividing the numerator of the dividend by the numerator of the divisor ; and the denominator of the quotient by dividing the denominator of the dividend by the denominator of the divisor. 5. Dividei^!^by'i^±2i^. x'—y^ x—y Wchave, 9^'-4?/' . Zx+2y _ Zx-2y x^ — 2/2 x—y x+y EXERCISE 53 Simplify the following: 12 a%^ . 9 a'b^ t^-t-12 . P-St-\-m 55 cW ' 22cHf ' 8^ ' 6^ 9.^2 _-i6 ,^ ' .. ^ ^v'-hv-Vl 4v2_io^-24 3a;+7 ^ ^ Zv^2 9ir^-4 8. 4c2+4cd+d2 ^ 4c2--4c(i-8d2 aH4 ' a2+2a+2 ^3 I /..S 10. 2 h3 • 4a:-h7 3ir-3 11. Divide 2 by 3 • ft '-3 5/+2V>-ri'!=--2Jn9+^+4 +2 y V <+2 FRACTIONS 101 COMPLEX FRACTIONS 131. A Complex Fraction is a fraction having one or more fractions in either or both of its terms. It is simply a case in division of fractions ; its numerator being the dividend, and its denominator the divisor. I. Simplify • "-I ° - • -ax-^(il80)- «* 7_c bd — c bd—c bd—c d d It is often advantageous to simplify a complex fraction by multiplying its numerator and denominator by the L. C. M. of their denominators (§115). a a .,. ,.„ a~b a+h 2. bimplity • b a a—b a-\-b The L. C. M. of a + b and a-b is ia + b){a-b). Multipl5dng both terms by (a + b){a — b), we have a a a — b a + b a(a + b)—a(a — b) _ a^+ab—a^+ab _ 2 ah b I a ~b(a + b)+a(a-b) ab-hb^ + a'^-ab a^ + b^' a — b a-\-b EXERCISE 54 Simplify the following : 11 !_§_? a?— y x+y , r s 7 b - ^ V I. 3. 5. ^— • 1 _ 1 , 9 a^ a:+i/ x—y r s 49 b X y c+— ^ — - + 2+-— ru c^ ^ 3 c 2 a 2. . 4. . 6, . 1+1 i+?l JL+i- r u c 3 c 2 a 102 ALGEBRA 7. 12. i^ „ 2m2-5m-h3 i;2_7^^12 13 4m2-l ^_ 5'y2-72 6m2-9m '?;2^3i;-18 x+2 x+3 \ ^ 14, a;+- +4 x+2 V a:y a:~2 _ x-\-2 , x^-y^-2yz-z^ a: + 3 x-3 9t^-25u^ 15. . _^ x_ (x+y-\-z)(x—y—z) x + 3 x-3 6t-{-10u 2 be 8fe^-6M-35F "' a^-\-b^-c\ ^' ^ ' 5/l^-ll/^A:H-2F ' 2 ah 4,h2-hk-U¥ MISCELLANEOUS AND REVIEW EXAMPLES EXERCISE 56 Simplify the following : 3a— 26 , ,^ , >. . x4-l X— 1 (a-2 6). 4. ^ ' 8c"-f27 c 28. Translate into English: 4^-^ 5.V2a-?-^V ^2a+6 29. Translate into Eniglish : 3 m— 1 ^ o 4 m— 1 4-T-ii • 2a-l 3a-l 30. Translate into English: ^3 m— 1 4 m— IN ~ 3a-l/ ,2a-l 3 1 . State algebraically : The sum of 3 a and b divided by the difference between 3 a and b : Multiply this quotient by the fraction whose numerator is the difference between 9 or and 6^ and whose denominator is the sum of 9 a- and 6-. Seduce your statement. X. FRACTIONAL EQUATIONS. RATIO AND PROPORTION SOLUTION OF FRACTIONAL EQUATIONS 132. If a fraction whose numerator is a polynomial is pre- ceded by a — sign, it is convenient, on clearing of fractions, to write the numerator in parenthesis, as shown in Ex. 1. If this is not done, care must be taken to change the sign of each term of the numerator when the denominator is removed. This is readily understood if one remembers that the line between the numerator and denominator acts as a vinculum. FRACTIONAL EQUATIONS 105 1. bolve the equation =4H — - — • ^ 4 5 10 The L. C. M. of 4, 5, and 10 is 20. Multiplying each term by 20, we have 15 ^-5- (16 ^-20) =80 + 14^+10. Whence, 15 ^-5-16 ^ + 20 = 80 + 14 <+10. Transposing, 15^-16^- 14 ^ = 80 +10 + 5- 20. Uniting terms , — 1 5 < = 75 . Dividing by —15, ^=—5. Verify the result. 2 5 2 2. Solve the equation — „ — =0. ^ x-2 x-\-2 x'-4: The L. C. M. of x-2, x + 2, and x^-4: is x^-4:. Multiplying each term by x^ — 4, we have 2(a; + 2)-5(x-2)-2=0. Or, 2x + 4-5a;+10-2=0. Transposing, and uniting terms, — 3 a;= — 12, and x = 4. Verify the result. If the denominators are partly monomial and partly poly- nomial, it is often advantageous to clear of fractions at first partially ; multiplying each term of the equation by the L. C. M. of the monomial denominators. c 1 ., .. 6^ + 1 2s- 4 2s-l 3. bolve the equation = • 15 7^—16 5 Multiplying each term by 15, the L. C. M. of 15 and 5, 6. + l-30£i:60^e^_3 7s-16 Transposing, and uniting terms, 4 = — ^~ •• ♦ 7s — 16 Clearing of fractions, 28 s - 64 =30 s - 60. Then, - 2 s=4, and s = - 2. Verify the result. EXEKCISE 56 Solve the following equations, verifying each result : i + -l_ = i5_J_. 3 5 X 15 3 a: 106 1+ X 2 3a; 3 5x 2 15 a; 7 30 4_ X 1 4ar 6 1 X 2 3x" 187 "168 3. o 5a:-f-15 4 a: 3 a; = — • 3 5 15 , 4i; , 2^-7 ,1 7 1? 5 4 12 15 ^ P 4/g4-7 . 5fi + 9 9. -- 3 8 2 Zm -1 5m4-l 9 8 m- ^-0 4 m '8m 40 5 m — w. 5 1 5(11-3 1/) 1 7 -91/ 5 u 21 6 w 2i/ 7m b{x -1) 2(a;+2)_ = 4- 5 a:- 15 6 3 4 6gr+4 3gr-4 , ^_5(7+8 8 2 9 2 i;4-l _ 6 v—4: _ i v—5 7 9'i;+l~ 14 8a:H-1 5 11 a:4-15 13 ar + 29 _ 14 a: + 66 3 5 10 15 2^ + 9 ^-11 3x-l a:-hl '4. — — 7 = :— — —' 15. - 13. '4^ + 1 2^-15 18a:-19 6a:-7 16. -^-?^ = 1. (See§ 103.) x—2 X t-^5 t-3 3 5-R S-R 19. x—l x—2 x—S x—4 x—2 x—S x—4 x—5 (Reduce each fraction to a mixed number, then see example 21, Exer- cise 55.) FRACTIONAL EQUATIONS 107 2x ,2x+l 3a;-f2 X+l X x^+x 2 1 . What number added to twice its reciprocal gives 3 ? 2 2. The sum of |^ of a certain number and ^ of its square is J. Find the number. 23. Make two problems similar to 21 and 22. 133. Solution of Special Forms of Fractional Equations. 1. Solve the equation h — — =2. ^ 2x-3 0:2+4 We divide each numerator by its corresponding denominator ; then l + -^ + l-4±i = 2,or -2 ^±1=0. 2x-S x2H-4 2a;-3 x^ + 4: Clearing of fractions, 2 a;^ + 8 - (2 x^ + 5 a; - 12) = 0. Then, 2 a:2-|-8-2 x^-5 x + 12=0; whence, a; = 4. We reject a solution which does not satisfy the given equation. 2. Solve the equation 1 = — • ^ x-S x-2 x^-5x+6 Multiplying both members by (x — S)(x — 2), or x^ — 5 a: + 6, a;-24-a;-3=3a;-7. Transposing, and uniting terms, —x=—2, or a; = 2. If we substitute 2 for x, the fraction becomes - • a:-2 Since division by is impossible, the solution x = 2 does not satisfy the given equation, and we reject it; the equation has no solution. 3. Solve the equation 1 = 1 • ^ x + 10 x+Q x+S ir+9 Adding the fractions in each member, we have 7a;+58 ^ 7x4-58 {x+10){x + Q) {x + 8){x + 9)' Clearing of fractions, and transposing all terms to the first member, (7a; + 58)(a; + 8)(a; + 9)-(7a; + 58)(a:+10)(x + 6)=0. (1) Factoring, (7 x + 5S)[{x + 8){x+9)-{x-\-10)(x + 6)]=0. Expanding, (7 x-f-58)(a;2 + 17a;+72-.T2-16 a:-60)=0. Or, (7x-f-58)(a;+12)=0. This equation may be solved by the method of § 103. Placing 7 re + 58=0, we have x =^ — -^' Placing a:+ 12 =0, we have a; = — 12. 108 ALGEBRA 134. If we should solve equation (1), in Ex. 3 of § 133, by dividing both members by 7 a: + 58, we should have (^-f8)(a: + 9)-(a; + 10)(a;+6)=0. Then, a^HlT a:+72-a;2~16 ir-60=0, or x=-12. In this way^ the solution x= —^^- is lost. It follows from this that it is never allowable to divide both members of an equation by any expression which involves the unknown numbers, unless the expression be placed equal to and the root preserved, for in this way solutions are lost. EXERCISE 57 Solve the following equations : 4a:4-ll 1 1 a;2+a:-20 x+5 x-4: ^ x+S x+4 ^+^-3 * x+2 x + S x+4 x + 9 x+4: x+3 ir + 18 2x+3 2ir-3 36 2a;-3 2 a;+3 Ax^'-d 2ar+5 3x2+24a: + 19 =0. ^' x-\-7 x''+Sx+7 , x^—2x-{-^.x^-\-Sx—7_(^ * x''-2x-3 x^ + Sx-\-l~ SOLUTION OF LITERAL LINEAR EQUATIONS 135. A Literal Equation is one in which some or all of the known numbers are represented by letters ; as, 2a:+a = 62 + 10. X x-^2b a^+b^ Ex. Solve the equation x—a x-\-a FRACTIONAL EQUATIONS 109 Multiplying each term by x^ — a^, x{x-\-a)-{x + 2h){x-a)=a^ + h^, or, x^-\-ax—{x^-\-2hx — ax — 2ah)=a^ + h'^y or, x^-{-ax — x'^ — 2hx + ax-{-2ah=a'^ + h'^j or, 2 ax-2 bx=a^-2 ab + h\ Factoring both members, 2 x{a — b) = (a — by. Dividing by 2(a-6), x = ^^^ = ^. In solving fractional literal equations, we must reject any solution which does not satisfy the given equation. Compare Ex. 2, § 133. EXERCISE 58 1. Find the coefficient of x in (b-\-xy + (2c-Saxy. 2. Find the coefficient of t^ in a(t-b)(t-b}-b(t-a^(t-a}-3at(2a+f), 3. + ■ =3. Solve for x. 2x+a 4x 4. -^4-— H =a + b-^c. Solve for -y. ab be ca m .15 1 7 9 ^9 Solve for t. ct bt cbt 6. =0. Solve form. 7. 2m-36 4a-56 u{a-^4b)-b^ ^u-b^u±a^ Solve for 2.. a^—b^ a + b a — b 8 a--6^6^^c-a^Q Solve for <. t—c t—a t 5 2 3^ 2s+5d Ss-4d 6^2^7^5-20^2 a b b'^-a' Solve for s. w—a w — b b^—bw Solve for w. £(£Z^ + M*zi)=a+6. Solve for a:. x—b x—a 110 ALGEBRA iv-^-Sn , iv — Sn 10 n^ cs i £ 12. \ = bolve tor n. v+2n 3n — v v^—nv—^n? Zx 5ax—2b a-{-Sbx ax-\-2a^ — 4b '^' T ITa 86 16 a6 2R±Sa^la±±b^ Solve for /J. 2R-Sa 3a--46 15. — 1 = 2h — ; — bolve for a\ bx ax . abx RATIO AND PROPORTION RATIO 136, The Ra.tio of one whole or fractional number^ a, to another^ b, is the quotient of a divided by b. Thus, the ratio of a to 6 is - ; it is also expressed a : b. We make no b attempt to define ratio. When applied to whole or frac- tional numbers, ratio is only another name for quotient or fraction. When the fraction - is called a ratio, its numer- b ator a is called the antecedent or Jirst term^ and its denom- inator b is called the consequent or second term. The ratios here spoken of are but fractions under another name, and have all the properties of fractions. If a and h are positive numbers, and a>6, -is called a ratio of greater inequality ; if a < 6, it is called a ratio of less ineqicality. (The signs > and < are read "is greater than " and "is less than" respectively.) PROPORTION 137. A Proportion is an equation whose members are equal ratios. Thus, if ^ and -^ are equal ratios, a ■L J a c a :o=c : rf, or - = -» b a is a proportion. (See example 14, Exercise 58.) RATIO AND PROPORTION 111 138. In the above proportion, a is called the first term^ b the second^ c the thirds and d the fourth. The first and third terms of a proportion are called the ante- cedents, and the second and fourth terms the consequents. The first and fourth terms are called the extremes, and the second and third terms the means. 139. If the means of a proportion are equal, either mean is called the Mean Proportional between the first and last terms, and the last term is called the Third Proportional to the first and second terms. Thus, in the proportion i =""♦ b is the mean proportional between a and c, and c is the third proportional to a and b. The Fourth Proportional to three numbers is the fourth term of a proportion whose first three terms are the three numbers taken in their order. Thus, in the proportion - =- » d is the fourth proportional to a, &, and c. 140. A Continued Proportion is a series of equal ratios, in which each consequent is the same as the next antecedent ; ^^9 a:b=b:c = c:d=d.e h c d e The definitions and explanations in §§ 138 and 139 refer to propor- tions written in the form ^ . i ^ . , a : o=c : a. Because of greater facility in operation, however, we shall use the fonn a c b~d' IMPORTANT PROPERTIES OF PROPORTIONS 141. In any proportion, the product of the extremes is equal to the product of the means. Let the proportion be 7 = 3" d Clearing of fractions, ad=bc. 112 ALGEBRA 142. (Converse of § 141.) If the product of two num- bers be equal to the product of two others, one pair may be made the extremes, and the other pair the means, of a proportion. Let ad=bc. Dividing by 6d, g = ^, or 2=^. oa oa a 143. In any proportion, the terms are in proportion by Alternation ; that is, the means can be interchanged. In § 142, had we divided by cd, the proportion would have been a_ 6 c d In like manner, the extremes can be interchanged. 144. In any proportion, the terms are in proportion by Inversion ; that is, the second term is to the first as the fourth term is to the third. It follows from § 144 that, in any proportion, the means can be written as the extremes, and the extremes as the means. 145. In any proportion, the terms are in proportion by Composition ; that is, the sum of the first two terms is to the first term as the sum of the last two terms is to the third term. Let the proportion be ? = ^. Then 9i±k = m d hours, how many hours will it take to do the work if all work together ? XI. SIMULTANEOUS LINEAR EQUATIONS CONTAINING TWO OR MORE UNKNOWN NUMBERS 151. An equation containing two or more unknown num- bers is satisfied by an unlimited number of sets of values of these numbers. Consider, for example, the equation re + 2/ = 5. Putting x = l, we have l+?/ = 5, or 2/=4. Putting x = 2, we have 2 + 2/ = 5, or 2/=3; etc. Thus the equation is satisfied by the sets of values x = l,2/=4, and a;=2, 2/=3; etc. An algebraic equation which is satisfied by an unlimited number of such sets of values, is called an Indeterminate Equation. If we agree, as in example 28, Exercise 41, that distances measured toward the right from a definite line and upward from another definite line shall be positive and that measure- ments in the opposite directions be negative, and also that the vertical measurements shall be y measurements and the horizontal distances x measurements^ a definite picture of the equation x-\-y = b may be drawn. On square ruled paper, choose a horizontal and a vertical line, X'X and Y'Y ; these lines are called the x-axis and y-axis respectively. 122 ALGEBRA Y Pi P. P p. x' P5 X M Pe y:j When a: = 1 , 2/ =4, laying off 0M = 1, and MPi=4, we lo- cate the point Pj. 1 and ^ are called the coordinates of the point Pj. MPi is the ordinate and OM the abscissa of the point Pi- Oistheongrin. Tak- ing other pairs of values of x and y which satisfy x-hy = 5, we may locate the points P2, P3, etc., obtained from x=2, y=S;x = S,y=2;x = 4,y==l; x = 5,y=0; x = Q,y= —I; etc. Connect these points. The line thus drawn is called the grah of the given equation. The graph of an equation of the first degree in X and y (§ 75), is a straight line. Therefore the equation is called linear. Coordinates are often written thus, (a:, 2/), the x coordi- nate being written first. EXERCISE 61 1. Locate the points (2, 5) ; (3, -2); (-3, 2) ; (-5, -1); (2, 7); (-9, 4). 2. Construct the graph of: x — y = 5, 3. Construct the graph of: 2x-\-y=^S, 152. Consider the equations I ^+ y= 5, (1) l2a;+2 2/ = 10. (2) Equation (1) can be made to take the form of (2) by multiplying both members by 2; then, every set of values of X and y which satisfies one of the equations also satisfies the other. Such equations are called Equivalent. Again, consider the equations ' rx+2/=5, (3) U-t/ = 3. (4) PLATE I SIMULTANEOUS EQUATIONS 123 In this case, it is not true that every set of values of x and y which satisfies one of the equations also satisfies the other ; thus, equation (3) is satisfied by the set of values a: = 3, 2/ = 2, which does not satisfy (4). If two equations, containing two or more unknown num- bers, are not equivalent, they are called Independent. 153. Consider the equations ir+z/ = 5, (A) .a:4-2/ = 3. (B) It is evidently impossible to find a set of values of x and y which shall satisfy both (A) and (B). Such equations are called Inconsistent. 154. A system of equations is called Simultaneous when each contains two or more unknown numbers, and every equation of the system is satisfied by the same set, or sets, of values of the unknown numbers ; thus, each equation of the system {x-{-y^b, (1) U-y=3, (2) is satisfied by the set of values a; = 4, y=l. In § 151, we found that one equation containing two unknown quan- tities was indeterminate. Notice that the graphs, Plate I, of x + y ==5 (1) and X — 2/=3 (2) do not have the same direction. If constructed on the same diagram these lines will cross at some point. Construct the graphs on the same diagram, i. e., use the same axes for both equations, and you will find that the coordinates of the crossing point are the same as the X and y of the set of values in § 154, i. e., x=4, y = \. The coordinates of every point on the graph of equation (1) satisfy equa- tion (1), also the coordinates of every point on the graph of equation (2) satisfy equation (2), but only at this point (4, 1) do the same coordinates satisfy both equations, hence the name simultaneous equations. In solving simultaneous equations we are simply finding the point where the graphs intersect. The finding of this point by means of graphs is somewhat slow and inaccurate, but algebra offers several methods by which the point may be readily found. mm A Solution of a system of simultaneous equations is a set of values of the unknown numbers which satisfies every equa- 124 ALGEBRA tion of the system ; to solve a system of simultaneous equa- tions is to find its solutions. 155. Two independent simultaneous equations of the form ax + by = c may be solved by combining them in such a way as to form a single equation containing but oiie unknown number. This operation is called Elimination. ELIMINATION BY ADDITION OK SUBTRACTION {5 ; 5a:~3i/ = 19. (I) x+iy= 2. (2) Multiplying (1) by 4, 20 x-12 y=76. (3) Multiplying (2) by 3, 21a;-f-12y== 6 . (4) Adding (3) and (4), 41 a; =82. (5) Whence, x=2. (6) Substituting X =2 in (1), 10-3 y = 19. (7) Whence, -Sy=9, or ij= -3. (8) Check this solution by substituting x = 2, y= -3 in the given equa- tions. The above is an example of elimination by addition. We speak of adding a system of equations when we mean placing the sum of the first members equal to the sum of the second members. Abbreviations of this kind are frequent in Algebra ; thus we speak of muUiphjing an equation when we mean multiplying each term of both of its members. 2. bolve the equations i ^ ^ ^ {l0x-7y==-2i. (2) Multiplying (1) by 2, 30 a;+ 16 y = 2. (3) Multiplying (2) by 3, 30 a;-21 y=- 72 . (4) Subtracting (4) from (3), 37 y= 74, and y = 2. Substituting 2/ = 2 in (1), 15x4-16=1. Whence, 15 x= -15, and :c= -1. The above is an example of elimination by suhtractioiu From the above examples, we have the following rule : If necessary, multiply the given equations by such numbers as will make the coeflacients of one of the un- known numbers in the resulting equations of equal ab- solute value. SIMULTANEOUS EQUATIONS 125 Add or subtract the resulting equations according as the coefficients of equal absolute value are of unlike or like sign. If the coefficients which are to be made of equal absolute value are prime to each other, each may be used as the multiplier for the other equation; but if they are not prime to each other, such multipliers should be used as will produce their lowest common multiple. Thus, in Ex. 1, to make the coefficients of y of equal absolute value, we multiply (1) by 4 and (2) by 3 ; but in Ex. 2, to make the coefficients of X of equal absolute value, since the L. C. M. of 10 and 15 is 30, we multiply (1) by 2 and (2) by 3. EXERCISE 62 Solve by the method of addition or subtraction ; verify each result : r6a:H-52/=28. {llx-lby=- 7. '* I4a:+ y = 14. ^' I 52/+ 9x=-23. 82/=-35. ^' l42i + 3'y=5. f2<-32/ = 19. ^ f 8m + 6A: = 9. 7< + 4w = 23. I 12m-9A; = 8. ELIMINATION BY SUBSTITUTION 157. Ex. Solve the equations \ y - \ ) ^ l82/-5x=-17. (2) Transposing — 5 a; in (2) , 8 y = 5 a; — 1 7. Whence, y^^J^nH, (3) 8 Substituting in (1), 7 x-^ / 5 x- 17 \ ^ ^^ ^^^ Clearing of fractions, 56 a:— 9(5 x — 17) = 120. Or, 56 a;-45 a; 4-153 = 120. Uniting terms , 1 1 a: = — 33 . Whence, a;= -3. (5) Substituting a; = - 3 in (3) , ?y = "^^"^"^ = - 4. (6) Verify the result. From the above example, we have the following rule : 126 ALGEBRA From one of the given equations find the value of one of the unknown numbers in terms of the other, and sub- stitute this value in place of that number in the other equation. ElilMINATION BY COMPARISON ISS. Ex, Solve the equations 1 ^~ (1) (2) Transposing —5y in ( 1 ) , 2x=52/-16. Whence, 2 (3) Transposing 7 yin (2), Sx = 5-7y. Whence, „5^. (4) Equating values of x, 52/-16_5-7?/ 2 3 (5) Clearing of fractions, 15 2/-48 = 10-14 2/. Transposing, 29 2/ =58. Whence, 2/ = 2. (6) Substituting y = 2in (3), 10-16__3 (7) From the above example, we have the following rule : Prom each of the given equations, find the value of the same unknown number in terms of the other, and place these values equal to each other. EXERCISE 63 Solve by either substitution or comparison, using the method that seems the easier. Verify each result. (Sv-5y=-5. ^ (I5t+ 8u = S. ' \x-9y=^7. ' [6y-llz==74. ^"1 t^'+/=12. ' I 67-11 r=-45. SIMULTANEOUS EQUATIONS J 126-11/=19. f Sm-I5v 127 8. 1 12/- 11 e= -27. 9w + 6'y=-16. 13w + 7'y=-22. 18. 12 m+ 6t;=-ll. Sh-3k = 35. EXERCISE 64 Solve the following equations, using any method of elim- ination you choose, and verify each result : 6<-7^=-12. ril'y+4w = 3. fl5r+ 8*=-14. I 8v- i x+2y=-2. [ \4x-7y=37. I ; + 9l/=~l(). 5. f2jr 3j. 3 4 6r + 12.^ = l. 7^ 2* 4 5 2 ' If the given equations are not in the form ax-\-hy=c, they should be reduced to this form before applying any method of elimination. 6. 8 ^+7/= 12. 4 3 ^-?=2. 2 2 3-2a: 4+5y __. 5 11 8. 7 4 3 a?- — ^ =2 y-4 3 ^ 8-2/^ 2i^-f-3 ^y4-3 5 4 4 * 1+42/ ^-f-7 11 = 3. In solving fractional simultaneous equations, we reject any solution which does not satisfy the given equations. 10. Solve the equations 2a;+32/ = 13. x-2 y-Z (1) (2) 128 ALGEBRA Multiplying each term of (2) by {x-2){y~S), we have 2/-3 + a;-2 = 0, or ?/== -x-{-5. (3) Substituting in (1), 2 a;-3 a;+15 = 13, or a;=2. Substituting in (3), t/= -2 + 5=3. This solution satisfies the first given equation, but not the second; then it must be rejected. 2 5 x+5 y+l 11. = 0. X- 11 x±5 3 = 5. = -3y. 13. 14. 15. w— 3 9 (2m-l)(2-4)-(m-5)(2 2+5) = 121. 4m-3z=-29. 048. 2/ = .478. 2x-Sy 4 x+6y 8 'v— 5 5 I .3 X— .35 v = A 2u-l Sv+i =0. x±n 7 a:-l +^i::^ = -4. 5 16. 2 rf-2n 3d-fn+3" d + 3n d+4:n-7 10 -45. 1 5* 18. 19. 5 a: 4-2 y 7 y — 3 a: 1^ 2' 39^ 10* ifc-4 2 r5a;-^(3a:-22/-h5) = ll. 11* '^- U(x-4i/)-|(a:-2/) = 16. 21_£+1 5i;-h2 63 .7- 130 t; 2 3 21 ' 14j7-fl 10t^-3 __o 3 5 Certain equations in which the unknown numbers occur in the denominators of fractions may be readily solved without previously clearing of fractions. (1) (2) 10_ -? = =8. X y ? + 15^ = _ 1 [a; y x+y=5 UB) 2X''y=4 (CD) 2x—y=4 X y 5 1 4 2 3 3 2 4 1 + 5 - 1 6 X y -4 1 -2 2 3 2 4 4 - I -6 -2 -8 Solving the equations for x and ?/, we have x = 3, y = 2. Note that these values correspond to the x and y coor- dinates of the point P. In general, in solving two simultane- ous equations in two unknown quan- tities, the real values found for the unknown quantities correspond to the intersection points of the graphs of the given equations. PLATE II i SIMULTANEOUS EQUATIONS 129 Multiplying (1) by 5, Multiplying (2) by 3, Adding, 74 ^-1^=40. X y X y =37, 74=37 X, and x = 2. 9 Substituting in (1), 5 — 23. 24. 25. 3 3^ X y 6_3^ X y 6 12 t w 1 2 9 =8, — =3, and y — y X 3 26. -1. 8 t 3 1^ 1? _ 9 w -^ = 1. = 7. 27. 28. -3. - 3 iy = 4. x-'^y^^T 6 r 2 6 :-7. 3m~ 10. o _l10 2 m + - - = k :4. 159.' In graphical work the drawings are much more effec- tive and pleasing if one uses a color scheme similar to that in Plate II. Cross-ruled paper and color crayons, for either blackboard or paper, can be secured at nominal expense. EXERCISE 65 Construct the graphs of the following equations, in each case comparing the coordinates of the intersection with your algebraic solution : a?+2/=8. X— y=6. h {2x^- 2/= 10. 1 ic+22/ = 4. r5ir-62/=-9. 1 3 x — 5 ?/=— 4. '•{ 8ar-3i/ = 47. 6a:-72/=21. x+22/ = 10. 2a; + 4 2/ = 30. {2x-^y = l2. l4a»-6?/ = 24. 130 ALGEBRA /3; M4. x+3y= 5. 9. a; 4-2/ =8. Find the area of the triangle formed by this line and the axes. r x—y=5, {x-y==S. 10. \ ^ II. ^ l4x-2 2/ = 16. I 3 07+2 2/= -6. An interesting application of the graph is the construction of the geometric picture of related data : 12. The enrollment of pupils in the Seattle high school for ten years is as follows : Year No. of Pupils Year No. of Pupils Year No. of Pupils 1899 592 1903 1213 1906 2312 1900 684 1904 1522 1907 2794 1901 800 1905 1960 1908 3500 1902 947 Choosing 1 year for the horizontal unit and 500 for the vertical unit, we have the following graph : Vertical Scale, 600=1 Unit Horizontal Scale, 1=1 Unit / / / ^ y ,^ ^ — ■ -^ L Nil ' i \ y Mil Horizontal Vertical 592 1 684 2 800 3 947 4 1213 5 1522 6 1960 7 2312 8 2794 9 3500 Construct the graphs of the following : 13. The enrollment in the Toledo high school is Year No. of Pupils 1898 773 1899 984 1900 1110 1901 1102 Year No. of Pupils 1902 1376 1903 1414 1904 1500 Year No. of Pupils 1905 1622 1906 1791 1907 1900 r SIMULTANEOUS EQUATIONS 131 14. Standing of the Chicago National League Baseball Team: Year Percentage Year Percentage Year Percentage 1898 .567 1902 .497 1905 .601 1899 .507 1903 .594 1906 .763 1900 .474 1904 .608 1907 .704 1901 .381 15. Enrollment in Cleveland high schools: Year No. of Pupils Year No. of Pupils Year No. of Pupils 1898 3378 1901 3595 1904 4491 1899 3460 1902 3796 1905 5001 1900 3589 1903 4151 1906 5070 16. Enrollment in Chicago high schools: Year No. of Pupils Year No. of Pupils Year No. of Pupils 1897 7847 1901 9661 1904 9936 1898 8432 1902 9627 1905 11208 1899 8830 1903 9488 1906 12024 1900 9190 17. Standing of the Detroit American League Baseball Team: Year Percentage Year Percentage Year Percentage 1901 .548 1904 .408 1906 .477 1902 .385 1905 .516 1907 .613 1903 .478 18. Enrollment in New York City high schools : Year No. of Pupils Year No. of Pupils Year No. of Pupils 1899 13731 , 1902 21461 1905 30340 1900 17018 1903 23701 1906 31949 1901 19013 1904 27794 160. Solution of Literal Simultaneous Equations. — In solving literal simultaneous linear equations, the method of elimination by addition or subtraction is usually to be preferred. ax +by =c. (1) ImL HjX, bolve the equations ^ , ,, , ■P ^ Xa'x^Vy^e, (2) Multiplying (1) by V, ah'x-\-Wy — h'c. Subtracting {ah' — a'h)x = h'c — bc\ 132 ALGEBRA ^ h'c-bc' ab'—a'h (m'x-\-a'hy=ca'. {oh'— a'h)y=c^a— ca'. c'a—ca' y=- (3) (4) Whence, Multiplying (1) by a\ Multiplying (2) by a, Subtracting (3) from (4), Whence, ^ ,, ab —ab In solving fractional literal simultaneous equations, any solution which does not satisfy the given equations must be rejected. (Compare Ex. 10, Exercise 64.) EXBBCISE 66 Solve the following: r5a;— 6i/ = 8a. imx —ny =mn. \4:X + 9y = 7 a. \m^x+n^y=m^n^ { 2ax-hy _^ 3. 8. + 9?/ = 7a ax-{-hy = \. cx-{-dy = l. 'aiX+a2y = bi, M,jX~aiy=b2. m _ n n+y m a 3a+2 6. 7. m—x n n+x 10. rrii mg m^ ^ 4-^ =1. Til ^2 ^3 bx — ay=b^. (a — b)x-\-by = a^. ax-j- by =2 a. [a^x—b^y=^a^-{-b^. '(a + l)a: + (a-2)z/=3a. .(a+3)x + (a-4)i/ = 7a. m-y (ab(a-b)x+ab{a-\-b)y=-a^+2 ab-b\ \ax+by=2, b - + - =c. X y X y 14- 15. 13. a^ , b _«+&, bx ay ab b a^b^-a^ ax by a^b^ m(x+7j)+n (x—y) = 2. m'^{x^-y) — 'n}{x—y)=m'-n. f (a+6)x4-(a-6)2/ = 2(a'^ + 62). b a ^ [ x — a — b y — a-\-b SIMULTANEOUS EQUATIONS 133 i6. 17. xj X _ ^ cib a—b a+b a^ — b^ bx-\-ay = 2, ab(a+b)x—ab(a—b)y=a^-{-b^. SIMULTANEOUS EQUATIONS CONTAINING MORE THAN TWO UNKNOWN NUMBERS 161. If we have three independent simultaneous equations, containing three unknown numbers, we may combine any two of them by one of the methods of elimination explained in §§ 156 to 158, so as to obtain a single equation containing only two unknown numbers. We may then combine the remaining equation with either of the other two, and obtain another equation containing the same two unknown numbers. By solving the two equations containing two unknown numbers, we may obtain their values ; and substituting them in either of the given equations, the value of the remaining unknown number may be found. We proceed in a similar manner when the number of equa- tions and of unknown numbers is greater than three. The method of elimination by addition or subtraction is usually the most convenient. In solving fractional simultaneous equations, any solution which does not satisfy the given equations must be rejected. (Ex. 10, Exercise 64.) Qx-^y- 7 2 = 17. 9x-7y-mz = 29. (1) I. Solve the equations (2) . 10x-5y- 3z=23. (3) Multiplying (1) by 3, 18x-12y-2lz=: 51. Multiplying (2) by 2, 18a:-14if/-32 2:= 58. Subtracting, 2y+llz=- 7. (4) Multiplying (1) by 5, 30a;-20 2/-35 2= 85. (5) Multiplying (3) by 3, 30X-15?/- 9 0= 69. (6) Subtracting (5) from (6) 5 2/ + 26 2=-16. (7) 134 ALGEBRA Multiplying (4) by 5, Multiplying (7) by 2, Subtracting, Substituting in (7), Substituting in (1), 10 y + 55z= -35. 10.V + 52 2=-32 . 3^=- 3, or 2=-l 2y-ll = - 7, or 2/ = 2. 6x-8+7= 17, ora: = 3. EXERCISE 67 Solve the following : 4x-3y = - 5. *i. iy-3 2=-13. .4z-3x = 18. 4 a; — 5 1/— 6 2=0. .-r— 2/+ z = l. Ox + z=8. 3a;+ y— 2 = 14. x-\-3y— 2 = 16. x+ 2/-3z = -10. ^+ h~k = 2i, (3x+5y= 5. I 9ar + 5 2 = 55. l92/+3z=-30. 5m— ?/+4i;=— 5. 3m + 5i/+6'y=-20. I m-\-3y-Sv=-27. l2x-5y=-26. 7 x-\-Qz=-33. 3 4 7. i^- 2+2 2x+42/- 2=~ 2. 8. . 18a:-8 2/+4 2=-25. 10ar + 42/-92=-30. 3p-f4 5r+5r = 10. 4p-59~3r=25. l5p-3g-4r = 21. 4 u—11 v—5 w= 9. 8 w -f 4 i;— ^^ = 11. i 16 2^+ 7i;+6i/; = 64. (Sx-^-iy-h 32 = -52. II. -{ 5a:- 2/ + 12z=-52. [9a:-h72/- 62 = -36. 6r- 5+3^=42. 10 r- 5 s- f= 2. 6r-175+4/=-46. 2a;+52/+3z=-7. 13. I 22/-42 = 2-3ir. .5a; + 9y = 5+7 2. 5 8 X 14. = -3. = 1. y 25 2 + — = 2. 3x * Eliminate 2/ from (1) and (2) you then have two equations in x and 2; or add the three given equations. SIMULTANEOUS EQUATIONS 135 15- 2 +'- 3x y = - 3 1 = 4y z _ 7 "30 2 X 1 'l2 10 ' i6. ax+by = by + CZ = cz-\-ax = abc b'+c' abc abc PROBLEMS INVOLVING SIMULTANEOUS EQUATIONS WITH TWO OR MORE UNKNOWN NUMBERS 162. In solving problems where two or more letters are used to represent unknown numbers, we must obtain from the conditions of the problem as many independent equations (§ 152) as there are unknown numbers to be determined. 1, Divide 81 into two parts such that three-fifths the greater shall exceed five-ninths the less by 7. Let a:=the greater part, and 2/= the less. By the conditions, x -{•y = Sl, (1) ^=^+7. (2) a:=45, y=SQ. 2. If 3 be added to both numerator and denominator of a fraction, its value is | ; and if 2 be subtracted from both numerator and denominator, its value is ^ ; find the fraction. Let and and Solving (1) and (2) By the conditions, and n=the numerator, d = the denominator. n±S^2 rf+3 3' n-2^1 d-2 2* Solving these equations, n — 7, d—12; then, the fraction is — • 3. A sum of nloney was divided equally between a certain number of persons. Had there been 3 more, each would have received $1 less ; had there been 6 fewer, each would have received $5 more. How many persons were there, and how much did each receive? 136 ALGEBRA Let a: = the number of persons, and 2/ =the number of dollars received by each. Then, xy=the number of dollars divided. Since the sum of money could be divided between x + S persons, each of whom would receive y—l dollars, and between x—6 persons, each of whom would receive y-\-5 dollars, {x+3)(y—l) aad (x- 6) (?/ + 5) also represent the number of dollars divided. Then, (x + S){y- 1) =xy, and {x- 6) (?/ + 5) = xy. Solving these equations, x = 12,y=5. 4. The sum of the three digits of a number is 13. If the number, decreased by 8, be divided by the sum of its second and third digits, the quotient is 25 ; and if 99 be added to the number, the digits will be inverted. Find the number. Let X = the first digit, ^ 2/ = the second, and 2 = the third. Then, 100 a:+10 y + z=the number, and 100 2+ 10 y + x = the number with its digits inverted. By the conditions of the problem, x-\-y-\-z = lS, 100a;+10?/ + z-8 _og y + z and 100x + 10y-{-z-{-99 = 100z + 10y+x. Solving these equations, x = 2,y=8,z—3; and the number is 283. 5. A crew can row 10 miles in 50 minutes down stream, and 12 miles in 1| hours against the stream. Find the rate in miles per hour of the current, and of the crew in still water. Let a; = number of miles an hour of the crew in still water, and 2/= number of miles an hour of the current. Then, a: -h 2/ = number of miles an hour of the crew down stream, and X- 2/= number of miles an hour of the crew up stream. The number of miles an hour rowed by the crew is equal to the dis- tance in miles divided by the time in hours. Then, .r-f-v=10^ -=12, 6 and a--?/=l2-^-=8. 2 Solving these equations, a;=10, ?/=2. SIMULTANEOUS EQUATIONS 137 6. A train running from A to B meets with an accident which causes its speed to be reduced to one-third of what it was before, and it is in consequence 5 hours late. If the acci- dent had happened 60 miles nearer B, the train would have been only 1 hour late. Find the rate of the train before the accident, and the distance to B from the point of detention. Ijet 3x = the number of miles an hour of the train before the accident. Then, x = the number of miles an hour after the accident. Let 2/ = the number of miles to B from the point of detention. y The train would have done the last tj miles of its journey in ^ hours ; y but owing to the accident, it does the distance in - hours. If the accident had occurred 60 miles nearer B, the distance to B from tlie point of detention would have been y—60 miles. Had there been no accident, the train would have done this in — — hours, and the accident would have made the time — - — hours. T,,e„, Vz:60^,/^^j Subtracting (2) from (1), ?5 = M ^^^ ^^ ^ = 4; whence, 5: = 10. X S X X Then, the rate of the train before the accident was 30 miles an hour. Substituting in (1), ^ = ^4.5^ or ^=5; whence, y = 75. EXERCISE 68 1. If the numerator of a fraction be decreased by 1, the value of the fraction is |^, while if 7 be added to both numer- ator and denominator, the value of the fraction is ^^ ; find the fraction. 2. The sum of two numbers is 7. The ratio of their pro- duct to the product of three times the first number and the second increased by 2 is |. What are the numbers? 3. The sum of the two digits of a number is 14; and if 36 be added to the number, the digits will be inverted. Find the number. 138 ALGEBRA 4. Nine shares of N. Y. Central stock and 7 shares of Illinois Central stock cost $1702, and 5 shares of I. C. cost $35 more than 6 shares of N. Y. C. Find the cost of one share of each. 5. Find two numbers such that the ratio of the first num- ber to itself increased by 3 is equal to the ratio of the second number to itself increased by |^; and the sum of the two numbers is to twice their difference as 7 is to 4. 6. If 3 be added to the numerator of a fraction, and 7 subtracted from the denominator, its value is ^ ; and if 1 be subtracted from the numerator, and 7 added to the denomi- nator, its value is f . Find the fraction. 7. Find two numbers such that one shall be n times as much greater than a as the other is less than a ; and the quotient of their sum by their difference equal to 6. 8. A wheat field is 80 rods longer than it is wide, and the distance around the field is 1^ miles. Find the length and br-eadth. 9. In plowing the long way of the above field, a farmer finds he can turn 33 twelve-inch furrows a day. At $3.50 per day for a man and team, what is the cost of plowing per acre ? 10. C's age is three times the sum of A's and B's. Three times B's age added to A's is 12 years less than C's, and if 8 years be subtracted from C's age and this difference be divided by B's age, the quotient will be 4. Find their ages. 11. A rectangular mirror is 6 inches longer than it is wide. It is surrounded by a frame 3 inches wide, whose area is 216 square inches. How much wall space will the mirror and frame occupy ? 12. A man had $3000 in a savings bank which paid him 3 % interest. He drew out a part of his money and invested it in municipal bonds which paid him 5 %. His annual SIMULTANEOUS EQUATIONS 139 income from the entire sum was then il26. Find the amount left in the savings account. 13. If we consider y = kx sl proportion, what is the ratio oi y to X ? Give k some definite value and make the graph of the equation. If perpendiculars are dropped from the graph to the a;-axis, triangles are formed by the perpendicu- lars, the graph, and the ir-axis. Are the triangles alike in form ? Is the ratio of the altitude to the base the same in each ? If k is given a different value from the one chosen, what effect does this have on the graph and on the triangles ? 14. A rectangular field has the same area as another which is 6 rods longer and 2 rods narrower, and also the same area as a third which is 3 rods shorter and 2 rods wider. Find its dimensions. 15. Find three numbers such that the first with one-half the second and one-third the third shall equal 29 ; the second with one-third the first and one-fourth the third shall equal 28 ; and the third with one-half the first and one- third the second shall equal 36. 16. The circumference of the large wheel of a carriage is 55 inches more than that of the small wheel. The former makes as many revolutions in going 250 fefet as the latter does in going 140 feet. Find the number of inches in the circumference of each wheel. 17. A man having $4500 in a savings bank which paid him 3 % interest withdrew the money, investing a part in Rock Island 5% bonds for which he paid #80 (par value 100) ; with the balance he purchased Pennsylvania Railway 5 % bonds at par. His annual income from these invest- ments was $255. Find the amount invested in Rock Island bonds, and their face value. 18. A number consists of two digits. If the first digit be divided by one less than the second digit, the quotient is 3. 140 ALGEBRA If the first digit, increased by 3, be divided by the second digit the quotient is 3 ; find the number. State your prob- lem, then compare § 152. iQ. The sum of the length, breadth, and height of a rect- angular parallelopiped is 20. The dif- ference between the length and height is I the sum of the height and breadth, and three times the breadth added to the height is 8 more than the length ; find the dimensions. 20. If the digits of a number of three figures be inverted, the sum of the number thus formed and the original num- ber is 1615 ; the sum of the digits is 20, and if 99 be added to the number, the digits will be inverted. Find the number. 21. A train left A for B, 112 miles distant, at 9 a. m., and one hour later a train left B for A ; they met at 12 noon. If the second train had started at 9 a. m., and the first at 9.50 A. M., they would also have met at noon. Find their rates. 2 2. A boy has $1.50 with which he wishes to buy two kinds of note-books. If he asks for 14 of the first kind, and 1 1 of the second, he will require 6 cents more ; and if he asks for 11 of the first kind, and 14 of the second, he will have 6 cents ov^r. How much does each kind cost ? 23. The difference between the length and breadth of a rectangle is 6. If the length were diminished by 3 feet and the breadth increased by 3 feet, the area would be in- creased by 9 square feet and the figure would be a square ; find the dimensions. Have you more conditions than you need? Are your conditions inde- pendent? (Compare § 83.) 24. A number consisting of two digits is such that if the digits be inverted the number formed is 27 less than the original number. The product of the digits is to their differ- ence as the second digit is to | ; find the number. SIMULTANEOUS EQUATIONS 141 25. A man invests $10,000, part at 4^%, and the rest at 3|%. He finds that six years' interest on the first invest- ment exceeds five years' interest on the second by $658. How much does he invest at each rate ? 26. A man buys apples, some at 2 for 3 cents, and others at 3 for 2 cents, spending in all 80 cents. If he had bought I as many of the first kind, and | as many of the second, he would have spent 99 cents. How many of each kind did he buy? 27. An annual income of $800 is obtained in part from money invested at 3|%, and in part from money invested at 3%. If the amount invested at the first rate were invested at 3%, and the amount invested at the second rate were in- vested at 3| %, the annual income would be $825. How much is invested at each rate ? 28. The contents of one barrel is | wine, and of another | wine. How many gallons must be taken from each to fill a barrel whose capacity is 24 gallons, so that the mixture may be I wine ? 29. A boy spends his money for oranges. Had he bought m more, each would have cost a cents less ; if n fewer, each would have cost b cents more. How many did he buy, and at what price ? 30. A vessel contains a mixture of wine and water. If 50 gallons of wine are added, there is J as much wine as water ; if 50 gallons of water are added, there is 4 times as much water as wine. Find the number of gallons of wine and water at first. 31. A man buys 15 bottles of sherry, and 20 bottles of claret, for $38. If the sherry had cost f as much, and the claret | as much, the wine would have cost $38.50. Find the cost per bottle of the sherry, and of the claret. ■p 32. If a field were made a feet longer, and b feet wider, its area would be increased by m square feet ; but if its length 142 ALGEBRA were made c feet less, and its width d feet less, its area would be decreased by n square feet. Find its dimensions. 33. If the numerator of a fraction be increased by a, and the denominator by 6, the value of the fraction is — ; and if n the numerator be decreased by c, and the denominator by rf, the value of the fraction is — . Find the numerator and de- nominator. 34. A certain number equals 59 times the sum of its three digits. The sum of the digits exceeds twice the ten's digit by 3 ; and the sum of the hundred's and ten's digits exceeds twice the unit's digit by 6. Find the number. 35. A piece of work can be done by A and B in 4| hours, by B and C in 2| hours^ and by A and C in 3 hours. In how many hours can each alone do the work? 36. The numerator of a fraction has the same two digits as the denominator, but in reversed order; the denominator exceeds the numerator by 9, and if 1 be added to the numer- ator the value of the fraction is |. Find the fraction. 37. A man walks from one place to another in 5| hours. If he had walked | of a mile an hour faster, the walk would have taken 36| fewer minutes. How many miles did he walk, and at what rate ? 38. A man invests a certain sum of money at a certain rate of interest. If the principal had been $1200 greater, and the rate 1% greater, his income would have been increased by $118. If the principal had been $3200 greater, and the rate 2% greater, his income would have been increased by $312. What sum did he invest, and at what rate ? 39. A crew row 16| miles up stream and 18 miles down stream in 9 hours. They then row 21 miles up stream and 19j miles down stream in 11 hours. Find the rate in miles an hour of the stream, and of the crew in still water. SIMULTANEOUS EQUATIONS 14:^ 40. A man buys a certain number of $100 railway shares, when at a certain rate per cent discount, for $1050 ; and when at a rate per cent premium twice as great, sells one-half of them for $1200. How many shares did he buy, and at what cost ? 163. Interpretation of Solutions. 1. The length of a field is 10 rods, and its breadth 8 rods ; how many rods must be added to the breadth so that the area may be 60 square rods ? Let .-5= number of rods to be added. By the conditions, 10(8 + x) = 60. Then, 80 + 10 x = QO, or x= -2. This signifies that 2 rods must be subtracted from the breadth in order that the area may be 60 square rods. (Compare § 11.) If we should modify the problem so as to read : '' The length of a field is 10 rods, and its breadth 8 rods ; how many rods must be subtracted from the breadth so that the area may be 60 square rods? " and let x denote the number of rods to be subtracted, we should find x = 2. A negative result sometimes indicates that the problem is impossible. It sometimes indicates that measurement is taken in an opposite direction (Ex. 28, Exercise 41). 2. If 11 times the number of persons in a certain house, increased by 18, be divided by 4, the result equals twice the number increased by 3 ; find the number. Let a: = the number. By the conditions, 1I^±1? = 2 a:+3. 4 Whence, 11 a;+18=8 a;+12, and x = -2. The negative result shows that the problem is impossible. A problem may also be impossible when the solution is fractional. 144 ALGEBRA XII. INVOLUTION AND EVOLUTION 164. Involution is the process of raising an expression to any power whose exponent is a positive integer. We gave in § 88 a rule for raising a monomial to any power whose exponent is a positive integer. 165. If an expression when raised to the nth power, n being a positive integer, is equal to another expression, the first expression is said to be the nth. Root of the/ second. Thus, if a^=b, a is the nth root of b ; if 5^=25, 5 is the square root of 25. Evolution is the process of finding any required root of an expression. 166. The Radical Sign, \/, when written before an ex- pression, indicates some root of the expression. Thus, V a indicates the second, or square root of a ; y/a indicates the third, or cube root of a ; Va indicates the fourth root of a ; and so on. The Index of a root is the number written over the radical sign to indicate what root of the expression is taken. If no index is expressed, the index 2 is understood. An even root is one whose index is an even number ; an odd root is one whose index is an odd number. 167. A Pov^rer of a Fraction. We have, ('«Y = ? x?X? = «^''-^« = «': \bj b b b bxbxb b^ and a similar result holds for any positive integral power of ". b Then, a fraction may be raised to any power whose exponent is a positive integer by raising both numer- ator and denominator to the required power. INVOLUTION AND EVOLUTION 146 EXERCISE 69 Find the values of the following : Ic'dy ^'\ h'y J' \ nY J' 9 mn^y f ix'^y . ( a^m^ y 168. A Root of a Monomial. To find any root of a mo- nomial which is a perfect power of the same degree as the index of the required root. 1. Required the cube root of a^lfc^. We have, {aWc'Y =a^V'c\ Then, by § 165, -i^oW =-aWc\ 2. Required the fifth root of —32 a^. We have, (-2a)^ = - 32 a\ Whence, ^ 128m^V^ ,,. V29r6. g J/ 27 a« ID. '^-343a:^+y^ 15. \/30625. 125 6« II. ^625 a^«^6^\ 16. \/86436. 17. V25 . 36 . 196. 20. ^4 a6 • 144 b^c • 24 aV. 18. ^27 . 64 . 8. 21. ^252 a^ • 245 ti^ • 150 c\ 19. V25O . 32 . 45. 22. ^59049. 23. ^112 . 168 • 252. 24. \/(a2-5a+6)(a2-f2a-8)(a2+a-12). 25. \/(2 a^+7 a- 15)(8 a^-^2 a~21)(4 0^+27 a-h35). 169. It may be proved that 2 -b a^-2a^b+ ab' - a% +2 a62 (a-6)3 = a3_3^2^^3a62_j,3 * The values of examples (1) and (2) are of frequent occurrence and are important. INVOLUTION AND EVOLUTION 157 That is, the cube of the difference of two numbers is equal to the cube of the first, minus three times the square of the first times the second, plus three times the first times the square of the second, minus the cube of the second. 1. Find tKe cube of a +2 6. We have, (a + 2 by=a' + S a'C2 6)-f 3 a{2 6)2+ (2 by ==a^ + 6a^b+l2ab^ + Sb\ 2. Find the cube of 2 x^—5 y^. (2 x'- 5 y^y = (2 x^y- 3(2 x'yi5 2/^) +3(2 x%5 y^- (5 y^ =8 x^- 60 xV+ 150 xY- 125 y\ The cube of a trinomial may be found by the above method, if two of its terms be enclosed in parentheses ; and regarded as a single term. 3. Find the cube of x^—2 x— 1. (a;2- 2 x- \y =[(a;2- 2 x)- If = (a:2- 2 a:)3- 3(^2- 2 x)2 + 3(a;2- 2 a;)- 1 =x«- 6 a;5 + 12 x^- 8 x'- Z{x'- 4 a:3 + 4 a:^) +3(a;2- 2 x)- 1 =x^- 6 ^5 + 12 a;*- 8 a:^- 3 a;^+ 12 x^- 12 x^ + S a:^- 6 x- 1 =a;«- 6 a;5 + 9 a;*+4 a;3- 9 a:2- 6 a;- 1. EXERCISE 77 Cube each of the following : 9. a + 6+c. 10. a — 2 6--3 c. -Sy\ .^ II. 3rf + 4c2+ifc. -4^^ 12. 3a^+2 6^ 183. Cube Root of a Polynomial. The cube roots of cer- tain polynomials of the form can be found by inspection. Ex. Find the cube root of Sa^-36 a^'b^+^^L ab^-21 b\ We can write the expression as follows : I. 2. a'b-ab\ a;+4. 5. 6. 5+3 3. c-b. 7. 2 m- 4. 3a:2 + l. 8. 9x'- 158 ALGEBRA (2 ay -3(2 a)2(3 6^) +3(2 a)(3 6^)2 -(3 by. By § 182, this is the cube of 2 a -3 6^ Then, the cube root of the expression is 2 a -3 6^ EXERCISE 78 Find the cube roots of the following : 1. x^+Qx^' + Ux+S, 2. 27 a^-27a^ + 9a-l. 3. m« + 15m^+75m2 + 125. 4. a^-12 a^b +48 ah'' - 64 b\ 5. l25x^ + 150x''y + 60xy''+S7j\ 6. 216a3-^08a26 + l8a6--6^ 7. 27a;«-135a;H225a:*-125ir\ 8. 64t^-144Pu + 10Stu^-27 u\ 9. S h^+m h^k-{- 150 hk'' + 125 k\ 10. l-18a:2 + 108a:^-216a;^ XIII. THEORY OF EXPONENTS. IRRATIONAL NUMBERS 184. In the preceding portions of the work, an exponent has been considered only as a positive integer. Thus, if m is a positive integer, a"*=aXaXaX«'« to m factors. (§ 6) The following results have been proved to hold for any positive integral values of m and n : a^Xa^=a^+'^(§5()). (1) (a^)«=a'^'^(§85). (2) 185. It is desirable to use exponents which are not posi- tive integers; and we now proceed to assign to them the most convenient definitions and then prove the rules for their use. New meanings are conformed to the old laws. Thus our new exponents are to obey the old index law oTxa^'^-a'^-^''. (1) (a'^)'» = a'"". (2) THEORY OF EXPONENTS 159 Let it be required to find such a meaning ioT fractional^ negative and zero exponents. 186. Meaning of a Fractional Exponent. Let it be required to find a meaning for a^. If (1), § 184, is to hold for all values of m and n, Then, the third power of a^ equals a\ _ Hence, a^ may be defined as the cube root of a\ or a^= ^ a'\ Consider the general case : p Let it be required to find the meaning of a^, where p and q are any positive integers. If (1), § 184, is to hold for all values of m and n, ^ ^ ^ ^ ?+?+^+... to, terms ^X? a^Xa'Xa'^X - to ^factors =a'' ^ ' =0*^ =aP. Then, the q'th power of a^ equals a^. p p Hence, a*^ must be the ^'th root of a^, or a*^ = \/a^. Hence, in a fractional exponent, the numerator denotes a power, and the denominator a root. For example, a^=-'^a^] h^z=\^y ^i^^,^^. ^^c. This statement indicates that in expressions affected by a fractional exponent, both a root and a power are to be taken. EXERCISE 79 Express the following with radical signs : 1. a^. 3. 7 m^. 5. a^fe^. 7. 8a^m^. 9. x^y^z^ \ 2. x^ 4. 5x^ 6. x'2/'^'. 8. 10 nV^. 10. 20*^6^^^. Express the following with fractional exponents : 11. Vx\ 13. V^. 15. ^y^. 14. ^^^ 16. 4^. 18. i^^'Vf, 19. ^a^fe^. 20. i/^\/^>yz7 160 ALGEBRA 187. Meaning of a Zero Exponent. If (1), § 184, is to hold for all values of m and n, we have a'^xa^=a'^-^''=ar. Whence, a° = — = 1 . m a' This meaning may be illustrated as follows : Arithmetically, ~3~-^' Algebraically, a^^a^=a^, ' Therefore, a^=l. (§4, ax. 4) We must then define a^ as being equal to 1. 188. Meaning of a Negative Exponent. Let it be required to find a meaning for a~^. If (1), § 184, is to hold for all values of m and n, a-3xa^=a-3+3=a^=l (§ 187). Whence, a~^=— • Consider the general case : Let it be required to find the meaning of a~*, where s represents a positive integer or a positive fraction. If (1), § 184, is to hold for all values of m and n, a-*Xa*=a-*+*=a^=l (§187). Whence, a~* = — • a* We must then define a~* as being equal to 1 divided by a*. For example, a"^ == — ; a~^ = —; 3 x'^y'^ = — -; etc. 189. It follows from § 188 that Any factor of the numerator of «a fraction may be transferred to the denominator, or any factor of the denominator to the numerator, if the sign of its expo- nent be changed. THEORY OF EXPONENTS 161 EXERCISE 80 Express with positive exponents: 1. x~^y^. 5- a~^m~^, 9. wT^n'^, 2. aV^ 6. 7ri"^^^x\ 10. 8a~*6-^V. _1 1 . _4 . _9 _7 3 3. m %^. 7. 4 a ^71 ^. II. 6 m ^n ^a;^. 4. 3n~^a:. 8. b x^y~^z-\ 12. 7 a"%-^a;~i Transfer all literal factors from the denominators to the numerators in the following: ,1,1 ^ 2"^ la'b-^ 13. — • 15. r ^7. -3 — 19. — j-^- a^ . 3 o 2m^ 14. — -• 16. • 18. Ifi ax * 5np ^ 4:n~^y~^' Transfer all literal factors from the numerators to the denominators in the following : 21. • 23. • 25. . 27. 3 2 c"" vi5 X^y ^ 22. — • 24. -^ — • 26. • 28. — — — — • y^ ^ ^~' 5c-«tr^ 190. Since this is an elementary course, the student is only expected to read §§ 191 to 194, then use § 196 in apply- ing the principles involved, 191. We obtained the definitions of fractional, zero, and negative exponents by supposing equation (1), § 184, to hold for such exponents. Then, for any values of m and n, a^Xa'^=a'"+". (1) The formal proof of this result for positive or negative, integral or fractional, values of m and n will be found in the Second Course. 162 ALGEBRA 192. — = a^~" for all values ofm and n. a"" ^ "^ By § 189, — =a'^Xa-"=a'"-^ by (1), § 191. a^ The proof of this result in the case where m and n are positive integers, and m>n, is given in § 63. 193. To prove equation (2), § 184, for any values of m and ri, considering three cases, in each of which m may have any value, positive or negative, integral or fractional. I. Let n be a positive integer. The proof given in § 85 holds if n is a positive integer, whatever the value of m. V .... II. Let n = -, where p and q are positive integers. Then, by the definition of § 186, p mp (a'^y=^=>y2^=24(§ 186) = 2i=\/2. Ex. 2. Reduce v 16 to its simplest form. We have, ^16=[(20i]^ = (22)1 = 4* = ^4. In many radical forms, operations are more simple when the quantities are reduced to forms with fractional exponents. * Note that we do not define irrational number. The two most impor- tant irrationals, — tt and e (the base of a system of logarithms), — have been proved not to involve surds. 168 ALGEBRA EXERCISE 82 Reduce the following to their simplest forms: I. \/25. 5. V^49. 9. V^243. 13. v'216 a V. 10/- 16/ 15/ 2. V4. 6. V81. 10. \/343. 14. \/64 a«6^«. 3. V^m. 7. V^64. II. V^144 a:y. 15. ^Sa^\ 9/ 10/ 6/ 12/ 4. V125. 8. V81. 12. \/27n^x'\ 16. V625a:^y. 204. Case II. When the expression under the radical sign is rational and integral^ and has a factor which is a perfect power of the same degree as the surd. I. Reduce ^54 to its simplest form. We have, -^54 = (27 • 2)i'=27^ • 2i = (33)i • 2^ = 3 • 2^=3^2. 2. Reduce ^3 a^6 — 12 0^1)^^-12 a¥ to its simplest form. \/Sa^b-12a^b^+12a¥=-\/(a^-4:ab-\-4:b^)3ab = \/a=-4a6 + 4fe2\/3a6=(a-2 6)\/3a6. We then have the following rule : Resolve the expression under the radical sign into two factors, the second of v^hich contains no factor v^hich is a perfect power of the same degree as the surd. Extract the required root of the first factor, and mul- tiply the result by the indicated root of the second. If the expression under the radical sign has a numerical factor which cannot be readily factored by inspection, it is convenient to resolve it into its prime factors. 3. Reduce '?/l944 to its simplest form. ^1944 = '^23X3'^ = (23 • 35)^=(23 • S^)^ • (32)-^=2 • 3 • (32)^=6^9. 4. Reduce \/l25 Xl47 to its simplest form. Vl25Xl47=\/53X3X72=\/52x72xV5X3=5X7X\/l5=35Vl5. IRRATIONAL NUMBERS 169 EXERCISE 83 Reduce the following to their simplest forms : 1. (45)i 4. ^/75. 7. "^54. lo. V^4. 2. (12)i 5. vi^. 8. (128)*. II. \/500 a^6«. 3. (96)^ 6. (27)^. 9. Vl92. 12. \/98 a:«2/-49 ary 13. [(a + 3)(a^-9)]^ 14. \/(2 a2+2 ay-4:y') (3 a-3 ?/). 15. [(a;2+9)a:]l iQ. ^63 x^^/ • 75 a^^z/^ • 98 2. 16. \/(ic2+6a; + 9)a7. 20. \/98 • 196. 17. V(a2+2a6+4 6>2^ 21. (5i45)*. 18. \/a2+4 a6+4 fr^-ft). 22. V3 a3-24 a2+48 a. 23. \/l8a36 + 60aW+50a6^ 24. \/(6 a:2 + 5 xy-iy"") (3 ^^^-2 a:2/-8 i/^). 205. Case III. W^e7^ ^Ae expression under the radical sign is a fraction. In this case, we multiply both terms of the fraction by- such an expression as 'will make the denominator a per- fect power of the same degree as the surd, and then pro- ceed as in § 204. Ex, Reduce ^ — ^ to its simplest form. a Multiplying both terms of the fraction by 2 a, we have \Sa^ \ 16 rt* \16a^ \16a* 4 a^ EXEBCISE 84 Reduce the following to their simplest forms : I. \^. 3. V^-. 5. v^lf. 7. ^i 9. v^f 2- ^. 4. V^. 6. \/ll. 8. V^. 10. V^. 170 ALGEBRA ^- "■ Vl!=-'- -3- V"- -ab + b^ 2a+b ^ a+6 1 / 2a^-a-15 ^ a:^ / 3x^-18x + 27 '"^^ a-3^ a-3 * '^* a^^-S a:4-6^ a:^ 206. To Introduce the Coefficient of a Surd under the Radical Sign. The coefficieut of a surd may be introduced under the radi- cal sign by raising it to the power denoted by the index. Ex. Introduce the coefficient of 2'y3 under the radical sign. 2-^3 = ^8X^3 = ^8X3 = ^24. A rational expression (§ 198) may be expressed in the form of a surd of any degree by raising it to the power denoted by the index, and writ- ing the result under the corresponding radical sign. EXERCISE 85 Introduce the coefficients of the following under the radical signs : I. 2\/3. 3. 6V6. 5. 2^5. 7. (2a + 6) 5\/2. 4. 5^3. 6. 3\/i. ^ 4 a^-b^ 2. 5V2. 4 4-. 8. (3x-2?/)v^. j„ a-3ja^+a-6 a + B^a^-a-Q (> "+'W^J- «+^' "•2^^:^^4a^-9 6-^, ADDITION AND SUBTRACTION OP SURDS 207. Similar Surds are surds which do not differ at all, or differ only in their coefficients ; as 2'Vax^ and S'^ax^. Dissimilar Surds are surds which are not similar. 208. To add or subtract similar surds (§ 207), add or subtract their coefficients, and multiply the result by their common surd part. IRRATIONAL NUMBERS 171 1. Required the sum of \/20 and ^45. Reducing each surd to its simplest form (§ 204), V20 + V45 = V4X5 + ^9X5 = 2 V5 + 3\/5 = 5\/5. 2. Simplify V|+\/|-\/|. 2 3 4 3 4 We then have the following rule : Reduce each surd to its simplest form. Add or subtract the similar surds, and indicate the addition or subtraction of the dissimilar. EXERCISE 86 Simplify the following : I. Vi2 + \/48. 2. VEb-VTs, 3. 2V2 + \/l28-V98. 4. Vi25 + \/l80 + \/80. 5. ^2+^16. 6. ^iO + i^lSS. 7. V^32-\/l62. VTtS \/ri2 \/44 ^^- ^- J~ 2- 16. 7v'27-\/75-24VX-27VT. 17. n+jj^' 18. 6V8 a^6+a6V50 a^b^-a^Vl2S ab\ 19. \/3 x3+12 ic2 + 12~x + \/27 a?3-72 ir2+48 a:. 20. 2V12 x^'-^QO xij + 7^ //-- V^48>_72^2/T27^. 8. \/250 + \/490-\/l0. 9. \/44-\/275 + V89i. 10. \/32-h\/48 + \/80. 11. \/5 + \/245-V320. 172 ALGEBRA TO REDUCE SURDS OF DIFFERENT DEGREES TO EQUIVA- LENT SURDS OF THE SAME DEGREE 209. Ex, Reduce V2, ^3, and Vs to equivalent surds of the same degree. By § 1 86, \/2"= 2^ = 2t^ = v'2«"= V^. ^3=r3i = 3A=V^=v'8T. 4/- 1 3 12/— 12/ V5 = 5T=5T2^ = V5-'' = V125. Rule: Express the surds with fractional exponents, reduce these to their lowest common denominator, and express the resulting expressions with radical signs. The relative magnitudes of surds may be determined by reducing them, if necessary, to equivalent surds of the same degree. 12/ 12/ — 12/ — Thus, in the above example, v 125 is greater than v81 , and vSl than v'ei. ^ Then, \/5 is greater than ^3, and ^Z than V^. EXERCISE 87 Reduce the following to equivalent surds of the same degree : 1. V^ and ^3. 4- Va, \/6, Vc. 2. ^5 and V^T. 5- ^^h, ^^, y/a'-h\ 3. ^2 and v's. 6. Is \/2 greater than i'S? 7- Compare v5 and v7. 8. Write in order of magnitude V4, v6, v 15. 9. Which is greatest V3, V^, v'253? lo. Which is greatest i^3, V^, v^i? MULTIPLICATION AND EVOLUTION OF SURDS 210. I. Multiply Ve by VlS. \/6X\/i5=\/6Xl5=\/2X3X3X5='\/82X2X5=3\/i0. 2. Multiply \^¥a hy i^fo^. IRRATIONAL NUMBERS 173 Reducing to equivalent surds of tlie same degree (§209), \/27x^ra2 = (2a)ix(4a2)-3==(2a)tx(4a2)|==v^(2a)3x^(4a2)2 = \/23 a^ X 2* a* = \/2' a« X 2 a = 2 a\/2a. We then have the following rule : To miiltiply together two or more surds, reduce them, if necessary, to surds of the same degree. Multiply together the expressions under the radical signs, and write the result under the common radical sign. The result should be reduced to its simplest form, 3. Multiply V5 by v'S. By § 186, \/5 = 5i=5t = \/5^. Then, Vd X \/5 = \/5^ X ^^5= \/5^=5t=5J= ^5^ = ^25. 4. Multiply 2V3+3\/2 by 3V3~\/2. 2\/3 + 3\/2 3V3- V2 18 + 9\/6 -2\/6-6 18 + 7\/6-6 = 12 + 7\/6. To multiply a surd of the second degree by itself simply removes the radical sign; thus, \/3X\/3 = 3, or si - si=3i+i=S, 5. Multiply 3\/l+^-4V^by Vl+x+2Vx. 3\/r+x-4\/:r \/l+x + 2\/x 3 {l+x)-4:\/x + x^ -{-eVx-^x^-Sx 3 {l+x) + 2\/x + x^-Sx = S-5x + 2\/x+x\ EXERCISE 88 Multiply the following : 1. \/3 by ^27. 4. (108)^ by (192)^', 2. ^^36 x^y by ^6 xy\ 5. ^72 by '?/l2. 3. 10* by 30*. 6. VI by \/|. 174 ALGEBRA 7. \/^ by \/|. II. ^7 ab^ by Vfab. 8. \/a2-62 by V^Tft. 12. ^5^ by Vl25 aft. 9. ^3 by V2. 13. 2+3^ by 3+2^ 10. V^byVy. 14. 5 + V7.2 + \/5. 15. (5 + \/7)(24-v/5). 16. Expand (Va+Vby (§91). 17. Expand [(a)U7(6)^][(a)^-5(6)^]. 18. Expand (^/3a + ^/4T)(^/3a-^/46). 19. Expand [3. 2^- 2. 3*f. 20. Expand (\/i + V2 + \/3)2. 21. Expand <2\/3)^ (3i^2)^ (4V^)^ 22. Expand (vl2)3. (V^)^=(12)* (§193) =12i=2\/3. 23. Expand (\/l2)i2 24. Expand (\^ a^j^s 25. Expand (iVa^-\-S\/a+b)\ Expand and express result in the form a 4- 2^6: 26. (5 + v^3)2 = 25 + 10V3+3 = 284-2 • 5V3=284-2\/75. 27. (5^-3^)2. 30. (\/6-2\/s)\ 28. (2^+3^)2. 31. (7 + 4\/3)2. 29. {VsWiy, 32. [3 + (2)if. 33. (2. 2^- +3 -3^)2. Supply the missing term in the following trinomial squares: 34. 4 + 2Vi2+? 35. 7 + 2\/l4. 36. 14+2(14)i. 37. Extract the square root of 7+2n/12. 38. Extract the square root of 5+2(6)^ IRRATIONAL NUMBERS 175 Note that in squaring a binomial^ one or both of whose terms are affected by the exponent J, the square reduces to a binomial surd if both terms of the binomial to be squared are numerical. (Compare Examples 27-36.) Also the part 2( )^ corresponds to the middle term of the examples in § 91 . In 2(a&)^ if ab can be so factored that the sum of these factors is equal to the other term, the square roots of these factors connected by the sign of the irrational term will be the square root of the binomial surd. 39. Find the square root of 8+2(15)-. 15=5 • 3 and 5 + 3=8. Hence by the above rule, V8 + 2(15)4=\/5+\/3. 40. [17+6(8)*]*=[17+2(72)*]* = [17 + 2(8. 9)i]5 =9^+8^ =9i+(22)i^.2i = 3 + 2\/2. Find the square roots of the following binomial surds : Remember that the coefficient of the radical must be 2. 41. 3-2V2. 4S. 23+2- 132i 49. 37-640*. 42. 11+2(30)^ 43- 14 + 6\/5. 46. 29 + 2-54*. 50. 4 + (15)*. 47. 55 + 3V24. 51. 5 + \/2T. 44. 24+2-140i 48. 12-\/l08. 52. 55-20v'6. 53. 44- -4(72)i 54- 53-\/600. DIVISION OF MONOMIAL SURDS 211. Whence, \/ab=\/ay.\/b. Va Rule: To divide one monomial surd by another, reduce them, if necessary, to surds of the same degree. Divide the expression under the radical sign in the dividend by the expression under the radical sign in the 176 ALGEBRA divisor, and write the result under the common radical sign. The result should he reduced to its sim2olest form, 1. Divide ^^405 by ^5? We have, ^ = jl^^i/si = ^273^3 = 3^3. 2. Divide ^4 by Vg. Reducing to surds of the same degree (§ 209), i/4 4 (22)1 \/2* e/ 2* 6/2" 6/ 2x3^ 16,- V6 62^ (2X3)^ V23X3^ M ^ X6 ^t 6 \ d 3. Divide Vlb by V^40. We have, \/io = loi = lot = (103)^ = (53 • 23)i Then, ^ ==(?L^\i= (52)i = 5* =^5. EXERCISE 89 Divide the following : 1. VeO by Vs. 10. 20\/l2 by 8^3. 2. (72)* by (2)*. II. (6a26)*by (96 fcc^)'"'. 3. Vis by \/32. 12. Vso^ by \/2^. 4. 75* by 60*. 13. \/27 ar« by ^36 a-. 5. 6.3* by 2.3*. 14. Vp by \/|. 6. V32 by V4. 15. (fl)^ by (5|)^ 7. 45* by 9*. 16. V^l by Vf. 8. \/i~28 by V48. 17. (|l)* by (||)'. 9. ^12 by VTg. 18. (II)' by 2.8'. EVOLUTION OP SURDS 212. I. Extract the cube root of ^27^^ ^(\/27Ta)=(v'(3^)5-[(3 .T)?.]^-(3 t)1=\/3^. IRRATIONAL NUMBERS 177 2. Extract the fifth root of Vg. Then, to extract any root of a surd, If possible, extract the required root of the expression under the radical sign ; otherwise, multiply the index of the surd by the index of the required root. If the surd has a coefficient which is not a perfect power of the degree denoted by the index of the required root, it should be introduced under the radical sign (§ 206) before applying the rule. Thus, \/(4\/2) = \/( V32) = V2, or \/(4\/2) = (4 • 2i)^=(22 • 24)^=2* • 2To=2ro . 2x^=21^= \/2. EXERCISE 90 Find the values of the following ; 1. \/(\/25). 5. V^(V^9a*^ + 12a-|-4). 9. \^(16a^v'3a). 2. i/(\/8aW). 6. ^(V^). 10. \/(2x\/4x'^). 3. \/(^13). 7. V^CSl^ie). II. ^^(Vsis). 4. \/(V^243a;'^).8. \/(2^3 a^ft). 12. ^^(2 n^x/lGr^^). REDUCTION OF A FRACTION WHOSE DENOMINATOR IS IRRATIONAL (§ 198) TO AN EQUIVALENT FRACTION HAV- ING A RATIONAL DENOMINATOR 213. Case I. When the denominator is a monomial. The reduction may be effected by multiplying both terms of the fraction by a surd of the same degree as the denomi- nator, having under its radical sign such an expression as will make the denominator of the resulting fraction rational. 5 Ex. Reduce 3. — - to an equivalent fraction having a v3 a^ rational denominator. Multiplying both terms by ^9 a, we have 5 ^ 5^9a ^ 5^9a ^ 5i^9^ 178 • ALGEBRA EXERCISE 91 Reduce each of the following to an equivalent fraction having a rational denominator : VS ^2 ^4 i/ZQb , 1 .5 , a" _ 9 2. —* 4. -• O. ' O. \/2 V? i^9a2 ^27 214. Case II. When the denominator is a binomial con- taining only surds of the second degree, 5— \/2 1. Reduce to an equivalent fraction having a rational denominator. Multiplying both terms by 5 - \/2 (5 — \/2 is called the conjugate of 54-\/2), we have 5-\/^ ^ {5-\/2y ^ 25-10\/2 + 2 ^^^ g^ g^. ^ 27-lOV^ 5+\/2 (5+\/2)(5-\/2) 25-2 ' 23 2. Reduce ^ ^—— to an equivalent fraction having 2Va-3Va-b a rational denominator. Multiplying both terms by 2\/a + 3\/a — b, S\/a-2Va-h _ {S\/a~2\/a~b){2\/a-{-3\/a-b) 2\/a-3\/a^b (2Va-3\/a^)(2\/a + 3\/o~-^) __ 6a + 5\/a\/a-b-Q(a-b) _ 6b + 5\/a^-ab _ 4a-9{a-b) 9b-5a Rule : — Multiply both numerator and denominator of the fraction by the denominator with the sign between its terms changed. BXEKCISE 92 Reduce each of the following to an equivalent fraction having a rational denominator : 3 ^ 8 4-(2)^ _ ' \/3-f\/2 * 3~(5)^- ' 4 + (2y- IRRATIONAL NUMBERS 179 a + Vb V1O-6V2 ^ (x^+u^)^+a_ '^' a-Vb ' Vi0 + 3V2 ' (a;='+o^)^-o 7. l + -7=^- 2+V"'^ 9. . 8. ^^-(^+y)\ 2-^ 3-(a-3)* 5.2*4-6* ,, Vx^ + \ 10. ^^ ^. II. -• 12. • 3 + (a-3)* 3.2*-6^ Va;-2+2 Add the following fra(itions : (The common denominator is more readily found if the denominators of the fractions are first rationalized.) ^ I 3 g + fc^ a~b^ 3i+2^ 3i-2i " ■ „_2 5i „+fei' 14 2\/6 + l , 5 + \/6 ^^ (a; + l)^+2 3\/2 ^3 + 2 V12 ■ (x + l)*-2^(2a;+2)i* 215. The approximate value of a fraction whose denomi- nator is irrational may be conveniently found by reducing it to an equivalent fraction with a rational denominator. Ex' Find the approximate value of — — — r to three places of decimals. ^~ 1 _ 2 + \/2 2+\/2 _ 2 + 1.414- ^ j ^q^. , . 2-\/2 (2-\/2)(2 + \/2) 4-2 2 The \/2 and the V^ are important values and are of frequent occur- rence in mathematical investigation. EXERCISE 93 Find the values of the following to three places of deci- mals: 3 1 1 V5 V 2 V3 180 ALGEBRA 8. Vs V5-1 \/io 1.5 1 5+2N/7 2\/3-4 V'6-2 216. Important Property of Quadratic Surds (§201). I. A quadratic surd cannot equal the sum of a rational expression and a quadratic surd. For, if possible, let \/a=6 + \/c, where b is a rational expression, and V a and Vc quadratic surds. Squaring both members, a = b^ + 2 feVc+c, or, 2 6\/c=a— 6^ — c. Whence, Vc = — 26 That is, a quadratic surd equal to a rational expression. But this is impossible; whence, Va cannot equal 6+ vc. II. If a+ V 6=^+Vc?, where a and c are rational ex- pressions^ and V 6 and Vd quadratic surds^ then a=c, and v6 = Vrf. If a does not equal c, let a=c+a; ; then, x is rational. Substituting this value in the given equation, c+x + \/b=c + \/d, or x + Vb==Vd, But this is impossible by § 216. Then, a=c^ and therefore V6 = Va. That is, an equation of the form a + \^ = c-\- Vd may be written as two equations, a=c, b=d. 217. Solution of Equations having the Unknown Num- bers under Radical Signs. I. Solve the equation (x^ — 5y — x= — l. IRRATIONAL NUMBERS 181 Transposing —x, {x'^ — bp=x — \. Squaring both members, x^ — b=x^—2x-^l. Transposing, 2 x = 6; whence , x—S. (Substituting 3 for x in the given first member, and taking the positive vahie of the square root, the first member becomes (9~5)i-3 = 2-3=-l; which shows that the solution a: = 3 is correct.) Where no sign occurs before a radical the positive sign is understood. Also, in equations of the type, \/x-\-S + \/x'^ + 9 — \/x—S = \/x, usage requires that we regard only the given sign before the radical rather than the double sign that naturally belongs to a radical. Rule : — Transpose the terms of the equation so that a surd term may stand alone in one member ; then raise both members to a power of the same degree as the surd. If surd terms still remain, repeat the operation. The equation should be simpHfied as much as possible before per- forming the involution. 2. Solve the equation \/2 rr- 1 +\/2 x-f 6 = 7. Transposing \/2 x- 1, \/2 x-\-Q = 7-\/2 x-1. Squaring, 2 a;-f 6 = 49-14\/2 x-1 + 2 rr-1. Transposing, 14\/2 a;-l=42, or \/2a;-l=3. Squaring, 2 a: — 1 = 9 ; whence, x — 5. Substitute a; = 5 in the given equation to verify the result. EXERCISE 94 Solve the following equations, verifying each result : (Any radical may be changed to a form with fractional exponents and the exponent form is often more easily solved.) 1. (2a; + l)i-5=0. g. _2 2z_^(2-2z)K 2. V2"^T7+5 = 8. (2+8)^ {2-2 z)i 3. (4<2-19)*=2<-l. 6. \/5m-24 + 4 = V5m. 4. VM^ri+M = ll. 7. (i;)*+(v-3)*=3. 182 ALGEBRA g^ (3a:+24)^-(3a^)^ _l ^^^^^^^ (3 ir4-24)*+(3a;)* ^ 9. Vu-S-Vu + 21 = -2Vu, 10. {Sx^+S6x'')^-S=2x, V a; + a + V a; — a (Rationalize denominator, or use "• V^^-V^^a ' ^''''^ 12. (^2_5^-2)i + (^2^3^ + 6)*=4. 13. Vx + 15-Vx + S==2Vx. ^^ (2x-\-S)i + (2x)-^ _^^ (2a:+8)*-(2a;)* 15. ^x-2 a-Vx-da = 2Vx-5 a. IMAGINARY NUMBERS 218. If a number involves an indicated even root of a negative number it is called imaginary. Such numbers de- pend upon a new unit, V — 1 or (—1)^; as v — 2, V — 3. In contradistinction, rational and irrational numbers (§ 199) are called real numbers. 219. An imaginary number of the form v — a is called a pure imaginary number, and the sum of a real and an imaginary is called a complex number; as a4-6V^--l. 220. Meaning of a Pure Imaginary Number. If Va is reaZ(§218), we define Va as an expression sueli that, when raised to the second power, the result is a (§ 165). To find what meaning to attach to a pure imaginary num- ber, we assume the above principle to hold when va is imaginary. Thus, V — 2 means an expression such that, when raised to the second power, the result is — 2 ; that is, (V — 2)^ or (-2*)2=-2. _ In like manner, (V— 1)2 = (— 1^)2= _i ; etc. IRRATIONAL NUMBERS 183 OPERATIONS WITH IMAGINARY NUMBERS 221. By §220, (\/^y=={-5^y=-5. (1) Also, (VW^y={\/5y(V^y=5(-i)==--5, (2) or (\/Z5)2_(5iy2. (_.li)2_5(_i)__.5^ From (1) and (2), (\/-5y = (\/~5\/-l)\ Whence, V^5^VW^, or 5*(- 1)*. Then, every imaginary square roo t can be expressed as the product of a real number by ^—l. It is advisable to re- duce every imaginary to this form before perforjning the indicated operations. \/— 1 is called the imaginary unit ; it is often represented by i. 222. Addition and Subtraction of Imaginary Numbers. Pure imaginary numbers may be added and subtracted in the same manner as surds. 1. Add V^ and V-36. By §221, \/^ + \/^^ = 2(-l)i4-6(-l)i=8(-l)i. 2. Subtract 3-\/^ from l + V^MG. In adding or subtracting complex numbers, we assume that the rules for adding or subtracting real numbers may be applied without change. Then, l + \/^^-(3-\/^) = l+4\/^-3 + 3\/^ = -2 + 7V^. EXERCISE 95 Simplify the following : 1. (-9)^ + (-25)4. 2. \/-5 + V^^^. 5. V-(a; + 2)2-\/--a:2. 6. (-a:2)i + (_2^2)^(_22)|^ 7. 3\/^^+2\/-144 + 'v/^^. 8. 2(-16)*-5(-49)*-8(-121)^. 9. \/-16a;2-\/-9a:'-V^-4a;2. 184 ALGEBRA 10. \/-4 a^-4: a6-62- V-9 aH6 ab-b\ 11. Add2+(-3)ito5 + (-27)l 12. Add 6~\/-64 to l-\/-49. 13. From 2-f V^ take 8-\/-25. 223. Positive Integral Powers of V^-1 or — 1-. By § 220, (-li)2=(-l)i. (-l)i=-l. (By adding exponents.) Then, (V^l)^ = (-li)2-(-l)i=-l-(-l)i=-\/^l; (\/^)^ = (-li)2.(-li)2=-i.- 1 =1; (\/iri)5=(-li)* •(-!)* = !•- li=\/^,etc. Thus, the first four positive integral powers of \/— 1 are \/-^, —1, — \/— 1, and 1; and for higher powers these terms recur in the same order, the sixth power being like the second, etc. 224. Multiplication of Imaginary Numbers. The product of two or more imaginary square roots can be obtained by aid of the principles of §§221 and 223. I. Multiply V^ by V^. By §221, -2i- -3^=2*. -1* • 3* • -1* = 2i.3i-(-li)2 = 6i(-l) (§223) =-V6. 2. Find the product of V^, \/^^, and \/-25. \/^X\/^^X\/^^=3\/^X4\/^X5\/^ = 60(\/^)3 = 60(-\/^) (§ 223) = -60\/^. 3. Multiply 2-f-5\/^ by i-SV^, In multiplying complex numbers, we assume that the rules for multi- plying real numbers may be applied without change. 24- 5\/5\/^ 4- sVEV^ 8+20\/5\/^ - 6\/5V^-15(5)(-l) S + lW-5 +75=83 + 14\/^ IRRATIONAL NUMBERS 185 4. Expand {V-5+2V-3y by the rule of § 91. (-5iH-2--3i)2 = (5i- -li + 2 • 3*- -li)^ = (5i)2. (-ii)2 + 4 • 5^- -li • si- -li + 22 . (3i)2 . (-li)' = -5 + 4.15i-(-li)2 + 4--3 = -5-4 • 15i-12= -17-4\/i5. EXERCISE 96 Multiply the following : 1. (-4)i by (-9)*. 5. Vile by V^^. 2. (-36)* by (-16)*. 6. (-9)* by (-18)*. 3. V^ by \/^. 7. 3 + (-3)* by 2 + (-2)*. 4. (-196a2)*by(-144a2)*. 8. 5+4\/^ by 2-\/^3. 9. 3\/^+2\/^ by 2\/^-t-3\/ir^. 10. SvC:?, 5V^, -3\/^, and 2V^, 11. Vrie^ \/-49, \/^^, and V- 100. Expand the following by inspection : 12. [2 + (-3)*p. 14. (5\/^4-3V33)2^ 13. (5-\/'^)2. 15. [6+4(-3)*][6-4(-3)*]. 16. (\/-3 x+y){V-3 x-y), 17. [(-5a:)*+74][(-5x)*+5*]. 18. (8V^+3\/^)(8\/^-3\/^). 19. (a+6V^)(a-6\/^). 20. Add (a+6\/^) to (a-6\/^). a + 6\/— 1 and a—h\/—l are called conjugate imaginaries. Note that their sum and their product are real. 225. Division of Imaginary Numbers. I. Divide V-40 by V"^. 186 ALGEBRA 2. Divide Vld by V^. \/-3 VsV-i vS\/^ \/3— n/— 2 3. Reduce -^: y=-^ to an equivalent fraction having a \/3_^.\/-2 real denominator. Multiply both numerator and denominator of the fraction by the conjugate of the denominator, i. e., V^— V — 2, that is, the denomi- nator with the sign between its terms changed. V^-x/^j ^ (\/3-\/^)^ (189) 3-(-2) ^^ ^ _ 3-2\/^-2 _ l-2\/-6 3 + 2 5 EXSBCISE 97 Divide the following : 1. -20Hy -5*. 5. -^/32 by -\/^. 2. V-is by \/-3. 6. (180)* by -(10)*. 3. -36* by -12*. 7. -v^^^ by V^^. 4. -(-12a26)* by (3a6)*. 8. -Va by \/^. Reduce each of the following to an equivalent fraction having a real denominator : 9. ^ _ > X, 3\/::^+2v^^ 2-\/-3 • 3x/Z^_2\/_6 xo. 2±i-:3)*. ,,, 2x/g-5 2-(-3)* 2\/-5 + 5 QUADRATIC EQUATIONS 187 XIV. QUADRATIC EQUATIONS 226. A Quadratic Equation is an equation of the second degree (§ 75), with one or more unknown numbers. A Pure Quadratic Equation is a quadratic equation in- volving only the square of the unknown number ; as, 2x^ = 5. An Affected Quadratic Equation is a quadratic equation involving both the square and the first power of the unknown number; as, 2x^ — 3a; — 5 = 0. In § 103, we showed how to solve quadratic equations of the forms ax^ + bx=0, ax^ + c=0, x^ + ax-\-b=0, and ax^ + bx + c = 0, when the first members could be resolved into factors. PURE QUADRATIC EQUATIONS 227. Let it be required to solve the equation ^^2-4=0, or x^ = 4. Taking the square root of each member, we have ±x=±2; for the square root of a number may be either -f or — (§168). But the equations —x = 2 and x=~2 are the same as x=—2 and a? = 2, respectively, with all signs changed. We then get all the values of x by equating the j^ositlve square root of the first member to the zb square root of the second. The graph of a quadratic expression, with one unknown number, a?, may be found by putting y equal to the expres- sion, and finding the graph of the resulting equation as in §151. In the equation a;^ — 4=0 placing that is, substituting 7/ = 0, and finding values for y by assigning v^alues 0, 1, 2, etc., to X, we have 188 ALGEBRA ~ Y G i D \ \ \ X' F^ 'c X \ 1 \ ^ 1 I i \ / A k \ J s S <" / A Y' -4 W) 1 -3 (5) 2 (C) 3 5 m 4 12 -1 -3 (E) -2 -0 (F) -3 5 iO) -4 12 Connecting these points (A), (J5), (C), etc., we find, that they form a smooth curve, that the low- est point of the curve is on the y-axis, that the curve crosses the a:-axis at ±2. That is, the curve a;^ — 4=0 crosses the line y=0 (the x-axis) in two points and that these points of intersection (2, 0), ( — 2, 0) correspond to the algebraic solution of x^ — 4=0 in § 227. In general, it will be found that the graph of every equation of the second degree (§ 75) in two variables (unknown numbers) is a curve. The above geometrical picture shows in a graphical way why a quad- ratic equation has two roots. Find the graph of : I. a?2_9=0. BXERCISE 98 2. 0^2^16=0. 3. a'2_25=0. 228. A pure quadratic equation may be solved by redu- cing it, if necessary, to the form x'^^a^ and then equating x to ± the square root of a (§227). 5 x^ I. Solve the equation i^x^+7= f-35. Clearing of fractions, 12 x^ + 28 = 5 a:^ -f 140. Transposing, and uniting terms, 7 x^ = 112, or x^ = 16. P^quating x to the ± square root of 16, x= ±4. Verify by substituting x= ±4 in the given equation. QUADRATIC EQUATIONS 189 2. Solve the equation 7x^—5=5x^—13. Transposing, and uniting terms, 2 x^= ~H, or a:^= — 4. Equating x to the ± square root of — 4, a;=±\/ — 4 = ±2\/^ (§221). In this case, both values of x are imaginary (§ 219) ; it is impossible to find a real value of x which will satisfy the given equation. Verify by substituting in the given equation. Make a graph of a;^ = — 4. In solving fractional quadratic equations, any solution wliich does not satisfy the given equation must be rejected. Thus, let it be required to solve the equation x^-7 ^ 1 1_. x^ + x-2 x-^2 x-\ Multiplying both members by (a; + 2)(a;— 1), or x^-f-x — 2, x'^ — l=x — \—x-2y or a;^ = 4. Extracting square roots, x— ±2. The solution .r = — 2 does not satisfy the given equation ; the only solution is a; = 2. EXERCISE 99 Solve the following equations and verify each result : I. 8 7;2^24-7'z;H25. 3. 3(2 <-f4)H4(3 ^-2)2=256. ^' Zx'' 5^2 60* "^^ 3 3 a: 12 a;* 2a:H4 3a;^-7 _ll ^'5 3 15' 6. ^^'""-i-^^!±^ = ?j::?. (§132, Ex 3.) 12 5xH4 4 ^^ ^ 7. X/10T^-VI^3^=2. 8. ijj!±3_8.^-l^l, 7 2 14 3 a ^+5 6 t-bh 3a+10 6 4x^-1 . 10. — + - =0. Solve for t. 190 ALGEBRA a t+a . t II. 1 t—a t+a a+c a—c a^c a+c (§ 133, Exs. 1, 2.) 12. Vx^+2=-x-- The following equations occur in the study of physics. Solve in the first six equations for the number which appears to the second power. mM 13. S=^\ge 14. E- -- 2 mv'^. i6. H = C'RL i8. R = 19 . If the square of a certain number divided by 4 is added to twice the square of the number divided by 32, the sum is 20 ; find the number. 20. One number is five times another, and the difference of their squares is 216 ; find the numbers. 2 1 . If one angle of the right triangle ABC is 30°, the hypothenuse is twice the shorter side. The side opposite angle B is 10V3; find CB and explain your negative root. 2 2. A ladder 25 feet long standing in a court, will reach a window on one side of the court 20 feet from the ground. If turned on its foot as ^^ an axis it will reach a window in the op- posite wall 15 feet from the ground. Find distance across the court and explain your negative roots. 23. Two camps, A and J5, are at oppo- site sides of a lake. In order to find the distance between them, a line BC was measured at right angles to AB. BC was found to be 441 feet. AC was measured and found to be 735 feet. Find AB, 24. In a semicircle, if the perpendicular DP be dropped from a point P in the circumference to the diameter AB, DP QUADRATIC EQUATIONS 191 is a mean proportional between the segments AD and DB, If the perpendicular DP is 6, find AD and Z)S, the radius of the circle being 10. If / k DP is 10 and the radius 6, what effect does / ^ q ^\ it have on your solution ? Draw the figure. ^ 25. When a body falls from rest from any point above the earth's surface, the distance, S, which it traverses in any number of seconds, t, is found to be given by the equation in which g represents the velocity which the body acquires in one second ; ^=32.15 feet, or 980 centimeters. A stone fell from a balloon a mile high ; how much time elapsed before it reached the earth? 26. In the equation t=7r\-, t represents the time required 9 by a pendulum to make one vibration, / represents the length of the pendulum, and g is the same as in Problem 25. Find the length of a pendulum which beats seconds. 27. If a pendulum which beats seconds is found to be 99.3 centimeters long, find from the above equation the value ^^ 9' B 28. The area of an equilateral triangle /j\ ABC is 16V3^ Find the altitude DB. 29. Two balloons start at the same time from St. Louis on a long-distance race. . One strikes a northwest current carrying it ^ 30 miles an hour; the other strikes a southwest current carrying it 25 miles an hour. At the end of the second hour each balloon is one mile from the earth. How far apart are they ? 30. Two automobiles start from A at the same time, one going north at 18 miles an hour, the other going east at 15 192 ALGEBRA miles an hour. How far apart are they at the end of the A first hour ? 31. Z) is due west of C, A is due north of Z), and the distance from C to D is 84 miles. At 2 p. m. a train leaves C for Z>, running 40 miles an hour. At 2 : 30 p. m. a train leaves D for A, running 44 miles an hour. How far Q apart are they at 3 p. M. ? /^1^ 2X 1 32. A window in the form of a rect- angle surmounted by a semicircle is found to admit the most light when its height and width are equal. If the area of this win- dow is 32.1372, find the width. AFFECTED QUADRATIC EQUATIONS What must be added to 0^^+40: to form a perfect trino- mial square? (Exercise 30.) What must be added to a;^-f 10a? to form a perfect trinomial square ? 229. First Method of Completing the Square. By transposing the terms involving x to the first member, and all other terms to the second, and then dividing both members by the coefficient of x^, any affected quadratic equation can be reduced to the form x^+px=q. We then add to both members such an expression as will make the first member a trinomial perfect square (Exercise 30); an operation which is termed completing the square, Ex. Solve the equation a;^ + 3 a; = 4. A trinomial is a perfect square when its first and third terms are perfect squares and positive, and its second term plus or minus twice the product of their square roots (Exer- cise 30). Then, the square root of the third term is equal to the second term divided by twice the square root of the first. QUADRATIC EQUATIONS 193 Hence, the square I'oot of the expression which must be added to oiy^-\-Zx to make it a perfect square is 3 x-j-2 a:, or |. Adding to both members the square of |, we have Equating the square root of the first member to the ± square root of the second (compare § 227), we have Transposing |, ^=~| + 2 ^^ ~|~'2^^ ^^ ^'^• Rule: Reduce the equation to the form x^'\-px^q. Complete the square, by adding to both members the square of one-half the coefficient of 05. Equate the square root of the first member to the ± square root of the second, and solve the linear equations thus formed. 230. I. Solve the equation 3 or^ — 8 a: =—4. Dividing by 3, x''-—^--. o o which is in the form x^-^-px — q. Adding to both members the square of - , we have o 2_8a: , /4\2 4 , 16_4 3 W 3 9 9 Equating the square root of the first member to tlie ± square root 9 3 3 _ . 4 4 2 2 Transposmg — » x=-db-=2or-- If the coefficient of x^ is negative^ the sign of each term must be changed. 2. Solve the equation — 9 a;^ — 21 a; =10. Dividing by -9, x^-^ ^= - ^9. ^ •" 3 9 194 ALGEBRA Adding to both members the square of Extracting square roots, Then, 7N2 6> ._10 , 49^ 9^ 9 36 36* ^-1-7 .3 ^^r^6- 6 6 3 3 EXERCISE 100 Solve the following equations and verify each result : I. ^2+4^=32. 6. 12A:2-A:=1. 2. u^—u=6. 7. 9^2-3^=2. 3. ^2^8'i;=~12. 8. 9^2_9^__2. 4. m2-2m=15. 9. 9/2+9^=4. 5. 4x24-4x=3. lO, 16s-^-8s=15 231. The graphs of afiEected quadratic equations can be readily constructed by the method used in § 227. Construct the graph or geometrical picture of a;2_a;-6 = 0. (1) Placing the first member of the equation equal to y, we have x^-x-6=y. (2) Assigning values to x, we obtain corresponding values of y. For example, Substituting a;=0 in (2), we have 2/= —6, Substituting a: = 2 in (2), we have y= —4:, etc. x'^ — x — 6 = y y Y \ \ \ \ 1 X' K X 1 1 r 1 H / c 3\ i .. y' 3 4 -i -1 _2 -3 -6 -61 CD -6 -51 (B) -4 (C) -21 (D) {E) 6 -51 (G) -4 (H) (7C) 6 QUADRATIC EQUATIONS 195 Solving x^ — X — 6=»=0 or (x-3)(a:+2)=0, we have, a; =3 or —2. In the graph of this affected quadratic equation note (a) that the lowest point of the curve is not on the 2/-axis ; (6) that the curve crosses the a;-axis in two points (a;=3, a:= — 2) cor- responding to the algebraic solution. The graph of every equation of the form x^-\-px=q or ax^ + bx+c=0 is a curve of the above form and is called a parabola. EXEKCISE 101 Construct the graph of the following equations and com- pare the points of intersection with the algebraic solution : 1. x''-x-2=0, 4. Sx^+6x=^-l. 2. x^'-S x-{-15=0, 5. 8a;2-2ii:=l. 3. x^'+Qx^-S. 6. 3a:2-17:r-6=0. 232. Second Method of Completing the Square. Every affected quadratic equation can be reduced to the form ax^+bx-\-c=0, or ax^-\-bx= — c. Multiplying both members by 4 a, we have 4ia^x'^'\-4abx== — 4:ac. We complete the square by adding to both members the square of or 6. (If the coefficient of x^ is a perfect ^/\^a square, the trinomial square may be completed by adding to both members the square of the quotient obtairied by dividing the coefficient of x by twice the square root of the coefficient ofx\ §229.) Then, 4 aV-f 4 a6a;+62=62_4 ^^ Extracting square roots, 2 ax-^b= ±\/6^— 4 ac. Transposing, 2ax= — b± \/¥— 4 ( Whence, ^^^b±Vb^-^ac^ 2a 196 ALGEBRA EuLE : — Reduce the equation to the form ax^ -f- &a5 = — c. Multiply both members by four times the coefficient of 05^, and add to each the square of the coefficient of oc in the given equation. The only advantage of this method over the preceding is in avoiding fractions in completing the square. I . Solve the equation 2 o;^ — 7 a; = — 3. Multiplying both members by 4X2, or 8, 16a;2-56a:=-24. Adding to both members the square of 7, 16 a;2-56x+72= -24 + 49 = 25. Extracting square roots, 4 a; — 7= ±5. Then, 4a; = 7±5 = 12 or 2, and a;=3 or \' EXERCISE 102 Solve the following equations using the second method ; verify all results : 1. 3mH10m=-^3. 6. 15 mH16 m4-l=0. 2. %e-lZt^-%, 7. 12a;2^11a;=~2. 3. 2r2^15r+25=0. 8. 6 o^^-j-ll a;=7. 4. 5 2^2+3w~2=0. 9. Qx\-T x^2^, 5. 4a;2+2a;~l=0. lo. 10^2 4-3 9=1. 233. Solution of Affected Quadratic Equations by Formula. It follows from § 232 that, if ax^-\-hx-\-c=^0, then ^^-Z^^^^K^. (1) 2 a This result may be used as 2i formula for the solution of any affected quadratic equation in the form aa^^-f 6a;H-c=0. I. Solve the equation 2 x^-\-b x— 18=0. Here, a=2, 6=5, andc=- 18; substituting in (1), -5i: V25Jrl44 _ -5dbl3 _o _ 9 J, ■ • — — ^ or — — • 4 4 2 QUADRATIC EQUATIONS 197 2. Solve the equation — 5 0:^+14 a; -f 3=0. Here, a= —5, 6 = 14, c =3; substituting in (1), -14jzv/l96 + 60 _ -14dil6 ^ _ 1 ^^ 3 -10 -10 5 3. Solve the equation 110 a;^ — 21 a? = — 1. Here, a = 110, 6=-21, c = l; then, ^__ 21±\/441-440 ^21jzl^ 1 Qj. _!. 220 220 10 11' Particular attention must be paid to the signs of the coefficients in making the substitution. EXERCISE 103 Solve the following equations by formula : 1. 4x2~7ir=-3. 6. 8x2-h2x=3. 2. 9 1*2^22 2^=~8. 7. 3<2_2^=40. 3. 8^2^10^=3. 8. m2-f7m=18. 4. 3'y2-8i;-3=0. 9. 28a:2~a:-15=0. 5. 12=23^-5^2 10. 5a;2-17a;+6=0. 234. The formula in §233 is important in determining the nature of the roots of a quadratic equation, also in de- termining the relation between the roots and the coefficients in the equation. In ax^ + bx+c=Oy , ^ -b-{-Vb^-4ac -b-Vb^-4ac by § 233, x= 2^ or ^- Call the first root Vi and the second r2. I. If 6^ — 4 ac h positive, Ti and rg are real and unequal. i:x.,x^-2x-H^0, 62-4ac=4+32=-f. Solving, a- = 4 or— 2. See Figure 1, Plate III. II. If 62-4ac = 0, Ti and r^ are real and equal. 198 ALGEBRA Ex.,x''-2x+l==0, 62_4ac=4-4=0. Solving, x = l or 1. See Figure 2, Plate III. III. If b^—4:ac is negative, Ti and rg are imaginary (§218). Ex., x''-2 x + S=0, 62_4ac=4-12. Solving, x=-^^^-y ^^ ^^ =l + V-2or l-V-2. See Figure 3, Plate III. The intersection of the curve with the x-axis is imaginary. Imaginary roots always occur in conjugate pairs (§ 214). Note that these three equations differ only in the third terms and that this difference seems to have the effect of raising or lowering the curve with respect to the a;-axis. Adding the values of n and r2, in _ -b+Vb^-4ac _ '-b'-Vb^-4ac 2a 2a r ^ -2b -6 we nave r, +r2= = — • 2a a Finding their product, 2 a 4 a^ a Hence, if a quadratic equation is in the form the sum of the roots equals minus the coefficient of ac divided by the coefficient of x^, and the product of the roots equals the independent term divided by the coeffi- cient of ay^. 1. Find by inspection the sum and product of the roots of 8 a;2^7 0^-15=0. 7 — I'S The sum of the roots is ~ , and their product — ; , or — 5. iJ O 2. One root of the equation fia-^+'H a:= — 35 is — |; find the other. PLATE III "-I QUADRATIC EQUATIONS 199 The equation can be written 6 a:^ + 31 a; + 35=0. 31 Tlien, the sum of the roots is 6 31 / 7\ 31 7 Hence, the other root is — — — { ~o )' *^^ ~ "^ + «» 6 \ 2/ 6 2 We may also find the other root by dividing the product of the roots, ^ by -^. 6'^ 2 We may find the values of certain other expressions which are symmetrical in the roots of the quadratic. 3. If Ti and rg are the roots of ax^-{-bx+c==0, find the value of ri^+r^ra+rg^ We have , r^^ -l- r^r'^ + rg^ = (r^ + r^) ^ — r^r^. b c But, r, + To = — , and r.Vo = — Whence, V + ^i^2 + V= ^ -- = ^^- a^ a or 4. Determine by inspection the nature of the roots of 2 0:2-5 0?- 18=0. Herea=2, 6=-5, c=-18; and 6^-4 ac= 25 +144 = 169. Since 6^ — 4 ac is positive, the roots are real and unequal. Since 6^ — 4 ac is a perfect square, both roots are rational. EXERCISE 104 Find by inspection the nature of the roots, the sum and product of the roots, and construct the graph of each of the foUovi^ing 8 problems : 1. a;2+8 a:H-7=0. 5. ^^+2 a;4-4=0. 2. :r2-a;-20=0. 6. 9 ^^.^G a;- 1=0. 3. 4a;2--a;-5=0. 7. 9 a;2+6 a;+l=0. 4. 6ic2+a;=0. 8. 25a?2--4=0. 9. One root of x^ + 7 a; =98 is 7; find the other. Note that your definitions §§ 39, 60 are involved in these examples. H| 10. One root of 5 a?^— 17 a? +6=0 is | ; find the other. ^K^ II. Is 5 a root of x^-^r^ ar + 5=0? 200 ALGEBRA If Vi and r2 are the roots of ax^-\-bx-\-c=0^ find the values of: 12. A-L^. 13. -+- . 14. — +— • 15. n^+rg^ [Hint: (x+2/)3+(a;-T/)^ contains but two terms.] EXERCISE 105 Solve the following equations by the method which seems best adapted to the example under consideration, verifying each result : (In solving any equation, we reject any solution which does not satisfy the given equation.) 10, 4 2/H ^ = 14. 2. 49 07^49 a;+ 10=0. 2/ + 1 3. 5h'+12h=~4, II, ^ ^^1^. 5 — 2 2 4 4. 32 y- 48^2^-3. 5. 9m2+6m=19. 6. 2r2-15r=-13. 7. 12a;H5a:-hl=0. M 12. \/3+a:-a:2=2a:-3. 13. ^5 6-+ll = \/35+l+2. ^-2_^+4___7 x-{-5 x—S 3 8. 10-21 k- 10 F=0. (Compare Ex. 19, Exercise 56.) 2^+3 2^+9 _Q a: + l a;+3_^8 ^* 8+< 3<-f4. * ^ ' x-\-2 a;+4~3 3 a?2 4-4x^-1 16. 17. 2a;2-x-l 3a;2-2ir + 7 2<^-4f-3 _ <^-4<+2 .8.^^ L_=i + ^. m2-4 3(m+2) 2-m 19. \/8y+7 = \/4i/+34-V^2 2/4-2. a;+l , a?+2 , x+3 ^ 20. -r 1 =0. x—l x—2 x—S (Compare Ex. 14.) QUADRATIC EQUATIONS 201 21. (a?-2)(x-f3)(a^-4)=0. 23. ^^2^x4-1=0. 22. lx-S){2 x^' + lS x+20)=0, 24. x^=l. . 1 j^ 1 1 7 25. = — . 1-^2 i^i i_^ 8 26. 3- 1 ^ ^ x+2 2(2a;-3) (a:+2)(2x-3) ' ^ a:+l . x-\-2 2ar+13 27. 1- = . x~l x—2 x+l 28. t^= — S. (The roots are the three different cube roots of —8. Compare Ex. 24.) Vv Vv+2 5 29. \/v+2 Vv 6 2 . 2 m2+3m~16 30. 1- = m— 2 m— 5 m^— 7m+10 31. x^+ax— 6x— a6=0. We may write the equation x^-{-{a — h)x=ab. Multiplying both members by 4 times the coefficient of x^, 4 x^ H- 4(a — 6)a: = 4 a6. Adding to both members the square of a — 6, 4x^-{-Ma-b)x+{a-by = 4:ab + a^-2ab + b^ =a^-h2ab + b\ Extracting squ^^re root, 2 x+{a — b) = ±{a + b). Or, 2x==-(a-b)±(a + b). Then, 2 x= -a + b-\-a + b=2 6, or 2 x= —a + b — a — b= —2 a. Whence, a: =6 or —a. If several terms contain the same power of x, the coefficient of that power should be written in parenthesis, as shown in Ex. 1. For the solution of literal affected quadratic equations, the methods of § 232 are usually most convenient. The above equation can be solved more easily by the method of § 103 ; thus, by § 101, the equation may be written {x + a)(x-b)==0. Then, x + a=Oy or x~ —a; and x—b=0, or x = b. Several equations in Exercise 105 may be solved most easily by the method of § 103. 202 ALGEBRA 32. Solve the equation (m— l)a?^— 2 m^x=--4 m^ Multiplying both members by m — 1, and adding to both the square ofm\ {w-l)V-2m2(m-l)a; + m*=-4w2(w-l) + w* Extracting square root, {m — l)x — m'^=±{m^~2m). Then, {m--l)x=m^ + m^ — 2 m or m^ — m^-\-2 m = 2m{m — l) ov 2 m Whence, x — 2mov — —- m—l 33» x^—mx=m^; solve for x. 34. x^—mx=m^; solve for m. Solve the following for x : 35. x^—2ax= — Qa + 9, 38. x^—m^kx+mk^x=m^k^, 36. x^—(a—b)x=ab, ^/ — ; — ^ /7r~ 2 a ^ ^ - 39. Va+ir--v2a^ = --^:^- 37. x^+nx+x=—n. \/a-\-x 40. (a+6)a?H(3a+fe)a;=-2a. 41. Va:~a+\/2 a;-h3 a=5 a. Solve for / : 42. V5a-f^+V5a--<=2\//. 43. ^^'^ v^2 / + !=<- 1. 44. V/+9 a-f V25 a-f= V2 ^+32 a. 40. — -4 = 1. 47. = . l-at 1-i-at t + h /+« 2 ' (Compare Ex. 14.) 48. Solve for cr: t=7r\~' 9 49. Solve for s : V== \^2 gs, d 1 £ 2n— 3 a . 3n4-a 10 50. bolve for n : = — -• 3n+a 2n-3a 3 51. Solve for 7M S= ~[2 a+(n- 1)4 QUADRATIC EQUATIONS 203 PROBLEMS INVOLVING QUADRATIC EQUATIONS WITH ONE UNKNOAVN NUMBER 235. In solving problems which involve quadratic equa- tions, there will usually be two values of the unknown num- ber ; only those values should be retained which satisfy the conditions of the problem. 1. A man sold a watch for $21, and lost as many per cent as the watch cost dollars. Find the cost of the watch. Let a; = number of dollars the watch cost. Then, a; = the per cent of loss, and X X > or = number of dollars lost. 100 100 By the conditions , = a: — 2 1 . ^ 100 Solving, a: = 30or70. Then, the cost of the watch was either ^30 or $70 ; for either of these answers satisfies the conditions of the problem. 2. A farmer bought some sheep for $72. If he had bought 6 more for the same money, they would have cost him $1 apiece less. How many did he buy? Let n = number bought. 72 Then, — =number of dollars paid for one, n 72 and = number of dollars paid for one if there had been 6 more. By the conditions, — = -^ + 1 . n n+6 Solving, n = 18 or— 24. Only the positive value is admissible, for the negative value does not satisfy the conditions of the problem. Therefore, the number of sheep was 18. If, in the enunciation of the problem, the words " 6 more " had been changed to " 6 fewer," and " $1 apiece less " to ^' $1 apiece more," we should have found the answer 24. 3. If 3 times the square of the number of trees in an orchard be increased by 14, the result equals 23 times the number ; find the number. 204 ALGEBRA Let a; = number of trees. By the conditions, 3 x^ + 14 = 23 x. Solving, x=7 or §. Only the first value of x is admissible, for the fractional value does not satisfy the conditions of the problem. Then, the number of trees is 7. 4. If the square of the number of dollars in a man's assets equals 5 times the number increased by 150, find the number. Let a; = number of dollars in liis assets. By the conditions, a:^ = 5 x + 1 50. Solving, a; = 15 or— 10. This means that he has assets of $15, or liabilities of $10. EXERCISE 106 Verify all results. 1. What number added to its reciprocal gives |^| ? 2. Divide 17 into two such parts that three times the square of the greater shall exceed twice the square of the less by 115. 3. Find three consecutive numbers such that if the square of the second number be subtracted from the sum of the squares of the first and third, the remainder will be 38. 4. The sum of two numbers is 3 and the sum of their cubes is 7 ; find the numbers. 5. Two rectangles have their corre- sponding sides in the ratio of 5 to 2. In the greater the ratio of the length to the 375 breadth is |^. The area of the greater is 375 ; find the area of the less. 6. A farmer bought a certain number of sheep for f 300. Having lost 7, he sold the rest for $2 a head more than they cost him, and gained $44. How many did he sell ? 7. A rectangular field is twice as long as it is wide. If 20 rods were subtracted from the length and the same amount were added to the width, the field would be square and would contain 22|^ acres. Would this change decrease or increase the area of the field ? QUADRATIC EQUATIONS 205 8. A fast train's schedule from New York to Chicago is 12 miles an hour faster than a slow one, and requires 5 less hours to travel 960 miles. Find the rate of each train. 9. If the product of three consecutive numbers be divided by each of them in turn, the sum of the quotients is 107; find the numbers. 10. The area of a trapezoid is equal to the product of one- half the sum of the parallel sides and the b c altitude. Find the sides and altitude of ^^\ \. trapezoid A BCD in which u4Z) is 8 feet ^ e d more than B (7, and EB 2 feet less than B (7, the area being 55 square feet. Are there two such trapezoids ? 11. A merchant sold a bill of goods for $24, making as many per cent as the goods cost dollars. Find the cost. 12. Find two numbers whose difference is 4, and the differ- ence of whose cubes is 3088. 13. The area of a certain square field exceeds that of an- other square field by 1008 square yards, and the perimeter of the greater exceeds one-half that of the smaller by 120 yards. Find the side of each field. 14. A and B set out at the same time from places 247 miles apart, and travel toward each other. A's rate is 9 miles an hour ; and B's rate in miles an hour is less by 3 than the number of hours at the end of which they meet. Find B's rate. 15. A man buys a certain number of shares of stock, pay- ing for each as many dollars as he buys shares. After the price has advanced one-fifth as many dollars per share as he has shares, he sells, and gains $980. How many shares did he buy? 16. The two digits of a number differ by 1 ; and if the square of the number be added to the square of the given number with its digits reversed, the sum is 585. Find the number. 206 ALGEBRA 17. A merchant sold two pieces of cloth of different qual- ity for $105, the poorer containing 28 yards. He received for the finer as many dollars a yard as there were yards in the piece ; and 7 yards of the poorer sold for as much as 2 yards of the finer. Find the value of each piece. 18. In a circle with centre at (7, the tan- p^ gent PT is a mean proportional between ^^ the whole secant jPZ> and the external part PJS, If the tangent is 8, and the diameter UI) is 12, find PK 19. A and B gained in trade $2100. A's money was in the firm 15 months, and he received in principal and gain $3900. B's money, which was $5000, was in the firm 12 months. How much money did A put into the firm ? 20. The formula for the volume of the frustum of a cone is F=| TT yl(^- + r- + i?r), in which r is the radius of the upper base, jff the radius of the lower base, A the altitude and V the volume. If F=872 tt, R--^ r=10and ^ = 6; find P. 21. A square garden plot containing 144 square feet has two walks of equal width intersecting at right angles to each other and to the sides of the garden. The area of the walk is one-half the area of the entire square; find the v^idth of the walk. 22. A square piece of tin is to be made into a rectangular box by cutting a square out of each corner and folding up the sides. The pieces cut out are 6 inches square; the volume of the box, 1944 cubic inches. How large was the sheet of tin ? If a cubical box had been cut from this 1 I ! X 1 , 1 X-12 I QUADRATIC EQUATIONS 207 sheet of tin, would its volume have been greater or less than that of the first box formed ? 23. In a right-angled triangle, ABC^ one side is 5 more than the other, and the hypote- nuse is 5 more than the longer side. Find the dimensions. Draw diagram explaining your solutions. 24. If a body is thrown downward with an initial velocity, %, then the space it passes over in t seconds is found to be eriven by the equation A stone was thrown downward with a velocity of 40 feet per second from a balloon a mile high; g is 32.15. How many seconds elapsed before the stone reached the earth ? 25. In the equation jP=— — , M and m represent the masses of any two attracting bodies, as, for instance, the earth and the moon, d represents the distance between these bodies, and F the force with which they attract each other. If the moon had twice its present mass and were twice as far from the earth as at present, how much greater or less would the force of the earth's attraction be upon it than at present? 26. In the equation E=^ mv^^ E represents the energy of a moving body, the mass of which is m and the velocity is v. Compare the energies of two bodies, one of which has twice the mass and twice the velocity of the other. 27. When a bullet is shot upward with a velocity, v, the height, S, to which it rises is given by the equation v=V2~gS, Find with what velocity a body must be thrown upward to rise to the height of the Washington monument (555 feet). (See Problem 25, Exercise 99.) 208 ALGEBRA 236. Equations in Quadratic Form. An equation is said to be in the quadratic form when it is expressed in three terms, two of which contain the unknown number, and the exponent of the unknown number in one of these terms Is twice its exponent in the other; as, x^-6x^=W; aTHx^--72=(); etc. In equations in quadratic form, the simplest method for the beginner to apply is to let some letter represent the lowest power of the unknown quantity in the given equation. 1. X®— 6a:^=16. Let ^=x*\ Then, ?/-0 2/-16=0. Wlience, y=8or— 2, x8=8or -2, a: = 2or -^2. Verify these roots. 2. 2 a?+3Vx=27. Let y=x^ or V^. Then, 2 y^ + Sy =-27, (2 2/ + 9)(?/-3)=0, ?/=3< »-!• \/i = 3 rr=9 "f Verify these results. 3. 2 s-'' -85,v- -^48^ = 0. Let X =s-*. Tlien, 2x2- 35 X : + 48 = 0. Whence, ^« 3 s-< = 16or ^ . 1=16 or ?, a* 2 1 *^ .s<= or "i Hi 3 1 ^ */2 2"' ^3 J=:t.>r f-V^^ QUADRATIC EQUATIONS 209 EXERCISE 107 2. 2/-H19r^==216. 6. 32a?»+l = -33. 3. S/ + 14V/, = 15. 7. 6m-J-^5m-J = 6. 4. m'-3m^==88. 8. 4\^i?'-h6 = ll\/:r2. 9. (2 x^-S xy-^2 x''-3 x) = 9. 10. (5m+12)-5(5m+12)* = -4. FACTORING 237. Factoring of Quadratic Expressions. A quadratic exjjression is an expression of tlie form ax^ + bx-\'C. In § 94 we showed how to factor certain expressions of this form by inspection ; we will now derive a rule for fac- toring any quadratic expression ; we have, ax^^-hx'{■c=a(x^^ h a a a \2 a 4a2^aJ Vf , by ¥-Aacr\ A"^2^)^-4^J by § 89. But by §233, the roots oi ax^-{-bx + c=0 are b . Vp^^ac J 6 Vb^^4ac and • 2a 2a 2a 2a Hence, to factor a quadratic expression, place it equal to zero, and solve the equation thus formed. Then the required factors are the coefficient of x^ in the given expression, x minus the first root, and x minus the seqpnd. 210 ALGEBRA 1. Factor 6 x^ + 7 x-S. Solving the equation 6 x^ + T a;-3=0, by §233, .^_ -7±V49 + 72 ^ -7±ll ^l _3. 12 12 • 3 2 Then, 6 x' + 7 x-3 = Qfx-^fx^^^ =3^a;-|^ X2^a:+ 1^ =(3 x- 1)(2 x + 3). 2. Factor 4+13x-12a;2. Solving the equation 4 + 13 a;-12 x2=0, by § 233, -13±\/l69 + 192_ -13dzl9^ 1 ^^ 4. -24 -24 4 3 Whence, 4 + 13 x- 12 x'=-12^x+ ^(x- -^ = (l+4a:)(4-3a:). 3. ¥3iGtor2x^-3xy-2y^-'7x + 4:y+6. We solve 2 a;2-x(3 2/ + 7)-2 2/^ + 4 .v + 6=0. By § 233, x = ' ^y+7 + \/(3y+7)^+16y^-32.v-48 4 _ 3y + 7ib\/25y^+10y+l _3 y+7 + (5 i/+l) 4 4 4 4 ^ 2 . Then, 2 a;2-3 a:2/-2 t/2-7 x4-4 ?/ + 6 ==2[x~(2y+2)]rx-=:^l = {x-2y-2)(2x+y-3). EXEBCISE 108 Factor the following : 1. 4x2-12 0^-7. 4. t^-^t + l, 2. a?2-fa;-12. 5- 6/H3/-f2. 3. 25 0-2-100?- 11. 6. 36?w2-5ryf~l. I QUADRATIC EQUATIONS 211 7. 20a;2-13x-fl. lo. 6-c-2c\ 8. a2+2a-f2. ii. 8t>2 + 18t;-5. 9. x*+x, 12. a^ + 4a-hl. 13. a2+a6-6 62+a4-13 6~6. 14. 2 a^^—x?/— i/^+3 a;+3 2/— 2. 15. 2 x""-^ xy+x-Q y^+lS y-6. 16. 6a2+7a6-4a-3fcH5 6-2. 238. We will now take up the factoring of expressions of the forms x^+ax^y^+y*^ or x^ + y*^ when the factors involve surds. (Compare § 96.) 1. Factor a^-f 2 a262+25 6^ a* + 2 a^b' + 25b' = (a* + 10 a'b^ + 25 b') - 8 a^b^ = ia^+5by-{ab\/sy\ = (a^ + 5b' + abVs)(a^ + 5b'-ab\/S) = (a2 + 2 ab\/2 + 5b%a^-2 ab\/2 + 5 b^. 2. Factor x^+l. x*+l={x*-^2x^-^l)-2x^ =(x'+iy-(xV2y = (x^-\-x\/2 + l){x^-x\/2 + l). KXERCISE 109 , In each of the following obtain two sets of factors, when this can be done without bringing in imaginary numbers : I. x'-7x^-hi. 4. 4aH6a2+9. 2 a' + h\ 5. 36 0:^-92 a:2+ 49. 3. 9m^-ll?^2^1. 6. 25mH28mV-|-16n^ Solve the following : 7. x^-\-l = 0. (The three roots are the three different cube roots of -1.) 8. x'+2x'^ + 4=0. 9. x'-\-Sx^O, 10. Find the three different cube roots of 27. (Compare Ex. 24, Exercise 105.) 212 ALGEBRA XV. SIMULTANEOUS QUADRATIC EQUATIONS 239. On the use of the double signs ± and T- If two or more equations involve double signs, it will be understood that the equations can be read in two ways ; first, reading all the upper signs together ; second, reading all the loiver signs together. Thus, the equations x= ±2, z/= ±3, can be read either x=+2, t/=+3, or 0?= — 2, 2/= — 3. Also, the equations x= ±2, ?/= =F3, can be read either a;=+2, 2/== — 3> or a;= — 2, y=-\-Z. 240. Two equations of the second degree (§ 75) with two unknown numbers will generally produce, by elimination, an equation of t\iQ fourth degree with one unknown number. Consider, for example, the equations {x^-\-y=^a, (1) \x+y'=^h, (2) From (1), y=a'-x^; substituting in (2), x+a^—2 ax^+x*=^b; an equation of the fourth degree in x. The methods already given are, therefore, not sufficient for the solution of every system of simultaneous quadratic equa- tions, with two unknown numbers. In certain cases, however, the solution may be effected. In the present work we shall consider only five simple types. 241 . Type I. When one equation is of the second degree, and the other of the first. Equations of this kind may be solved by finding one of the unknown numbers in terms of the other from the first degree equation, and substituting this value in the other equation. ■ ^I^^^^H ■ ■ ■ ■■ ■ ■■■ ■ ■ ■Q 1 ■■ ■ ■E ^ s 11 ■ ■ *' / ^! ■ i J f '■ ■ ■ >_ 1 II ■ ■ ■ ■ ■ ■ ■ 1 ■ ■ ■ Q ■ i ■ 1 ■ 1 ■ ■ B" li : ) ■■ ■^ i ■^ I HI n ■ ■■ ■ I i^mH^H I (1) (3) y—2x=—4 X y y4 1 1 2 2V4 3 4 4M) \i -1 1 -2(fi) 2^4 -3 4 -4 JC y -4 1 -2(5) 2 3 2 4 4(^) — 1 -6 -2 -8 The points A and J? are the only points common to both curves. Their coordinates, (4, 4) and (1, -2), satisfy both equations and corre- spond to the two algebraic solutions. In general there are two solutions of a quadratic equation and linear equation in two unknown quantities. PLATE IV i SIMULTANEOUS QUADRATIC EQUATIONS 213 \y-2x=-^. (2) From (2) y=2x-A. (3) Substituting in (1), 4 a:2-16a;+16=4 :r, 4^2-20 a:+16 = 0, a;2-5x + 4 = 0, whence, a; = 4 or 1. Substituting in (3) , ?/ = 2 a; - 4 = 8-4, or 2-4 = 4, or -2. The solution isa: = 4, 2/=4; ora; = l, 2/=— 2. Verify by substituting in the given equations The graphs of these equations are given in Plate IV. 242. Type IL When the given equations are symmetrical with respect to x and y ; that is^ when x and y can he inter- changed without changing the equation. Equations of this kind may be solved by combining them in such a way as to obtain the values oi x-\-y and x—y* Ex. Solve the equations \ ^ ~ ' ^ 1 xy^^l (2) Multiply (2) by 2, 2xy==- 14. (3) Add (1) and (3), x^ + 2 xy-\-y^=SQ, or x-\-y=^ ±6. (4) Subtract (3) from (1), x^-2 xy-\-y^ = 64, or x-y= ±8. (5) Add (4) and (5), 2 a: = 6±8, or -6±8. Whence, x = 7, —1, 1, or —7. Subtract (5) from (4), 2 2/ = 6=F8, or -6T8. Whence, y==—l,7, —7, or 1. The solution is x= ±7, y=Tl; or, x= ±1^ y=T7. Verify by substitution. In subtracting ±8 from 6, we have 6T 8, in accordance with the nota- tion explained in § 239. In operating with double signs, db is changed to =F, and T to ±, whenever 4- should be changed to — . The graphs of these equations will be found on Plate V. Note the symmetrical arrangement of the points of intersection. 214 ALGEBRA 243. Type III. When one equation is of the third degree and the other is of the first degree. Certain forms of systems of first and third degree equa- tions may be reduced to Type I or Type II by dividing one equation by the other. Ex. Ix^^f^lS. (1) \x+y==S. (2) Dividing (1) by (2), x^-xy + y^ = Q. (3) Use Type II, squaring (2) and subtracting the result from (3), -Sxy=-S. -xy=-l. (4) Adding (4) to (3), x''-2 xy + y^ = 5. _ (5) x-y=±V5. . (6) Solving (6) and (2) by addition and subtraction : S±V5 ^j. S-\/5 y- 2 ' 2 . 3-\/5 3j±V5. 2 ' 2 The solution is x=^^. ?/= ^~^^ , or 2 ' ^ 2 Verify by substitution in the given equations. 244. Type IV. When each equation is in the form In this case, either x^ or y^ can be eliminated by addition or subtraction. I. Solve the equations Sx''+ 42/2 = 76. (1) ' 3i/2-lla:2= 4. (2) Multiply (1) by 3, 9 x^^ 12 1/2=228. Multiply (2) by 4, I2y^-Ux^== 16. Subtracting, 53 0:^ = 21 2. Then, x*=4, and a;= ±2. Substituting x=^ ±2 in (1), 12 + 4 2/* = 76, or 4 2/* = 64. Then, 2/^ = 16, and i/= ±4. The solution is x = 2, 7y= ±4; or, 0*= —2, y= ±4. |B B m ^■■1 ihhhiiihh ■■■nBi ■■■■■■1 ^■■■■^l ^In ■■■■■■1 IHHHI^^H ^|m ■■■■■■I IHHHI^^^H ■9 ■b Slkii laiH ■■■in II ^In ■■■■■■ ■Ibih ■■■■■^^H ^|m ■■■■■■1 ■Hims^^H ^■m ■■■■■■ ■■■lifli ^^H (1) (2) xtf=—7 X y iSN/I ± I ±1{A) ±2 ±\/46 ±3 ±\/4T ±4 ±6V3 ±5 ±5 ±6 ±Vl4 ±7 il(0 jr y 1 -7 2 -% 3 -% 4 -% 5 -% 6 -% 7 -1(0 -I + 7(^) -2 + 72 etc. In equation (1) since both x and y appear only in the second power, the double sign occurs in each substitution, so that for every pair of numerical values we obtain four points on the curve. E. g. (±1, ±7) gives the four points A, B, G, D. The graph of equation (2) is in two branches. (See Ex. 4, § 245.) In general two equations of the second degree in tw^o unknowns give four solutions. PLATE V SIMULTANEOUS QUADRATIC EQUATIONS 215 In this case there are four possible sets of values of x and y which satisfy the given equations : 1. x = 2, 2/ = 4. 3. a;=-2, 2/ = 4. 2. x = 2,2/=-4. 4. x=-2, 2/==-4. It would not be correct to leave the result in the form x = ±2, ?/ = ±4, for this represents only the first and fourth of the above sets of values. The method of elimination by addition or subtraction may be used in other examples. 2. Solve the equations \ J^ [ 7 x^-\-%y=^66. (2) Multiply (1) by 3, ^x^-V2y = Ul. Multiply (2) by 2, * 14x^+12y= 66. Adding, 23a:2 = 207. Then, a;2=9, and a: =±3. Substituting a; = ±3 in (1), 27-4 1/= 47, and y=-5. It is possible to eliminate one unknown number, in the above exam- ples, by substitution (§ 157), or by comparison (§ 158). 245. Type V. When each equation is of the second de- gree, and homogeneous ; that is^ when each term involving the unknown numbers is of the second degree with respect to them (§ 59). Certain equations of thi$ type can be solved by the methods of §§ 242 and 244. The method of Type V should be used only when the example cannot be solved by Type II or Type IV. Ex. Solve j^'-2a:.v= 5. (1) |a:H 7/2^29. (2) Dividing (1) by (2), ^=^'-|^ or 29a;2-58x2/=5x2 +5 7/2. Then, 5 2/' + 58a:?/ -24 x^^O, or (5 2/-2 a:)(?/4- 12 a;)=0. 2 X Solving for ?/, 2/== -^ . or - 12 x. o Substituting these values in (1) we have a;2_ 1^^=35^ or a:2 + 24a:2 = 5. 216 Whence, ALGEBRA x=±5, or x= ± — —' \/5 x= ±5 was obtained through 2 X y= — , whence y= ±2. 5 x= ± — — was obtained through 12 y= — 12x, whence y=T — =• V5 The solution is = ±5,?/=±2, or x = V5 \/5 3a;2+2i/2 = (3x \9x' 2 + 5 1/2 = 3x^5 y^== EXERCISE 110 = 66. = 189. -116. 12. 7 a;+4 2/^ = 121. - ?/= 7. x^—xy+y'^ = l24:. x-\-y = S. (4t^-\-u^^ 61. R +S =3. M = \. xy=^25. x-{-y=U). S_yS._ 3 l^'-l ' [^ -I = 159. -117. -3. (x+y=2. U2/ = -15. (Type II.) 8. 122. -10. 26. -I 22 + i;2: z +v = (x^+k' kx=5. u—v= 4. 2 i^i; = 42. J^ -S = 3. 15 1 6. 17. 18. 19. 20. ' [L- ■I lxy=24. • l2ar-«= 9<2-5m='=205. 4fi+9u^ = l36. 4A^+ 7ik2=32. 3/!.2-ll F=-41. 3 2 2a; Sy ^ 36 5r/i = 10 f xy=ab. [ x—y=a—b. 2 1 . From i; = ^< and S= - gf, find i; in terms of S and g. SIMULTANEOUS QUADRATIC EQUATIONS 217 22. From C= - and £C= — ,findif intermsof C,/J,and^ R t 23. From E = FS, F = ma, S=-af, and v=at, find E in terms of m and v, . x^—xy= 4. r 5^2- 2/2= 1. ^^ f2x2-ar2/ = 28 I; 24. 25. 26. p2+p?-5g2=25. p2_^4 72=40. Uy-3 2/2=-10. .0:2+2 2/2=18. GRAPHS 246. Consider the equation x^+y^=25. that for any This means point on the graph, the square of the abscissa, plus the square of the ordinate, equals 25. But the square of the ab- scissa of any point, plus the square of the ordinate, equals the square of the distance of the point from the origin; for the distance is the hypotenuse of a right triangle, whose other two sides are the abscissa and ordinate. Then the square of the distance from of any point on the graph is 25; or, the distanjce from O of any point on the graph is 5. Thus, the graph is a circle of radius 5, having its centre at O. (The graph of any equation of the form x^ + y^=a is a circle.) graph of (1) Plate V is a circle. 2. Consider the equation i/2=4 a;+4. Ifx=0, 2/'=4, or2/=±2. _ {A, B) Ifa: = l, 2/' = 8. or2/=±2\/2. (CD) Ifx=-1, y =0. Etc. (E) The graph extends indefinitely to the right of YY\ (Fig. 2.) If X is negative and < — 1 , y^ is negative, and therefore y imaginary then, no part of the graph lies to the left of E. Y B ^ ^ ■^ N / c / / y y X' / X A \ \ \ / ^ ^ f^ y Y' Fig- 1. The 218 ALGEBRA (The graph of Ex. 2 is a parabola; as also is the graph of any equation of the form y^=ax or y^=ax + b. The graph of (1) § 241 is a para- bola.) 3. Consider the equa- tion x^+4y^=i. In this case it is con- venient to first locate tlie points where the graph in- tersects the axes. (Fig. 3.) If2/=0, a;2=4, ora;=±2, (A, A') If a;=0, 4?/2 = 4, ory=±l. {B,B') Putting x=±\, 4 2/2 = 3^ 3 Y ^ ^ ;^ ^ ^ ^ y^ ^ X /" It: A / / y f / X' E X [ \ \ \ B s \ P. rv s. V s^ < v^ ^ *^ Y' "^ y .•ory=^^. Fig. 2. (C, D, C, DO Y B ^ — ■— -^ ^ ,y x' k, / \ A' f \ A X X' i / \ / s. •»s ^ X' ^^ ^^ ^ ^ ^ V B' y' Fig. 3. If X has any value >2, or < - 2, yMs negative, and y imaginary ; then, ) part of the graph lies to the right of A , or left of A \ SIMULTANEOUS QUADRATIC EQUATIONS 219 If y has any value >1, or < — 1, a;Ms negative, and x imaginary ; then, no part of the graph Hes above B, or below B\ (The graph of Ex. 3 is an ellipse ; as also is the graph of any equation of the form ax^-\-hy'^ = c.) Y s /" \ s^ y / S ^' B / \ / \ / \ A \ X' \ A' A X / / / \ / \ / \ / c c \ / / S \. A ^ S _J Y' _j Fig. 4. 4. Consider the equation x'^—2y'^=l, r2-l Here x^ — 1=2 !/2, or 2/^ If x= ±1,2/2=0, or 2/=0. {A\A) (Fig. 4.) If X has any value between 1 and — 1, 2/^ is negative, and y imaginary. Then, no part of the graph lies between A and A'. 3 If = ±2,2/2 ■y=*4 (B, C, B\ CO The graph has two branches, BAC and B'A'C, each of which extends to an indefinitely great distance from 0. (The graph of Ex. 4 is a hyperbola ; as also is the graph of any equation of the form ax^ — 62/2 = c, or xy—a.) The graph of (2) Plate V is a hyperbola. KXEBCISE ill Find the graphs of the following sets of equations, and in each c£ise verify the points of intersection by comparing with the algebraic solution: "^ = 4. f x^—4:y= — 7. ra;2-f4i/2 \x-y=-l 2x+Sy=4,. 220 ALGEBRA ^' \xy=10. Sx-y^8. 5. 6. ^/^ — 3 x= — S. x+2y='-2. 4 x—9y=6. 247. In solving problems which involve simultaneous equa- tions of higher degree, only those solutions should be retained which satisfy the conditions of the problem. EXERCISE 112 1. The sum of the squares of two numbers is 34 and their difference is one-fourth of their sum. What are the numbers? 2. The sum of the squares of two numbers is 52 and their product is 24; find the numbers. 3. The sum of the sides of a triangle, ABC, is 18 inches. The sides AB and BC are equal, and the side ^C is 17 less than the square of the side BC. Find the length of each side. ^^ .^^c 4. In a number consisting of two digits, the first digit is equal to the square of the second, and if 5 times the first digit be divided by 3 times the second, the quotient is | less than twice the second digit ; find the number. 5. If the length of a rectangular field were increased by 2 rods and its width diminished by 3 rods, its area would be 70 square rods ; and if its length were decreased by 2 rods and its width increased by 3 rods, its area would be 110 square rods. Find the length and width. 6. A tangent TP is a mean proportional between the whole secant DP and the ex- ternal segment EP, If EP equals the radius of the circle and TP is 3\/3, find the area of the circle. 3V3 SIMULTANEOUS QUADRATIC EQUATIONS 221 7. The perimeter, a + b+c+d^ of a rect- angle is 36, and the area of the rectangle is 80. Find the sides. 8. A farmer bought 15 cows and 20 sheep for 1720. He bought 3 more cows for $320 than he did sheep for |30. Find the price of each. 9. The sum of the numerator and denominator of a frac- tion is 7. If the numerator be diminished by 1, and the de- nominator be increased by 1, the product of the resulting fraction and the original fraction is ^-, Find the fraction. 10. If 7 be added to the numerator of a fraction tlie value of the fraction becomes 7. If the square of the denominator be subtracted from the square of the numerator the result is 7. Find the fraction. 11. The area of a triangle ABC is one- half the product of the base, AC^ and the altitude, DB. The area is 48 square feet. BC is 10 feet and its square is equal to the sum of the squares of BD and DC, AD = DC, Find AC and BD, Can more than one such tri- angle be drawn ? 12. A triangle ABC has the angles B and C equal. The angle A is 60° more than the square of the number of degrees in the angle B, The sum of the three angles is 180°. Find the angles. 13. A travels from C to D. Two hours after he leaves C, B starts out to overtake him, traveling 3 miles per hour faster than A. Had A traveled 1 mile per hour slower, B would have overtaken him 12 miles nearer to C. Find A's rate. 14. In a triangle with a right angle at C, the altitude drawn from C to the hypote- nuse is a mean proportional between the segments, a and 6, of the hypotenuse. We ^ 222 ALGEBRA know also th-dt EC' = h^ + b'\ li AC =12, CB^% and ^J5 = 15, find a, h and h. 15. The sum of two numbers is to their difference as 7 is to 2. The ratio of their product is to the product of their sum and difference as 45 is to 56 ; find the numbers. (Is the statement or the solution the more difficult?) 16. In a right cone, we know from geometry that S=7rRH, /\ and V=\^R'A, / aI \h where S = lateral surface, J? = radius of base, / j \ F= volume, // = slant height, .4 = altitude. A 'lUlrrX If S = 60 TT and £r=10, find F. (Remem- ^-— ^Jl-il^ ber that because of the right angle at Z), H^ = A^+RP.} XVI. THE BINOMIAL THEOREM POSITIVE INTEGRAL EXPONENT 249. A Series is a succession of terms. A Finite Series is one having a limited number of terms. An Infinite Series is one having an unlimited number of terms. 250. In §§91 and 183 we gave rules for finding the square or cube of any binomial. The Binomial Theorem is a formula by means of which any power of a binomial may be expanded into a series. 251. Proof of the Binomial Theorem for a Positive Inte- gral Exponent. The following are obtained by actual multiplication : (a + xy = a^ + 2 ax +x^; (a+x)3=aH3 a^x + S ax^+x""; (a + xy=a*+4: a^x-{-6 a^x^+4: ax^-\-x^; etc. In these results, we observe the following laws : 1. The number of terms is greater by 1 than the exponent of the binomial. THE BINOMIAL THEOREM 223 2. The exponent of a in the first term is the same as the exponent of the binomial, and decreases by 1 in each suc- ceeding term. 3. The exponent of x in the second term is 1, and in- creases by 1 in each succeeding term. 4. The coefficient of the first term is 1, and the coefficient of the second term is the exponent of the binomial. 5. If the coefficient of any term be multiplied by the ex- ponent of a in that term, and the result divided by the expo- nent of X in the term increased by 1, the quotient will be the coefficient of the next following term. 252. If the laws of § 251 be assumed to "hold for the ex- pansion of (a+xy, where n is any positive integer, the expo- nent of a in the first term is n, in the second term ?^— 1, in the third term n — 2, in the fourth term n— 3, etc. The exponent of x in the second term is 1, in the third term 2, in the fourth term 3, etc. The coefficient of the first term is 1 ; of the second term n. Multiplying the coefficient of the second term, 7i, by 7i— 1, the exponent of a in that term, and dividing the result by the exponent of x in the term increased by 1, or 2, we have ViJlZLJ as the coefficient of the third term ; and so on. 1-2 Then, (a + :rf =a^+na^-^ar4- ^'^^~'^^ a"~V 1 • ^ , n(n—l)(n — 2) «_, «, ,-x 1 • 2 • 3 Multiplying both members of (1) by a+x^ we have 1*2 1 • 2 • 3 1 .2 Collecting the terms which contain like powers of a and x, we have 224 ALGEBRA r n(n-l)(n-2) n(n- 1) 1 ,. I 1.2.3 1.2 J 1 . 2 L 3 J Then, (a + a;)^^+ ^ = a'^+i + (n + 1 )a^a; + nf^^-la'^- 'x'' ^ ^ 1.2 , (n+l)n(n— 1) „_, 3 , /o\ 1.2.3 ^ ^ It will be observed that this result in equation (2) is of the same ybrm in ?i+l, that equation (1) is in n, and equa- tion (2) was obtained by multiplying equation (1) by a + x; which proves that, if the laws of § 251 hold for any power of a 4-07 whose exponent is a positive integer, they also hold for a power whose exponent is greater by 1. But the laws have been shown to hold for (a^-a:)^ and hence they also hold for (a+x)^; and since they hold for (a-fa?)^ they also hold for (a+xY; and so on. Therefore, the laws hold when the exponent is any positive integer, and equation (1) is proved for every positive integral value of n. Equation (1) is called the Binomial Theorem, In place of the denominators 1-2, 1 • 2 • 3, etc., it is usual to write |2, [3, etc. The symbol |n, read "factorial-n," signifies the product of the natural niunbers from 1 to n, inclusive. The method of proof in § 252 is known as Mathematical Indicction. THE BINOMIAL THEOREM 225 253. Putting a=l in equation (1), § 252, we have (1 +x)«= 1 +nx+ "(^^-i-Un n( n-lKn-2) ^3+.., 254. In expanding expressions by the Binomial Theorem, it is convenient to obtain the exponents and coefficients of the terms by aid of the laws of § 251. 1. Expand {a + xy. The exponent of a in the first term is 5, and decreases by 1 in each succeeding term. The exponent of x in the second term is 1 , and increases by 1 in each succeeding term. The coefficient of the first term is 1 ; of the second, 5. Multiplying 5, the coefficient of the second term, by 4, the exponent of a in that term, and dividing the result by the exponent of x increased by 1, or 2, we have 10 as the coefficient of the third term; and so on. Then, {a + xy = a^ + 5a^x+ 10 aV+ 10 a^x^ + 5ax^ + x\ It will be observed that the coefficients of terms equally distant from the ends of the expansion are equal. Thus the coefficients of the latter half of an expansion may be written out from the first half. If the second term of the binomial is negative^ it should be written, negative sign and all, in parentheses before ap- plying the laws ; in reducing, care must be taken to apply the principles of § 88. 2. Expand (1-xy. {\-xy=^[i + {-x)f = V-\-6'V'i-x) + 15'V'{-xy + 20'V'(-xy + 15 • 12 . (-xy+6' 1 • (-xy-\-{-xy = 1-6x4- 15x2-20x3+ 15 a;*-6a:^ + a;«. If the first term of the binomial is an arithmetical number, it is con- venient to write the exponents at first without reduction; the result should afterwards be reduced to its simplest form. If either term of the binomial has a coefficient or exponent other than unity, it should be written in parentheses before applying the laws. 226 ALGEBRA 3. Expand (Sm^-^n)'- = (3m2)* + 4(3m2)3(-n4)4-6(3m2)2(-ni)2 + 4(3m2)(-ni)» + (-ni)* = 81 m«- 108 m«ni4-54 m*n?- 12 m^n + ni EXERCISE 113 1. (c + dy. 5. (ab + c^y, g. (2a'-5b'y, 2. {x + iy. 6. {x + 3yy. 10. (a-3-2 6i)^ 3. {a-by, 7. (2a-6)^ ii. (a:* + 2 6^)^ 4. (m-ky. 8. (4/?+3A:)^ 12. \l-x^y. 13. (2a* + 3 5*)«. 15. f3iC-^--^Y , .■ \ 2x^J X4. (2ai + 3a-ir. ,6. (3 a-^+^)«. 255. To find the rth or general term in the expansioii of The following laws hold for any term in the expansion of (a + ir)% in equation (1), §252: 1. The exponent of x is less by 1 than the number of the term. 2. The exponent of a is n minus the exponent of x. 3. The last factor of the numerator is greater by 1 than the exponent of a. 4. The last factor of the denominator is the same as the exponent of x. Therefore in the rth term, the exponent of x will be r— 1. The exponent of a will be n— (r— 1), or ?i— r+ 1. The last factor of the numerator will be ?i— r+2. The last factor of the denominator will be r— 1. Hence, the rth term _ n(yi-l)(n-2)>»(n~r+2) ._,^^^,_^ ,^. 1 .2.3...(r-l) * ^ ^ THE BINOMIAL THEOREM 227 In finding any term of an expansion, it is convenient to obtain the coefficient and exponents of the terms by the above laws. Ex. Find the 8th term of {^a^-h-y\ We have, (3 ai-6-i)'' = [(3 a4) + (-fe-^l^^ In this case, n = ll, r = 8. The exponent of ( — 6~^) is 8 — 1, or 7. The exponent of (3 ai) is 11 —7, or 4. The first factor of the numerator is 11, and the last factor 4+ 1, or .'^. The last factor of the denominator is 7. Then, the 8th term = ^^ • 10 • 9 ♦ 8 - 7 - 6 - 5 ^3 ly^_^^-,y 1 . 2. 3- 4- 5- 6; 7 ^ = 330(81 a2)(- 6-0 = -26730 a^h-\ If the second term of the binomial is negative, it should be written, sign and all, in parentheses before applying the laws. If either term of the binomial has a coefficient or exponent other than unity, it should be written in parentheses before applying the laws. EXEKCISE 114 Find the : 1. 5th term of {ci-dy, 5. 6th term of (a"* + 6-2)io, 2. 5th term of (x-\-\Y, f 6/ — ^Kh ^\ ',, 6. 10th term of Vm^- -^ 3. 7th term of {a-\-2h)\ \ 2 4. 8th term of (a^ + ft^yi y 51-^ ^^j,^ ^f ^^4^3 ^f^iz 8. 4th term of (c-^-5 cdy\ 9. Middle term of ^3 a'-f — V' . THE METRIC SYSTEM Linear Measubb The standard unit of Linear Measure in the Metric System is the Meter. It is determined by taking one ten-millionth part of the distance from the earth's equator to either of its poles, measured on a meridian. It is equal to 39.37 inches. 228 ALGEBRA The problems in this book make use of the following sub- divisions of the Meter : 10 Millimeters (mm.) = l Centimeter (cm.) 10 Centimeters = 1 Decimeter (dm.) 10 Decimeters =1 Meter (m.) Measures of Weight The Gram is the unit of weight. It is equal to the weight of a cubic centimeter of distilled water at its greatest density. The following multiples of the gram are used in problems in this book : 10 Grams (g.) =1 Dekagram (Dg.) 10 Dekagrams =1 Hektogram (Hg.) 10 Hektograms = l Kilogram (Kg.) XVII. HINTS ON CHECKING 256. It is sometimes desirable to check a result by nu- merical substitutions. Any number may be substituted for the letters involved in the problem, but since all powers of 1 are 1, a substitution of 1 for a letter above the first power is not an accurate check. It is best not to use a numerical check when other means are convenient. In addition : Check : Let a=2, 6=2, c=l. a + 26-3c 2+ 4-3 = 3 -2a- 6 + 5c -4- 2 + 5 = - 1 -3a-6 6 + 7r -6-12 + 7 = -11 9a-46- c 18- 8-1 = 9 5a-96 + 8c = 10-18 + 8= The horizontal and vertical additions being identical is a fair, not an absolute check. In subtraction : Check : Let a = 6 = c = 1 . a+ 26- c= 1+ 2-1= 2 - 4a+13 6 + 4c =- 4 + 13 + 4 = 13 5 a- 11 6-5 r= 5-11 -5 = -11 Or add the difference to the subtrahend. The sum should be the minuend. HINTS ON CHECKING 229 In multiplication : Check: Let a =6=* 2. 2a - b 4-2= 2 3a +46 64-8 = 14 Qa^-Sab + 8a6-462 6a2 + 5a6-4 62 = 24 + 20-16 = 28 If the multiplicand and multiplier are homogeneous, the product will also be homogeneous, and its degree equal to the sum of the degrees of the multiplicand and multiplier. The Illustrative examples in § 53 are instances of the above law; thus, in Ex. 2, the multiplicand, multiplier, and product are homogeneous, and of the third, first, and fourth degrees, respectively. The student should, when possible, apply the principles of homogeneity to test the accuracy of algebraic work. Thus, if two homogeneous expressions be multiplied together, and the product obtained is not homogeneous, it is evident that the work is not correct. Multiplication may be checked by using the multiplier as the multiplicand and the multiplicand as the multiplier. In division: The product of the divisor and quotient should equal the dividend. If the dividend and divisor are homogeneous^ the quotient will be homogeneous, and its degree equal to the degree of the dividend minus the degree of the divisor. In factoring : The product of the factors should equal the given expres- sion. In fractions : Since fractions involve the four fundamental operations, addition, subtraction, multiplication, and division, the four checks above given will suffice. In equations : Reject any root which does not satisfy the given equations. INDEX (Numbers refer to pages) Abscissa, 122. Absolute value, 11. Addition, imaginaries, 183; literal coef- ficients, 26; monomials, 15 ; polynomi- als, 17; positive and negative terms, 11; similar terms, 16; surds, 170. Affected Quadratics, 187, 192, 202. Aggregation, symbols of, 7. Algebraic expressions, 8; addition of, 14; definition of, 8; division of, 35; expansion of, 47; multiplication of, 28; subtraction of, 14'. Algebraic symbols, 1. Alternation, 112. Antecedent, 110. Approximate square root, 155. Arrangement of terms, 18. Axioms, 1, 44. Base, 6. Binomial, cube of, 156; defined, 17; divi- sion of, 57, 68, 70; square of, 58; theo- rem, 222. Braces, 7. Brackets, 7. Cancellation of factors, 35, 86, 97; of terms, 45. Checking, 228. Circle, 217. Clearing of fractions, 45. Coetficient, definition of, 15, 167; literal, 26; numerical, 15. Common denominator, 90; factor, 71, 76, 80; multiple, 83. Complete divisors, 150. Completing square, 192, 195. Complex fractions, 101 ; numbers, 182. Composition, 112. Conjugate imaginary, 185. Consequent, 110. Continuation, sign of, 28. Continued proportion, 111. Coordinates, 122. Cube, of binomial, 156; root, 69, 157, 158; root of unity, 201, 211. Decimal Equations, 117. Degree, of an equation, 44; of surd, 167; of term, 34; with respect to a letter, 43. Denominator, of fraction, 85; lowest common, 90; rationalize, 177. Difference, 20, 58. Dissimilar, surds, 170; terms, 15. Dividend, definition of, 35. Division, binomials, 57, 68, 70; definition of, 35; algebraic expressions, 35; of fractions, 99; of imaginaries, 185; law of exponents, 35; of monomials, 36; of polynomials, 37, 38; proportion, 112; rule of signs, 35. Divisor, complete, 150; definition of, 35; trial, 150. Double sign, 145, 212. Elimination, 124. Ellipse, 218. Equality, sign of, 1. Equations, clearing of fractions, 45; con- taining surds, 180; decimal, 117; defini- tion of, 1 ; equivalent, 122 ; fractional, 104, 107, 132; inconsistent, 123; inde- pendent, 123; indeterminate, 121; in- tegral linear, 42, 43, 44, 48; literal, 108, 120, 129 ; members of, 1 ; numerical, 42 ; of condition, 43; of identity, 43; prin- ciples involved, 44; quadratic, 187; quadratic form, 208; root of, 43; satis- fied, 43; simple, 44; simultaneous, 121, 123, 129, 135; simultaneous quadratic, 212; solution of, 3,45,48, 76, 104, 123; statement of, 48; to verify, 46; trans- forming, 45. Equivalent equations, 122. Evolution, 144. Expand an expression, 47. Exponents, definition of, 6; fractional, 159; law of, 28, :i5; negative, 160; of powers, 54; theory of, 158; zero, 160. Expression, fractional, 85: integral, 34; mixed, 88; quadratic, 209. Extremes, 111. INDEX 231 Factors and products, 54; common, 35, 71, 76, 80; definition of, 14; hints on, 74; monomials, 55; quadratic, 209, 211; removal of, 35, 36, 86, 97; solution by, 76; (a'-i — 62),55; {a^ ±2 ab -\- b^), 57, 5S; x2 4- (a + 6) ic + ab, 61 ; {x^ + ax + b), 62 ; {ax^ + to + c), 64 ; (x^ + ax^ -j- y^)^ 67 ; (as ± ^')» 68 ; (a« db ^)» 69 ; (ax + ay + az), 74. Finite Series, 222. Formula, quadratic, 196. Fourth proportional, 111. Fractional, equations, 104, 107; expo- nents, 159. Fractions, addition and subtraction, 92; algebraic, 85 ; clearing of, 45; complex, 101 ; division of, 99 ; multiplication of, 96; principles of, 85; reduction of, 86, 88,90; signs of , 85, 89, 92. Gram, 228. Graph, 10, 78, 122, 129, 187, 1.94, 193, 199, 212, 217, 218. Highest common factor, 80. Hints, on factoring, 74; on solution, 48. Homogeneous, polynomials, 34; terms, 34. Hyperbola, 219. Imaginary numbers, 182, 183. Inconsistent equations, 123. Independent equations, 123. Indeterminate equations, 121. Index, 144. Induction, mathematical, 224. Insertion of parenthesis, 26. Integral, equations, 42; expressions, 35. Interpretation of solutions, 143. Inversion, 112. Involution, 144. Irrational numbers, 158, 166. Law of exponents, division, 35; multi- plication, 28 ; powers, 54. Literal coefficients, addition and sub- traction of, 26. Literal equations, 108, 120. Lowest common, denominator, 90; mul- tiple, 83, 84. Mathematical Induction, 224. Mean proportional. 111. Means, 111. Member of an equation, 1. Meter, 227. Metric system, 227. Minuend, 20. Monomials, addition of, 16; definition of, 14; degree of, 34; division of, 36; evolution of, 144; H. C. F., 80; involu- tion of, 144 ; L. C. M., 83 ; multiplication of, 28; power of, 54; rational and inte- gral, 34; root of, 145; subtraction of, 20. Multinomial, 17. Multiples, common, 83. Multiplicand, 13. Multiplication, algebraic expressions, 27; law of exponents, 28; monomials, 28; of fractions, 96; of imaginaries, 184; polynomials by monomials, 29; polynomials by polynomials, 30 ; posi- tive and negative numbers, 13 ; rule of signs, 27. Multiplier, 13. Negative, exponents, 160; signs, 11. Numerical, coefficient, 15; value, 8. Numbers, complex, 182; cube root of, 68 imaginary, 182; irrational, 158, 166 known, 1; negative, 10; positive, 10 rational, 166; real, 182; square root of, 152; unknown, 1, 43. Ordinate, 122. Origin, 122. Parabola, 218. Parentheses, insertion of, 25; removal of, 24; use of, 25, 26, 32, 33, 41. Physics problems, 117, 190, 191. Polynomials, addition of, 17, 18; arrange- ment of, 18; cube root of , 157 ; defined, 17; degree of, 34; division of , 35 ; H. C. F., 81; homogeneous, 34; L. C. M., 83; multiplication of , 30 ; rational and integral, 34 ; square of, 147, 148 ; square root of, 148, 149; subtraction of, 22. Positive signs, 11. Power, arrangement of, 18; of binomi- als, 58 ; of fractions, 144 ; of imagina- ries, 184; of monomials, 54; of num- bers, 6; of polynomials, 147; of powers, 54; of products, 54. Products and factors, 54; power of, 54; ia + b)(a — b), 55; (a±6)2,57; (x-f-a) (x -h ^), 60; (mx -j- n)(jox + q), 63; (a i bXa^ q= aft -f 6«), 68. Properties of surds, 180. Proportion, 110. 232 INDEX Proportional, fourth, 111; mean, 111; third, lU. Pure Quadratics, 187. Quadratic, affected, 187, 192, 201 ; factor- ing of, 76, 209, 211 ; pure, 187; simulta- neous, 212; surd, 167; theory of, 197. Queries, 26, 42, 79. Quotient, 35. Radical sign, 144. Ratio, 110. Rational, denominators, 177; monomi- als, 34 ; numbers, 167 ; polynomials, 34. Real numbers, 182. Reduction, fractions, 86, 88; L. C. D., 90; mixed expressions, 89. Removal of parentheses, 24. Root, cube, 68; definition of, 144; of equation, 43; of fraction, 145; of mo- nomial, 145; of unity, 201, 211. Rule of signs, addition, 12; division, 35; fractions, 85, 89, 92 ; multiplication, 27 ; powers, 54; subtraction, 21. Satisfy, an equation, 43. Series, 222. Signs, aggregation, 7; continuation, 28; double, 145, 212; equality, 1 ; fractions, 85, 89, 92; negative, 11; positive, 11; powers, 54; rule of, 12, 13, 27, 35, 54,85, 89, 92. Similar terms, 15 ; addition of, 16. Similar surds, 170. Simple equations, 44. Simplify an expression, 33. Simultaneous, linear equations, 121, 123. 129, 135; quadratic equations, 212. Solution, by formula, 196; hints on, 48; in- terpretation of, 143 ; Lost, 108 ; of equa- tions, 45, 108; principles involved, 44. Solve an equation, 43. Square, completion of, 192; of binomial, 58; of polynomial, 147, 148; perfect trinomial, 58, 174. Square root, 148, 149, 152, 155, 175. Subtraction, definition of, 20; imagina- ries, 183; monomials, 20 ; polynomials, 22. Subtrahend, 20. Sum, 15. Surd, 166; addition and subtraction of, 170; coefficient of, 167; degree of, 167; dissimilar, L70; division of, 175; equa- tions, 180; evolution of, 176; multipli- cation of, 172 ; properties of, 180; quad- ratic, 167; reduction of, 167, 172; similar, 170; square root of, 175. Symbols, algebraic, 1 ; of aggregation, 7. Terms, arrangement of, 18 ; cancellation of, 45 ; definition of, 14 ; degree of, 34, 43; dissimilar, 15; homogeneous, 34; negative, 14; positive, 14; rational, 34; r<'s 226; similar, 15; transposition of, 44. Theory, of exponents, 158 ; of quadratics. 197. Third proportional, HI. Transforming an equation, 45. Transposition, 44. Trial divisors, 150. Trinomial, 17; square, 58, 174; x- -f- ax + 6, 62 ; ax^ + 6x -)- c, 64 ; x* -\- ax^y^ + y*, 67. Unit, imaginary, 183. Unknown numbers, 43. Value, absolute, 11; numerical, 8. Verification of results, 21, 23, 31, 39, 46, 47, 228. Zero exponents, 160. 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