IMAGE EVALUATION TEST TARGET (MT-3) 1.0 !.l ^ m 122 2.0 1.8 125 1.4 1.6 ■^ 6" — ► V. <^ /} '<3 % % '<3 V #> <^ Ci^ 'V -u ^/' exercises — the student may first take a beginner's course, and afterwards complete the work for a more thorough course. As the book is specially intended to bo a drill-book for ruriLS, the answers to the examples have not been included ; they will, however, be published in a cheap, se])ar.ate form, Avitli hints on the more difficult (juestions — for th'> use of teachers and private students who may desire to con. ult them. For higher work .and applications in tlie Theory of Divisions, Resolution of Polynomials, Higher Eijuations, Determinants, Series, etc. , teachers and students are respectfully referred to the University Jxajebra, published by Ginn & Co., Boston. CONTENTS. PAOR. - 1 28 - 36 46 - 54 67 CHAPTER I. Alokukaic Notation. - . . , " • • CHAI^TER II. CoMI-OtlNl. KXPKKSSIOXS. ADDITION'. SrHTKACTION. CHAT'TER III. Easv Equations and 1'koblkms. Rkmovai. of Huackkts CHAPTER IV. Multiplication. Detaciikd Coiokkiciknts. CHAPTER V. Division. Hokneks Method. CHAPTER VI. •SiMi'LE Equations. J'hoheems " " - I CHAPTER VII. S,.ECIAL FOKMS IN MuLTIPLICATn.N. INVOLUTION, EvoLUTION. 76 CHAPTER VIII. Symmetry and Its Applications. . CHAPTER IX. Resolution into Factors. Exact Division. CHAPTER X. Hi,iiiEST Common Factor. Least Common Mr Miscellaneous Exercises in Measures Ind Mu CHAPTER XI. Fractions. Ratios. - . . . _ CHAPTER XII. Miscellaneous Fractional Equations. Problems. CHAPTER XIII. 'oifR Methods ok Elimina - 204 [5] 90 97 TLTIPLE. MULTIPLES. 131 148 188 .SiMtrLTANEOUS EQUATIONS. Fo TioN. Problems. 6 CHAPTKH XI v^. t^LTADHATIO EQUATIONS Vr.,v. (\ CHAl^lER X\'. SiMULTANKOUS QuADKATIO... Pi,o,.;.K.M.S. CHAPTER XVI. Thkokv oi.- QuAUKATU- EgnATIONs Rmm. Maxima and Minima. . .' ^'' '^^^ <^"^!''PrciENT.s. CHAPTER XVII. I'AOR. 235 269 2{)0 302 ELEMENTARY ALGEBRA. CHAPTKK J. AUJEBRAIC NOTATION. 1. Symbols of Quantity. —A quantity is a dcfinito portion of any magnitudo ; thus, any definite niinilxT of net, y a nnmher we take a certain por- tion of it as the unit of comparison, and then determine how manij times this unit is contained in the ([uantity to l)e measured. To measures distances, for example, we take a foot, oi- a yjii-d, or a mile, etc., as the unit, and then find how many times tliis unit is contained in tin; distance to l)e nu'asured. When, for example, we call a certain distance five mih-s, we mean that the unit, one mile, is c(mtained five times in the given distanc(>. So of time, we call a certain definite length of time five hours, and mean that it contains the unit, one hour, five times. 3. Symbols are the marks, or cliarai (ers, nsed to denote the number of such units in any giv.'i. .luantity. The marks on a scor- ing-sheet in a game of cricket are the si/mho/s used to denote the number of runs in the match. The (igtircs 7. 12, 4, etc.. are the commcm symbols used in Arithmetic to denote the numbers seven twelve, four, etc. In Algebra, wc ns<' more genend svmbols t(') denote the number of times the unit of comparison is" repeated such as a, b, c, x, y, z, cc, /9, j', d, etc. ' 7 8 ALOIcnUAiC' KOtATIOK. 4. A comploto oxprcssion of (nmntit;, by mciiiis of Tiiimbcrtj flicrcforc iiicliidcs two tiling's : - ^ 1 . The unit of measurement ; which iiuikI, of cunrsc, he of llic siimt' l;iii(l lis the (iiiiiiitily to oc mciisiircd. T. Thr number of such units in the ^'ivcn (lUiiiitity, 5. Ill clt'iin'Mliiiw Al,i,'('l)i\'i, we consider all <|iiaii1 if ics ,'is cxprcuscd iiiiiiiciMcull.v in terms of s(»nie unit, and the s.vnd)ols of (|nanlity repi'cHCiut oidy the />///>/// hhiik rictil /xtrts of such (|nantities. in otlujr words the symbols denote what ai-e called in Arithmetic (tbstrc siihlrdctcd, the rcmoiixhr in'// fxj/f/i//' the; miinber of miles, lhon,ti;h not f/Z/W/// ^Hven, may l)e found from the (/nerally used to represent knonui numl)ers, while z, y, iV, H\ etc., commonly stand for iDiknouni numbers. SYMBOLS OF QIANTITY — WOKD-SYMMOLS. 11. When Miiy Iri Irr, ;is ./'. is \\st'(\ ill llic course of a calciiliilinn, it (Iniolcs llic saiiir miiiihci' tiirdiinhoiil. WC may Jilso rcpi'cscnl (lilVcrcnt nuniltcrs l»y the same Idler witli marks allixed ; thus, instead >A' writiiiLi; '/, /*. <■ I'ur three dilVrreiit iiiiml/ers, we may re|)reseiil tiiese iilllilbel's l)y f/ , , i/.^. l^.^, ur l»y tlie symlnds ii\ it", It'" (read an; Ifdni/, and (> and 11 ntimerii'itl n wo liuve, = , read rf//nt/s. is lyitii/ to, ii'ill hi' iqitiil to, etc. >• read is (jnittir tlntii, thus !>>4. <, i'(>a:l is less (Itiiii, thus 4. :, ::, the si;;ns of ])roportion, as in .Vrithmetic, thus It : /> :: (■ : (/, or a -.b — r. : i/, is read It is ii) /> as (■ is to i/. The pili'e if()/-i/ si/i/i/to/s are, .•. read, tliiiij'i>n\ miisiiiiii itlh/, Ihiki' read, Inntiisi', siiti-i'; tlub '/ = /', and h — r \ .-. rr = ^, is read "Since a e(pials h. and (> e(pials r ; tlien;- fore It ('(pials c. " . , the symbol (»f continuation, is read lonlitittiil Inj the sitnii; l expression of a (luantity. How is this ajjplied in arithmetical Alj^ebra ? (iivo an example of exi)li('itly assii^ned. implicitly assi 7. 2. 12 < 20. :}. l> > c. 4. .r < //. n. •.• I = 1, .-. i)f, =: 10. 0. If ^/< 1 and t>>\, .-. a. 7. Let a = l> and l> > 1 2, .-. a > \'2. S. 0< ^/ < 12. 0. 0< 1 . ;} X 12 -4 X <>. 11. \U( :6 ■.-.r-.d. ad !) ::4 : 12, ')<: 12. i:j. 7 + :5 = 9 = 7 ^- 2. 10, 7 = 10 - 3. 14. Signs of Operation. — The fundamental operations of Alge- bra are Addition, Subtraction, Multiplication, Division, Involution, .and Evolution. A n)ark used to denote that one of these ojx'ra- tions is to Im^ performed on a quantity is called a .siynof operation. These sii/ns <>/ operation will now be explained. 15. The sign of Addition is +. read />///.s\ As in Arithmetic, it denotes that the (puintity before which it stands, is an addend. Thus a + h, means that b is to be added to will represent (5 + 4, which =- 10. Hence, The sunt of ati:/ itandter of qnantifics is e.rpressed Inj writing tlieni in a row with the sign -f- between evert/ two of them. EXERCISE II. 1. If a = ',], h — Ty^ and <', = H. tind the value of i. a + f). ii. h + r. iii. <■ + a. iv. a + f> + c. V. a -\- b -^ a. vi. a -f b -|- r -|- /; -|- a. 3. If a* = 4, 1/ = 7, and >■ — 10, prove, by finding the value of each of these e\pressi(ms, that x + y + z = i/ + z + x = z+t/ + x — x + z + y = 2 + x + y. SIGNS OF OPERATION. H 3. V^^ad th(; following statcinciits : i. (I + b -{- (• — ('. ii. f, ^ r + a -\. I =a + e. iji. (, + a + /> + (> = e + 2. iv. a + b> r, v. n ad nu'nKs. Plaeed before a quantity it denotes that that quantity is a subtrahend. Thus (' - />, read a minus b, indieates that the number r(>presented by /> *s to be subtracted from the number represented ])y a ; so that, If a r(>presents 6 and b represents 4, a - b is equivali'nt to « - 4,' wliieli = 2. Henee, to subtract a quantity, sav a, from anotlier (luantity, say .r, wrUe the subtrahend after the minuend with the sKju, — , between them; thus, x — a. EXERCISE III. 1. If ^/ = 2r, Mild A = 7, what is the v.'ihieof i. a - b. ii. f/ - /, _ /,. iij b -^ I, -\. I> -^ -; _ a. 2. Read the following; e.\i)ressions : \.u-\-b~,'. W.a-b — i-. iii. ^/-/, + r. 3. Find the value of tiiose thnv exi)ressions if a = 12, b = r> and 4. Write the expressions i.i (2) ;.t full j.'ii.irth in words. 5. If a ..15 b = 7, r = fi, -nid d = 4, prove, bv findin- the valu(> of each expression, that (i-b-c + d = a-b + d-f;=:a + d-b~r = a-c + d-bz=;a-c-b+-d = a+d-c~b. 12 ALGEBRAIC NOTATION. G. Usin^ the si 10, .-. X > ;] ; •.• y — 5 < 7, .•. y < 12. V. •.' + b — b = 0, .-. a = a + b — b. PLUS AND MINUS. 13 2. Write tlio moaning of iii and v at full length in words. 3. AVrite with proper signs of operation : — i. a minus m plus 7 minus b minus 6 plus c. ii. X plus h minus a minus (/ i)lus h minus 6. A. \t a = 25, b — 10, and r = 8, tind the value of i. a + b — b ; b — r: -{- b + a ; b + c + n — r + a. ii. a + b — c; a + b + a\ a — c + b ; — b -\- a + b - r -|- (/, iii. (/ — 7— 5— c; &+10 + C— «; c + a— 10— 6; a—c+U)-0 + h. ;■), To /x add I and from the result take q. (). From (( take b and tluMi <' from the remainder. 7. Expr(\ss in syml)ols : — i. The difference of p and g ; the sum of 7>, q, and r. ii. The numl)er by whieh a exceeds c. iii. Tile uunil)er l>y which the sum of d and c exceeds (\ iv. Tlie statement lltat the sum of d and b is gre.iter than r. V. Tile statement that the ditferenee between <( and b is less than c vi. The number by whicli the sum of u; //, and z exceeds w. EXERCISE V. ] Express in symbols : — i. That a* diminished l)y 3 is equal to four. ii. That x is four more than 3. iii. That x is equal to the sum of a and b diminished by c. iv. That if x, y, and 2 be added together, the sum will be 20. V. That if iji be taken from thu sum of x and «, the renuiinder will be 5 more than c vi. That if m be taken from th.; excess of x over n, and // be then taken from the remainder, the tinal remain- der will Ijc zero. 'a. Express the following numbers algebraically : — i. The number that exceeds x by y. ii. The number thnt is .") l,>ss than the sum of x and //. iii. The uujiiber which is less by in than the excess of .r over I/. u ALGEBRAIC NOTATION. iv. The number vvhieli if increu.sed by 4 would be equal to c diiiiiiiish(Hl by 5. 3. James has a marbles and John has 7 more ; how many has Jolm ? 4. John is X years old now ; how old was he 5 years as innl)er lie ' exeli:in,i!;e in .r eents ; .ctors, and 5e faetors. of opera- nto : thns, . le product 3S the con- e manner, l&. by h'tters, d, and tlie imJicatinif it Cusually) d t of the 22. In fin expression such as lax + r,6y - '6cz - Aur (read seven ax phis hve bu minus tliree cz minus four fz-y) ^/ie mnltiplimtions are. to be performed before the additions and subtractions. EXERCISE VI. 1. If X denotes a certain number, express :— »' i. DouWe the number ; five times the number. ? ii. Three times double the number ; 2a times the number, m. The s/tm and the difference of three times the number and four times the number. IV. A number seven less than five times the number. 2. If a .= 13, find the value cf i- ha - 3rt ; ?J x m ; 1 • 2 • 3 . 2rt ; 4a — 3a. ii. 1.2- 3a -20; 3a > 2a - 10 ; 4a. 1.2. 3. ?. Xead 'la + '^b + 3^- ; 4a^ - Tuic f 'ibc, ; 3a.c — by + z-^ Tula — 366 — 2cc. 1. Write at full len,:,-ih in wordr, the mt^inin,!,' of the exjiressionb in (juestion 3. 5. Express : — 1. X dollars in cents, and y dollars in dimes, ii. X dollars and y cmts in cents, hi. X yards, // feet, and z inches in inches, iv. a cwt., b pounds, and c oz. in ounces. 23. A quantity which multijaies another quantitv is called a Coefficient of that other quantity. A eoeffirient ( Iiti-raU v x\ how much did he gain ? 1(5. John had m marbles ; he gave n of them to James, twice as many to William, and the rest to Thomas. How many did Thomas receive ? 17. John's father is not three times as old as John by four years ; his mother is three years younger than his father ; what is his mother's age if John is x years old ? 25. The sign of Division is ~, read by, and denotes that the ([uantity immediately following it is a divisor. Thus a-^h (read (I by h) means that a is to be divided by h ; so that if a represents 12 and 6 represents 4, a ^ 6 represents 12 -f- 4 ; that is, 3. Division is also indicated by position, i.e., by arranging the (luantities in the form of a fraction with the dmdend for numer- ntoi\ and the dinmr for denominator; thus, a-^h may be written ^ , nr ,, «.r ^ 6y may be written ", etc. by 26. In an expression such as ^'^ + ^ _ ^, the nmltinlimtiims in n (J ' and dimsions are to be performed before the additions and mb- traetions, so that in this expression the (piotient of ax by m is to be increased by the quotient of by by w, and the sum diminished by the quotient of oz by q. 18 ALGEUKAIC NOTATION. EXERCISE VIII. 1. U X denotos a certain number, express : — i. A (juarter of Ave times the number; three (juarters of the number. ii. The sum of 2i times tlie numl)er and a quarter of the number. iii. Twelve divided by the number, and the quotient dimin- ished by two tliirds oi three fifths of the numl)er. iv. Twelve divided by the number and the quotient increased by three divided by the number. V. Seven less than seventy divided by six timrs the number. 3. liead the following expressions and write their meaning at full length in words : — i- o 5 7^ ? 7^^ '■> l • ''♦^^ ; a« -i- 3 ; Ha -=- 36 ; 10 t- 3a. 6 12 2a 4 a 1 „ ^, 11. . + . ; « - 2i ; - + a 3A; 3ia-l.-; 3^-3^. 5 5 ' 6 "' 9 a ^ ' ^ 8 3 ' 8 a Z. If a ■=■ 24, /> = 5, c = 2, x = 3, find the values of the expres- sions in i of question 2. 4. Read the following expressions and write in words : — 2ah 36o Am x y z 3 4 ^^ a b c .. ax hx ex \ 2 1 oca aa ab bb m n h h ,c a . 111. a-^ — c — h«-; « — b — ('-• a c G a b •5. If a = 10, 6 = 6, c = 1, m — 3, n = 20, and x — ij — z = 30, find the values of the six expressions in question 4. 6. Express : — 1. x cents in dollars ; x inches in feet ; x oz, in pounds ; x feet in miles ; x minutes in days ; x pounds y oz. in cwts. ; x yds. y ft. in miles ; x ten-cent pieces in quarter-dollars. » EXERCISES— on A L AND WRITTEN. 19 7. Write Willi alg^'bruit; sij^n.s : — i. Tiireo a by .r, ininu.s four h hy //, plus fivo c by z. ii. Throo (tx by b<\ plus five hij ])y ra, minus sovcu m l)y ab. iii. Seven a^y by m, plus six it'// by //, minus 8. 8. Express ulgebraieally :— i. Six more tlian tlie mi\\ \n\vi of .r, inerensed by four less than the a{\\ \y,iYi of ii.v. ii. Ten more than the exeess of the fifth ]»art of a over the third part of h. iii. The number that exceeds h by one-fifth of m times x. iv. The number that falls short, by twelve, of the sum of the ath part of x and the bi\\ part of y. V. That six more than two-thirds of x is ecjual to one-third of X diminished by three, vi. That five less than one-third of five times x is ecjual to eleven more than one-fifth of thrice x. 9. If rt = 2, 6 = 3, and c = 4, wiiy is not a- = 2|, and ;^=1? c *' be 34 10. What are the numerical coefficients in '6ax 'Sx ax X 1 Hnx %' a"' T' a' 4'"T 11. What are the coijllicicmts of x in the five expressions in question 10 ? 12. U x= i/ = z = (r, find the value of ■^ + -v + - // ^ .t" ax (12 ay z y x w. y X z X y 1 zx 2 a 3 a 4 a 13. What are eggs apiece at ic cents a dozen ? 14. Find the price of 9 ducks at x cents a pair ; and of x eggs at y cents a dozen. 15. John had x dozen marbles ; he gave half of them to James, half the remainder to William, and one-third of what then remained to Henry ; how many did each receive ? , 20 ALOKliUAU; NOTATION. 10. .Tamos had w marbles; 1h> ^avo lialf of tlumi to Joha and a third of them to VVilliaiii ; how many had he left ^ 17. Ih^nry walked 12 miles in t hours : How many miles an hour did he walk ? How l(m<^ did he take to walk a mile i IS. HeJiry walked d miles in ii hoiii's : How niiiny miles jin hour did he walk ? How lonj^ did he take to walk one mile? 19. How many posts a feet aj)art fi'om centn! to centre; would ])c required for a fence alon^ the front of a lot x feet wide, a post to be at each end years more than a tifth of his fatlier's ago ; his father is ,r years old ; how old is John ? 21. A man cut a lield of wheat in x days ; how much of it did he cut per day ? 28. John could cut a field of wheat in x days ; his father could cut the same in y days ; how much of it could both together cut in a day ? 23. A pipe could fill a cistern in .r hours ; how much of the cistern could it fill in one hour? In tuv hours? In //re hours? In n hours ? 24. Two pipes let water into a cistern and a man pumps it out ; one of the pipes could fill the cistern in .r hours, the other could fill it in y hours, and tlu^ nuin could empty it in ^ hours. If the i)ipes let in more water than the man can pump out in the same; time, and if both i)ipes are opened and the man sets to work to pump, how much of the cistern would be filled in one hour ? In two hours ? In five hours ? In ii hours ? 27. A Power of a quantity is the product obtained by using that quantity a certain number of times as a multiplier, starting from unity as first multiplicand. The oj)eration of forming a power is called Involution; the quantity used as a multiplier is called the Base of the power ; the number of successive multiplications is called the Degree of the power ; and the number indicating the degree of the power is called the Exponent, or Index, of the power, n and a rould bo vt wide, fatlu'r's it did he er could together e cistern hours ? s it out ; le other y^ it in ^ man can » opened e cistern e hours ? )y using starting a power is called lications iting the e power, POWERS — DIMENSIONS. 21 and is written in small characters to th(! right and a little above the line of the base ; thus : — 1 X ^/ X (I is represented l)y f/^. reid n square; hero a ia the base, 2 is the exponent (or indejc) ; the (piantity itself, vi/. «', is of the semnd deytre, and is called the second /H)trer of a. 1 •(•'('•(; is repn'sented by r", read c ciihi' ; Ikto <• is tlie base, ',\ is tlie index ; and the (|Uiintity, f•^ is of tht; tltinl dcgret;, and is called th(! third power of r. In i^, read x to tlie Jifth, x is the ))ase, 5 is tlx^ cx])onent ; and the (puintity, a;*, is of the Jifth degret', and is called the fifth power of X. 28. C'oniparing powers among themselves, the second power is said to be Itiyher than the first, the third higher than the second, etc. 29. The dimensions (sometimes called degree) of a term are the number of literal factors it contains ; this is indicated by the mm of the exponeids; thus : — Sa^feV is of seven (=2 + 3 + 3) dimensions. This term, 3«"^6V, is said also to be of tiro dimensions in rt, of two dimensions in 6, and of three dimensions in c. The dimensions of a ^;o/^/e dimensions, Yjccause its highest term (3rt6c) is of three dimensions. A polynomial is said to be homogeneous when all its terms have the same dimensions ; thus : — x^ + 'ix^y + 'ixif + if is homogeneous. 30. Since the exponent denotes how many multiplications by the base are to be made, the first to be performed on unity ; it follows that a\ the first power of a, represents 1 +1 +1 + to a terms = 1 x a, or simply a. So also, a +« + n + "a And a' + «' + «'+.... "a a'^ + a'' + «' + etc. " = 1 X a X rt a'. " = 1 X a X a X a a*. etc. 29 AUi KHUAIC NOTATION. Urnco a", th(> zero power of a, denotes Hint no multipliention l)y o is to !)(> made, or in otiier words, lliat the unit-inultiplicimd is ii(»t to l)(^ multiplied by >• expression such as Aa'^b'* -^ c" (read 1 a si/aare b cube by r square), tlio involutions are to be ))erformed before the nud- tiplieations and divisions ; thus the factors are 4, a'\ and //^ In an expression such as H^/V — Ab.r^ 4 SkV -r- (P — 7e* (read Jire a square x to the fourth, uiinus four b x cube, j)l us three c square X square : by d square, minus seven e to the fourth), the order of performing the opei'ations is, — 1', involutions; 2 , niulti- plications, and dioisions ; 6% additions iiml subtractions. EXERCISE IX. 1. Find the values of l'-'; 1 '; V; 2"; .'i'; 4\ 2. Find the values of the following: 1", for /( — 1 • 2", for it — '2 .T, for */ -3; 4", for u = 4:— i. 1". ii. l"+2". iii. 1" +2" +.".". iv. 1" +2" +3" +4". o. Find the values of . . ' 1111 i. ^.+ ^,+ ^,+ 4,. iii. :r + 4'+r)'-f)''. V. .3 X 4- 4 X 3-. ..1111 iv. 3'- -5'. vi. 3. 2* -3-. 8. 4. Read the following : — u--; x*; 4x'; 3x* ; (fx; 2o.e'' ; 3a'a-; 4a"x^h-- aV-^b". 5. Write with exponents : — i. XXX. ii. aaaxxxx. iii. 4axaxax. iv. aabbxxxbax. MlXi:i> i:\AAI I' LKS. 23 V. ,,, • VI. • VII. '\ s(|ii!ir(\ pins h < iil)<\ ii. Seven ./• lo file I'onrtli, niinns five ./• euhe //, plus three .»' s(piiire // s(niare, niinns live .r y cnlu', pins seNcn // to the fourth, iii. ./• to the ni, i)lus ./■ to th(> /;( tninns n y to the //, plus // to the m plus n. 9. Kxpress jil^'ehraically : — i. Seven more than the scpiaro of u' ; seven less than tlio cube of X. ii. The excess of llr' over seven ,v. iii. The nuiiil)ei' which excc(>(ls the cube of ,r by twice the scpiare of ,r. iv. Tlio nuni})cr which is less than twice the sfpiari^ of x by half \\w cube of .r, V. The nnnil)er which is greater by the cube of x than tlie excess of X over the square of a. vi. The snm, and the difference, of twict lonji; and xy feet vrido? How many square yards in each case i 11. A rectangular room is x feet long, y feet wide, and 2 fe(>t in lieight to the ceiling ; find i. The area of the walls, ii. The area of the walls, ceiling.- and floor, iii. The volume of the room. 12. What would be the results in the last question \i x — y = z'i 24 ALGEHUAIC NOTATIOX. 33. Like Q,liantities ^<"" Terms) arc sucli as contain the same letters, tlie (lorrcspondin^ letters in the several — Qta''b + 15ac', the sum of the coefficients of the positire terms in a'-b is 8, and the sum of the coellicients of the negtdire terms is 10 ; the difference of these is 3, to which we affix the literal i)art a''b, sign — , getting — 2a''b ; similarly combining tlse terms m ac^, we get +2ac^, and the whole expres- sion is simplifed to — 2a'^b + 2ac^, or 2a< ^ — 1- Hence a dirisor may be re[)lac(Hl by its fractionally-expressed reciprocal as a multiplier. If, for example, the product of a and b is to be divided by m, and the quotient divided by ;^ this may be 1 1 , ab represented by ab -^ m -j- w, or by a6 x — x , or by -. Wi 7t III ' 'I EVOLUTION. 25 Hfjh ; EXERCISE X. 1. i. Dofino like and vnliJie quantities, ffivinj? examples of both, ii. State the rule for the addition of lilce quantities. iii. What is meant l)y the dimensions of a term? Of an expression ? What is a homo,i,'eneous expn^ssion ? 2. i. Group toj^'ether the like (piantitic'S from the followinjjj.— 3.f, 5(/, 7rt.r, -3a", 2.r, ~\,m, -3«au-, '^xx, Oa;", 3aa;, 4«, — 5ic, 8a;a, 4^. ii. Simplify each group obtained in (i). If a = 4, i = 3, c - 2, and e^\, find the value of o. a" ah (lb ah' (,,,,, , — + P ; — (^ ; — ; (r + (>' + (' (', c c ah-'. Wti.v' + 2()aa;'. 4. Simplify '^a'^x — "ntKv - Hf/^r + lOr/^/' f). Write in the form f — ,y + 1 + if + if. lii. And in Ascending powers of //, x* — x'l/'' + xf — x^i/" — a-'^ + y + l. 37. It has been S(H'n that involution is the operatirms : — i. a + h\ a — h\ a — b + (\ ii. Does a change in order make a cliJing(> in value ? iii. What are complementary terms ^ What simplilicatiotr may be mad(^ in an expression having a pair of such terms ? Give thnv' (examples. 5. Writ(^ in as many ways as i)ossible, by changing the order of the factors : — MISCELLANi:OUS EXAMPLES. %1 10. 11. (2. i;?. 14. i. ah ; ra -^ b ; abc ; r/'// -r- ^'. ;i. Docs a chanfjc in order niako a chanf^o in value? iii. Express algebraically "The (luotient of one (luantity l)y another miiltij)lie(l by tlicir product equals thes(iu;ire of the first quantity." What is the difference between an exponent and a coefficient ? Which is the coetlieient and which the exponent in (ufl What is the value of this if u- = 1, a — 3, n = T) i Su])stitute 8 for x and find results in What principles are illustrated by the following : — i, (I -\- f) — (' = a — c + It = h + ti — <■ = — c + (I + b. ii. (d -i- c = (t -^ (' X b ~ b(t -e- f = 6 -f c x a. If I) be a whole number, express : — i. An even nunilx'r. ii. An odd nuniber. iii. The three consccutivi^ nuin))ers of which n is the middle one. iv. The three consecutive eren numbers of which 2/^ is the middle one. v. The three consecutive (xhl numbers of which -in + 1 is the middle one. Two numbers a and b are to b«! added together, th(! sum divided by x + y, and the quotient subtracted from ni times k to the power of u. Express this algel)raically. A grocer mixcHl n pounds of tea worth x cents a pound, with b i)ounds worth // cents, and c pounds worth ^ cents ; what is one pound of the mixture worth i Two men start on a journey of c miles, the first going . Kx|)rcss ai,!;:i'l>rai<'ally and calenlale : — i. 47 diniinislird by lli<' excess of 22 over 15. ii. 10 anj^nicntcd by 1 lie excess of 15 over H, and liic suni diininislicd by 11. ii. Express in woids : — i. a — .)■ + ii — //) ; (f — (.V + <■) — // ; (t — (.r + c — y). 7. If a = f)0, b - 10, (' = 6, and d ^ 2, find the values of i. a (h — o. + (I)\ ah — c + (l\ a {b — c) + d\ (a — b) (c — d). ii. (a — b) <;— d\ (I —b{c — d) ; a — be — (/ ; {if — b + c} d. iii. 3 (a — b)c — d: ('2« — b) {a — d}\ 4 (2a — 36) r - 1 OOr/. iv. ^\'^a-\Q{b-<:)d\\ ^\ {fi-r)b)'-d{c-d) \ • ^' m' + ,V'»- 8. Express al^jjebraieally that i. The sum of .v and (t is to ])e diminished by c. ii. Six times ,v is to in) diminished by tlie sum of (( and c iii. Thrice \\h'. excess of .<' over cube root of tlie '.'X- cess of x s(iuare ov(T a scpiare. 40. An expression havinij: a part enclosed in brackets may itself be enclosed in brackets to foi'in part of a lotKjcr expression : this apiin may be enclosed in brackets to form part of a still longer expression ; and so on to any extent. When s«'veral pairs of brackets are tluis used it is usual to make each pair dilVerent from the others in si/e or shape. Thus, 1". If it were re(iuired to multiply « + b into the sum of a(.r + ij) and b {X — //), the result would by exi»ressed thus, {a + b) \a(x + //) + b{x— >j)\. 3°. 1(W — \T)b — ;4r; + 2 CM) — a)\ |, — read, " 10 a ininus liracket 06 minus bracket 4c' plus two bracket ;>6 minus a^ close all brackets, " - 30 COMPOUND EXPRESSIONS. den()t(!s 1", that a is to bo subtracted from Hft, 2°, the remainder to be (loiibkfd, 8°, the product to be added to 46*, 4°, tiie sum tt) bo subtracted from 5/>, 5", the remaimler to l)e subtracted from 10a. Hence if a = 15, 6 = 0, and c = 1. we sliall have U)a — [56 - ]4r + 2(36- «)fl 1 Kl\ [30- |4r' + 2(18- ^-m 1 50 — 30 — j4 + 2 X '6\\ 150 — [30- H + 6U 1 rt/j [30- lOJ 150 — 20 130. ^■! EXERCISE XIII. ;. Multiply the ditference of the quantities a and 6 by d and subtract the product from c. 2. Write down the expression which indicates that h is to be subtracted from a, that remainder' from c, that remainder from (I, and that remainder from c. 3. Multiply the result in (piestion (1) by x + y. 4. Exi)r(^ss algebraically that the; square of the difference be- tween a and b is to l)e added to 2a6, and the sum subtracted from the s(piare of the sum of a and 6. 5. If 6 = 3 and a = 2, find the values of i. '6\h+yia-h)\. 11. ( 36^ — 2a'' )\a — A (h + a -2b)\. iii. a' + b'' — \ b (a + b) — a (a — b)\. iv. 7(1 — [56 - H« — (^« — 26) [ ]. 6. Write the expression which means that 1", 5a; is to be taken Jrom (ix, 2°, the remaindc^r sul)tracted from 4x, this remain- der to b(! taken from iix, and the result from 2x. '^. F.xpress in words : — fi i. 8 (6 — c) — I — (a — 6) — 3 (c — 6 — a) }. ii. _3[__2|_4(-a)H-t 5|-2{-2(-a)n. COLLECTI SO rOEFFiriENTS. romaindeif sum to bo from 10a. m by d and b is to be remaindo" erence be- subtracted o be taken lis remain- Tl- the sum of the nnanHties ^ ^:.::Z^TZ'Z^'7T' 42. If some of the teruis iu the e\i)n'ssinn< f,. k i i , of'^-tenn!is;;':n;^:'ir^^'' "<'- ^'- -"> of eoetficient. Sum of eoeffieients of the negative terms is (4 + Oj = 10 ; •■• ^^^' - 1<^)«/ = -w, is the result. Kx. 3 . Add Sa'x - 3^*^^ + o^. _ o^. , ,,, ^ , ^ ■-—.'«'., and 5,..+,/;,, ^.ZiZ:'"^"' -"""'^ 4a'x + Ahy' _ 'Sx Sum = \Oa-x-. ^by^ + 5^ N. B.-Eaeh new quantity is written to the right in a new column. EXERCISE XIV. Add the following quantities :- 1- -^ aud 7 : 5 nnA ^^ . ^ , 2- |.ry* and -o^y . j^^,^^, ,^^^^ ^loa'^c. ^- 4(f/'^ + 6') and 3f-.?^ + 6»). 7/^, .,, , 7 Va; and - 9 ^/x. — & ) I 32 ADDITION. 4. 7 (a' + &' — c») and 3 (a" + h^ — c'). 5.-8 (^c" + 2/' + 2"') and - 4 (a;'^ + if + ^'). 6. «" + Z/' and ^^ — y' ; 3rt — 26 and — 5i«. 7. 3.^ and 4-''; a; a; 7 ' i and — 3. — • a — o (I — b 8. X + y + z and x — d — z \ '6ab, — 7^/6, and Haft. 1. nx^ — 'Sx'y — y\ -liix' (ft) .. ,,.. ^^ ff y, ...x^y + 2xy\ 4x'y-xy'' — '^y\ and Sxy\ 3. 2 (m + w) + 3 (a + 6), (a -f- b) — 3 {ni + n), {a + h) — {m + /?.), and — 6 (m + n). 3. 7a' — 20" + dax, — a"" — a} — ax, — Ga" + 3a' — 2ax. 4. 2(tn + nf + x, 3(m + w)" — y, 4{m + nf—2x, —{ni + iif+x. 5. a;" - 4^7^ + 60;^", 2a;V — Sxy' + 2y", y" + 'dx'^y + 4xy\ 6. ^a' — 2a'b - |6», |a' — fai' + 26^ — |a' + a6' + ^6'. 7. ?_3!!i,3^-4^\3^-3!^,4^-4^. y n y n y n y n V 43. When several terDis contain a common literal part multi- plied by different coefficients, the coefficients may be collected and their aggregate prefixed to the commo)i literal part ; thus : — as 7x + Sx — 4a; = (7 + 8 - 4) a;, BO «a; + 6a^ — ca; = (a + b — c) x. In like manner. mnx + 2by + pyx — 4by = ()nn + pq) x + (26 — 4b)y = {mn + pq) x — 2by EXERCISE XV. Collect the coefficients in the following : — 1. ^x — 2y-\- 66a; — 4y + 7aa; + m + n. 2. ax — dy + mx + ny + x + y. 3. ^ay — 2x + ^by 4- 6aa; — 36a: + y. 4. 2dx + 3ey + ^g — 2fx— Zdy + ^ez. ah. and Hxy^. -(m + n\ IX. ni + ny+x. vy\ purt multi- llected and lus :— k'e manner. 46) y COLLECTING tOEFFIflKNTS— SUBTRACTION. 33 5. |aV - 4^. + rV - _^,/3^ - 7y + y + (a + b)x. 6. aic- 6// + (a -b}x- {a + 6)y + in + b)x-(b- n) y. 8. ia-^x-' + (6-c)^' + («-a)^- + (h-c)x-' + (.-a).' 9. ^'"-«^"-36^-9a^-" + 76r + (a+6-c>-+10«y- + ^-3a6cr. 44 Subtraction.--77*6 Algebraic Difference of two quantities is found l,y c'^«;,^^,,^ the siym of all the terms of the mbtrahend or conecvrng hem to be changed, and combining ^Am, as in addi • tion, with the terms of the minuend. Ex. 1 Suppose that 7, which is 3 less than 10, orriO-3) is to be subtracted from 15, the work is indicated thu^ : lLriO-3 Subtract 10, the result is 5, which is k,o small by 3, since not 10 uir tnn^: :^" '- -^*--^- ^^-- -alll^t 15 Minuend. lOj- 3 ^Subtrahend. •'i + 3 Remainder. and'i ' ; ^^f^ i!/' f' 5/^P^^'^^"^ «"y 0"antities whatever, and & - . IS to be subtracted from a ; then, we have a-ib-c)- subtractuig 6 we have a - 6, which is too small by c, since not i' mt . units less than b is to be taken from a. -. The true remainder is a - 6 + o ; or • . -lutj irue « Minuend. ^ — c Subtrahend. a — b + c Remainder. Bx. 3 . From 5x' - ixy + ny' take 2a;» - ^xy + ly\ nx"" — Axy + 5y' Minuend- ~ ^^1+J_^±:z2^ Subtrahend, with signs changed. Sx' ■\-2xy-~ 2y» Remainder, found as in Addition. 34 K\. 4 . From take subthactioi^. a.*-' — hi/'' — 'icz' px' — q\f -y I'f (ix^ — px^ + qu^ — hif — rz' — 2^3"' Or, oolloctinj^, as in addition, the coi^dicicnts of tlu; like powers of a;, y, anil z^ we hav<; (a — p) x} + (v — ?>)//' — (/• + 2f •) z"". \\. EXERCISE XVI. 1. Subtract 3 from 12 ; — ;} fioni 12 ; — 3 from — 12. 2. Subtract Sa? from 12a; ; — \\x from Vix ; — !!/• from — 12a;. 3. Subtract — 3a;'^y from ia;"// ; from — lia;'// ; from xif. 4. Subtract 6^/a; from 12y'a; ; —'a^x from — ^^x. 5. Subtract from T^x ; 5a; from ; x — y fi-om 0. G. Subtract — 5 {a + h) from 4 (c/ + h) ; from — H (r/ + 6). 7. From 26 take a ^h — a ; and also ^(t — ^h — ^r. 8. From (t — 6 take i. a- — //■ — r", and ii. —7 {.r/y - -kjf take - f r'^ + .r// - //'. 7. From \a' - '2if,f' - Jf/z'.c tak<« y.r + {//•' _ ^„y'. 8. From (5./-=' - 7^-^ + 4a7/'^ - 2/f - rw' + .nj _ 4>/' - 9 taki3 Sw' - 7.r-'// + .ry-' - //•' + ju-'^ - .,■,/ + Oy'^ '_ 4. 9. From a'b''—a'(K: — hr'—(UA:* take 2a7>r- — Tir/^y-V- + Saic'-' — nbh-'. It). What mimlxM- mii.st bo takoii from a' + 2.,Vr + h* to lmvo the romaiiKh'r 4r/'''6''? 11. From rtV + /y'^/y-' + (.^^.^ tako 6'V- r'^y'^ + a'-'^-'. 12. lorr - i.y/' - r» + .v/" tak(^ - <)fr + 2i'>'' + r,-" - rxA. 1 ;{. From (U-' -b.v- + .r - 7 tako if>.t=' - ,j.v' + rx - 8. 14. What mimbor must ho taken from 2a' - (5 + (Sab'' - '>6» to leave a' — 7a-b — iib' i 15. From what number must % + y,, _ 2^ be subtracted to .nve 1«. From 5 (x,/z - h.v + ajY _ 1 ,t^ (, _ y + .,.^)_o^ (,,,. + /,^._,.^) take2| u-ijz - bx + cyf + ^ (, _ y^ax)-2\ ^ax - c'z + by'). 17. Add together the ei-ht foHowin- expressions ; from the sum subtraet the first expression ; from the remainder su},traet the second exi)ression ; and so on, till notliin- r(>mains :- 2{a + \-\b-^i-lr-\xd + x\e; 2yi - A-lb - (liy - 7"./ - 5^^^ ; ii. iii. iv. V. vi. (5#a in - 7|6 + 7|.'- + (;i,^ _ 4^^, . np 'Hf> -., IT' vii. 7ir/ - 5|.6 - Of r _ 7^d ~ '^e ; viii. d^a + S\b + 4^e + 6lf/ - 4^e. 18. What do the following expressions become, when x = al- ,. (x^ + ar--4ax(x^-ax + a^); ^(x^^ax + a^, ^(x^ + ax^ + a^. u. (x^+ a^ + ay- (x^ -ax + ay ■ ^|«3 ^. ^3 ^ 3^ ^ (UIAPTKR III. EASY KQl'ATIONS AND IMIOIJLKMS. — IIHMOVAL OF HUACKKTS. 45. We sliiill now ^Ivc ii few cjisy |)r()l»l('iiis showing (i.) how Al;^«'hr!i is applied in liii'ir soliilioii, and (ii.) how ccrlain al,n('l>rai<^ expressions may l)e read as prol)leins. In the ai'ithiiielieal solu- tions the sii^ns +, — , ai-e used. In the alj^cbraic solutions, .*• irill rcpresott the unmher soiigltt, tiie aritlunetieal eqiiatious are plaeed on the left hand, and tin; alj^ebraie, on the ri;th = 10 ft. Or /^leni;th = 10 ft. tV It^'njJth = 10-:-5 = 2 ft. length = 2x1 2=24 ft. X = f r + f r + 10 = -,V'> + io. ,-,x — ^x= 10. Or ^^.'r = 10 -rV= 10-r-5 = 2. u;=2 X 12 = 34. AIU'IIIMITK'AL AND A IJ) KlMtA I( ' SOI.l'TIONS. 'I* ACKETS. (i.) how il^('l)rai(! wil Holu- js, X frill I'e placed I'diiaiii- • NH. 10 T). 5 ; (111(1 ho 12. water, iiid tlut 2. N24. Ex. 4 . A piii-se and flie inoiiey if coiitairiH are worth |24, and the money is wortli 11 times the value of the purse. What is th« purse worth '. Value of purse + 11 times value of |)Ui'se is iis'M ; Or 12 times val. of purse = $24 ; '' =124 -5- 12 =$2. It b I .<■ + 1 l.r - 24 ; Or 12.r = 24: .■..r = 24 -^ 12 =2. Kx. "» . Divide $<.>."> amoii^' .\, li. and (', so that A shall have |10 less than I'., and (' $1.") more than \\. We have: -IVs share ; and A's shan; is IVs " -*1(). '• ("s '' " IVs " 4- ll"). .-. B's-i-iVs- 1 4- H's+ 15 = $!».-,. Or W limes IVs share -f- $•") = >j?!>"). and ;i times IVs rrS!).-,— $5 — $!)() ; .-. IVs share - pM) ~ '.i -- $;U), etc. Let X denote B's share ; then .r— 10 = A's, and X + 15 =: ("s. .-. aJ + a*— 10 + ;r+ 15 = 95. Or :Vr f 5 = !>5. and ;{.r = {>5-5 = y0; .-.a; =90-;-;} = :}(), etc. Ex. () . (Jive the expression 4.*; — 2 = :U' + •{ iis a word problem and solves it. " If 4 times a eertain number be diminished by 2, tin' re- mainder v.ill be e()ual to :{ niore than :{ times the numi)er." Kind the numl)er. 4 times Xo. — 2 is :} times Xo. 4-IJ ;' .•.4 times No. — :{ t imes No.— -2 is ;}. Or N( 2 is :{ ; Ex. T .-. No. = ;{ + 2 = 5. X X A Ax — 2 = :J.r + W ; .•.4.r-;Vr — 2 = :V Or it- — 2 = a ; ..x = \\ + 2 = 5, + '•\. ''If a (piarter of a eertaiTi numl)er b(> 2 4 5 subtracted from half of it, the remainder will be 3 more than ono- tifth of the number.'" ^ the no. — I the no. is ^ the no. + 3. Or \ the no. is -J- the no. +3 ; {\ — \) of the no. is 3. '^'" yV f>^ the no. is 5i ; Tho no. is ;} x 20 = 60. X X X 2~4-5 + ^- Or i-,r = \x 4- 3 ; .-. (|-|);r = 3. Or y' ;.r = 3 ; .i; = 3 X 20 = 60. 3g EASY EQUATIOKS AXD PKOKLEMS. '-Hs^^":;:;::';^;^ - ^^oso, in w,.ic.h the „„.. Ul Find EXERCISE XVII. (<0 •^ in tlio following :— 1. x-H = i); ,--12 = 0; ;r-CA = 0: iv + a ~ 0. J = 0; a; + 4z=0; a--« = 0; •r \- ;^'.. 5. 10a; = ^; 4./ = '^_ ,. . -,. n (I - h. 7. 5«. = + o,. . 1 , ,. ^ ^j ^ ,^^^ ^ ^^ _ ^^^ _ ^^^ ^ ^ _ ^^^ a (I in 5 .. •'^ in ; the first has throe times jis much as tli<' second ; liow miicli has each ? I)ivi(h! $70 ))etvveen A and B, givin*? B thn'c-foiirtlis of what A receives. John lias 40 cents less than thr(>e times what James has ; how much has each when both have |1 ? Three hank notes are worth !i!l(> ; the first is worth |2 mon' than tiie second, and the third, $8 more than tlu; second ; find the vahu; of (>ach note. Three numbers amount to GO ; tlie first is one-fifth of the third and two-thirds of the second ; find all the nunil)ers. A man had a sum of UKmey at interest for 8 years at 8^ per year; he spent |;}0 of t]i(> interest and had $90 left; find the sum at interest. Divide $420 amoii,<>: A, B. and C, so that B shall have $17 more than A, and (^ $20 more than B. A certain num])er increased by its half, its third, its fourth, and its fifth part, amounts to 274 ; find the number. John can copy a dictation exercis(> in a (piarter of an hour ; with B"s helj) he (h)cs it in 10 minutes ; how loiij; would it Take B to copy it? A youtli spends .\ of his sjilary for board, ^ for clothes iiiid books, and ^'^ in other expenses, and has $;508 left ; find his income. The ,u:reatcr of two nunil)ers is ecpial to six tim<>s the smaller, and their sum is 49 ; find the numbers. A, B, and V nmt a gra/in,;,^ farm for $112. A puts in 8 cattle, B, 9, and C\ 11 ; find each man's share of the rent. N. B.— Lt?t X = nmt due for each head of cattle. 40 EASY EQCAT10N8 AND PROBLEMS. 13. In an orcliard .', of the tivos arci of the same kind, }, of another ' "4' I. kind, I, of a tliii'd, j'^, of a fourth, and thore are 5 odd trees Ix'sides ; how many trees of each kind are there ? N. B.— Let 24a; = the wlioh' number of trees. 14. Two persons jjjained pilit in a j»artnershii), to which (me con- tributed .t'"><*<>, Hud the otiier, 1850 ; divide the profit be- tween them. N. B. — Let X = i)rofit dii(> to if^l cajjitaL 15. After payinj^ an income tax of Td. on th(( £, I have £1165 left ; lind my iiji'oss income. N. B. — Let 240.1' = <,'ross income. 16. In a c(n"tain l)iisincss. .$'20()() produces .$50 profit in 3 mos. ; how lon^ would it take for .$3000 to produce -f 175 profit i N. B. — Let d- -— no. of months. 17. Two-fifths of A"s money is c(|Ual to B"s. iind seven-ninths of B's is e([Ual to C's. In all they hav(^ $770 ; what hiis each ? N. B.— Let 45,r = A"s money ; .-. lH,r = B's, etc. 18. A hare is 50 leaps ahciid of a hound, and takes 4 leaps to his 3 ; but two of the hound's leaps are as louij; as three of tlu^ hare's; how many leai)S must the dog take to catch the hare ? N. B. — Let '^ >• — length of hare's leal), '^'i^^ •'• '^■*" = It'ngth of hoiin is, etc. 19. A gamester lost | of his money and th<'n won $10 ; next he lost .^ of the remainder and won .$3 ; lie now had $03 left ; how imich had he at first ? N. B. — Let \i},v — the sum ; .-. 12.r -f 10 = 1st rem., etc. 30. Find a number such tliat if | of it be subtracted from 20, and i\ of the reuuiinder from \ of the original number, then 12 times the second renuiindt'r shall eipial half the original number. N. B. — Let 88x = the requin^d no. ; .• 20 — 33a' = etc. RULES FOR REMOVING 13UACKETS. 41 '>f another » oUd trees one eon- profit ne- ve £1165 3 nios. •rofit :' 46. Eemoval of Brackets.— An his 3 ; -' of the teh the ni?th of icxt he 3 l(>ft ; 'te. 0, and len 13 "iginal EXERCISE XVIII. Remov(> l»raekets and eolleet tenns when possible in the following; eases : — 1 . ,r + {tr - r- — d') ; re<-edinif rale. It \f.\ ijsual to be(/ia with tlie innermost brjiekets. Ex. 1 . ll.,'_|T.r_ jH.r— (9.r-(5.r)f| = ll.*--|r.r— regarded as a hlnoniial, the first term lu'ing 2a and the seccmd all that | | incloses ; so | } incloses a binomial, viz. 3c, and all that ( ) incloses ; similarly ( ) incloses two terms, and finally the ~ (vinculumj incloses two terms, 2c + 36. Hence, 1°. The — before | changes two Signs, that of 36 and that of {. 2°. The sign before { being now +, the tav signs in • do not change. 3°. The — before ( changes two signs, that of 3a and that of the . Th(^ sisrn befon^ the being now +, its quantity does not change signs. . . .Tn using this shorter method the learner may at first write down the sajiis alone, and afterwards insert the proper tenns. Ex. 1 [% '5.y - 2z [2x - 2z + 8y)l 4- .... Writing signs first. 22 — 2.V + 3^ + 8.y — 0. Ex. 3 . 8a' - |0a- - ■;S6^ - (9c'^ — 3a-) } ]. — + — 4- ... .Writing signs first. /. e., 8a' — (5a-' + H6' — 5)(r + 2 brackets and collect terms in the following :— 1. (a + ,r) - [h - X) -('2a- 36] - (a -h- x). 2. ;V/ - .; (// _ -Sr) -{20- r.) } + {-({-/>- <;). ■A ^•'o-^-<',iaO-fr)\-ia''-i4af, + h')\ + {21)' - [a' ■- ab)\. ' 4. \'^■>•-i^>/~^)\-]!/+^2.r-^}\ + >■i^-^.r-2>/)\-\2x-il/-^,;. :.. I-;i-(l-4.n[ + \2x-{;i-nx)\ + (- 7^r + 7). i). {2a - m + . _ 2j-{ \x - -Aij) I \ . 10. my - \x + % + |3my - 3 (x - ,j) - Aah\ + 5). 11. <'-2ih-n-[-\Aa-b-c-2ia + b + r)\\. \2. -2{-\-(^x-y)\\ + \-2\-(x->j)\\. 1-.. Find vah... „f a - (6 - r, - ]6 - ui - oj- - [« -|26-r .-c)\\ when r-.= i|o_.;o_^^_o_;,.,|.,_ Hi. If r, = 2, 6 = ;}, .,.==6, !j = r,, find the v.due of 49 ro/,r.nsr/y, i,y the rules of Art. 4«, th(> terms of a onantit; may !,<> enclosed within brackets with the sign + or - nnreding I lie bracket. i fe r. If the .sign + precedes, there is no change of signs. 2° If the sign - precedes, the sign of every term (.nclos(>d in brackets is changed. is t tri'f! *'; ^''';;' ^'"^r '^^ '""^^^'^'^'^ '^'^ «'^'" ^^^ '^^^ ^^-^ which 18 to Stand hrst 'Athin the bracket. 44 REMOVAL OF BRACKETS. Ex. \ . x — i/ + ^-n = (x - y) + {z — a) = (X + 2) — (11 + a) = (X + 2) + (— U -d) = x— (// — ,r 4- (1) = x + (2 — !/ — (I) = x—{a—2 + !/) = — {-X + !/} — { — 2 + (J). Ex. '"♦ In ox* + hx- — -f 2bx — r.,r'- + -'•<' — '*^'V, bracket tin 1* *!1(h K lis of lik<^ powers of x with + before all the l)raekt t.'^. {a + 2j u;' + [b — 5) x' + {'2h — ;}) x — (5. Ex. 3 . Place in brackets r — lie + 2(1 + (5/— 7;/ + 2)i — 6.r +t/. I. iM (..Jrs. with three pairs preceded by the sij^n — ; ((; -: ^.7) - (tie — (\f'} — (7.7 — 2n) — {(ix — //). ii, Wth/o^ '■::"s in each bracket, and tjie sij;n — before l!, 2 ! ;;i, (/; + 2(1 — ;](' 4- ()/') — (7,7 — 2)1 + C}X — //). iii. All but lirst two in brackets, ))rece(le(l by the sij^n — . c + 2(1 — {'.ie — ()/ + 7.7 — 2 II + Hx — y). EXERCISE XX. 1. Reduce tlie followinjj; exi)ressi()ns to tlie form x — (l)racKctod (piantity : — i. X — (I — h \ — a -\- X — Wh + 2y. ii. X — 2)11 + 2)1 ; — 86 + x + 2<- + Tk/. iii. X — 2))i — V^a — 2b) ; x -V d — (b — <•) + ())i -■ )!). Iv. \2 — X — b + (■ 4- ./• ; X — (d + /y) (/> — (j) — 0)1 — Ii). 2. Collect in brackets the coefficients of x, y, and ^ in the foUowinjiC : — i. 2ax — l\(fy + 4b2 — 4bx — 'Scv — 'icy — acz. ii. ax — by + cz — bx + cy — a2 + ex — (jy + b2, with a . . tirst in each quantity. iii. \2(ix — \2ay — \by — \2bz — 15ca; — Ocy — 3c2r. BH ACK ET8— K X KRCISKS 45 '^'i', bracket th( II thv ])ra('k. is. - f). (' sign — ; - !/)■ '■ «ign - before tlie ,siir,i __ — (bracketed '^ ■'• s.. that + shall stand before each bracket :- 1. 2 - l,r^ + :],u'- — 2r.r + (),f.f' + 7,r _ lu'' 4. lu .hT> following expressions collect the coctlicicnts of the powers ot w with the sign - prefixed to all brackets :- 111. ,50 .(, — Ox _ f.,,. _ ^,,.4 _ ~^^^ ^ ^^_3 5. Simplify and collect coefIici(>nts : — x^ - u;^ 1. a.f=' 2kv' + '^ + hx~ r.r' - ^ X' + .*;=> _ r/rf;" + ex. 6. ir./=25, b=M .=4, ^/=K find the values of the following:, 111. ^irun ~ ] 2 .^ilib) + ^(2r) - 4 ^d\ ; 7. Express in its simplest form «. CoIIoct coeHicients, and arrange in ascending pow(Ts of .r, y- 'Simplify, and arrange in powers of .1; ciiAPTKii rv. MULTIPLICATION. , 50. When two or more quantities arc to bo multipliod together, the product may be in(U. -ax+h^ ^-''> + <-'0 + (-.„ to /> terms ~ + ( — " — a — (( ti »i )=z4-( ; _ lip., are to b^ changed. T,;;: ''^ "' '" ""'"■'"""' '■ -' "" "» I aiiise, ^//e si{/n — if theij are unlike. EXERCISE XXr. Multiply :-_ '**) [1- "Kv />;,,=/, by «6^ 3.,y^,,,. 4^,^^, '^^' by 5^.3y ; 2abni by y«-V,,"^ ; _ ^.^s ,,y 3^^ 48 MLLTII'LK'ATION. ii. — 3 by 4 ; -8 by —4 ; --.Inh })y 4 ; \iab by -4 ; —ah by —al. 4. na'jr by —IJa^aj" ; T^/^" l)y — (/=7> ; Tu-yby— ;ir; ;W/> by — (W/// 5. f^6 by ^i'0^\ 'da^b^c* by — 46Va^ g//*"// by — f//7- b>' —upq. (ft) 1. Tf « = — 2, & = 3, f = — 1, fiiid tbc values of :— 3rt'6 ; Sahc'' ; — Tic" ; - h''(i' ; — l<(%i', ; - .k/'^^^j^ 3. Write out tho followinj? products : — i. mx X my x m2\ ax x hx x ex \ 2' - 7aY - 7a* y by 8rt6. —4 ; —ah by —(il, ^.r\ \\(tH)hy —{\,t(, ^y - ^'n'l/' ; 'Sp,f of :— r}a'bV. X '^f(h X (— 4ah). I)'!/-) X c-z' X <(h<'-. A'ljz X {-r^x-y'z). 1, y = 1, find tli( 'ih' + '7nj* ■ a'~h\ X ax ; I from ArithiDctic, 8 = (56; 8 = 80. >i'; find niiul •.-Multi pi >j Her, • ;s : — a' + 6'bya&. V by 8ff6. POLYNOMIALS BY POLYNOMIALS. " ^\f \. Multiply x-\-y—z by xyz; —x—n—z by —U\ 1 +./• by ~.x\ \. r>a' + ;{6'^ - 2c' by 4a'^6c'=' ; Oa" - r)r/'^6 + 7f/6'^ by Ho»6». (») Perform tli(! multiplications in the following' cases: — _ ,rhr + h" — r^ by — ah ; '.ix — 2y — 4 by — Ix. _ ;{„,,■ (-by- ':ti;z — 5) ; ('to' _ 4//''' + 5^'^) x '.Wyz'. ). C^x' + 'lry-^xy' + y-') x (-oj-^) ; {-^i-2ah + a"h'') x {-a'). ( _ ,r-^,f^ -z^ + Ax'y-'z') x (- '6x''y''z) ; (fa - ).b - e) x ^ax. '*■'■' - ->■!/ + y - l-ryz) X Ix'y'z' ; - ^a'x' (- f«» + ax^^x\ \{a + b)' - {a + b)\ X -2 (a + b) ; \^ {a - b) + » (a - f))H X - (a — b)\ \i»i"->'f + {"i'-n)\x{m'-nr- \ii(a+b)+2(a + b)*\ (a+b)\ jp. {[a + br + {a + br\ X (a + b); {(a-b)~ (a - 6)'f x (a - ft)-. I 56. Ill Polynomials by Polynomials. — We may multiply f + 4-;5) by (8 + 3) thus:— + 1 - ;5) X 8 = 8 X 7 + 8 X 4 - 8 X 3 = 8 times the multiplicand. + 4 -8)x 8 = 8 X 7 + 3 X 4 — 3 X 3 = 3 *' " Addin,!,' these products we have the complete product. Similarly, with algebraic quantities :— (// + 6 — c) X {III + n) = (a + 6 — (') m ^- (a + b — (') II = {ma + mb — mc) + {iia + nh — ne). = ma + mb — me + na + nh — nc.. So also: {a + 6 - c) ( m - n) = m {a + b-c)-n{a + b- cj — {ma + mb — inc) — {na + nh — nc) <'., II times the Multiplicand is to ha subtracted from m times he same : .-. the result is ma -f mb — mc — na — nh + nc. Ihnee, Multiply every term in the multiplicand by every term in Ike multiplier, observing the rule of signs as in the previous cases. 50 MILTI PLICATION. Kx. 1 . Multii)ly '.le - \y by J.r - ",»//. 8a; - 4// Ax — 21/ nx^^l&xy. . . . = 4a"tini('s the niiiltiplicjind. 2// subtracted. \2x' — 22./// + H// (•fitid. a prodnet := {4,r— 21/) times tlic imiltii) Ex. 2 . Multiply Tj" - i\.r' + Tu- — 4 by iVv' - 4.v - 5. Ta;" — 6a;'' + ^^.r - 4 3a;' — 4a; 2la;*^Sa^TT5a^ - 12.^' ... = 3a;' times themiilti)lcii(l. — 28a;'' + 24.r'' — 20.r' + lO.r. . . ==— 4a- — 42a''' 4- 3<;.r- — :U).r + 2 4. . =—i\ 21a''' ^^(Ja* — ;Ja;= + Ax' — 14a* + 24. . = retiuireil product. Ex. 3 . iMultiply x' —• x — a; — f a;^ — :ether, find, first, the product of two of thcui ; thou, of that product and a tliird ; and so on. EXERCISE XXIII. Multiply : — 1. 2a; — 1)1/ l)y 3a; — 2//. 2. 3x""- — h by 5a; — y. 3. x^ — '.i.r'. 1 T. ./•" - ;{./••' + Ir — i by y - ;{,*•- - ;]. IH. (i- + //-' + r,'''' — ah •+- /><■ f- ra by r/ + /> — r-. 19. .r- - ,/■// + ./• + ij- + // 4- 1 l)y .r -(-,/_ 1. 20. <|.r" _ ,,•■• f o,,.4 _ o,^.n _^ o,.. ^ ,,,,. _^ ,. ,,^. .5^., ^ ^^^. ^ J 21. .<• + // + ,? + ^/ hy .J- + ,j _ y _ „. 22. 4 - 2r - r/. 23. 4^/ - Wh + 2r — il \)\ \a - \]b — 2r + . f/- + 4/>' 4- Oc- — 2(/6 4- f + 2u;y. 28. f r- - lx\ 3L u:'" 4- •r"'//'" 4- .y'-'"" by .*••" — y/"*. ^2. (.r ~ (/h) X (.r 4- r//^) X (,<•- 4- ^r/r) x (./•" 4- a*!)*). 33. (.V — 1) X {X 4- 1 ) X (,/•- 4- 1 ) X (,/•' 4- 1 ) X (x*" 4- 1). 34. {x'' 4- ax 4- '/■-') (.r' — (tx + tr) (x* — tr.r + ((*). 35. {X + y + l){x-!/- 1) (.r 4- y-" 4- 2/y 4- 1). 57. Detached Coefficients. — The l;ii)<)iii- of nuiltiplyini,' may bo lesscnofl by using only the coefficients. Tlie (iiiantities must l)e arranged in ascending or descendinf/ powers, and the coefficients used in correspond in (/ order, those of tlu^ mnltiplieand in a hori- z(mtal line, those of the multi])]i(T in a vertical column to the left. If anypoivers are wanting, these must be supplied by Zero terms. ONTARIO COLLEGE OF EDUCATION 6-Z MULTIPLICATION. Ex. 1 . Mullii)ly 7x'' — G.r-' + '>J- - 4 l»y :U'- — \.r — 7 - + r, - 4 . . . . 3 — 4 — G eocilliciciits of iuultii)li(,'an(l. 31 — 18 + 15- 12.. .. " " " into 3 _os + 24 — *2o + k; " " " " — 4 — 42 + ;j() — :{() + 34 " " " " -0 2i~-r'4(71-" •!"+ 4— 14T34 " " i)ro:r' by 1 — 2.r + 'lt-\ We have 1 „_ 6 + +0 —c — 2 — 2rt + 2h _0_0 + 2r' , + 3 + 3a —■•ih + {)+ {)—:)!'. n—{2u+b) + {iia + 2l» — 'fM)—r + 2(;—'Sc, . . coolficionts of the pro- duct, — which is, therefore. a — (2a + b)x + (3a + 2b)x^ — 36x» — ex*' + 2cu;* — 3ca;*. DETACHKI) C'OKFFK'IKXTS— KXAMPLKS. 5:3 0. ilti])lio}in(l. " into 3 " " -0 .-t•^ the 2cl, x\ line question, multiplicand, refore supply 1 — V 3 )f product. li(n'(>fore i>ro- )f course, the It of the 1st tmltipUcand i huvo tsof the j)ro- 3ca;* EXERCISE XXIV, UsiuK detached coeHicicuits find the products of tlu; following 1. ,/•' — ;U'- + Wa' — 1 jind ,/•■-■ — 'Ic + 1. L'. ,?••' + ;i.r- + ;>,.!• -|- t ;,nd ./•=' — ;u--' + ;!,/• - l. U. .i:^' — Ix" + 5,^! + 1 and !>./•- — 4,/' + 1 . 4. x" - 2.r + 3,r - 4 and 4./'^ r :5.'- + 2./' + 1. 5. T.r' — .■).<•■■■ + 2 and ;5,/'' + >2j'- — T.f + 1. 0. Mr* — 2,i--' — 2.f + 3 and .r' 4- 3,r - 2. 7. 2a"' - 4.<- + 2,*' — 3 and 2.r-' - 3. S. ,r^ - ^r"- + 1 and .r' - ,<•-' + :',. !). .,» + .-),»■■-' — 10,/' - 1 and .«•" - .-),/• • - KU' + 1. 10. G^-" - ,v' + 2x' - 2.1- + 2.V' + l!).r + G and 3.r^ + 4.v + 1. 11. .(•= - 3.v-a + dxa'' — if and x^ + 'dx'a + 3.w' + a\ 12. 1 + X + X- + x' + X* + x' + x' and 1 - x. 13. 2 - 3,7 - a^ + 2a' and 2a' - a" - M + 2. ' 14. 1 + .;• + x"- + .r' + x" and 1 — x" + .r^* — x'' + .r' — u-'^ + ^-'V I"). /.-,<= + /.<•- -I- /;iu; + n r.ud ao;^ + 6.t; + c. (6) 1. Show that Kn^h)'' — (/> + <•)= _ (n-rf = 3(a + /;) (A-fn (a -<•). 2. What does a 4 ^/' — r^ -f- Wnlx- become when <■ -n -^ I, { 3. Wliat does x' + \2x'' -f 4T,r- + (Wlr + 'js become wlim ij ~ W \^ sul)stitm('d for a-? 4. If 2.V = . Find the result when (t — „ — j, i.-, put f,,,. .r in y' — (2^/// — )})x 4- (Hn — 11). 7. What does a" — 6» — c-' — 3«&c become when a — h — (={)'i CHAPTER V. DIVISION. 58. In Multiplication wo have tivo factors given to find their prodwt; in Division wo have the product and one of the factors given to tind the other factor. In Algebra, as in Arithmetic, the given product is called the Dividend, the given factor is called the Divisor, and the requir<'d factor is called \\\v Quotient. The division of one quantity by anotlx-r is represented ])y writ- ing the divid n^ (f" -^ (f« — ^'^' ^ ^Q ^^^ ^'^^^_*:^'^ _ (i-(ij_ ?i, ,,..". . ;.,.„_ «-^-«'*'» to m factors a ^'^-^ -6T^-..T^o-/rfact.Tn^ = 6-^-^-- ■ ^^ '"-« factors, Or, if in < n, a 1 f) XX to n — in factors* In Division, therefore, we dinde coefficients and snhfrnd exponents. (8ee Art. 54.) X. \\.—,i- ^ ,/» = 1 . )),it (r ^ a" = f/»-" =r ,/» ; .-. ) = + mx ; .•. + mx -^ ( + in) — + x. 3^. — a; X ( + m) = — mx ; .•. — ntx -f- ( + /y/ ) = — x. 'i". -\- X X {— ni) — — nix ; .-. — nix-^ (— m) — + x. 4°. —XX ( — ni) = + mx ; .*. + m-^ -i- (— m) — — x. EXERCISE XXV. Perform the following divisions : — 1. iix^ -^ x" ; 21.c^ -f- lix* ; — 2U' -^ 7x' ; — n^x" -5- 7x*. 2. (fV^— aV; V -^ - i'i%''(i\ 3. 33a'/;ja: -t- 11 /;u' ; — 'Z\x"yz -i- Vixyz : 4^'"' -f — 2.r'. 4. _ 27a'6'c -=- - 9«=6' : - .r V -v- - f a^// ; - Vir-^-'y "^ kV- 5. - Ua^hi-" -f- 13a6c ; a"^' -r- — a"" ; r/^"+' -^ .^-l e. Ga^i" -^ — 3^/6 ; — r/" -j- a"-' ; ^/^z'^" -=- fr+' ; Cu** '-' -4- — 3,*-". 7. _-[H^,"'y>»-; (If/"/) : 44a''(.^' — /y)'*-^ Ih/^'i.r— //)' ; {a + ^>)" H (a + />)". 8. 4a'/y<*a''' -r- — m^'m^x ; — 3a-/>V* -^ 4r/6'^f** ; 3f/''6V X \n'l)' -^ (/''6^(r». 9. — (th-hc-<-(i -f- 6" ; ^z^^" -i (ih" ; .r"//^ -^ .*•■•//", (;/ > 4). ■_0. v/j^t'"* ' '// -f- y/.r//" ' ' ; «/ • ah • 6c' • abc -i- {ax ■ hx • cx). 64. II. Polynomials by Monomials. -When the divisor is a nioiiwinia! and the dividend a jjoIn noinial : Dirific ecvry tvnn (\f the (liridentl hy the mianmual dirisor ; the reqaired quotient uili he the snm of the partiid quotients. For since {a + b — r) x ni = ma + mit — mc, .'. (ma + nih — me) -i- /// = a -\- h — c. The ,sit/ns will be determined as in the last case. I'OLYNOMIAL I)IVI80KS. 67 EXERCISE XXVI. Divide : — 1. x'^ — 2xy by x\ x^y — {/"x by — xy ; a^b' — d^b'^ — ab"*. 2. 2d' — io(ex — 8a V by 2(V ; nab.r - Ua'b'x' -^ — V2ubx\ 3. a'^—ah—ac hy —a ; (("^—a'^b—d'-b- by —(r ; ^x'^y''—^xy by 2xy 4. 21amH'"—14a^m*xf + 2)" -~ 3 id — b)' ; a" + a"-' by f/l 0. 4x*y* — 8.r'y^ + 6.1^ l)y — tixy ; — |,/'2 + f^-y — J^x l)y — ^.c. 7. a"" + .ra" by — a" ; a'"^'" — r/'-'V'-" + a"x"' by a".<'". ^ ,,. ,.„ ISa^rV" lOf/Vw^" ir)««^^y^ KJa'x'w^ 9. 8.i;*y — 4,r='/y2 + I2x^y* — 4.rV '>>' — ■l^'y*• 10. '2a2Z^2 — ;}/A-8 + 46«c2 l)y r)a«6«r« ; IG,*'"* — 4.r'"-» + G.i'"-'* by ^r**- ". 11. .r«^2 — .r//^? + B.r^s by - 4xj/z^ ; o^e - a\v^ -f f/».j-« - a.r* l)y x*a. 12. — 3aa;* + rm^x^ — 6(f\i^—((* ])y —2(i\r'^ ; (a— />)"-• })y (a -6)""-". 13. 27 (a + bf — 18 {a + i)" + 9 (a + /^l^ l)y 18 (a + ^>j2. 14. a""!)/ — /;)" — a"(a — /'>)"' by a"(V/ — //)". 15. 10 (.I- + yr (X — //)" — T) (.*; + .y)P (x — //)'' by 5 (,/• + //)2 (.V — /y)«. 16. (a + 6)™ (// — 6j" — (a + 6)" (// — /^)"' by (ii + 6)-' (a — b)\ 65. Ill Polynomials by Polynomials. —Tins is similin lo " Loiiiir Division" in Aritiiiiictic, where we. in elTect, sepanile the (lividoncUnto parts, obtain the (inotients of the sevei'a! parts and add these partial (piotients for the conipleto quotient. Ex. 1 . Divide 805 by 23. 23 ) 805 ( 35 115 115 58 DIVISION. i hi this opc'ratioii what wo hav(i really (lone is this : — 805 690 + 11.-) r.90 IIT) 23 "^ aa 30 + 5 = 35. 23 23 Similarly, in Algel)ra ; since 1 \ (.r -f- 4) X (X + 5) = (x + 4) X X + (x + i) X T) = a^' + O.r-l^O ; ./;" -f- dx + 2() _ (x + 4)JX: + 5) _ {X + 4) x x (x + 4) x 5_ _ ^ X +~4 ~ " x+~i ~ X + 4 '^f +T ~'' 2\ i'2x — 3) (3.f — 2) = (2.r — 3) -3^ — (3a;— 3) -2 = 0^-'— 13.r-f f, ; x'—Uix + (i _ (2cc— 3) Cix—2} _ (2x—ii)-'6x {2x—'S)-2 OJ /w. 2a'— 3 2;c— 3 2iC— 3 2.r— 3 3°. And, f/ettei'alli/, (a + h) (c + d) = { the parts x^ + 4.f, /. e. (x + 4)-.r and Tu- + 20, /. e. {x -f 4)-r) : lliese parts correspond exactlij with the jxtrts in Kv. I. which (j iind th(! m(?thod are exactly similar. The divisor and the quoticmt are written under ('acli other for convenience in multiplying. Dividend. Ex. 2. '6x* - \x' + 'Ir + :},/• - 1 .r' - j- + 1 . Divisor. 3^-* — a^-" + 3.< '- .u-' — ,v—:. Quotient. . . 1st HemaindiT. -x^ — x' + 'Ax , - .»;=> + x" — X • — 2x' + \x - - 1 — 2./- + 2x - -2 2d I'i'mainder, 2^" + 1 3d Remainder. Here the first term, 2.r, of the 3d remainder does not contain x' exactly, and the division cannot be carried on without fractions ; we write tiie result just as we do in Arithmefj;' when tiiere is a remainder, thus : 2x + 1 3u' X-2 + X' —x + \ Ex. 3. Divide 4ah^ + r)Ui'b'' + U)a*—^Sa^b—iryb* by 4ah-rur + 'Mr. Here we first arrange both dindend and dirisor in descending powers of a. 10«*- ^uH)— iurf)'' 2rK/V/^ — 2(w//' — i.y/ 2rK/V>' — 20a/>='— 15// - 2,/« + Hili^lU? 66. Hence for Polynomial Division : — Arramje dindend and din'sor in asce/nlin;/ or descend inr/ powers of the same letter. Diride the first term of the diridend by the first term of tht divisor for the first term of the quotient. lU GO DIVISIOX. Multiply the dioisor by tliis first tcnn and imUnui the prod ncX from the dirideiid. Consider the remainder as a ne>r dieidend and proceed as before. The KuLK OF SKiNS is the same as in tin former maes. \ i 11. ITu'^ + 14.f — S by Tu' — 3. 12. lO.r' - IT.r + :5 l)y ■i.r — ;{. \\\. .*•' — //' I)y .<•■ + //. 14. .Sir'- 1()//' l)y !)./■■-' -I//-. 1 .-). — Vvy— 1 5 //- + Wu-' I )y 1 ti.r —. •-.//. 1(1. 7.r' + mx" — 2H.V by 7.r — 2. 17. lOO.r' - '.].)• — \:\.r- by :? + '2.u'. 15. .r' + 'l.v- — ;2r- 1 bv .r - 1. EXERCISE XXVII. I)ivi(l<>: — 1. .r- + Vie + ii") by .r + ."). 2. (r — \\a + ;5() by // — ."i. 3. ().r''— i;5,r + (I by 2,r — ;5. • 4. ^/'- — 49r/ + OOO by r/ — 2."). 5. G,r- - 7.C — ;} by 2.r — :5. 8. n.r^-.r- 14 by.}- + 3. 7. I2.f^ — .V — by 4.V — '). 8. Vlv- +.r — (} ))y ;U-~3. 9. ().r- — l:}.ry + G// by 2.r - :'.'/. 10. a' - 3^/ + 1 by ^/ — 1. i(). 7./-'' — .■)()./• + 7 by 7.V — 1. 3(1. ,i'\r' - 1 by f/.r — 1. )ll. "Zah + {\(ihi- — Sr/^v/ by 1 + We — Aid. 22. 2^/^* — 'Mia' -i\ha'' + WVi'-ir — (5//V^ by 3./ — Mi. 3:5. .<■' + 4// by ,r + 2//" — 3./'//. 24. ir-* — 1 ;?.<•'' + ;5(i l>y .c- + :>,,• 4- ('.. 25. .<:« — .-),<•■' + 1 l,r- — 1 1 Ir + I'y -r — ;5.r + ;J. 36. 48.-i;=' — 7«Kr.r- - <54^/'.r + lO.")^/-' by 3.r - ;5^r- by i^ - h\ 30, Zx* f 14r' + '.\i' + 3 by r' + 'w \ I. act the product u(l proceed as /■se.y. by ").*■ _ 2. by 2.r-n. %r' - \,f. \.r'h\Vl.r—:u:. \x I,y 7,,_ -3. \.v'- by;} + '2 .'),/■. - 1 by./-- I. ■■' + 'Sp'Y - '^l>y x' + mx + //. 3.-,. .<•"'*'+ ,<••"// + .ry" by u.-" _ ^» 30. .r'" - 3.f^"^ + ;},<-^'^» _ ^^o j^^. ^.„ _ ^^^^ :{:. „-'.|.:' + ,/ . _ ^^aLv' + b\c' + cv'b' - iia*b by ao; - bx + a' - ab, 67. Sometimes the work van l)e .shortened by using brackets. Also, by iisin- detached coefficients as de.seribed under multipli- cation. (Art. or. ) Ex. 1 . a + (6* - be + c') (b'' — be -{ e')a -^d'' + c^) (6" — be + ,-')a + (,V + <•■') Ex. 2 . Divide A-'+.r^+.,-+.r'+.,. + i by .r- + ^-^ ^.^ ^'^ a; + 1 . Write til., euefficieid.s, in.serting ^ero for missing powers, thus: 1+0+1+1+1 + 1+0 + 1 1+1_+1+ 1 + 1+1 eoeme's of div'r. 1+1+1 +1+1+1 -l+() + () + 0+\) + o -1-1-1-1-1-1 l-l + l quo'ent. 1+1+1+1+1+1 1+1+1+1+1+1 .-. the (juotient in x'' — x + L - Hry' hy EXERCISE XXVIII. Divide : — 1. a' + b' - c' + Sabc by a + b - c. 2. X' -(a + b + c) x' + {ab + be + ea) x - ahc by x-a 3. / - mif + nf - „y' + my - 1 by y - l. 4. p' ~ (/ + r^ + ;}^;y^. )j^ p-y A,i' - 2.r' + ;ir' - .r — 1. 68. Aiiolbrr method may ot'lcii be used, which is tiic inverse operation of miiltiplieation with detaelied coenieieiits (Art. 57), and m called Homer's Synthetic Division. Kx. 1 . Midtiply x' — 'ix — 1 by x* |- 'Ax — 'A. u" - 'Zx - 1 + 'Ax — 3 X* — 'Zx' — X' + 8.r'- Gx'' — 3iC . . — 'Ax' + 6.'C + 3 . p. . First partial product. .JSecond " " .T/tird " . Co/nph'fe product. Now, P is the sum of the three horizontal rows ipi, Pa, Ps) between the; lines ; .•. by siibtravtlmj the second (IikI flit'nl roirs (P',, p^) from I* ire (jet tlie frst wtv{p^)\ and this row di tided by thefrst term {x') of the inultiplier will yire the multiplicand. Hence to divide P by x- + Ax — 3, proceed thus : — x^ — 'Ax + 3 X* + x' — lO.^-" 4-3^+3 - Ax'' + Qx^ + 'Ax ^\ - "---. 4- Ax- — (iX — 3 .r' — 2x — \ . . . . Quotient. 1°. To the left of the dividend writ*' the divisor with all its sifj;ns changed except that of the tirst term. This is done to sub- tract the two rows correspond iny to p^ and p^ above. 2°. Divide the lirst term of the dividend by the tirst term of th<' divisor {.*•'-), gettinjif .<■*, the first term of the ({uotient, and niul tiply x' into — 'Ax and 4- 3, plaeinj^ the products in the obliquf J column, — 'Ax^ 4- 3.^-. 3°. Add the second rertical column, — 3.r' 4- x^, divide by x^ a*, before, getting — 2x, the second term of the (quotient, and mul- tiply — 2x into — 'Ax + 3, getting the oblique column, 4-6.^'' — 6.1'. + Hif + 6x!/ by -1. + 8.r''-./-— 1. Ii is the iiivcr.sc lients (Art. 57), partial i)r()duet, h'te product. nid third roir.s n'.s row di tided mi(ltiplihinms adch'd «;ive e.neh zei'o. thcrcfnre tlie division is exact. I5y iisin^ (h'taehed eoellieients tlie labor is .still further les.seueil. Kx. 'i, . Divide ./•' + 4./- + K! bv .r — 'Ir + 4. 1 + '2 1 -1- + 4 + + H) + '-3 + 4 + H - 4 _ 4 _ .s - 10 1 + 'i + 4 + + /. f'., .*■-'+ 2.v-\- 4 (Quotient. K.\. 3 . Divide (')./••' + ■"m* — 1 :.*••' — (I,/- + Kir— -2 bv ~V + ;U-l. o (5 +5 — IT - (i + 10 — 2 - ;? — \) + + r.' - (5 + I + :i - -J - 4 + 2 ;{ - •,> - 4 + '^' + + /. e., ;}.r^— '^.r- — 4.r + 'I (Quotient. Ex. 4 . I)ivid(^ Tu* — A.v"- + :'.,<•'- — 'Iv + 1 l>y .r — 'Xv + 5. 1 1 D _ 4 + ;; - 2 + 1 -I- :{' +1.-) + :}:{ + m •) — ,.) — .).> 1 55 5 +11 +11;— 24 — 54 Kcniainder. /. p., 5.r + 1 l.r + 11 (Quotient. .1* + .r' = j;'^. There will l)e, therefore, time terms in the (juo- lii'nt, and the remainder is — 24.r — ."•4. NoTK. — If (I rtrficfd lino dv dniirn irith as nnnii/ nr/ind i()/Kiiins to the I'iffht <»f it i tlitin tin' niinthi r of ti r))is in tlie tlirisor, it irill nku/: trh< re the remainder tteyins In he formed. Thus, in the last example, tiu; vertical line would cut off — 24 — 54 from the quotient line in the solution, and tiier*'- fore the remainder is — 24.i- — 54. Horner's Method is of special use when the divisor is a binomial. 64 DIVISION. M EXERCISE XXIX. Divide : — 1. 2a;* + Ix* + 30a;» + UOa;' + Mx + 35 by a;' +2x + 5. 2. 2j'' - Ix^ + IBx-" — U\x + 8 by 2a "^ - \i.v + 2. :J. rtx" — IHa" — Ha." + 20a: — 5 by x" + 2.r — ',i. 4. x" — r)X* + lOa;" — lOa;' + nx — 1 by x' — 2x + 1. 5. 4a;' - 7a-* + 2r)a;=' — 153;' + 8a; + 10 by x' - x + 5. (). 5a* - Ix^ + 15a;'' — 12a' — 80 by a;" +x + :}. 7. 5a;* - Ux"" + 20a; + 42 by x' — 2a;' + 3. 8. 10a-'" + 10a-" + lOa-'' - 100 by x' + a-=' — x + I. • 9. 1 + iix" + 5a-" by 1 + 2a- + a;'; and a" — 6(/ + 5 by ^^'— 2rt + l 10. x*—4xf + ^i!/* by x''—2xy + tf\ and m* + 4m+3 by in' + 2ni-{]. 11. .r"— 2.c'' + l bya-'-2a-+l; and rt" + 2a-y/ + 6" by (/''4-2«i + 6'. 12. 3a;" + 7a-'' - 12a;* + 2.r' — 3.*-" + 13 r - (5 by x' + 3./' - 2. 13. x' ~ 3a-" + 4a-'' + 18a'" — 7a; + 12 )jy a--" — 3a;'' + 3a; - 1. 14. a;" — 3a-' — 5a;* + 2a-* + hx"" + 4a;' + 1 ))y a-" + 2x — 1. I 15. lOa-" - 1 la-* - 3.C* + 20.c» + lOa-'^ + 2 l)y 5.t--' - 3a;' + 2a; - 2. 10. x'' + 5a-* + lOa-^* + 15.t-' + 5a- + 1 l)y a; + 1. 17. 4.r* ~ 20a-" + 32a;' — 22a; + 3 by 2a;'' - «.i; + 1. 18. a-''' + 2a-'' + 1 by a-* + 2.*-'' + 1 ; and a-* + 1 l)y x + 1. 19. x^ + (ia + h) a-' + (. dh'ecfhj c(nitritrii^ uv oppositi', dire(tion, /. e., west; so that { + (1) + (— tt) ={), and represents tiio point from which measun;- mclil be;.jins. If /represent tile mechanical efTect of a certain force, then +/ will mean that this force acts in re th(^ zero point in a thermometer, then — J^ means a fall of the same number of de- i;rees lieloiv the zero point. If the mercury rises d° and afterwards falls s in (me. e( tion thronj^h the point C, we may call the anjjle de- scribei' BAC, and —7^.16' will denote the angh' described when li \\'\ es the same distance in the o/>/>o*<^e direction. 66 DIVISIOK. It is thus plain that the alyebrak, ov symbolic, sum of two quan- tities may be eitiier positive or iKifdticc. For in the statement, a man walked yds. + 7 yds., if he walked six yards ((istirunl and then seven yards instivitnl, his position will he repre.sented by (+ () yds.) + (— 7 yds.) = — I yd.; /. <^, he will be one yard to tlie West of the starting point. But if the man walks (> yds. we.st and then 7 yds. east, his jxisition will l)e r('i)resente(l by (—6 yds.) + (+ 7yds.) = + 1 yd., /. <., (me yard to the east of the starting point. In l)oth ea.ses the ((rithmcficti/ sum is yds. + 7 yds, = i;} yds., the total distanee walked, without regard to dii-eeti(tn. Thus w<' se(i that the algrbi-aic sign of such a (plant ity as { 6, then (a -f- />) is positive; ])ut, if d ) is negative; thus if a = 6, h = — J), nt + f>) = — :{ ; but if (( = 1(5, b = ^- 9, ((( + b) = +7. Similar observations hold good with regard to tlie arithmetical and the (itm-bntir. (tiffereuce of two (juantities. This opposition {ovcoiitntriitf/) serves to reconeile some apparent anomalies whieh aris(^ from the more general meaning of algebraie symi»ols as eompared with those i. I'd in pure Arithmetic, e. preceding fundamental rules, the signs + and — have l)een used in their ai'itlnuelieal meanings, lint all the operations coMsidered an; true whet', er the (piantities coiu'erned are essen- tially positive or ueijatir' . Thus in the proof given for the stau- ment tliat a — (b — v) ^= n -^r b — i\ b whs assumed to be greaver than >' ; but it is triu> if f> < c : i. c. if b — r is negative. Suj)i)()se it is re(piired to subti'act : i. + b from (/, and li. — b from a where a and l> represent any (plant it ies. — b and -f b of course denoting the s) idcs and the , and a > 6, is negative ; iS,h= ^9, 'gard to (he 1 ies. rui apparent of algeln-aic ic, e. y. «+» here mnlti- ro factors ; a. Tliu.s, nd, so gen- 1 ^ n factors* nd — have operations an* essen- r the staie- be greaier Suppose )ni a where 5e (h'noting he signs + n'oni this -b and +6 CHAPTER VI. SIMPLE EQUATIONS. — PROBLEMS. 70. An Equation is a statement, in the langu{.'ge of Algebra, tlmt two expressions are ecpial. The two exi»ressions are called members, or sides, of the ecpia- lioii. the on(!,to the left of the sign of ('((uality, the lej't member, or side, the one to the right of the sign of equality, the ritjJtt mem- Inr. oi" side. Thus, 'i.r — 7 = ;{ + .r, is an equation ; 'Ix — 7 is the left side, and ;! 4- •'■. the ri(//it side. 71. If an ecpiation involves arbitrary numbers (Art. 9), that is, if it is true for all Vidaes of the syml)ols it contains, it is called an l(h idind Equation, or simply an Identitij ; thus, 2r + 4 — .*■ + ,r + 4, and {X +a) (X — a) = .i:"" — a'' are identities, for they are true for (i/l, or any, luilaes tluit may be assigned to the letters. 72. If i»n ecjuation is true only for some particular value, or viilurs. of the unknown quantities (Art. H), it is called an equation, n/cotidition, or simply an Equation. To this meaning, the word is generally restricted. Thus, the eipiation ^.t- — 7 = :} + x is ti'ue for x = H, and for no I'f/ii rfnite value of x. 73. Any valiu' of tin* unknown (piantity for which an ('(piatioii is true is called a root of the ecpiation. 74- The process of linding any such root is called solriuf/ the c<|uation. When a root is substituted for the unknown ((uantity in ail ('((nation, th(* ecpiation Ix'couics an idrnfifi/, and the root is said to be rerifed, and the e(|uarion. sitdsfied. lb- A simple ('(piation is one which contains only the Jirst pninr of th(! unknown quantity ; thus. (J.r =1 24, ax + b = r, are simple ('(piations. An e(|nation in which the highest power of the unknown quantity is the second. SIMPLE EQUATIONS. is called an equation of the second degree, or a Qnadratic Equa- tion, thus a:" 4- « = 0, .f* + ax + 6 = are Quadratic equations. 76. The followiiij; AXI(>MS, which are obviously true in Arith- metic, are true also in Aliijebra, and are of use in solving ('(jua- tions : — 1. Things which are e([ual to the same tiling' arc ecjUiil to one another. 2. If equals be added to (ir subtractcnl from ('(juids, the results are equal. 3. If e([uals be multiplied or divide e(pial. 5. The same roots of e([uals are ee solved by reducing it to the foi'ui f/x = b, and dividing botii mem- bers by a. 78. Transposition. — If the equation has no fractions, it is re- duced to this form simply by the transposition of its terms, whicli c(msists in removing all the terms involving the unknown (puin- tity to one side of the ecpuition, and all the remaining terms to the other side. Any term may be transpose^<'lve the e(|uations : — I. .-,(K) _ '.]2X = 12 + 'Wx — H. ;!. r» ( ,r - :}) _ 7 (0 _ „., + ;{ = 04 _ ;« ,« _ ^), •• X(x-U + 1 7 (./•-;{).=: 4 (4.r_!M -(- 4. .-.. 1 (.,. + r^f - ISO = (O,. + 1 ,-^ 4. ;{ ,.,; _ r„_ 6. 7. H. 9. 11. VI. 1:5. 14. 15. 1(). IT. 18. 19. eo. 21. SIMPLE EQUATIONS. {X ■\.\)(x + 2) (a; + 3) — (x + 4) {x + 5) (.c - 3) = 84. (« + 1 ) (.f + 2) (X + ()) — 9a;'^ = x' -\-\ (Ix — 1). 7^- - r, |.r - 1 7 - Pu- + 1H[] = 3j; + 1. .375*- — 1.H7.-) = .12u; + 1.185. a (X — (() + h (X — 6) + x" — (X — a) (x — b^. (IX + b = tax + n + (l)X — ttf — l)\v^ — ex. ah + (tx = he — ex — b'^x — (<: — dfx. {X — a) (X — h) = {a: — a — b)'\ X'^ — ((X — ')X + (■ + ^ {a — h) X — ^ ((• — a)x. 23. 2-1. 2"). (./• — <{)" (x + a — 2b) = {X — bf (x — 2*]^H-(^ + ..4)[. Ix — 8 3.r — 2 _ 1(U- + 3 7./- — 3 _ //-' — 36u;_ .3 _ ?u; ishx — T^d" iU- + Aa ii" ~ a ^2(i' ~ ia" 3 -.125.' .... 3 + .1875j^ + .16 = .083. II EXAMPLES WORKED OUT. n 80. Solution of Problema— When u problem is proposed for solution the unknown nunilu'r is rehited toeertain K'von numbers : these rehitions, expressed in the hm^'ua^'e of Algebra, form an (•<|uati(>ii, by the soluti(m of which the unknown number is found. (See Chap. III.) PROBLEMS. Kx. 1. Kind a number sueh that if 8| be added to the (h)ubh' o«' it. live times the sum isecpud to four times the number diminislied !)y ;{. and the remainder divided by 6. Let tlicn. .r = number recpiired (2x + ^)r, 4.r - ;i 6 Multiply throu^di l)y 12, /. e., L.C.M. of 4 and 6. {2,v + :{f)($() = H.r \2{U+ 2L>r) = H.r - (5. fanspose like term!- 12(M— H.r — — 6 — 2*j; or i. e., 112.r — — X = — 2:{1. ill this case the root iafrdrffonul and He(fatim. Ilx. 2. Find a number sueli tliat if k l)e added to the double of it. live times the sum is e(iual to four times the number diminished t'V ir. and the remainder divided })v 0;;/. Let iIhmi. X = the re(|uired number ilx + Ji) 5 z= Multiply throu^'h bv fim, lim Tr mspose (2u- + k) mm = 4.r — ,'/• = {'A)tii.r + lM)mk. 4x—mmx= ir + 'M))nk = .;(4 — i\Oin), w here the coefficient of .v is 4 — OU/y; >r 4- \\[)iiik 4 — mm mmmt n SIMPLE EQUATIONS. Ex. 3. A smuggler expected to sell a cargo of coiitrahaiul brandy for !|'9t> ; but after lie liad sold 10 gallons, the oflicers seized on«'-tliird of the remainder, and ('onseciuently ho received only $81 for tlu^ lot. Find the nund)er of gallons he had, and the price per giillon. Let iV — no. of gallons. X value of one gallon, and (x— 10) \ — no. of gallons seized, and this was worth $1)9— |H1 ^flH. $UU Thus we tind ^ (x — 10) galhms @ to be worth $18; that i.s, v. (K) iU'-lO) ^=18. Midtiply through l)y 8a*, (,r— 10)J)9 = ri4x. hivide fhi'ough by l>, and lU — 110 := «{./• Transpose, and ')./•— no, .-. .r = ',>t> gallons ; iind .-. price = III — I r= $4..")0 per gallon. EXERCISE XXXI. 1. A boy was told to divide half of a certain Tiumber by 4, and the other half by () oz. more than if it were all silver, liud tiie actual weight of the lump. M .V person bought eggs at litJ cents a d«t/.en ; ha(> cents; at the end of the time he received foO. How many (hiys did lie work ? 14. Sold j,'oods at a uain of 10 jjcr cent ; had tliey cost me .t'-30.'~0 more, the same selling' |)rice would have I'csnltcd in a loss of 1*^^ per cent. Find the cost of the j^ocxls. 1"). The lenj^th of a field is twice its breadth ; another field which is r>() yards lonj^er and 10 yards broader contains (JHOO s(iuan! yards more than the former. Find tlu? si/e of each. 16. A nnml)er consists of two mixed with 80 gallons o, spirits which <-ost 1.") shillings ji gallon, so that by selling the mixture at 12 shillings a gallon there may be a gain of 10 per cent. 15. A wa^ks ;i| miles an hour, and starts ISA minutes ]>efore li ; find H's rate if he overtak<'s A at the ninth milestone. IJ). Two persons, A and M. could do a W(»rk in o days; they worked togcthei- f) days, when A was called otf and H finished it in •• days. in what time could each do the work i 20. A grocer expected to realize £1> IS.v, on a cask of syrup; after selling 10 gallons, he lost a third of the remainder by leakage, and so he received altogether oidy t'H 2.v. Find the luimbei- of gallons he liaH00 Hir size of nrd by in- fil number. fifnllons ui t by sellin<; a gain of 2'1. Find tliree eonsccntive nninbers such that a third of lht> t;realest mav be 'J less than a fifth of the othei- t svo together. '2'-^. Find tliree numbers in the i)ro|)orti(m of 3, I], 0. sneh that when each is diminished by 4, the products of the first and third and of the first and second may l)e. toirether, '.V2 less than the ])rodiict of tiie second and third. ,'l. .\ numl)er has three dii^its. tlie units' figure beini; ".' nu I'e than tile tens' liufure. and :') more than tiie hundreds' liirure. if the order of tlie di^i'its were inverted and <>•,> subtracted, the result would be double t he orii^inal mimlMM*. Kind it. J"i. A person j^oin.i; at the rate r)f a miles an hour, finds himself h hours behind time when he h;is yet c miles to ;;o. Il(»w much must he increase his speed to reach lutme in time^ OH. Thei'c ai'c three consecutive numbers whose product is e(pial to '21 times their sum. Mud them. •JT. A and B can do ;> piece of work in /> d;iys. B and (' in . A number consists of three diirits which are consecutive numl)ers : if 15»s be sul>t racted the remainder will be the number formed by inverting the digits of the given number. Find the number. ;>u. .\ man b(»uglit oriiiigcs at t,i cents a do/.eii ; had he received // more for the same money, they would li;i\e cost him p cents a do/en less ; how many did he l»uy ^ ;>1. Divide 100 into four parts such that if the first be im-reased by '2, the second diminished by '2, the third multiplied by 2, and the fourth divided by '2, the results shall all be equal. yw CHAPTER VII. SPECIAL FOUMS OF MILTII'LICATION. — INVOLUTION. — EVOLlTIOJf. 81. Ccrtuin rvsuWs (fnnnu /(is) in Multiplication arc of ;;rc'at use ill shortening labour. The xtudent must theref(»re memorise these results, and aeciuire ease and rapidity in ai>|)lyin«; them. 82. Wu obtain by actual multiplication : — (x + a) ( X — a) = .*•'■ a' (A) In tlieso results tlm letters rei)resent any algebraic quantities whatever. Thus in (A), wi'ite — n for a. and we j;<'t (.r — (/)■' ~ x' —2nu- 4- a\ 111 (IJ;, write {a + b) for a and (<; + d) for x, and we have : — [(c + ii) + ia + h)\ [(r + , bv ( A ). Ex. 2. rlv - 111- Kv (A). rlrr '),o ri.r) -11 + 11-':= -Ix- — 44.r + 121, Kx. 8. {r\a'' - ili'f = ( ,V'-'»'-' - '-3 h'.""') i¥'') + (¥'') n 4 !• ,/^ 10 II' ,rfr + llh\ by (A). Ex. 4. {X -f- 'i) U - <5) = .r - <5' ^ -f' — ;{<'». by (H). Ex. 5. (7x -t- 11 ) (T.r -11)-= (T.r)- — 1 1'^ =r 4S).r- - 121, by (B). Kx. 6. (a'-\-(th^lr) {,r—ah + fr) — |(\ \{'■') — >iff\ Ex. 7. (?/' + a; — y) {w — x -\- ij) = \ tr + (.r — //)) | tr — ix — i/)] = ir'' — ix—i/)'^ = tr'^—x^—!/^ + 2x[/. Ex, 8. {a + h — r — d) (a — b + c — d) — \ (a — d) + {b — (■)\ x ^^(a — d)-d> — n\ = {a — df — (b - cf. INVOLITIOK. n 83. Tli('s<' foriniiliis nmy he rxiJicsscd in words thus :— \. — T/H'.s(/ii(in'/ ,;• _ 1 ; ^Ir + '.) \ I'../' - 1 ; 4.r - «. // ; - r + :i.r ; 2 - :?./■ + U : u-^ - y« <•• .. + •*• ; .. - •'• ; .. + -'' ; .> - '^*' ; •^•'" - ., ; -■'" :{y. — ' ; '2it.f — :{// ; (i.r — 1 ; <^*' — // ; 1 — % ; % + '3. S. i.< li- — >r ; " ,r'^ + //'' ; f'tf + •'■'•' ; "•'" — f>!/ > •'>■'" — • : ^ — Pf ♦. T"'' + ^ ; J-i- + ^y ; K - V ; ('' + ^^) + '• ; <•'•-//) + =^- Obtain tlu^ products of the followinjj: (piautltics : — 10. ./• - 1, .r + 1 ; .*' - ;{, r + :{ ; 2.v + 4, 2.r — 4 ; 55.r 4- 4, 8.r - 4. 11. a + 2/>, (/ — 2h ; 1 + u\ 1 — .r ; 2.r + /y. 2,r — //. 13. .f - 3, a; + 3 ; J- - 7, ./• + 7 ; .r - 10, .r + 10 ; 1 + 7u-, 1 - Ix. 1 1 i:5. 2x + 3, 2a; - 3 ; Ax - h, 4x + h ; , + 4i, - 4/j 1 1 14. X - \\(i, X + l\fi ; 3.r + .., 3.r - . : 1 - 'r, 1 -|- Jx 15. 2a-!, 2.*+ ! »l,a aO - I, (/6 + 1 ; 1 — irh\ 1 + a'b '^ii Sl'r,( lAI- KOItMS or M!ITII'|,I( AIION, « 10. Wy' + !j\ Wx' - >f ; 1 + //'. 1 - //' ; 'Zii' - f,\ -^d* -\- h* 11) 1 I — '2(if>i\ I 4^ '2afK t> ..ft ■r + !/", .f IH. a' + />, ii' - 1) ; //■ - a\ h" f >f' ; //' +- 1. /''' - 1. 1 15>. (I* — h\ ". 20. d' - h\ //", a.i'"' — % 21. (^A -f- f» + '', ('/ + /') — '• ; (.*• 4- //)' + z\ (./■ + //)'■' - z'. 23. (1 + X)* f !r\ ( I + .<•)•' - if ; (2 + .i-)^ - //\ (2 + .m* + y\ Write down tlir H(|ii;in's of 1. i:{.r — 2^/; l.M-Jr/; 21.n/4-:i.r; V2ah'' - (\a'h(: 3. 1000 + 12 ; 1000 - 2 ; 100 + 1 1 ; 70 + 5 ; .50 — 1. .•id , r"',,'n a''' + SHb" C2.r-,\,i/')(2.r' + -^\jf/'} 4. Mi^'b' + dirb'' ; f/"'' Find tlio i)rodu('t.s of 5. {i;{.r + 2)(13.r-2); (i'+ J„)(^''" Jo)^ 6. (7a-' — U;.rv) (7.r + l<{.r//) ; (fr' + i*-//) {\.r — ^*'//) ; 7. (1000+ 12) (1000- 12); (100 + 11) ( 100 - 1 1 ) ; (70 + 5) (70-r)) ; (50 + 1)(50-1). 8. («"» - ft'^'"') (^/"■■' + ?/■"•") ; (5f/'^7/- - {Wl,"') {rm''b''' + 0f/"7>") ; (77r/" + 886"") (77f/" - HHh""}. 8inii)lify tlu' following «'Xl)r(^s^i<»ns : — 9. 'Ma — 2.r)'+ 2(f/ — 2.r) (f/ + 2.r) + (;U' — (/) (f/ + ;}.r) — (2/r — :}.<•)». 10. {(l + /> + f) (V/ + /* — r) ; (.»• — // + .? ) (.<• — // — z). 1 1. (2(/ + /> — ;}/•) ("la — /> + :V') ; (2.r + // — ;5?) (// + '^z — 2.r). 12. (f/' + ./• + // + z) ( //• — .V + I/ — Z] \ ir + s' + t + a) (s + t — a — r). la. (a — 2/> + '.ir + if) (a — '.]r + 2/> + f/) ; (2A' - .*• + 3// + z) (X — 2A + ay + ^). 14. (;/j + 2/> + a.v — A-) (;;j — 2/> + /» + -U) — {ni + 3.y — k + 2p) (2/> — y/t + 3* + A). INVOLT'TIOX — S(^TAI{i; ' + f») = wut' + //' + '•'» - !(" — '')' + 'l = i 1 (r/ — />)■' + (/-' — '•/' + (/• — itf + « ((ih + /yc -H ('(/) |. 84. Tirsimic forniulu (A), Write (I + l> for II, and we havo (.1' + (I + h)'* — y + 'Iviif + f>) + { (A) and (i) and (ii) in /hrm, i. i\ the «'X|)an- sion will contain only tuv /iind.s of fcrnt.s, vi/. perfect .s(jfi(tns liko .r\ (i'\ li\ (■-, ete. and (JdiiJiIkI /)r<>(fK<-/s, like 2f/;r, 2f>.r, "ilMf, etc. Ue are thus enahled to «reneralise (A), (i), (ii) into one formula e\|)ressinj^ the sciuare of any al^r<'hi'aie polynomial : ui -{- h + (■ -{- .:..)■•'= ((!• + h' + (■- +, etc.) + V^(lf> + 'itlr +, etc., + 'ihc 4- 21x1 + etc., + 'led + etc., all possihle (louhle products). (( ). If the terms contain different letters, it is convenient to writo the .squares first as shown ahovc, and then form tlie douhh'd pro- ducts ; thus : — 2a (h -\- <• -\- etc. all the following,' term.s) + 2b (c -f ri 4- etc. all tho followin*; terms) + ete. Ex. 1. Cl(t^-^M,^Ar)'= [2(1)''+ {•\h)'+ (Ac)''Jr AaV.\h^Ar) + ^h{Ar\ — 4,/'i + \)(;' -I- i('»r'^ + \2(d> 4- Kim- + 2Ahr. Ex. 2. {^(1 + \h 4- \r)' = ^(r + -j^r + ;„r^ -f a i\h + ,\r) 4- V'-'^'- - I"' + tV''' + :.'«'■' + \'fh 4- i(i<- + i\f>': Ex. :{. ((( — 2h 4- 'ir - A(h'' = uir + (— 2f>i' + OV-)' + (— 4r/)' ^if({-2l) + '^r— id) — 4h{:)r — 4d) 4- <5c(— 4r/) _ „■ 4- Ah- 4- !>*-■ + Km/'- — Adh + (Uir — Sad — \2hi- + H>hd — 2A(:d^ Ex. 4. (l—2x + Sx''—Ax'f^{-A.i+i>,r-Hj-^ • • - 4-4,i'--12u"'4-H).^'' 4- 9.r''-24.*'>4-lfi.r" _ 1 _ 4.,- 4- 1 0.r" - titU^" -I- 'i.u* - 24 j"* 4- 1 <>u'.. HO SPKCIAL FORMS OF Ml'LTIl'LirATION. 85. '!''»« fonnulii (C) may thus he expressed in words :— C — The square o/ a pohjuomial is equal to the square of each lerm, Uxjtther in't/i twice each term info all the terms that follow it. EXERCISE XXXIII. Give the f^juares of tlie foUowiiij; : — 1 . a + b -\- c \ a \h — r ; a — h -\- r\ — (i + h + c. 2. \ +h + r; 1 + h — r: \ — f, -\- r ; — \ + f, + a. 8. a — h — c; l — b — r[ 1 + a + 2/> ; \ —a + 2b. 4. 2 + .*-// : ii-u.\-y;4 — i/ + x; 1 + 2.1; - //. 5. J- + 21/ + 82" ; a- — 2/y + 8^ ; 2// — .r + liz ; 1 — 2.r — 2y. (5. 1 + ./• + a-* ; 1 — ;r + .r' ; 1 + 2.r + n" ; 1 — 2.r + ./■'. . 7. 4 + . - 2// ; 5 _ // -Hz- 1 - .r - .r'^ ; a'^ + //- + .f\ H. 1 -I- J-'' + !l.r=' ; 1 - y + '.U-^ ; 2 - // 4- 2// ; 2.r + // - 1. 0. \ -X + 2x'' ; I + .*• - '.\y' ; 2•« + (^y + 'Z^)" 4- (nx — hw 4- <^ — dyf + (a// -- cir — hz 4- r/j?)" + (az — div + by — cxf = {(V + W f r-' + ^7") (M>» + x^ + y + ^''). to. Find witliout actual multiplication tlu; continued product of (d S-h v.) {h + '• — a) (c 4- a — b) {a + b — c). 86. The formulas (A) and (B) are particular cases of the follow- ing more general result, which maybe proved liy multiplication: — {X 4- (i) (X + b}= x:* + X (a 4- b) + ah. ( D) Tf we writ(> 4-« for 6 in this result, we get formula (A) ; if we write —a for b we get formula (B). Ex. 1. (X 4- 1 ) (X + 2) = .r'^ + J- ( 1 4- 2) + 2.1 = .r' 4- '\x 4- 2. Ex. 2. (}\x 4- 1) (3ir 4- 2) =r r^x)' + :i.r( 1 + 2) 4- 2.1 = {).r''-|-!>.r + 2. Ex. :5. (.« -If a){x - b) = x'' 4 .♦• (a — b) — ab. Ex. 4. {X — (t]{x — b) —x^ + x{—a—b)+nb = x'*—x{a-^b)-\-ab. Ex. T), (.*• 4- " 4- b) {X ^- a — ',\b) = {X + fir 4- ix + (I) {b — 3ft) 4- (b) (— 86) = x"* + '2r 4- c(i ) 4- (tin-. ( R) Thus \\v may write down (tt siijlit the proihict of tliree binomial factors wh»»se Jir.st terms an' tin- fniNit., hut the last tvt'niH itilftrctit. Kx. 1 . (.«• + \)(x + 2) (./• + ;{) = «* + :«•'( 1 +-' + •{) + .*•( 1 • '2 + t2 . :{ + :m ) + 1 • '2 • :{ = a;' + x' (1 + 'i + '{)+.*• c-i + <5 + :{) + n = a;=' + iSx' + \\x ■\- «. Ex. 2 . CU- + 4)(:?.r + r. )(:{.;• + fi) — (:i.n=' + rXrf-iA + r> + <',)+ :{./•( U»() + :10 4 '^Mi+ fit) 88. Simihii'ly, we may obtain by multiplication :— (X -f f/i (.r + li) [X + «•) (./■ f it) = X* + .*••' in + h + r + it) + X- [III) + /"• -f- III + '/'/ -1- III- + 1x1) 4- X {ilhr 4- /ycf/ -f fY/nnluit of nil the smmil firms uf tin- fiuturs, II, I), i\ ill-. y . Thi iiijffhient uf thv smmil fi rni is the sum nf . r, I'lr. 'V . Till rnrjfirirhf nf flu fhiril fi riii is flu sum it/ fin prmlii-ts It/ II, f), r, iff., mii/*i/)lii'it tiro unit tirn foijrthir. A". Till' i-oiffiiii'iit 0/ fin ni .rf ti rm is formnl In, tiil>iin/ fin 111 thrvr mill thrvt- toiji'tlur, unit so on. 89. This law is perfecll.v ;rcinr'iil. .nul may be bi-ielly written for 11 factors : — (x4-r/,) (.r4-'/,) (x -[ ii„)^x" ^i\x" ' 4- '•,/•"-- + 'V'"-'' 4- . . 4-'',. (F) where r, = a^ +a^+ vie. . +11,. ; i\ = fiiiu-i-ii.jii, 4 etc. ; c, = n^n/h-\-ii^!^ii -f Ai.'., and c„ = f/,'V/». . . t/,. niNOMIAL TFIKrtKEM. 83 90. Tf a = h = r = (1 =z etc., in fonimlas (D), (E), or «, = a, = t'tc. = f/„ in (F), we j,'«'t (a- + f/i' = u;"' 4- -'/•/• + '/■'. (x + 'O^ = u'" f ^'^ (-^o + .^•'•' (<»'/' I + jr(\a') + a} = X* + 4(10-'' + Vut'u-' + 4«'^- + a*. (.r + a)" = x"- + r,(i.r* + lO^V + \i)(i\f' + ruf*.r + a*. (./• 4 (if — .<•" 4- (}f/j'» 4- Vui-'u'* 4- 'l<)ij '.y nuini riial coejfirient, and iliridiiiij Ijy the iiinnber of that ' ' n from the heijinniiKj of the expansion. Thus in {x 4- a/, tho (•oenicieut '-^<'< =; 4 X IT) And X 1 (Jx U*4-'n' == u"' 4- '. i 4 n -I- 4 a ■'^•* 4- ' -~ ' (I *x^ 4- etc. r)x2i , , 4x:ir) u I i tr.r i f/V* 4- -^- \ -l >\ 4 =: /»' 4- i'/^'" + 2 ! it'x'' 4- '.\'ui\r* 4- :{.")f/\/' 4- :.' 1 '/ V 4- 7r/".r 4- a\ 2"'. .i//*'/" //h' miildle term the .sit me eoelfirient.s recur, and there- fore do not require to be caleulattd mjain. The siijns of the teniui (in iiieliided. Kx. \. {X — a}" =z .r^ — nax* + KWV - \U,iV + r>ii\r — ii". Ex. •.». ('2x — 5)* = (',»./•)* — 4 (2^')=* ( '>} + <; (',>r)- (5>'— 4 CJ^-) (Ti)^+ (Tt)* ~ UU-' — KMU-* 4 «;(M»./-^ - lOUO.r \- WIa. Ex. a. (./-'^/y)'" = r'" - n);/''" {'■III) -I- 171.r" (!>//)•- - IT X r)7.r'"(2//|' 4- etc. = ic"' - :{Hx""y -f 6H4.r"y^ - :T.V,>.r'V' 4 «tc.. \\\ nw.re terma. Ex. 4. qa 4- |/>)' = (^r/)' -I- U(^}» r= ^«» + \a'b 4- 2,- 4, X + 5 ; X + 5, ./• + ; X + fi, X + 7. X -f 5, X — 1 ; X — 4, X — 5 ; .c — 5, ./■ — (} ; x —'0, x — 7. X + a, X — 4 \ X + 4, X — \i ; X + H, j; — (5 ; X + 10, .r — 9, X — (5, .r + •{ ; x—7, X + r> ; .r — lo, r + 4 ; x — J), .r + «. J- + 1 1. .. — 12 ; X + 13, ^- — 1 1 ; X + 10, X — J) ; x + '2(i, x—d. X — 3rt, ^ + ort ; X — 9a, x + 10a ; x — H' 4- 3^, x — 4// ; ^ -f- 7f/, x—H)a. 2x + I, 2x + '3, 'ix— \, :{.r + 7 ; 4^-— ."►. 4.r + 7 ; Tu— . x^ — «, x'^ — 4 ; .i-^ — 2a, .^'^ + 5 ; 2x- -- 3f/, 2x' + 4. Expand the following':, by Art. 90 : — {X + y)' ; (X - yr ; (1 + ir)» ; (1 - x^ ; (a + 2)' ; (a + 3)». (a + 2by' ; (a - 26)" ; (2a - 1)' ; (1 + 2a r' ; v2a + 36)". i,i 4- h)* ■ {a - b)* ; (1 + a)* ; ( 1 - a)^ ; (2 -\- )' ; (5^ — \x)\ 14. Write down the hist three terni.s of (a + 6)" ; (a- + y/" ; (\\a— 1)". 1.'.. (live the mithUi^ terms of {2a + ih)* ; (f^/ - 21))" ; (a + 2)'. H). At compound interest the amount of s^l for 5 yrs. at 4^^ is represented hy (1 + -f^)^ E.xpand this expression, expre.ss the result (k'eimally. and thus find the ex + fr ; a — ft is the cube root of ti^ — fi* — '.\ + (W/'V/- — 4^///' I //. Kx. 1. Tlie cube root (»f f/'' is a ; of —a^ is —refore, that to extract any root of a monomial we i'.rfnicf tin roof of thi' coi'lfiiit nf , oik} (firidi the trjiont of of corli htfvr liy the indr.r of fhe r(n>f . i. c. by 2 fur the s(iuare root, 3 for the cube root, 4 for the fourtii root, etc. Kx. 2. The scpiarc root of J)./-'- — I'i.i;/ + 4y'^ is :ir — '2i/ ; for we see that \).r'- — {■irr. ;i id 4^'' = Ciyr , (hey therefore correspond to (r and //■' in o'' + '■i'fh + h". Also Vixy = 3 {'.i.r) (2//). and tlu' mhtH.s sign allows that our factor is positive and the other nci:ativ(*; thus. :{/• — *-2// is the s«piiire root, as it — ft is the S4pi.»r> root ut a' '2aft 4- //-'. Kx. :{. U//' — }2of> -f- 24f/c — ](U>r + 4fr + \i\r". To find the s^piare root of this (|uantity. notice that U(r. Ifr. ITk- arc the «.i(iiares of :\o. 2/;, Ir' respectively, and that \2m4c). We have then fore only to <»b.scrve the signs of the doubled products to get the jH'opef s(iuai> r»K(t, i\ii - 26 f 4i;. KVOLITION — KXKUrrsKS. 87 +a-) a) ti icin until CJllh'd is tlir > ; (»r •u' f"-/;' imial it of root, >r We l(j to inns ivc ; .t of 11 . -I re •8 of l>l('(] fore Ex. 4. The cub(! root of i -f. a,- + i (1)^ + Vlxf + )\ (1; rZx) (1 + 2.f), HOC ((;j, (2) r' + Hx^ IS 1 + 2^, for it \, for it is =('ix'f ~ :\{ > VJi'J r' + lU-'-;U'+l = (2.. ./•+!)=' terms i^wv us 2.f^iinl 1, and tl middle term. V) ix) +, ..te.+ (l,^ The lirst find tile last le second term shows that-j- is tht EXERCISE XXXV. Find the product of 1. {X — .■)) (X + (5) m U— 7) (.»• + H). (.'• f ) .-)- (.,. , /j and {X + f/)'-' — (.r + -f 1(1 III- Ixl. 4. f/-' -f Ir + r- — he — (I — III) and <( + /> f- .). ,/• (/>— l),r' + (7-/> 4 h./-'-' — ( />— h.r + 1 and <;. (.<•■-• + .<• + \). H. U - !jf + (./• + nr^ 4- :! Mr - //)'■' (./• 4- //i + (r + {>. (./• + // + ^)'' — ix f //)'-'(./•-//)[. // - yy^ - (r _ _y + ^,a _ ( y + „^ ._ '•)■ 10. (.r4-//4-,?) ,,r4-/y4./r) 4- (./•4-,j-4- 11. (. "') (H + ?+ir) — {,r-^x + i/ + rf\ + .'/ + ^r — (.r + //)•' - (// -\-sy' - (.? f 12. {X + 1/ + ^}{x + SJioNv that I) ~Z){x\2 ~ ij) {yJ^. (h) ■ -i'), when ./■'■' + //-' — ..-■- t:{. „,/, f />,. 4_ ,.,/,=• = a'/,^ 4- /A-=' + rV' f ;{„/,,. (./ f /> ^ n (,//, + /,,. 4.,,,, 14. (r/// hxr 4- (/'.r _ f/^-)-'' ^ ,/,;• _ ,.y,-i = {(r -f- Ir f- r-) ix' + if- 4- /• (ax f % 4-<'^)'' 15. ix"- + i/'~ 1)^ + (//-^ +/.• It' + 2 (//•./■ 4- ///.)' = ('/• + U-' - 1 )•' 4- (// -f k' - I )'^ + 2 (.r/y + ^/-^k)' 88 KVOMTION. 1«. (X - y)' + {y -2)* + {2- .1-}' = 2 (J- + y' + /' - .ly - yz-2.r)\ 17. 2(j-y + yz -k-zxy' — 2x''y''z' = i^ + y)' iy + z)' {z-\-.ri'— .r* iy + z}'— y* (^ + ,^)•-— z* {.i'-\-y)\ IH. (a — b)" + {b — ry' + (r — (/)''= -i ()' ; JUWV/. a. 4frVA;'; 35(/V/' ; ^Tu^y^z'; 10(kV>V ; UHd — b)*. \)aKr* 4'x'" t ()2.u^.y'' 21«WV>-* Find IIm' sqnaro roofs of : — 1. n'' + 2./A + //-' ; I — 2.r + .r" ; 1 + 2./'' + .r*. 2. a* + AitV,' 4- 4/;^ ; Aa* - Au' + 1 ; x' — Viah + m,"". 3. 1 - yiv' + Ml* ; 4 — 12a' + l>.r - , — x' + .r\ 4 4. \)a' - 'ZAdV,'' + 1(5// ; .v' -I .m- + "'* ; n' - a" + 1- 4 4 5. .»■•' + 2 + I, ; .r' + 2 + \ : hi''' - '2o"ii* + pi" 6. (/' + 'iiifi + f>' + 2Ar 4- '•'' 4- 2ar ; (,/+/>)'> + 2r (r/ + />) f r'. 7. rt" + %tb — 2ac + // — 2/>/' + f' • ^» + 4y'' + 9^''' — Uy+\2yz-iijn. ■4!.ry + jy>' //'•- Oiv. KVOMTION — KXKRCISKS. 89 8. a* — Aa^h + {\a'(r — 4ah^ + f,i y. I + 4j' + I(u-'^ 4 i*,).,"' 4. !,j.' Find tin' cuhu niots of :_ 13. .r'+ !•.»./•» + 4H.r + 04 ; Hj-' _ 14. (;w_,/=')'^4. (1 —wa^f^ ''' ; »«' - H4r/V> f •J!)4«6^ - y4;j//. ^••' - O^-'* + ir,u.-^ - 2{U--' + mu:' - (U- 4- 1. Ast'crtiiiii the ('•) square roots of 1. da' :.M ~'2 + -•") ffM' + :{,„.-■ 4. MA 4.M//'_ lo//V4-«6c''^_,,.8 0. --•-!:_ 07 + 2,r- 7 +'«_•-'• + .) ,.J ■f + 4./- 4- 8' 1 10. If « 4. ^ = ;C jj1)0W thi a It «' + -=./•».-. y^. 8t'uArt.Ul((i a ) (3). (MIAITKIl VIII. SVMMKTin. 93. An oxprossion is said lo he Symmetrical with respect to ///"(> (»r its U'llers when these call Ite ililerchan<,'esii»n. Thus , ir -i (i() + />'-, f/-' + a'-li + (dr + //' arc svuiniclricai with respect to a and l>, for on sultslituliii;,' f/ fur /< and !> t'nr n. these expressions hci'oine l> + a, Ir -\- tni -\- ir, //' -|- /ra -f /iir -| ti'\ which ditVcr from the jriven cxpressi(tns oid.v in the order of tlieir terms and of their factors. So also .r' 4- //'"•'' ^ i/f> + ''"'• is svm- mefrical with respect to // and f>, for on sul>stitntin;,' // for /> and // for//, it becomes ./•'•' + (r.r f % -t //'•.<•, wiiicli is identical with the ^iven expression. On inferchan^^inLT ./• and //. .r f f/'-.r f //// -t- fr.r iMconies //-' -f .»'\i/ + .ih 4- //■// ; this is iinf the same as the ori.:rinal cNpres- sion, which is. therefore, //o/ symmetrical with respect to .rand//. in the same way it may l»c shown that this expression is not sym- metrical with respect to ■>■ and h. 94. An expressi(»n is symmetrical with respect to f/inc of its letters tr fi, (\ if it remains the same when itito '•, and '• into a. it becomes h"r -f- '■''(/ -f irr, which dilVcrs from the oriirinal expres- sion only in the order of its terms. So also (//—/<)' -f (/' — ')' + {(• — af is symmetrical with respect \o)'■ -f (./• — In {h — i-r 4- (./• — <•] c- — ay is symmet- rical with respect to into r, h into A*, and A* into d. Thus, alt -f /"' 4- <(l 4- '/" find /' httvr.s. Thus It'' -f A" + '■■' — Wiihr is synitnetric with respect to n, h and c. it will llierefore remain syniniclric when written in any (tther form, as lor instance, in the form J (^/ + // + '•) \{ti — if + {It — rf + {<■ - (n"i. 96. 'I'Ik' ii,'e of any pair of letters alters the value of the expressions. It is clear, then, that an expi-c.ssion may he symmetric as deHned in the preceding article (/. r., cyclo-symmetrio with respe, t to ji set of letters, and at the same time not \)i' rninplrtdi/ symmetric. Hilt rrrri/ i-rprrssion irliii-h is <-onij)trtrli/ si/iiiiintrir w'llh reference to a set of letters is inri'ssarilij i-ijrli)-si/niiin trir with i'espc<'t to the .same letters. Thus n'^li + Ire -f- i-'-it is, as we have seen, cyclo- synunetric, but not completely .symmetric ; hut ./'•' + //' + ^' - li///^, wliicli is i-onijiltt(ff/ symmetric, is also cyclo-syminetric. We use symmetry us = eyclo-symmetry in the following i)ages. '!t IMAGE EVALUATION TEST TARGET (MT-3) 1.0 I.I IIM ilM ilM mm IIM (■>0 2.0 1.8 1.25 1.4 1.6 -• 6" ► Photographic Sdences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. 14S80 (716) 872-4503 ^^ fc tP &>. t^- (/j ^ 1)2 SYMMITRY. 97. In an expression wliieli is symmetric with respect to certain letters, any term of a (jroiij) of terms which are formed according to the same law, may l)e called a fi/j)e of tliat group, or simply a type-term. Thus with respect to the set a, 6, r;, ab may be called the type or type-term of t lie jjroducts of every pair of the letters, i. e., of ah, he, ca. So also a'^h is the type of the group a'^li, h'^c, c^a. Again, with respect to four letters, a, h, c, d, ah is the type- term of ah, ac, ad, he, bd, cd ; and ahc, the type-term of tlu> products form(;d by every three of the lett(>rs, that is of aljc, ahd, bed, eda. Such symmetric groups are, for the sake of ])revity, fivuiuently indicated by prefixing the Greek letter ^^ (sigma) to a type-term o! the group ; thus instead of a-\-h + c-\- . . . .k, we write 2>/. With r(!spect to the four letters a, h, c, d, ^abc denotes ahc + ac(f -(- hed + cda. Tue symbol (2") may be read " ///e sum of all such terms as/' Thus Sa = the sum of all such t(!rms as a. Etc., etc. a'h. EXERCISE XXXVII. Write the following in full :— 1, ^a'h ; :£a {a + hf ; l^ah {h — c) ; ^Vhr ; I'a {h + c). 3. l^ia - h) {h - c) , ^'a- (h - c) ; v-a (/> - rf ; :^(.r -a){h- c)\ a. :i><^ i7/-7>c; ^a'h\ ^[a-\-h)\ ^ah ; 2:a'(a—h)[ I'iu^hf,-^ each with resixH't to the letters a, h, c, d. 4. v(a — hf {h - cf ; Si-v - a) (h - ef ; and ^"6* (c - n) —each with respect to the letters a, h, c, and d. Show that the following are synnnetrieal : — 5. a-^ + .v^y + .vy"^ + y'\ with resjx'ct to a' and y. 6. (x -\- yf + {:r — yf, with respect to x and y, and also with resj)ect to x and — //. 7. a" + ?>" + r' — i]ahr, and (V/ — hf + (h — r)^ + (c — (')\ with res()e('t to a, h, and r. S. U- + // + ^f — (.r' + .v' -f- 4-'). and {x — yf + (y — 2f-\-{z-x)\ each with respect to x, y, and z. TYPE-TERMS — EXERCISES. 08 ,-\« 9. a (b - cf + b(c - af + c{a - f-)\ with respect to a, b, and r. 10. (fn- + b(If + (b(;-a(h'\ with respect to a' and b\ and also with respect to exact divisors of a^ + b^ — r'^ + '.\- ^vpc-tcnns are a:% u:% ^ijz. Hy Kx. 2 wo know tl.at tlir ...xprossion contains x\ ^x^u + //•' + :5/A~ (~~ + .V) + // (.^^y^-), in whieli // is nuini'rical. By putting .r = .y = ^ = i w(> jr,>t U + 1 + 1)^ = 1 + 1 + 1 +8,1 + i, + :5,1+1) + :j,,^1)^,^^,j. , ^^^^^ Ex. 4 . Simplify (n + h^v)- ^^u^h-n' -^(h^r-ay + (i■^.a-h)\ Tlie type-terms are a' and a6. We -et f roni the several brackets (i"" + a' + r/- + a' = ia^ '"i fju-tor r('(|nired is ' ({•' + b- + c- — ab — be - ea. Ex. 5 . Expand (a + b-\- ,> + d)\ From Ex. r, we get «" + 3f/^ (6 + c + (1) + 66cy/, and &=> + -^b'^ ((' + (l^a) + 6cda c' + He- {(t + rt + b) + (k/(/6 ^'+ 8r/-(^/ + 6 + r) + 6a6f;. Ex. 6 . Obtain the simplest form of ((f + b + r.){x + y + 2) + (a+b-c)i,x + y-z)^(a-b + r)(.Y-y+z) + (—a+b + i-){—x-\-y + 2). We liave to calculate only the type-terms ax, ay, az. We get ax from each of the four products ; /. e. , 4^/.^. ay -\-ay — ay — ay from the four products ; /. e. , 2ay - 2ay = 0. az-az + az-az " " " ^^ f. e, 2a2 ~2az = 0. Hence the given expression = iax + iby + 4.cz. SYMMETRY. EXERCISE XXXVIII. Simplify the following : — 1. {n-h— cy + (6 _ - af + (c — a — hf a. a (b — c) +b{c — u) + e (d — b). ;j. (X + y + zf — x(x + i/ — z) — // (// + z — x) —ziz + x — y). 4. (« - 6 - 2cy + (b-c— 2ay + (c — a- 2b)\ 5. {X - yy + (y- zy + (z - xy. 6. (a + 26 — 3c)' + (6 -f- 2c — 3^/)^ + (c + 2« - 3b y. 7. (a + 6 + c + rO' + (n—b—r + dy + (// — 6 + c—dy + (a + b—c—dy. 8. {ax-^by-^czf + (ax + by—czy -f (by + rz—axy+ Uv + nx—byy. 9. (c/. + 6 + '")' + (« + 6 — '•)' + (6 + c — rtj^ + (c + (/ — 6)'. 10. a(ffl + 6 + c— r/)f6(6 + c + r/— a)+c(c + rf + rt— 6)+(?('^^ + rt+6— c). 11. (a + by + (a + cy + (a + dy + (b + ey + (b + dy + , c + dy. 12. (a' — be) (b — c) + (6' — en) (c — a) + (c' — rf6) (ci — 6). 13. rt (b + ey + b(c + af + e (a + by-c'' (a + h) —¥ (c + a) -rt" (b + c). 14. {ab + &c + C6')' — («''6' + ^'c" + e'^a') - (a'^be + 6V'rt + Cab). 15. (aj + y + ^-j* + {X + y-zy + {y + z — xy + {z + x — y)\ 16. Show that (a" -h 2/>cj'' + (6* + 2c«)'' + (c' + ^ab)^ — 3(aH 26c) {b''+ 2ca) (c*+ 2«6) = (a' + 6=" + c" - 3a6c)'. 17. If 2s = a + 6 4- ^, show that (.9 — a) (s — b) + {s — b) (s — c) + {s — e) (s - a) = s» — ^ {a'' + 6» + c"). 18. If p = 6 4- c + ff — ff, c = *', where 2* = a + 6 + c. CriAPTEH IX. RESOLUTION INTO FACTORS. 99. In Multiplication wo have to determine the product when the factors are given ; the converse process of determining the fac- tors when the product is given, is called Resolution into Factors. Some of the methods of doing this will now be illustrated. 100. Monomial Factors.— When every term of an expression contanis the same monomial factor, this may be separated by dim- aion. (See Art. 04.) Ex. 1 . a--ah^a{a-b); crb + ah- + a = a(nb + b" + l). Ex. 2 . (cx' + frJrx' - a'bx\ Here drx' is contained in every term we therefore divide by this factor and write the quotient in brackets as the other factor ; thus, Ex. 3 . ah''<' - uHx' + r' =z c(nb'~a''b + c). Ex. 4 . 6,rf/ - 9.i'->-= + 12^.^3 ^ 3^^ ^^^ __ 3^^ ^ ^^,^ Ex. 5 . a\f' - n\r" = a^x* (1 — a'^'^xf-*)^ if y^ > 4. Ex. 6 . 6 (a + )) x^y - 18 («, ^ b) xif = %xy {a + b) (x ^ %). EXERCISE XXXIX. Resolve into factors : — 1. a^ + a ; a^ + at> ; a-a^ ; a^-a^ ; a^b + ab"- ; x^y-xy* ; «-«.&. ; rt*6* + (lb ; 77i'p — m''p\ 2. ab-ac-be-h7',b^~ba-c^-ca- ab^-abc- a + «^ + «» • rt^ - a' + a* ; a'b - at>' + a' ■ x + x' - a''. 3. ax^ - bx^ ■ a^f - ab^ ; xy - xy^ + xhj ■ ab - aVj + a«6' • •^IZ-f + y; d'b' + ¥(r - a'b'c\ 4. a - a^xy ; b'^xy - c^xy + a'xY ; a^b'cy + a'b^y + ab\'Y ! 08 liESOLT'TION INTO FAf'TOUS. 5. Chix'' + Qa-x - n^v-- ; (rV.r + (rV.r' ; 7 - \4(fb + 2lab - h*(f. 7. r)x' — lO.r'Vr' + \r)a\v'' ; !} — JWV" + Ua'U-' ; — /A/. - 8. (a — />) m — id — />) // ; 5 {x + ij) <■■ — 10 (./+//) f/" ; '' + 11/,, /. , ;i ■ •>/.. i ,2 f.;:'«: [ft ])'(({ — b)" — ,n.,3n a-"^" (2f/ — /y) + j'"y (LV> — r) — .t-"//* (2r' — a). 101. By the same principle an expression of four or more terms may soTiu'tiiiics ])v resolved by takiiij^ it in {)arts, (Art. 48. ) Tims, Ex. 1 . Just as Qx + Qi/ + 4x + -it/ = (« + 4) x + {6 + 4) y = (6 + 4) {X + y). So ax + — be. 7. 8(M- + ay + 3bx + 6y. 8. a — be — ax + bcx. 9. /' + g/" — 'dps — g*. 14. 1 — a — 6 + rt6. 15. G.r^ + iixy — 2ax — ay. 16. a^ + y'' _ ,i^yn ^ ,3^ 38. (t — b + r — (lb ~ br + b^. 1. 34. 2ai)q - 8f//y/- ;2y;Y.,; _|. ^hfx. 85. ( 1 + X') (1 _ .,.«^ ^,j_ ^,^.. _ ^j^, ^ ^^ 86. 2y>"/- - 4/;".v" + 64^ - 8ry"/-". 87. ad + (lb + (je - ae + bf - ,f + df - ,.,/ _ jg_ 38. 1 - « + 6 + y> + (/ - + /y> - aq + 6^. 102. The expression for the square of a binomial (Chap IV form A) helps us to find the factors of a trinomial which is a perfect square. We liave merely to connect the square roots of the two squares in the trinomial by the siyn of the other term and the result will be one of the e(,ual factors. In a similar way we may find the equal factors of any polynomial which is a perfect square (Art. 85). E.x. \ . a^~ Aab +4i» ^ (a - 2b f = (a - 2b) (a - 26). E.\. 2 . SCya'b' - 24abc + U-' = {Gab - 2c) [Qab - 2c). Here tlie scpiare root of the first square is CuiIk of th<> second 2r, and the sign of tlio other term {-24abc) is minn.s. .-.Connect eab and 2c by the sign -, and we have one of the equal factors (6a6 — 2c). Ex. 3 . 4 - 16/;^='//' + Wn^n* ={2- 4nrn') (2 - 4m'n'). Ex. 4 . {X + y + 2f ~ r^a + b) (x + // + ^) + 9 (a + bf = \i^' + U + z)-'i (a + b)\\u + y ^2)-'i (a ^ /,^}. Ex. 5 . Aa* + Ab* + \c* — 8aW — 8^/ V + 86V. In this exami)le we see three squares, and tliriH' double products.- it is, therefore, probably the square of a trinomial. Take the square 100 I{i:S0|,(TI()N INTO FACTOFtS. r(H»t of ('a(;li sV .-. 6- and c- '• ''alike. Hence we have : — {2//' — '2r') (*,V/'^ — '2/r — 2e-), Kx. fi . 4a'' — 12^/// + 2Ah — Wui + <>//- + Hi. The w/iians ai'e 4^/'', 96'^ and l<5 ; /. the fmiis ai'e t2^/. '.U/. and 4. 'iiid the expi'ession = (:2a — '.i(j — 4) C2a — ',ifj — 4). EXERCISE XLI. Resolve into factors : — 1. a- — 'lab + b" ; 1 - 'Iv + .v" ; 4 — S// + 4y' ; 9 - fi.r + .r'-' ; .7-' — \Uij + //'- ; 1 — 'in + if. 2. y — 4.V}/ + Aif ; 1 — (•).>' + !).<•- \ a' + '2a"b- + b* \ 1 — 'i.r + j* ; 4.y^-4//- + 1. 3. 4r/' - I'^f/^y + i>//-' ; '/■' + l'~''//> + 'M]b' ; 1 - '.>./•" + .r'\ 4. i>r/' + \'2a + 4 : K),/-'//- + '.M.r//,?- + 9.?^ ; 1 — Tw/ + T)'//. T). ^/- + r.V^/yr + //-r- : 9,r'- — i'ui/ + if : 1<»'/^ — '-Mrr'.r' -|- 9.r''. (i. 1 + 4//- + Af ; ',».■) - 40.r + 1 <).*''■' : '/Vr + 4r/'V/'r- -^Ab'v'. :. 49.f'"' - 42.r='//i''- + 9y'-i-'' ; 9(/-.r- — (u/.r + 1 ; , - '/ + y'. H. .v^t/ — xhf + • -1,/^ + l(5//-'r-' — \(C-br ; ^^/^ — \(rlrir + V'^C- 4 !). ia — /o'-' + '^ [a'- — ?/■') + (^/ + b)" : .<■- — 2.r iz — i/) + (z —yY. li). (/>+'/ + '■)'■' — -•»• (y' + 7 + /•) + -s'" ; f'.r — 'Ziu'if + //V; ^far,.„-s. K.v- 1 . J--' - 4„= = I.,. + o,„ ( ,. _ 3„ , . 1^., _ ,^^^, . =<3.i-+*n, ■.,■-*„; ,-,,,..-,1 +..,.,, I--.,, Ex. 3 . ^n„Hr -. 4.,.v« = ,4,,,, ^ ,,,^, ,4,,,, _ ,,,.,^^ • • ' (if 102 HESOM'TION' INTO FAPTORi^. Kx. '.] . {'ill + It)' — (■' = \rlil + b) + <:\ \Cl(i + b) ~r\ . = i^ia + 6 + c) c:ia + b — c). Ex. 4 . {\ -x + iv-'y - (1 + x + '.)j--')-' = (2 + 4ur') (- 20- — 2x^). IhMv tli(! roots uro (1 — ^' + x'') and (.1 + x + !).*•'•';. (Vtiun'ctiiij,' Ihcsc liy llic sij^ti +, and rcdiiL'iiii;, \v«> havo 2 4-4x'' ; and coiiiicctinj,' llicin by Ww sij^n — , and reducing;, we liiiv«» - 2.r - 2^'^ Kx. 5 . {.(■•' + iff — (.<•- - !j"f — (./•■- + y- — /-y- : Take I lie first two t<)<;('tli('r, thus : — {x' + ff - (.<■' - !/■)'' = (x' + if + x' — f) {x"" + if - ,r- -f //'■') = {"ix') (2f) = 4x'f ; now take this n'siilt witli tlio third s(iiiaro in th(; jjfivcn expression, thus : — 4x'f - (./•- + f - z' ) - = ] 2.r// + ( .r + //-' - 2')\ \ 2xi/ - {x^' + f — z' ) \ = \ 2XIJ + x' + f - 2' \ \ 2x1/ — X- - if + z' \ . TIjo first of tlieso factors = (,r+ //)" — ^* = ix + i/ + z) {x+ ij — z) '* second " " = z-—{x — ijf =. {z-\-x— ij){z — x+ij); liencc C.K^ fj;iven expression = (X -\- jj + Z) (X + n — Z) (Z + X — //) (Z — X -f- //). Ex. () . AOTui* — SO// =: r> (HW — 1(5//) = 5(9a'' + 46*) {%(f — 4//) = T) (<)^/'^ + 4//) C^a + 2b) (Jia ~ 26), since 9a'' — 4// = V^(t + 2b) (^(t — 2b). • This is an example showing that sometimes we luive lo separate a monomial factor, in order to apply the formula. Ex. 7 . Uix' — Six"" — 10.*;^ + 81 : This expression * = x^ ( 1 Gx* - 81 ) — (Uix* - 81) = ylVu* — 81 ) {x^ — 1) = i-ix- — 9) (4x' + 9) (.i;« — 1) = {2x + ;}) {2x — 3) (4^-- + {)) {x^ — 1). Tliis exanipkf shows that sometimes we must group certain terms to get the difference of two squares. EXKHCISKS IS FA(T(M{IX(}. 103 EXERCISE XLII. (fi.) Rcsolvo into fiictoi-s :— .1. 99^-1; lor_i; .!,v_,p ,^._^ ..,_ ,„._„,. 2. ./-^ - y^ ; .,.^ _ H5 ; 4./- - Uf ; 4./-^ - I ; \ - U>/^ ■ U,,'^ _ .^.. 4. 9^- - f ; ,,'^ _ 1,5//^ ; 49m^ - ^^ Ml _ ..-^ ; aV/^ _ c,,.^^ r.. 4^^ - y' ; 1 - 4^-'' ; 4.r'' - 1 ; 1(5.,- _ ^^ ; 9,,v - 1. «. a- - i-y- ; a:«y« - 4 ; |<,^ - 4 ; a^b\/' - ^- ; i _ 4y;,'^.^^-. r. ir,^YV--60«W; mr^-48y^ 3«^6 - 4«a6- 5 - «Ua W 1. (« + ^*)' - C ; 4 Or + !/y , .» ; .,^ _ ,y + ^,. . 4 _ „, + ^,.^ 2. (y. + 2y)» - r' ; lOu:^ - (a + 3b)^ ; 4m= - ( o - y/^ • 3. ]_(/._ ,.y. ; (,^ + 6 + .)■-■ _ ,:-^ ; 9 - , 1 _.,,'^ ; (/>_.,'^_„,_^,.. 4. 12 (a^ - bey - 8 (h'^ - ,,.,'^ ; (a - M,f - 1 ; 1 _ (., _ „ + ^>. . 5. (a _ 2hy - (2« _ m + 4c)'' ; 1 _ cla - u' + 6^)» • (3 + 7^)''-r5^_4)». i^'' - U' - 2''f - A,f2\ ^ 7. (.r^ - //^ + .^ _ ,,^, _ ,0,^ ._ o^,, ,. . ^,. _ ^, ^ ^ _ _ * ~ // — z — 2ijz. a;^-a.'^ + 2A--L 1. .y ^ - -^ ^ , 9. ^^ + 2^,- + ,,^ _ y. -2yz-z^, a"- - 2ar - }? + 2/W + .^ _ ./^ • 10. a^ - 2ab + b^-^^- 2,'!/ -i/^;a* + o.,3 ^ ,^. _ /^. ^ ,^^3 _ f,. 104 RKSOLITIOX INTO FACTORS. hi 11. a^b — ha? + (I'x — .f" ; a'' — I? - r^ + d' - 2 {ad — he) ; (t'U' + 2ahr - (M - h' _ /;-' + i)<» . a' + 2(w -V + (^ - 2hd - (P ; a' + 2■■• — (./•' + y-)- — x'\f = (X- + y- + xy) (,<•"' + y' — xy). Ex. 2. .<•" - \')x' + 9 ^ .*■" - (Sx' + 9 — ^x"" = (X- — ',\ -r ;}.<•) (.*•' - :5 — ;3,r). In this ease we had merely to sepai'ate the middle term — 15j;' into two i)arts, viz., — (J.r- — 9,r■^ etc. Ex. 8. x' - :.r/ + U' = •'■' - "T-'V + .'/' + 5>'''7/ - '^I'lf = x' + 2.r// + .V ' — t>.<7/ = {.r + if)' - i».rV/- EXAMPLES— a;* + ic'y" + y*. 105 = 4x' + 1 2x^}j- + 9//*-49j''^y-' = (2a;* + 3y^)--4$>.r-y =^ ( 3.f^ + 3//- + 7u\t/) ( 3,r- + '\f/''~7,ri/). Ex 5 Or tlio last Ex. may hr resolved by xepamthw Ihe muldle term -37..^ into - ,o,y _ o,,.^. . ',„.^ KxpriLion = 4;r* - 1 2.vhf + 9//^ - 25,.^^* = < o,. _ 3^.^. _ ^^. = {2x' ~ 'Sf + ru-i/) {2x' - 3y-^ - 5a-y). Ex. 6. X* + Aif = .r" + 4//" -I- 4.r^y^ _ 4.^^^ - <•*'■"' + 2y- + L>.r//) (.r- + 2f/- — 2.ri/). Ex. 7. a' + h' + Ut + /»■' r== a* + // ^ ^,..y^. _^ ^^^ ^ ^^^, _ ^^,^^, = Ur + /r + «^,) (a^ + /,-• ._ ,,^,, + j^^, ^ ^^ .. _ ^^/^, ,^^^ ^ ^^ .., ^ " I". + ^ + f ) («; + ^— ''^) + \ra terms of tlie re- fimred faetors ; /. v. fake roots of u- and y\ ris. .r and //. 2". P>om twice the product of these roots take the middh. term (x y-), and the stpiare roots of the result will be the t/iini t(>rnis of the required factors ; /. e. + xy and - xy. EXERCISE XLIII. Resolve into factors :— 1. ^x' + Tyr-y- + //♦. 2. \(Sa* - \'^a'(r + b\ 3. 9)n* — \\)a%-' + b\ 4. 2r)/«* — 9w'//* + Ifiz/A 7. .r' + 2Tui'- + 625. 8. f/' + a- + . 4 ^ 16 9. a-* — tlr'-y- 4- y\ 5. .r^ + .r^ + 1. 6. .7^ + t.r-' + 16. 1 6 *> 'ifi 10. .r^ + V + ^*". 9 ^ SI 11. <1* — Vury- 4- /y*. 12. in* + n* - I8m*n\ IOC RESOLUTION INTO FACTORS. 13. 81a* + nr/Vr + h*. 14. Ifia* + I'" -'2Ha''b\ 15. 35// — 41//V + 16?'. 16. H\x* — ;U.r7/ + .y*. 17. 4x* — 8a;' + 1. 18. Ix* + y*. 19. X* -\- 4aY- 20. 4a« + y*- ^Y- 21. .r" + x*i/* + 1/ ; ;r*' + x* + 1. 22. a*;r« + cr^a;* + 1 ; a" + 4//^ 23. (« + h)' - 7c* (a + hf + <■*; 1 + 4a;*. 24. 16a;* + 4 (y - ^-j* - 9a;* (// - zf ; 1 + 2"* + 252*. 25. 1 + 4a'' ; a* + 81&* - 63aV>*. 26. 1 + a* + (1 + a)* ; a-* + ^ + ^ • 27. a;* + A ; «* + A ? ('' + ^^ + (" " '')* + (^' - &")' 4y* a* 1 28. r-* + 4(a + 6)*; -^ + 7-,+ 1 9 3 1 a" tifl ' a'fj a * a»6' ■•■ 6*' 105. Application of Formula (D).— (.r + (f) (a- + ?>)== a-* + ( f . pio into factors : — 1. X- + ^rx + 0. 2. x" + 7.r +12. 3. x" + 'dx + 20. 4. a;* + KU- + 24. 5. a;" + 1 8.r + 72. 6. x' + \\)x + 84. 7. x" + 17.r + 00. 8. x" + 2 ."■>.<• + too. 9. a-- + ll.r + 24.. 10. .r- + '^x + 2. 11. x' + 7.r- + 12. 12. a' + 22^/ + 105. i;j. .<■" + 14a-' + 49. 14. a;" + Sru-^ + 150. 15. a-^y* + 17.ry + 72. 16. a'6V + 49(/6c + 180. IT. 18. 19. 20. 21. 22. 2;i. 24. 25. 20. 27. 28. 29. 30. 31. 32. /;/" + 40;//=' + 399. a\ir + M)/ + 361. x" - 40.r + 400. f _ 100// + 2500. ^•••_ -JO//^ + 125. m' — 22 m« + 85/^*. EXERCISK.S IX THIXOMIALS. 33. ,,.'' ~ 26x1/ + M59/. 34. X* - JU'^y-i + 20^4 35. a' — 2\)(tf} + 546*. 30. 12-7X + .V'. 109 37. im — ni ah + «;VA 88. a" - iiki + ;}75. 39. X* - ;f.t'^ + 1. 40. x" - 85.r=' 4-216. 41. nx' ~ 'M)x'' + 4Sx. 43. ax"" — lUix + Wa. 1. a* o r/- _ o J. a-" + f/ — (). 3. .»'^_,,;_0. 4. x'-rSx-iS. 5. a.-* 4- 5,r — 84. C y' + 7^ - 60. 7. rf- + 13^^ - 140. 8. r?* -f 13^; 6 - 'm)b\ 9. .r« +.v^ 132. 10. a;^_8.f— 30. 11. 2/' - 5ay - •'>0a«. 12. a'^b' - ^ab - 4. 13. 3aV — 39^/^- — 43. 14. a" - \r)a' — 100. 15. rr/A,'^ f 9.//yr - 23. 16. f?V/ - 37r/'V;- _ 90. 43. a-^- 169.1^' + 3600. 44. x^" — Wx" + 359. 45. [a + 6)'^ — 7(a + 6j + 13. 46. 143 - 34(/.t; + irx\ 47. 1 - 59.fV^ + 408.r•y^ 48. (/•' — 54r/ 6 + 7396'''. 49. «V.- 20«6.r=' + 756»^*. 50. nr — mm + 361. 51. p- — Upq + 729 — r/V>\ . 24. 204 - 5m - ;;/'. 25. 3r/^y + 36r//;.r// — ^Wx'ij. 26. 4.i'-' + 34.^.- + 35. 27. 9.f^ + mx + 35. 28. Xx'tf - ^x-'dz'' - 42z\ 29. 49./-- 112^/6 + 64/A 30. 4 //-./• — I2bx 1/ + r)xiji 31. m.r*//'- — 16(l,,Y^ -(- l()0.rV. 33. a* — uruiV/' — 2(H)b\ 1.10 RESOLUTION INTO FACTORS. 33. \2lx* — 2mx''y f 109//'. 37. i»yrt' - ^ab - 566». 34. 9 {a - h)* - 3r/' {(I - h)"-VZc\ ;JM. '* , + ^* - 03. x'^ X 35. 64a'"' — 8«"6" — '26''"'. 36. \x' + i^-' - 42. 39. ^^\ — ^- 380. 40. "ihx' - no.r'-y - Gr)ly». 106. In the examples of the preceding article the trinomial to be resolved is formed from factors of the form {)nx + « - m -. Hence ttXs.^,:; ' •"^' '"" '" '"•' '"■■«- "'"»' "^ »"'- "e -. = 38.*;- - r)3.r + JJ.U- - 65 = 4^;(7.r- 18) +5(7j;-13) = (7^--13)(4^- + r,). Ex»,'5 . 4 fa; + 3)* — 37a-* (x 4- '^\^ m q^-* . f k for (a; + o.. , ,. ., ' ,-^ + ^l + ^*' ■ ^"f" f'onvenieuee write Hence we have the expression ^ ^ ' = 4A-*-36Au-^- 7.^^+9^.4 , = 4A-(/--9a-*)-a--'(A-9.r*) = (A: - 9a-^) (4A- - a;^). Restoring the vahie of A- and r(^solvin^. by formula (B) we get ^(3-^"+l)(l-^')(3a; + 4jU- + 4). m RESOLUTION INTO FACTOKS. EXERCISE XLV. 1 . 4j;' + 8r + 8. 2. 4J-' + UU- + 8. 3. \nx' + 27a; + 12. 4. 0.k'^ + 7a; + 2. 5. 6a;'' + 2;}a- + 20. 6. Sx"" + Mxy + 21 y\ 7. 4a» + Via + 9. ■8. 7 + lOm + 3m'. •9. 4a;' + 23a' + 15. 10. Sa;" + 2.S.i; + 14. 11. 12a;^-a--6. 12. 12;i;' +a;-6. 13. 12a;'' + .y — 35. 14. 12a;' -a— 35, 15. 6x^ +x — 2. 16. 10a;' -17a- + 3. 17. 15.«' + l4.a;-8. 18. 7a;' - 50.a; + 7. 1^. 15a;' — 19a;y- 10y\ 20. a* — 2r/' — 323. 21. 6w'-17w -380. 22. 6a' + 22a — 380. 23. 12.r' + 13.r// - 35//'. ?4. 15 -27a;- 132a;'. 25. 20a* + .r'-l. 26. 15a' + 224a — 15. 27. 24a-' + 22.r — 21. 28. 18- ■ 'i-^!/ + 5y'. 29. 24a' ._ 29a// - 4?>'. 30. 24- ■ 37// - 72//'. 31. 28a-« + 115a'- 125 82. 56.r' — 3()a-// — 20//'. 33. 56a' — 68a-// + 20//-. 34. 56a;' + 36a-// - 20//'. 35. 56a' — 67a6 + 20//'. 86. 56//' — 558// — 20. 37. 72y' — 19yz — 40^'. 38. 36//' — 33a//- 15a'. 39. 56a^ — 276a'//- - 20// 40. 56a' — 229a// + 20/y'. 41. 39a' -16a-.// -48//'. 42. 39a-' -20a-// -11//', 43. 39a' + 13a-// — 26 44. 60a-'' — 31a-// — 104//', 45. 1 — ' >a;' — 143a;^ 46. a="- ■ 2//"a"- 143&'^", 47. r^a-" — 5 !./•='- 231. 48. 17a;' + 288.r-17. i'<)I.VN(,MlAI.S_EXA.IPLKS WOKKED OIT. ^g m cuol, «,.„„„. K«,- s,„.h »i„.,,. ,.„„,., „,„,'^ ,, f ^;" •"■'"- ''/'/'"/■.• ir, either ,■„.„, tluMiR,). ip in wWeh it L f "' ' "'"'" '««^„/„ «,-, u,„,u,., Tlius,- ' '" "•'"'■■ "■■""■" ""«'• E.V. 1. Wl,..n .■ + ,„.+ , i, ,„„„i„,j^.j ,^ „^.^^ wu„,u,tl„.^, ?> X (X' + HA' + 1 ) ■,.'./ X ix' + f/.r +1). i'V + a) X (x' + (tj:) ^'''^ (^ + a)xb. , . Ex. 3. If ..^ + ,,, + ^ i, „,,,^.pji^^^ ^^ ^. ^ ^^ ^^ ^^^^^^^ ^^^^^^^ (•« + 6) X (a-^ + b). and _ _ _ Now,, i" Ex. ,. The gr,.u„ (^'+™+,, j, ,„, ,„, „„,., ,^„^, ^,^^ ^ ^^^ " Ex. 3. " " Ex. 3. " " if 4- (I (x -f- 6.) a n a i t a respect to either letter. Take/ Then theCZsit "" "'' ^ I - 4ftir + 2a6 = _ 26 (2a; - a) + 2x^- ax= a',(2x-a) ••• the factors are (2x -.a)(x-. 26). 114 RESOLUTION INTO FACTORS. l...w..r, thc.r,.f„r. ,„k,. .,.n„ri,w; ,„„.,,"'"'" '' ""•"" '" '"" "'« Ex. .3 . mx^ — (m 4- //> >•» . (,, 4- ./. + a _ -.„^.^ _,,_,„. . . flu. r,..n'n'^.tonn.s grouped. Hcneo tlic; factors are (m.^- _ u) (u'^ - .r _ „). power of « occur.,. IIcuco wetm., ' ''"'•" ""'>' """ And till! factors are (x _ «) (^.. + ^^. _ 3^,^ -.her ,1,.. „-„.„„» ,„■■ ,1,0 4.,«;l. iv^'i, ™:f """'• "^ •""•■ «'•"""• } ^ -h ( w + ^ ),r + J6 f . . , ivni. terms grouped The bracketed quantity is at oiu-e seen to Imve the factors X + 2 and x + b; hence the given quantity = (x-u) Ix^ + (I + o),. + 2?4 =:(^-a) K.r + 6)(^-+ 2)f = {x- a) {x + 6) (a; + 2). EXERCISE XLVI. Resolve the following into factors : 1. a-='-(2 + a)a,-» + (2r/-l)a: + «. 2. x^ - f^ + jD) a;^ + (,^ + ,,^,) ^. _ ^^^ 3. may^ + (m6 - na) if ~ imc + nh) y + nc. 4. (26-c)a:'' + (26c-46^)a;-6c + 26^ I / ( . H. U. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. fcXKHCISKS IV POLYNOMIALS. 115 )i.t* — in + ii),!-'' — HI — (i).r -\- (I. ( m V 1 ) {Itx i- ((II) l>'\v'' — ( // 4- I ) ( tnhx + a) + a'') II — d^b. ./•' — Ut—h) ./•■-■ - (ah 4- 2/>') .<■ 4- 2«6'. .*•' — ( />'• + ;?7' I .'• + 2//-7 — 2v'. ./"' — ( — ;{) .r- — (f//; +,;?(/ — \\h\ X — Wnh. X* — (a — b-\-\) X- — (lib + b ~ (I) X + lib. 4x^ — 2 (a — 4) x' — 2 ( 2(/ + ',i) x + 3a. j>f -{p- q) If + (/>_,,,,/ + q, mpx^ + (mq — lip) x* — (inr -f ii(j)x + nr. tnax^ — ( in<- + mi) x' — (mb — tic) x + nh. Spx^ — ( pi: 4- 37) ./;" — ( pb — qc) x + qb. ax* — {up — b)x* + (aq — bp — c) a;" 4- {bq + cp)x — eg. 2x^ 4- (2// 4- 3c'j .x'" 4- (Sac — ib)x— 66c. 2j;' — (4r? — 3c'j .r'' 4- (()i — 6ac) x + 9bc. 2ap^ + (3« — 2b) p'q + {a — '6b) pq'' — bq'. dap" — (a + 'Sb)p''q - (2a — b)/}q'' 4- 2bq\ (icx^ 4- (be 4- ad) x'' + (bd 4- ac) x + be. 2ac,r' 4- ( 2bc — ar/) .r' — ( 3af; 4- ?>«'/) x — 'Mx-. 21 if - 9 (ah 4- iicj y^ 4- -^^^f + 3/> (a^' — He - 5rt) y 4-5a6*c, 108. A polynomial of the second degree which lias rational factors may he resolved by takhiij its terms in triiiinnial ij roups. Consider how such a polynomial is formed from its factors ; — Ex. 1. 2.* 4- y 4- 7^ X 4- 2// 4- 3^ 2x'' + xy + 7x2 + 4x1/ + 2y' -r Uyz Qx2 4- Sy2 + 2U* 2a^ + 5xy + 2y^ + \9xz + 17yz 4- 21«'. lu; HKSOMTION' INTO l< A< T(U{rt. 'I'liirt (•(•inplclc pfndncl is coniposi'd of tlir folic twiiij,' purls : — 2.r -h 5./7/ 4- 'J/ = ('i.r + //)(.'• + ••2// » . . . (1) ^\.l^^ -{■ T.*^= l;ir^=: ^'.r-a^ + .*-7^ .... (li) 2l2' = '.\2'!z (4) lict lis sec liow lli(> orijfiniil factors can l)c ;.c<'l from these jjarls, (1) (Jives directly 2j' + // and ' X + "Ijl i\\v first tint terms of llu) nHpiired factors, leaving; lix and 7^, to be fonnd. (2) + (4) (lives + ,r-l2 wliicli is clearlv formed from the factors and 2,r + 12 the binomials got by droppfnif the //-terms of the original qnantity. Hence, resolving 2x^ + \iix2 + 2\2\ we get i^.z and 7s, the required third terms. Hence, the factors of the given expression are 2x + y ■{- 72 and x + 2y + 'iz. Observe: — These t(M'ms ^2 and T? may also ))e obtained frorn (;{) + (4) that is, from the factors y + Ti- and 2// + '^z the binomials got by dropping the .r-terms of the original (piaiuity. Kx. 2. iSx'^ — 7x1/ + !/'• + iiru^ — 5//^ — V>2-. The lirst three terms give at once (i.r — y and X — //. Now drop the //-terms in th(> given expression and there results the trinomial ex"" + iir}X2 — 62' the factors of which are Ou- — ~ and X + iis hence — 2 and + 62 are the recpiired tfiird terms, and the factors are 6x — y — 2 and x — y + 62. ":XAMI'l,i:s OK IM)I,VN()MI.\I,M. 117 Kx. :{. r,V-" - liiuy + Wy' - -Ir + |:W/ - :,(|, The lirst llirvr Kriiis jrivd y.r--// Dn.p tli(> //-tonus and wo p't i -j.,- _ -».*• - 50, of wliich tin. fai'tni's iiro ;{./ _ 7 »n«l 4./' f «. Ilciicc. - T iiiid 4- 8 arc tlic rciiiiircd Ihinl Icrnis ami llic j;iv( 11 oxpiTssioii = (:{,/• — // — Ti (4./' - :i// f- H). We iiii;r|,t i,;,v(> ()l)iaiii('(l — r, ,111(1 f M. I)_v dropping; the ./•-tcnns in the fifivon oxprcssioii and rcsolvinjj: ;{//-'— i;{y _ :,((, oljscrv- in<,', however, /A^// Ihc sit/i/s nf tin f,iili,rs <>/ ;{//- iini.sf Ix — as determined by the signs of the //'s in ;{.r — // and A.v — ;{//. Iliiiii\ to fditnr such it jiolijiioiiiidi liaviii"^' (say) ./'-tenns y-terins, and i'-lernis. r. Omit all ^-terins, and fiictor tlie reniainiii-r ^M-DUp ; this will j^ive the tirst tin, terms of each of the n'cpiired factors. 2°. Omit ail //-terms and factor tlie remaining gronp ; this will give the tJiinl terms of Die re(piired f.ictors. 109. Another method of factoring such polynomials is lo change the given expression into the form a' — Ir. We select the ./'-terms (.say) ; multiply i)y 4 times the coenicient of .v s(//f(f/r, and hy aiht- /////certain (|Uantities form a compi.ktk syrAUK (see formula K. ), and if the rcuKtinini) terms nuilic >t nunph-tc s(jn(iri\ W(! have the ififfereme of two scjiKtrcs. Ex. 1 . 2.V" + n.,',/ + '2ir + i;i.r? -f- \:>/.? + -21^' -. Multiply hy H (4 X ti) then KU- + 4(U7/ + Id//-' + 104,r,? -f i;i(!//^ + HJHi'-' ; The ffrst term of the s<|uare trinoniial we need is 4./', the second 5// ('s?nce twi(!(> 4.r x rwy = 4<>.r//), and VSz will give; nearest sj - Viij-" - 5a - 15.y. 10. 4a;* - 15/ - 4a;y - 2U* - 36^^ - 8a;^. 11. 9a-* - 6a-y - 3//* + 8^* - \^xz + \^yz. 12. 6a-* — 13.r// + \^xz + 6//* - 13^2- + 6^'. 13. 5a;* - 8.r// + 3y* - 3a'^ + yz - 2^*. 14. 14a* - 20a6 + 41ac - 6c + 26* - 3c*. 15. 8a* — lOr/6 - 14ac + 106c + 36* + 3c*. 16. 1 + 4a; - y — 20y* — 21a;* + 43a;y. PRINCIPLE OF EXACT DIVISORS. 119 110. Exact Division.— We can sometimes discover by trial whether one quantity is a factor {i. e., an exact divisor) of another. The principle on which the process depends is a very simple one. In the explanations a stands for any one-dimension quantity, as a + y, a + b, a + b + c, X + y + z, etc. 1°. A quantity vanishes (i. e., becomes zero) when one of its (actovs, a, vanishes (i.e., becomes zero). 2°. C/^ ax"" + &.r + «, do not vanish when a = 0, because they do not contain a as a factor. So, a" {x - a) + b"- (.r^ - a') + cHx"" + a") does not vanish when a- — a = 0, since a* — a is not a factor of it. 3°. If a quantity does not vanish when a — 0, the result "s the remainder ivhich would be obtained by diridimj the quantity by a. Thus, put a = in ax + a'^y'' + bHf, and there results which is the remainder obtained by diridinrj the same quantity by a. So, in a' (x — a) + b"" (a*- - r/-) + (" {.r + a^), if we put x — a = 0. i. i\ ,r — a, we get + + c» (a^ + «') = 2aV•^ which is the remainder obtained by dividing the given expression by a; — a. n i 1^0 llKSOLt'TlON INto I-ACtOliS. «<• Ex. 1. Is X — 3 a factor of x" + a- _ ig ? Put a; - 3 = 0. .-. ,v = ii Substitute 'S for x in the expression ; tlien it becotnoS 3'^ + y - 12 = 0, .-. x—li is a factor, Ex. 2. Is .r + 3 a factor of ^•=' — 2.r^ — lO.r + iT) ? Let u- + 3 = 0, .-. x= — 3. Substitute —3 for ,r ; then tlie expression becomes (_3)='-S,-3)^-I0(-3) + lo = -27 - 18 + 30 + ir,z=(); .". w + 3 is a factor. Ex. 3. Find whetlier a + h-r is a factor of ,r' + fr'-r' + '.\ah': Let a + b~<- = 0, .-. c = a + h. Substitute this vahie of e in the given expression, and it becomes a' + b'- (a +hf + nab{a + b} = {a + br-— {(I + b)' = 0. .'. a + b — c is a factor, Ex. 4. Find tlie remainder wlien .<■' — 3.r- + 2x — 7 is divided by X ~ 2. Let X - 2 = 0, .-. X = 2. Sul)stitnte this vahie of x ; tlien the expression bc^conies = 8-12 +4 -7 = -7. Ex.5. What is the remainder on dividing .r' — (r7 + l)if + 2a— 1 by X — a + \'. Let .r-r/ + lz=0, ..f.^^f_^ Su))stitute this for x ; then the expr(>ssion = (a — \ )•-' - ( expression --. \b—ia-ifh)—rb)+a—b\\n^-b + «t + b)\+Habx -(a+ft) = \ — 2a \\ -2b \{ 2 {(I + b) \-Hab {a + b) -^ H(tb {a + h) —Hdb (a + b)=(;. The method of synthetie division is useful in finding renuihnlers and values^ esi)eeially when the divisor is a binomial, x ± y y + a. 9. When 8^"- - 16.r^ - 12.c - 10 is divided by 2x + r,. 10. When x^ - 9Hx* - 98..' - 100^^ + 98a,- + 100 is divided by iV — 99. ^ 11. When ^r" + px' + qx + r is divided by x - a. 12. AVhen x' - 3.r^ + r,x' ~ 40 is divided by x - 2. 13. When X* ~ x' + x'' - a- + l is divided by x + 0. U. Show that 0.-4 is a factor of a." - 5;r« - 2a: + 24 ; and find tlie other factors. 15. Find the fact )rs of fi.r' + 7^" — .^; — 2. 1«. Show that ^^ - 7x^a + Sxa^ + r,2a« is divisi})le by x + 2a. 17. Show that X* + 4a- - 5x' - 'S6x - 36 is divisible by a--a-6. 18. Find the factors of x' - 39.r + 70. EXACT DIVISORS — EX KUCISP:S. 123' Find tlio value of: — 19. X*— 19.r'' - 121, when ^ = - 8. 20. x'- — Ax^ — 2.r-, when x = — A. 21. .r"— l()2.r'' + lOO.r" + 102.?='— 99j-— 201,r, wlion ^ — 101 = 0. 22. ^x* + fiSOx^ + Ux- - mx 4 oH. when x = - 19. 23. What quantity must bo subtracted from x^ — /)x^ + ^ — f to tj;ive a remainder exactly divisible by x + a ? 24. Find tlu^ remainder when .r' + ^/' + i^ + U' + a) (x + b) {x + c) is divided by x + a + h. , 25. Show that ai/^ — {) y- + ^''^ is divisible by " 4- '-^ — 8r//w' is divisil)le by r/ 4- 6 + ^'. 28. Prove that a^ 0) — r) ^r h^ {r — d) -\- c^ (n — b) is divisil)le ])y f/ 4- '' + '*. 29. Find the value of \Otf - 1109//' - 109y- - 212//- 1111, when /y = 111- 30. Prove that ivx'' + a'" — 2//?>.r' + ^rV 4- '/'^' — 2a'6 is divisible by ((7 — b)(x^-^l). 31. Show that id — b) {x — a) {x — b) 4- (6 — /') (x — b) {x — r) 4- (c— a) (X — /•) (■*' — //) is divisible by a—b, b—(\ and c—d. 32. What nund)er added to Ax' + Mx* + Wx' + 2\x^ — 123u- — 41 will give a result divisible by 2,r + 13 :" 33. Show that a^ ib'' + <■- — a") + // (/•- + 4- /• = 0. 124-. RKSOF.('TI()\ INTO FACTOUS. n {.,:■ ■a; a9. What (Iocs ax' - (^/^ + f,) j-' + h" hccoinc, wlicn (u- + h = ()^ 40. What (Iocs (,r + i/) {,j + z) {^ + .;•) + .vyz bccoiiK;, when ;r + // + ^ = :• 41. Ts ./' + // — 1 n divisor of (./■ + ijf + W.vij (\ — .,■ _ //) _ i ? 42. What is th(; valiM* of .r' 4- (Uti — h").,- _ (^/ _ o/;^ („-' ^ ;}/;^^^ when a; — f/ + 26 ^rli)? 43. Show that (a - h) .r' - (x - /;j a" + {x - - a — 6 and by x + ^a + 6. 44. What must be siil)tra('t(>d from eacli of the (|iiaTirities x' — pl^q and ,r- — y>',r + ry', in order that l)oth result.s may l)t) divisible by x + a ^ 45. Show that x' - px' + (/x' - rx + s is exactly divisible b.y X — a, if a* —pa'' + q>r — nr 4- s — 0. 111. (Vrtaiu cases of exact division are of special iiui)orfaiice, and the results should b(^ carefidly memorized. These nre (x" ± rt") -^ (,T ± a). We find at once, by the last articl(>, that 1°. x" — a" is divisible by x — a, always. 3°. .r — + <•) + ^l> -<■ '■/"'• Ex. 2. u' — SI -i- a- + ;5. Tliis is X + o — ,,■' _ 'Sx'' + 9j' — 27, formula (II.)- Ex. 3. What factor nuiltii)lied into x* - x'l/ + d-\>f - .r/ + y* will give x" + //'? We find '^'-'-^^ -- - ^'^ - ^r'!/ + xir - .r>f + !j\ formula (III.); lienee .r + y is the reciuire'd factor. Ex. 4. Show that {a + h - 2rf + in ~ b - 2cf is exactly divis- ible by 2 (a - 2r). Th(! expression is the mm of odd powers, and is therefore (formula in.) divisi})le by the sum of the quantities, i.e., by i^a + h- 2n + (a -h- 'i'') ^vhich = 2« - 4r =:= 2 (^/ - 2r). Ex. 5. Resolves into factors a'- + (t'h + n'h' + u'h' + a// + 6\ By formula (1. ) we se(r at once that this is -Uf + (lb + ^>')(''' + '^') _ (n- + (((> + />') ('< + b) {(f — ab + 6'). V.X. 6. Prove that -;0(.r-l iT + (2.r^ + 2.r-4r-|2(a:' + 4.r-r, > -xactly divisible by 2x — 2 and by x + 5. The first two expressions are divisible by (.r - 1 ) + 2./- + 2,r - 4 := 2j;- -i- «.i' - 1 U = 2 (.<■•- f 4^ — ra ^^^* RESOLUTION rXTO FACTORS. Which divides the third quantity, and has the factors Ex. 7. (..- + .,-) _. (,,. + ,,.) . ,,,,,.,, ^.,, ,,^^^,^^ L Wy/ \nh} J • -,,^,,,7 - ; the d, v.sor = ^ ^, + ^' ; quotient = /-V- /-\' ^' ■ ^"'^ A'^^'-' /M = cd ub the EXERCISE XLIX. Divide : — 1. ;.« - 1 by .. - 1 ; ;,« _ ,,0 ,,^^ ^^ ^ ^, . ^, ^ ^.,^, ^ j^^ ^ ^ ^^^ 2. x^ + 1 by 1+ .r ; (nxy - l by ru- - 1 ; 1 _ i"Y Uy \ + I") 3. (a + b)' - 1 by a+b~l; x'^ + a' i,y .,.. ^ ^, '*• W " G) '^>' ^ - ^ n - ^)« + (1 + nxf by 2 + 2^. 5. (/> - 1)" + (^ + 1)^ by y, + y ; ,/-•« _ /;-., ,,_^. ^,„ ^ ^^„_ 6. 0-- + a- by x^ + a^;^- (,, _ ^,^4 ,,^^, ^ ^ "^^ _ ^ 7. x'+^hyx + l; a-^^ + 2x'' + 1hyx^ + 2x^ + 1, Resolve into factors : — 8. .r* + .rV' + //^ ; r,^ - u\h* ; (,r _ o.y)^ + (.r + 2//)^ 9. a' + a*b + ,flr + aV/' +.///' + /y ■ ,,.." _ -^o^^u,^ 10. (a-^ - /..,» + «6V^ ; (,r - 1 ,^ ,.,^ _ ] , „,4 + ^)^ 11 . ,,n _ ^,^.0 . ^, __ ^.3 _ ^^ ^^^, _ ^,^^ _^_ ^^ ^^^ _ '\ t.^^.r' 4- ls,//r, -02,/^ + ()6v'=') : ./^' Sr^* 1.5. S1m,w thnt Or^ _ :.r +-^y<^ (^,.. _ ;^,, .^ ^^ ; by '2.r- + X '- 0, s exactly divisible FAPTORIXO BY SYMMETRY. 12^ 14. Show that (;r and y = {(! — b)\ 1(5. Divide .r» — 1 by tho product of ./• +.<•+! and .r" + .r' + 1. 17. T)ivid(i f/'" + .r^/" + .r'" by the product of ir — + //)(// + z) (z + 0) = y-z + yz-' -{y'z + yz"-):^(). :.x is a factor ; and by symmetry, y and z are known to be factors, and .-. xyz is a factor. 8°. There can be no other literal factor, for xyz is of three dimensions, but there may be a innneriral factor, // suppose, so that (x + y + z) (xy + yz + zx) — (x ^- y) (y + z) {z + x) = nxyz, which is true for all values of x, //, and z : assume x = y = z = 1, and substitute; .-.(l +H- 1)(1 +1 + D-d +1)(1 +!)(! +!) = «• 1-M or 9 — 8 = 7il .-. 11 = t^ and X, y, z^ are the factors. 128 RESOLUTION INTO FACTORS. ,;« Ex. 2. a (b'' — c') + 6 (r» -a'') + c (//' - h''). 1°. This cxpiv^ssion is syiiiinctrical Jiiid of threo diinensions. 2°. It (l()(^s not = vvlu'ii (I z= .-. tlicn' is no monomial factor. 8\ Put a — b = /. e. a = b, and substitute hrcomcs a (a' — o') + (i (/;' — a'*} + c (a" - = a" — m'' + ac'' —a^ = tlio ox])r('ssion a') a — b is a factor, and therefore b — c, c — a are also factors ; their product (ii bec'oiiuj.s 1) sions ; tho al factors, it becomes ■ — a. and h a — b) is iression is icfor to be + 0; we -6). EXERCISE L. («) ♦iesolve : — ■ 1. {u.' + y -\-z)*-{x' + y' + 2^). 2. a' (6 — c; + 6' (c - «) + C (a — 6). ;}. be (6 — c) f ca (c — a) + a6 (a — 6). 4. a (6' - c») + ?' (c' - a") + o (a' - 6'). 5. {X - yy + (y- 2)* + (z- x)\ 6. {a' -by + (6' - c')' + (f - ay. 7. (a; + y + ^) (^y + y^r + zx) — ^-^'e. 8. a" (6 — c) + 6' (c — a) + c^' (a — b). 9. a (6 — c)» + 6 (c — a)' + c (a — 6)'. • lU. a;^ (y + 2") + y' (z + x) + z' {X + y) + 2ary^. 11. a^ic — b^) + b* (rt — c^) + c* (6 — a^) + abc (a'b'^c' — If, 13. a* {b — (■) + 6* (c — a) + c* (« — b). 13. «6 (a + 6) + 6c (6 -\- c) -\- ca (c + a) + «' -f b" -f- f'\ 11. a-" + y' + ^' + 3 U' + //) (y + z){2 + x). 1 5. rt' 1- ?/' 4- c' + ;M'^ + 6 + c) (^/6 + be + ca) — 'Sabc. 16. (a + b + <:f — (b -f cf - (<• + af - {n + h)'^ + (i- i- b^ + c'K 17. a' (6 - cf + 6' {c - «)' + c^ {a - h)\ 18. Show that (a + 6 + c)* - (6 + c)« - (e + «)* - (« + h)* + a* 4- 6* + c* = 12«6c (rt + 6 f c). 19. Show that {ab + 6c 4- caf - {a'b' + ft'c' + c'^a'j = 2a6c (a + 6 + c). 130 IlKHOLl'TION' INTO FACTORS. 5J. Sliow that (// + h + c + (i)* + (a + h — a — d)* + (a + (;—h— + (• — (f)* — {(I -^ h~r + (I)* — {(I — /> + c + (f)* — (—11 + I) + r + (/)* ^ \\)'2ii/m(/. Show thuf 8 ((/ + h + <■)'•' — {(I 4- /'»■' — (/' + ''f — {, -I- a)' = :J (2(1 + f) + <■) {If i- ',>/> + r) {tl -\- h \- 2r). ►Show that 2{ + n'- — {>. + a)''- .kiba = 8 (// + ft + c) {(lb + '"' H- '(i). A. Factor (./• + y + 2)^ — (.*'' + //* -f 2''). 5. Ka(!tor r/' + //' — /•" + Wnlx- ; — ^r' + //=' + '•' + :J'' ; and _ „;• _ /.a + ,fl — [iabr. C. What valiu) of m will make x' + \'.\x* + 2(U'' + 5'i.r + 7//j vanish when x + \\ = i 7. What vjiltu^ of in will make 2^;* — lO^-'^ + 4//m' + vanish. when x' — \ix + 3 = 0? 8. x^ + ax^ + bx + c vanishes for ;r = 3, — 3, and 5 ; d('t(M'inijie + r)' + (c + of + Ui + hf — 3 {a + b)(b + 1) {<- + a) ■ by a' + 6=' + r-" — 'Aabc. 13. Find the factors of a (b — c) (b + c — a}* + anal. + anal. 14. Show that (a — b) (1 + ca) (1 + cb) + anal.'+ anal. = {a — b) (b — c)(c — a). 15. Find all the factors of a(b — c^ y1 + ab) (1 + r/c) + anal. 4- anal. 16. Find all the factors of (6 — c) (1 + a^6j (1 + a'-'c) + the two similar terms. f + C — b-f/)* and I- r)2.f + 7m f 6 vanisl). dc^tormiiio dotonnino ivisiblo by h < ) (c + a) I rial. lal. 4- anal. + the two CIlAI'TKIi \. FFKiflKST roMMON lAc Toll.- I,|;ast roMMON XIM/ni'LE. 113. The Highest Common Factor dl. ('. F. > of two or nior» (xprt'ssions is that ('.\i)n's.sion wliidi is loi'iiird l»y niidtipIyiiiK tlin t^rcatcst common mcasun' (if there he any) of their jiiimerieal (NM'dieients into tlie factor of liifihist diinriisioiis wiiich will exactly divide each of the expressions. Tims, The 11. C. F. of X'u-'ifz and ^iiKrVV is tin- product of T), the ({.('. M. of 1.") and '20, a;", tlie lii;;[iest common fai-tor in x, y\ " " " y, and 2, " " " ^, rtnd is therefore Tw'^tfz. * 114. To tind the H. C. F. of two or more Monomials, th(>refore, we have un\\ i^Jintl and iniiltijdij ttu/tthir the (f'.c.Jt. of tlir innni'viml cocfflcitttts and the hiyhtst common factor of the literal parts. EXERCISE LI. Find the ir. C. F. (.f :- 1. 'i.v'y\ j'l/''; da''l>\ 27a^h^c ; - \{)aV,''c, r)Of/'/>'e=' ; 72l>, H4hc. )i. 2'Sah'c, ()daV/'\ — .V/, 2{)(rb; lHc'; aOu'"//", *.H)x'!/'z\ ',]. 42a-ljc\ 77a'h^c- ■ ]2x"''i/\ 4x\i/ \ ir),r"//'~'^"~\ U)ui"y"-'2'^\ 4. 44y;<='//V, 5-)///^//^^", and n(>m*nc' ■, (iO.r^yV', 105.^"//^, l«ru-y- T). 73.fY» S4x'y2, and W2Hxyz^ \ 'Ma*l>^\ \urui'(>'c\ U)4a^n^b''c. 0. 8rr/>", 24rr'"('>'", and 144a"'/>'' ; \2a''-'hf, \4a''l)^'c. 7. tM]aVrc\ Ua-h\-\ and 24^/"/*".' ' ; 2A(t^h^c'-, 24aVj*c\ ;V/VAr'. 115. In like manner when Polynomials can be resolved into factors, i?ie product of the comtnon factors (if any) will bt tlu H. (J. F. of the polyiioviialS. ONTARIO COLLEGE OF EDUCATION V]-i IIKHIKST roMMON FArTOR. Ex. 1°. Find tlic U. C. F. of Hii\v* - 24a'V + 16a' ami 12a.r''i/ — Vla.nj — 24(11/. Tlie first (^xprossi()n = 8«V' (.<•- — '.).)' + '2):^ S =(2x ^ y)(x — y). .-. II. C.F. =.r-.(/. Ex. n*^. Find II. C F. of xy — xz + hy — h? and ny—az—y'^ + yz, 1st expressi(»n — ./•(// — -? ) + ?>(.'/ — 4) =: (.*• + 6) {y — ^). 2nd " = y\(f — y)—2('i — y) ^«i — y)(y — z). .. II. ('. F. ^y-z. Ex. (}\ Find H. C. B\ of .r'+ 1 la'-'— 23,^ + 39 imd .r- + 9a'-52. 2iid expression -^ (x + I'i) {x — 4). EX A M PLES — EX ERPISES. V)^ 16a' and -9). -9). 1). -i/'+yz. 52. x — i is not in exact divisor of the 1st expression, since 4 is not a factor of 39. If. then, tliere is a common factor, it must bo a- + 13 ; tlii.s is a factor of the expression, for it becomes zero when .r +13 = 0. .-. II. C. F. = .r + 13. Ex. 7°. Find II. V. F. of 2.y' - (4^ - 3r') y' + {h - av) y + 9hc and 2y' + (2« + 3r) //'-' + (3(/c — 4b) y — 66c. In the 1st expression only one power of c occurs ; factor by Art. 107. 3,.ys — Gr/r// + %r = 3c' (^' — 2ay + 36) ~>f - ■^<'!f + <>% = ^.y (y' — 2a y + 36) .-. the 1st expression = (3c' + 2//) (y"" — 2ay + 36). Similarly, the 2nd " = (3t: + 2y) (y' + ay — 26). .-. H. C. F. = 3f + 2^. EXERCISE LII. (a) Find the II. C. F. of : — 1. (IX { a — .n, !>, .r" + lo.r + 9. 4. .r- - l.r + 10. .r-' - ,-)./■ + (5 ; ,v' + O.r + 14, ./■- + 3./' + 2. T). (a + 6)-', f/'-— 6-' ; ./•- + 4.r + 3, ,/-'+l : .r—j'—{\, .r- + 3./-— IS. 0. .r^ + {)./■ + 20, .<•- + ll.r + 30 ; .r - .c _ 20, .v' + ./■ - 12 ; (.*-■■ — 1 1'-', (,r + 1)-. 7. .r- - 17.r — (iO, .r + 23.j- + 00 ; ./•- - 22,/// + 121//", ,r — 121//^ 5. <{ (,r- + 0./7/ + IS//-'), i) {.r + 3,/7/ - \Hy-) ■ (,»' + //)-, (.,•- - y^)'^ ; a;"- 1(5, .r^ + 4,<-\ 9. .r^ + 5.t; + C, ./•'-• + ().<• + S. .<■- + 7.>' + 10 ; .'■'"' + ('/ + 6) ./• + tih, ,/•'- + ((I + n ./■ + (tf. 10. rf'^ — (a — (■) .V — r/r, x'^ — {a + r) ,r + f/r ; •^' ~ y'\ -<■' + (4 - //) .r - 4y. 134 HIGHEST COMMON FACTOR. 11. a;' + ^xz, x'' + {2y ■\-^z)x + Gyz ; (f" + {b + S)a + 3/>, a' + 5a + 6. 12. rt' + {b + ;}) // — {h + 4), a' — 01 — r) a + 2 — c x' — {a + 3).^' — [(t + 4), x"" — X (a + 4). i;i a' - 6* + 2/^r- - c\ a- + 2ah - c' + 6'' ; (« + /> + r;)^, a'^ - ^^'^ - c" - 2/; '<<:*. 14. {x + yf—2ah—a'—h\ —{y + h)''-\-x'' + a'' + 'iax\ x^-\-a\ x:"-^ a' a' .r- 4- Ttax — \S(i'' ; 24 (x"" — 9?/'), 16 U' — 5}^)' X. + ' + y* ; .r" + n\ (x:" + rtu- + a"") {x' + a") ; x* — y\ (x^" 15 O"-//"), Q(x + y)(x'-y''); \20i'-h*), 27 («» + 6') (« + ?')'. 15 (p' - q\ 25 (ir - q^ ; .i-« + u'^ + ^'i/' + yN (•«' - i/'O'- (ft) .r' + x'y + A-'//' + .r//' + .»'//' + y\ •<"' — 2J•^y' — 2.r-// + //*. :ir' — 5.r — 2, 6^-' + T)X + 1 ; U)x' + V^x — 3, 5.r^ — 1 l.r + 2. lOa' + 29f/ + 10, 4r7/' + 16a ,+ 15 ; let" + 33a/> - 106^ 7a"'' + 32a6 — 15^". ./•* + 6.r + 9, .r=' + 9,r(.r + 3) + 27; .r— 2.r+l, x:'-'^x(x-\)—\. a^ — h\ + 2aJr + li" ; a' + 2a'V> — ab' — 2b\ '■' - <-f'' + -'"y + >f)\ 4 (.f' - yy ; 1.12 a* a" a 6BNERAL MEtllOD— I'RIXCIPLES. 135 80. 3 + a« + l, a«-l. a' a" ..3 1 o „a - //. .<; + 2. x^ + 3^" - 18r - 15, x-" - fi.r - 55. ;?1. 2rtu;^ - (a - 2) x-\, ibx-" - (h + 2)x + 1 ; 'ix' — 22x + 32, x^ + 82.r - 28. 32. 4a' - 96' + 6/>r - c-^, 4a' + 9/;' - 2e' + 12a6 + 2a<' + 36c. 33. ('ix + 'iy - zf ^ ('ix + 7// + 2)\ 25 (.r' - 4^')'. 34. (m' - 3//1 4 2) x' + (2//r - Am ^-\)x+ m {m - 1\ m (m — 1) .v» + (2//i' - 1) X + m (m + 1). 35. 2ap* + (3a - 2b) p\j + (a - 36) />, 2a'x* - 5a\f' + Ba" - 26'.r^ + rmW - 3a*b\ 38. 6aaj' - 6a6' - 54a - 21. r' + 216' + 189, 606" — 12a - 216' + 42. 39. Ga'a;' + 4(ix^ — 10a xy — 3a-:r// — 2.r'// + 5y\ 10a'.i-' - 2au;'// - (la-r//- - 5a.i7/ + .<"//' + 3y', and — 4aa;' — Ga^bxy + 2ab'x^y — 2a6.r +2xy + lhfby--b-xy' + by. 40. x' - 3.«' + 3.r - 1, a-" - x' -x^\, x^- 2x- + 2x - 1 , and x^-2x^ + 2a'' — 2^'+ 1. 116. When two cxprossions cannot readily b(> rosolvcd into factors, their \\A\ F. may l)e found l)y a method which depends on the following' siniph^ principh's :— \°. If (lit ('xpt'essio)i has (t f(ut(>)\ (ini/ nudtiph^ of that cxpres- S1071 will have that factor ; thus, ax contains the factor a, and it is self-evident that any multiple of ax, as 3a.r, 5aj-, max, contains the same factor a. 2°. If tiro expressions hare a vnmmou ftcfor, then the sani or the differewe of any malt i pies of fhrsr ixpnssiiais /rift harr tlmt common faetor; thus, ax and ay have the conunon factor a, and it IS evident that max ± nay have the same common factor a, for VM) OKKATEST COMMON FACTOU. the expression = a (ma- ± n(/). Thus if A and B, any two expres- sions, have a common factor, that factor is also contained in 7nA ± nB. Tlie method referred to will l)e understood from the follow- ing examples. It is generally best to work with detached coeffi- cients ; we give exami)les in })oth ways. Ex. 1. Find the II. C. F. of the expressions designated A and B. x^ + 3.i;'' 4- 4a; + 13 a-" + 4a;* -I- 4a; -l- 3 A B B- A = C 4 X B = A=: (4B - A) ~ X =z D 3 X V = (I) - 3C) H- 13 == E 3 X I) = 4 X C = (31) -I- 4(^) -ir 13a' = F .-. E (or F), /. f^. a- -f 3 is th(> II. C. F. a;*- 9 4.r=' + 16.1-'^ X' + 'Sx' + 16a; -1- 12 -f- 4a; + 13 3a-' + 13a; 3a;' + 12 - 27 a- -f 3 9.r' + 39a; 4a'' + 36 - 36 a; -1- 3 1". We ojM'rate on A and B to (■(uiccl the tii'sttcrms and the last ; this gives us C and I). 3°. We operate on V and D in the same way ; this gives us E and F, which are identical. 3°. We introduce oi' strike out any fad '■.' which is manifestly no part of the recpiired II. (-. F. We are not ohlit/cd to operate on the I'lrst an following , ..pressi<,ns, A and B. Qj-^ + u---44.r -f 21 '^d;" — i;}.r " + 2;}./- — 21 8^/^ - 4.r _~^ 7 A B (A + H) ^ 8.i- =rc 213 == A = (A - 2B, ^ 9 =:_D 3a-^ _ lo^- + 7 (C + D) -=- 2.r = E Hx - , 7 ru'« - 2(m''^ + Mlf - 42 ^ii-' + a;' - 44a: + 21 =■ (llr-7){.( + l) = ('ix-7)U-\) (C - D) -- 2 ^1 ar 3.r - 7 is the H. (\ F Ex. 8. Find tlu. H. C. F. of the expressions marked A and B. A B 4^="- fi.r-- 4.r + 8 Ju-' + __ .r^ _ 1 12.^.3 _ Ts,?'^ - 127^•+~9' 12a-'' + 2.r^ —3 20.r^ + 1 2.r 4.r^ — C,r- n =(2j'-i)(]()^+n^ -3 4./- + 8 8 X A=: 2 X B = 2 B - 8A = 8 X B = A = (A + 8B) -I- X =J) 4 X C^ = 11 X I> = (llD-4C)-f-81.r:- E 1 1 X C = 10 X J) =: (110- ioi))^si == F ■ •• 2.r — 1 is the If. C. F. r. We mi^dit haa' obtained th(> re«jnired factor at once fromC since it^.s plain that only o;.. of its factors. 2.r- 1, ean divide A and B. 2°. Or, we might have stopped with 1), and obtained the re- quired factor by faetorin^j C and D. 22a!'^ — 3.^^ - 4 80.r' + 4H,r 242u'' - 88,^' - 44 - 44 2a- - 1 • 220,r- 4- 1 ;{2.,- 22oy-— ;}().?■ -121 - 40 2x — 1 138 GREATEST COMMON FACTOR. Kx. 4. Find tho H. C. F. of ',lv' + .v-2 and 3^» + 4x-4. .\ A li B - A == (3 2A =: (2A - B) ^ ^ = D 3 + 1-3 3 + 4-4 3 3 6 3 + + o 4 — 4 4 3 — 2 We usp detached coefficients and designate the ^iven expressions as A and B, and ^.' ■ resulting exjjressions as C, D, etc. Every co)., ,ion r.,.'tor of A and B must a|)pear in B — A i. e. in C. So every common factor must appear in 2 A — B i. e. in D ; but C and D are identical, and can therefore have in them nothing different from the common factor. .-. C or D, tha< ■- ;... ^i is the H. C. F. Ex. 5. Find the H. C. 1-. of r + lUN-81 and Ax" — 22;c* - 18. B A(A-B) = C 2A = 3A +9B = ^i-(3A + 9B)==T) D - 2 C = E 2 E = ^(D-2E)== F 21) = E = 1 3a; (2D-E) = G 4 + ■!- 0+0+81 4 -} i ; ... 33 + 0— 18 1 + + 2+0+ 9 8 + 33 + 30 + + + + 0+162 0—198 + 0-102 44 + 22 + 2+ 1 + 0—198 + + — 9+0 2 + + 0+1- 4+ 0+18 4- 9-18 2+ 1 + — 9 2- S-18— 30 „ 1 + 2 + _3_ 4 + 2+0-18 1-4-9-18 1 + 2+3 a;' + 3a; + 3 = 11. C. F. 5:^*0: EX A M PLES — METHODS. 139 expressions As before work on A and H to frot C and D ; then work on C n-1 I> to «et E anrs ^^t Z, "■ "''■•"'"^^^' '^''•'" ^'">' l-'-t -f the eonnnon f^.-tor A 1-4+6-4+1 1+0-2+0+1 1-2 + 1 .■.x^ — 2x+ 1 is the II. ('. F. a/"-;,,'''!",""' "■''•'• '" ■'■'-'"■■-"■■'■' -"V- 2,,' and BIO+ 3- :+ ;}_o A 8A = (;}A -.rH)_=_,, ^(1 1^ 1- 1- 1-^2 •^ - 7 + 3 - 8 + a^-_;r-_3^__ji - 6 4 - - 1 ry A = [ 1 - I _ 1 r~ 1 _ o R =r ! + 3 - 7 + 3 _ 2 (A - r/B) ^ .r = I) ~1 - 4 + Tir~7 41> ^ I 4 - 10 + 24 - 10 C^__lj4^- 6-1-6 f(C-4I)) = E| :!!;= |3- 12 + 18- 12 U (^_o^K2.,_,„ JC = : 8 — 1 2 — 2—12 ^ (2C' - 31), = F ;2^+ - J^^,^^_4^^ ,, + ,,^ ,,_,^,, • ■• Comparing E and F. w,> s.v that ./• - 2r/ is the fl. (". F. >"" '■^■.' '" ^-""V'^-'A^-sions .\and H. Divide A l,v H ,if ;." ■'" ' '":''!' ^'""-'--'- ^.■. 'v: m.k,. ,h. ,v,n;.ind, , • ■ , " . .■"•-'"'-:'! .1' '■..• //,v ..,/, ,1,,., ,..,^„ ,,., , 'in. way tdl there is uo renminder ; the last Uinsor is ti.e \ll R J 40 HIGirKST COMMON' FACTOll. required. This method depciuls on tlie same principles as the other. N. B, — 111 both methods, a factor may be VMU(»ved, and, to avoid fractions, a factoi' m;iy be introchiccd. at iiny stajre of the <)p«'riition ; also, all tlir siijns of any cxpi'cssion may be changed. Ex. 8. Find the II. ('. F. of 4.r' + 1 1.<' + -^l iind >,\r'-n.r-9. J'he riyht-luinti qnotU'iits (ire from riijIit-lHind (lioideiids ; the hft- hand quothnts from hft-hand dindends. QUOTIENTS. ■X 4.r'*+ lLr''+ HI 11)1 la;*- 22.r'4- 99 4 0,.2 x'^ 9 4a;* + 8a;'' + 36 4a;* + lla;^+ 18.t'+ 9a; + 11 -lla-='- 4 lOa-'^ - 9.r + 36 -44a;''- -44a;='- 4().r-' l'2l.t-- — 19S,r + 144 — 99 81)81a;^ + 162,r + 243 2a;" -U a;''- 9 o.y'*+ 4.r=' + 18a' Q1TOTIENT8. 2./; — 4.t•''-ll.^•■--18.r- — 4a''- 8a;*-12.i; — 3a;''- 6a;— 9 - 3a;'- 6a;— 9 -4a; — 3 a-^+ 2a- + .-. .r'' + 2.r + 3 is th(> H. V. F. 118. If ^li<' II- ^ • I'- of more tlian two expressions is to be found, find the 11. C. F. of two of them, then the H. C. F. of that and another of the ex[)r(>ssi()ns, and so on. Note. — The II. ('. F. of an al^jcebraie expression is not neces- sarily the (jreideM coiiimo/t mcasf/rc of th(> nd/ie of the exjjression wlu'U numerical values ai'e assi^^ned to the letters; e.;/., The alp'braie II. C. F. of x" — \ and (a- + 1 )■' is a; + 1. Suppose X = 11, th(>n the value of this II. C. F. is 12 ; but the (jreatest com- mon meastwe of the arithmetical nducs of the expressions, (11^ — 1) and (11 +1)- is 24. It is better to use the term H. C. F. ill al(/ebra, as it is more significant than greatest common measure. EXERCISKS. 141 ci|)los a.s the ^■('(1, and, to Sfil^'C of (ll(^ »n may be 'Is; the left' QUOTIENTS, ':-9 ;— 9 ;— 9 -4^ -3 D bo found, f that and not nocos- exprcssion e. (J. , Tlio Sup{)()sj 4 8, 3^=' + 17y» - 62// + 14. 6. \\,f 'i>f - 556// - 35, 2if - 17//' + 23// + 55. 7. ^x* - KU-^* + 5.r' — 18, 3.i:* — 12.r=' + 7,<' + 10^; - 12. s. 5.r' + 2./'^ — 15.»; — 6, - Ix^ + 4.f' + 21.<; — 12. 9. 2i\x* f .r' - 1, 25.r-' f H.r' ^ .<■ - 1. 10. 6,r* — ■>•'// — Wx-if + 3,/7/» - ij\ \)x* — '.Wi/ — 2./''//" + 'Xrif — if. 11. ./■■• — 2,r' + ./■- — S,r + S, 4-=" - 12./-' + 9.r — 1. 12. X* + (Sx^ + 1 l.r + 4.*; - 4, x' f 2x^ — 5,/' - 12.r — 4. 13.1 2a-'* - 1 2x^f + 1 2x-f — 'Axf, 1 a.-K" + 8a'*//- 1 Sa-^"//' — 6a'''//'' + 4a7/*. 14. ,,.3_y.r'^ + o(},f_o4^ a-''— l().r' + 31a-"30, a;=*— lla'^ + SSa;— 40. 15. x' — 10a' + 9, x' + 10a« + 20a'' — 10a - 21, and a-" + 4a* — 22a"' — 4a + 21. Find the H. 0. F. of the following oxpre>ssif)ns ; — 1. X' + 2a' — 13a + 10, 3a-=' + 3a' — 30a + 21. 2. 4a-' — 20a' — 396a; + 160, 2x^ — 12a;' — 172a + 70. 3. cv" + 2a' - 8a - 16, a' + 3a' - 8a — 24. 4. 4a-' — 32a' + 85a- — 75, 3a''' — 15a-' + 15a- + 9. 5. 48a;* + ^x^ + 31.<'' + 15a-, 24a'' + 22a'' + 17a- -i- 5r. 0. ^x" — 4a* — lla-" — 3a-' - 3a- — 1 , 4.r' + 2a-'' - ISa-* + Ik- — 5. 142 LKAST COMMON M (' I/I'l I'LK. m e,r2_4.r_ 1. 7. 6.r' + V4.r + l.V — '^'5, K)./-' + 2{).i'' + LMU- — r>{). 8. :}.f' - .1* — ;{./• + 1, >ir* + ./■•' + ,/•'-' -f J- - 2. J), a' — '.]c-", and rut^h'^c^d, must contain 3, 5, a, h, c, and d, and each in the highest jtower in wliicli it occurs in either of the (pwintities ; it is, therefore, \~)(i*/rf'\l. So in x — 1, x + 1, .r — 1, (,r + l)^ (.}'— l)^ the different factors are, .r— 1, and .(• + 1 ; the second power of x + 1 occurs, and the third power of x- 1 .-. H. C. F. is (x+ 1 )•-(.*• -1)1 120. Ifcnc(>. to find the L. C. M. of two expressions, we have only to ni/dfi/i/if to/dher t/ie different factors eai h in the highest power in wliicli it occurs. KXKUCISlCy. u;3 -'/ — 1. 4^- 1. + 270(1*. + r,Vu:\ wo or more c'h is exiivtlij >', or cx.'ictly <'i' ('«iit;iin, tlic rule for iHsf contain fx nI poiver. are 2^.ry-\\ (1 the L. C. wer, /. 6. it ^ •->. ^/, A, c, ■s in citiier - t .*■ + 1, '— 1, and ' power of > we have ''t hiyhest I I EXERCISE LIV. («) Write down, in factors, the Least Common Mulliph' of :— 1. 4a», 2«'; \x\ ' ; 7r/^ ;}/>^ iSah ; r//>, />r', fY/. 4. 2xy\ 8^V, 4^*V ; '^-i'//, 8a•//^ a^' ; -'-^'y. J^-i''//', 4^ •^y'' ; ^/.r», 6.ry. n. 2 ; 3^7/, 4.r'.//, 5^//" ; ;V>V, ^p(l\ 2/>=' ; f/^//, />y". 0. f<6(/', wV/'C ; loaVy, ruhv^y* ; ni'^n'^p, 3/;j>* ; yA/, «/7>. 7. iix, 'Ma — ^-j ; ;}«'V>, ;Jrt6(a + 6) ; a, () ; aV) ( y> + 7), ah'' ( p —7). 5). a, a + />, /> + r ; .*■ + 1, X ; .r - 3, x' ; u- — 1, .r + 1 ; a\ in — l))'\ 10. a {X + — y, (b — y)\ (a -^ x) (b -//)'; 2 (a - 1 1, 3 ui + 1), 6 (a* - 1)'\ J 144 LEAST (OMMON MTLTIPLK, ID. 2U-+ 1, 2j! - 1, 1 - ix' ; {x + y)\ (x - yf, .r' - y' ; a" (a - 1;, />" ( 1 + b). 2{). 2(r{h - 1 ), lih* (1 - 6') ; X + y, X - y, x"" - y"" ; x+ \, x- I. x'- 1, x' +1. Find th(^ L. C. M. of :— (ft) II. 4a''h - b, 2(1"" + a ; 0^;' — 2^-, 9a;' - Ilr ; x^ + 2.r, .r ').) a-' - 1, («- 1)' ; a-' - na; + 4, a;'-6a; + H ; x''—x-i\, .r' 4-^-2. !3. a;' + 8.« + in, a;' + 9a; + 20 ; a-» + a; — 2, x"" \x + 1. 24. 2(5. 07 2H. 29. ;u). ;m. 32. 33. 34. 35. 36. 37. 38. 39. 40. a' + a - 2, a' — 'M + 2 ; {x — y)\ (x + y)\ x' - //, x''y\ .r" + 2a; — 120, a;* - 2a- — 8 ; a-' — IHa- + WCk x' — 9.r - 3(5. ;?;' — 3a--4, a' -a- -12. a-'-9.r + 14, a-' - lla; + 28 ; !;, x'^ -\- {a — h) X — ah ; 1 +a', 1 — a', (a:+ 1/^ X -\- (ll -{- b) X ■\- ill), ^ "TV,"- — ^> ►»' — "" 1 * T** ? ' — ^1 K'' j'^ _ ,,'', a-" - (a + h) X + ah ; {a + />)' - r', fa + h + n'. {X + 2)», {X - 2)», (a;» - 4)' ; 2a-' - 7a; + 3, 2a'^ + 5./- - 3. %' + 7// + 2, y' — // - 6 ; 6a'' — 13a; + 0, ea;" + Tw - (5. (5.r - 13.r// + 6//", 4a'' — 9//' ; 9a-' + 3.r — 2, 9a-' — 3,r - (5. rHoa"— rm', 60«'// + 32a// + 4^; 40n^y—2a''y—2(n/, l(5r/'-1. 60a* + i " 7 i/ ' t/ ' *7 It/ «/ 3.1;' - a; ~ 14, 3.C' — 13a- + 14, a-' — 4 ; .t-'" + .y", a - y -. a-» _ r).r + 6, a-' — 7a- + 12, a-' - 9.r4-20 ; 4aa-'//-' + 1 la.r//' — 3a// and 24aa;' — 22aa- + 4a. .r + y, a- - //, a-' + y\ a-* - //*; 4 (x -f- 1)', 6 (.r' - 1), (x - 1 )'. a-' - a', (a; + a)\ (x - of ; 6a;, (a; + 1)», 4 (a;' - 1 ), ( 1 - .r). c (a + 6), & (a - 6), c (a' - 6') ; a;', (a; + 1)', (a;' - 1 )'. X + y,x-y,x'-y'-S(\- x), 4 (1 + a-), 12 (1 + a-'). a.r + by, ax - by, a^v"" + b^y" ; a + b, a - ft, a' - ft-, a* + b\ 4rt» {a^■y), 6a' (a — y), 2a' (a' - y') ; a;'-aa; + a', a;' + tt', a; + a. I OENKUAL MKTIIOD. 145 41. i-a\ I +x + j-\ l-.r: l-Hx\ l—U+U'; A'' + y\ (x + y)*. 42. i^-ic +\, x' +.r+ \, j-'+l ; x'-xti + y\ .r'+j-y + y\ x'-i/\ .\'.\. x''-y\x'-xy + y\x''-\-xtj + if\ (r4-4)% u''' + (54 ; .,'^"-l, .r'^-l. ■W. \ ^-x^- X \\ -X + x\ 1 +.r'4-.''' : x'-y\ x'-y\ x^-xy + y\ •15. (a - b) (a - <■), {h - r) {c - a), (<■ - u) (d-h) ; {x'-y") (y'-z'), (y' - z') (2' - x'), {2' - X') {X' - y'). 121. If ^^^'" oxprossions cannctt be easily fucforcd, their II. ('. F. is to 1)0 obtained as in Art. 1H5. Then the L. ('. M. can he f(ntnd by (UH(U)i(i either of the expressions by the H. C. F. and multiply- ing the quotient by the otin'r. Thns, lA't a and b have the II. (.'. F. m, and Let and a in .-. (I = niq ; — = (i\ .-. h = mq'. ni The L. C. M. of a and b is, of eourso, that o/ mq and ntq' = muq = — X mq = — x o ; ni m mq' b or -• = —- X mq = - X a. ni m 122. If tilt' L. C. M. of tnore f/iiin tiro e.xpressions is to be found, we find the L. C. M. of two of them, then of that L. C. M. and the third expression, and so on. EXERCISE LV. Find the L. C. M. of :— 1. a;' + G.r'' + ll.r + 0, a:" — 6^;" — 2.5a- + 1.50. 3. .r" + .1^ + a; + I, 6a-'* + .5.r* + Sx'' + ix" + 2x—i. 3. ,«=■ + lO-r" + 31a- + 30, a-" + 9a;'' + 26a- + 24. 4. X' + 12a;' + 47a; + 60, x* + 13a;' -f- 56a; -I- 80. 14G LEAST COMMON MULTIPLE. 5. 6. 7. 8. • 9. 10. 11. 12. 13. 14. 15. 2^;" +x''—Ux — \2, 6x' + ^x- — ^9x - 00. 4 _ t;u- + lU-^ - 2x\ 4x + 3.r- - ll./'' + 2x*. 2x* — Ax" — (U'' + Wx — 8, ',ix* — 1 .•)./•" + (\0x — 4H. 4a;'' + 18^;'^ + V.)x + 20, .4*'* + 2^;=' — 5U- — 4.r- 10. 'Sx^ + 2(b-- + ;5U- + 30, ik" + 2^;=* + 4./- — x + (I 4tt'' + 2;^/'-' + 24r/ + 45, 28^^ + 21a'-' + a-id'' + ^Cni^b + 21a% + 81^6. aa-' - Tyx"" + 5^- - 2, 2x" + ./•' - ./■ + 3, (i.r^' - x- - 11a; + 6. 5.*'' f 9.i; - 9, x^ - x" — 9.f + 9, x' — 4.t'' + 12j- ^x" + 13a'* + 2a=' + 2.r- + 1, 9a'* + 5a'' - x- - 1. x^ + Ga;" + 11a; + 6, a;" + Ta;" + 14.7; 4- 8, a;' + 8a''' + 19a; + 12, x^ + 9a;' + 2Ga; + 24. 3a;* - 10a;' + 53;" - 18, 3a;* - 12a;' + 7.1'" + 10a; - 12. EXERCISE LVI. ia) 1. If a; — 3 is a measure of a;" + 7a; + «, what is the value of a? 2. If a; + 4 measures x^ — x — a, find a. 3. Find vahies for a and 6, in order that a; — 3 may be a common factor of a'" — 7a; + a and a'" + x — b. 4. What vahie of a will make x — 7 a measure of a-" — ax + 21 ? 5. Find the values of a and h, when x + 7 is a common measure of a'^ — ' + 27/^ = 0. 5. If ax"* — bx + e and r/.r' — bx + r have a common factor, show that a^ — abd + cr/" = 0. 6. Find the H. C. F. of no.f* + il.r" + 1 and 125j-« + 24a; + 1. 7. Find the H. C. F. of ^x" -f 11a' - 2 and 81a-» + 1 l.r + 4. 8. Find the L. C. M. of 20.f« + a-' - 1, 2r)a-* + 5a;* - a; - 1, and 50a:* — 20a;' -»- 2a;. CHAPTER Xr. Fll ACTIO XS. 123. Algebraic Fractions arc similar to arithmetical fractions, and the saine (U'finilions, rules, and principles api)Iy in both cases. We hav(; seen that division may be indicated by writinjij the divisor licilow the dividend with a line between them. In the case of exact division, the work may be completed and the (piotient expressed as a s(^parat(>. number havint? no connection with Hk; 12 divisor ; thus, wo may noplace the indicated ([uotients and 4a'6 2ab by the actual quotients 3 and 3r/. But in the case of inexact division we do not get rid of the divisor by performing the opera- 1 1 ff 6' + c" tion indicated : thus, in the indicated qiiotients -r- and r — ■, 4t ab ^ 3 , ah'' + (P we g(!t, r(>spectively, - = — ^ = ^ + ^ = ,- -\ — - rzi 6 H — -, in which r<\snlts the divisors still ap|)ear. ab ab ab Th«' word "fraction"' is sometimes restricted to inexad division, but in both cases, and always, division is indicated, and to both cases, therefore, the word fraction is applied as denoting the " brc^aking " of a fiimntity into ])arts. Moreover, it nuist be remembered that inexact division is really inexact oiili/ irif/i reference to ihe unit /// wliirh the diridend is ex/)resse(f, and tiiat it may become exact divisicm by cJiatKjinn '/'^ diiyidend into an eqnirafent qua n fit)/ expressed in units of a loiver order; thus, ■ ' denot(^s inexact division in th<^ sense that the 4 (juotieul cannot bi^ expressed as an exact number of do/tars. But, changing the $3 into smaller units, as 2r)-eent pieces, or cents, we get an exact (juotient in terms of these smaller units, e. //.. $3 12 quarters .. . I3 300 cents 12 quarters ., , |3 - . = 3 quarters, or , 4 4 ~ 75 cents. It ,1 3£ DEFINITION — GENERAL PRINCIPLES. 149 !al fractions, 1 hath cases, writing tlie J» the fa.se he qii()(i(.nt on with tha "ts ^ and ■ of inexact the opera- 0-6' + [1 regards dividend, < '' also in inexact division. We, tlK^'efore, define a fraction tlnis : — A Fraction is an indicated quotient. In all tliat relates to fractions the student should keep the idea of division before him. 124. Then' is ".nother way of looking at a fraction, or (piotient, 4 feet tlius : take e. (/. ~ y - ; here we have divided 4 ft. into tiro e([ual j»arts and taken ane of them ; but we may obtain the same result by dividing cad unit into tu-o equal parts, and taking four of these parts. So, too, in ' -. there is either :} ft. -f-4, i. e., three 4 ft. divided mio four equal parts and one of them taken ; or, one ft. divided into four equal parts and three of them taken. A- C- E- H -B D -F Thus, in the diagram, let each of the lines AB, CD, EF repre- sent a foot in length, then ////v« -fourths of one foot will be rejjre- scntcd by the line A(r, while o;/c-fourth of three feet will be represented l)y AH + CK + EL, which is equivalent to AG. So, generallv, is a -r- f) — one-hf/i <»f a, i.e., a divided into b o ccpial i)ai'ts and one of them taken ; or. if is one divided into h equal i>arts and a of them taken, that is, , -f- + , . . . .to a terms. o ii b 125. From what the student has already learnelying the divisor by 3 we get = 2, 18 the same result. This is true for four times, five times, six times, etc. ; that is, // the dividend be diminished, or the divisor be increased, a certain number of times, the quotient is diminished the .same n u mber of times. (3) From (2) and (3) it follows that If both dividend and divisor be increased or diminished the same number of times, the quotient is not altered. (4) 24 24 X 3 72 Thus, —- = — , = — - 6 6x3 18 = 4, where the dividend and divisor have been increased the same number of times. So, .1 OKNEUAL PRIXCri'LES P'^MOXSTKATEIJ. 151 quotient, (i) ee times as the dlemi I by 3, or Jerid by :}, ^ we have 3ur times, at is, inished, a line Hum- (8) ? times as II, or tile dividend at is, •ased, n (' same (3) ^e(l the divisor 72 24 X 3 24 ,^ = a .y = -r = 4, when the dividend and divisor hav« Ijofh Itoen diminished th<^ same number of times. 24 12 + 12 12 12 We have -= — ^, = '^ + ,7 = 4. And conversely we , 13 12 24 ^„ . ''"'' 6 + 6- 0- = '■ ^''^^^ '«' If a (h'Hdend ho separntcd Into spvernl parts (adfltuds), the sum of the several quotients ?/v7/ be equal to the complete quo tient; and. ronrerseli/. if scrcrai dieidemls han'. the same dirisor, the sum of the several quotients irill he equal to the quotient of the sum i>f the several di rid ends. (6) It follows, also, from (4), that if the divisors are 7K)t the same, they can be made the same by multiplyin;^ both divisor and divi- dend in each ease, by suitable numl)ers. We have But also that is 3 6 _ >| X 9 _ 4 9 2^J = 4 9_ 2 ^ 3 " 2 X 3 2 X 3 = 6. 2x3 = 6; 4 X 9 _ 36 9" Therefore, The product of two indicated quotients w equal to the product of the diHdends didded by the product of the dirisors. (6j Suppose '- is to be divided by . Dividing by 9 we get, by (3), 36 6x9 6 ; but we have used a divisor three times too (jreat^ and this quotient. 36 6x9 , therefore, nuist be made three t\v(XG» a^ great^ and 36 X 3 the true result is .-. ^f,by(2). Hence, X li To divide by a quotient imdtiply by its divisor and divide by its dividend. (7) N. B. — We have merely to use numerator for dividend^ and denominator for divisor, and all tltese propositions apply U) fractions. The numerator and denominator are the terms of a fraction. 152 FRACTIOXS. The student should oljserve that the separatiiuj line in a frac- tion operates as a vim; alum. 126. Improper Fractions. —Mixed Quantities. -^V hen the uu- nienitor is not of lower diniensions tliau the (Iciiorninatoi', tlic fraction is called an improper fraction^ and it may •»<' redui cl (/. e. changed) to a mixed quantity by (louipleting the division. Conversely, a mixed n^imher may be reduced to an inipnipcr fraction by reversing the process exactly as in arithmetic (Art. 125, 1 ), /. e. by multiplying the integral (piotient by the divisor and adding the remainder to the product. 11 9 + 2 9 2 '^ -- = " ..— = ^ + ;^ = ;} +;^ == :5| (as it is generally written). ,, '^x + 1 ;3.c 1 ., 1 ^x" + 6 4^7" 6 „ G X X X x 2x -^ »• '> '• '^ '• Conversely 2^; 2.« 9 + 2 11 'Zx 8-8 '^■"' „ 1 ^x 1 3.C + 1 , 6 2.^••2.^• 6 4.r' + 6 3 + -= — - + - = -_:__ : 2x + -- = ^-- + - - = -^ — ■ " - " - 2.i: 2.K 2.r 2x Ex. 1 Ju JO •c' a:» - .1- + 1 X x — 1 ; on performing the division we get (luoticTii x, and remaintler 1, th(>refore tlu> given fraction = .r + X' Ex. 2. " ; here the ([UoticMit is ir + 1 and remainder —3 »/ ~~~ I .'. the fraction = (.*■ + t) — 3 xT^^l ., 5^::' — ir'* + 5 , 5a; + 4 Ex. 3. - ,- , ^='C—l+,r' . -■ r)x' + 4x — i 5.c"^ + 4jj — 1 1 Ex. 4. Express .c + 1 + as an improper fraction. We have {x + iyx + I _ x'^ + X + 1 ^ X ~ X -- // _(x + y)-l -\-x—y _ X + 1/ + X — y^ _ tx Ex. 5. 1 + a: + .V ^ + y -r + y -c + y -■44^ KKDUCTION — KXEKCISES. 153 \ie in a f,ac- ^Vlicn tlu! nil- nunntov, t|,;, Y be rt'diuc'i . 2 1 1 1. 1 + - ; 1 - - ; 1 + ; 1 - ; =5 4-^; y + - ; ^U - ' a X X X X X 2. a;+l4-^; ^/ - 1 - ^ ; ' + 1 a; a;' (/6 ^•^+.^1' ^-^3V ^^^-^.'Tl' ^^-'+^1 3rt6 2a6 a; + 1 3 a-— 1 6. .■ + 3. + ^^|--+i^ ;.-« + , + ,^--1-- ; 3x - 10 -;- /f^. r/" — a;' 7. 5^/ - 26 — , ; a' - r/a; + a;' + 5a — 66 a + .r a" a;' - a-y + // 1 , + a;' + .r// + y" ; ^ /- - '^ + a- - f/ + y ; a; + (t + a;" + a; + i. x-l 6x (a + 4.r) 656' 98.r - 37 a' + 2r/.r + x* - 3a; + t o^» . ^'f^'-Sx+Y a- + a" ^ 1 *■* ■— »ja; + 5. "^rj . «+l ar~3 a?-4 •^-4 ^._5' ^;'+~4:^«-:z-7i- '"^^ + r + rr nxZJi^r "''^'''±'^. 6..-^-7..-G# 7 o'-*' + w< «.^; + //. a ,.2 8. fiO^-^ 17^.-. __ 4^^ ^ J ^ • " , a--6 a + 2b + c 187. lowest Terms nf p>.^. fo'"«-/.™„,,,I,vi.l^'°^,lf?^f<'f-A fraction i, ,.«,„,„,,„ ,,roof '! '•!"'" '"'"« ""'"'^^'r- Of H.l. .i^\.'\'^^ multiplied or givori ff b« J ocing any fmet IS aij y quantity, pnnei])Ie the f„Ji '<»», wo liHvo to sJiow that "^ = '^ ow lllg 6~y/^' W''«l-« W that Quotient x d ifi. f/ '^•'^o'- = and to (Common 'ue of a >li<'d or blowing lere tn Multiply both sides by m, a But, • b ma mb a X nib = ma. X mb = ma [Art. 125, (1)J ; ma X mb = - , X mb. b mo Divide both sides by mb^ a b ma mb N. B. — It follows from this that the signs of both terms of a fraction may be changed without altering its value ; for this i8 equivjUeiit to multiplying both terms by — 1, Ex. 1°. Ex. 2°. Ex 3°. Ex. 4°. Ex. 5°. aa-g + .r//" _ xg (a + g) _ a + g axg xya cx + x'' a'^c + a''x ~ X ~ a' (c + X) {C + X) TMHt-Xnn 353 _ na^i a X a'' j-3i) _ aia — ^b) SOai" + 10\/'6"' ~ 10(i6= (37r+ 7o ~ W^'ibVa) x"* — 7^ J- 13 _ {x — 3) {X — 4) _x- 3 X^ — 9x T 20 ~ {x — b) (X —1) ~ x~b ' ic=-13.r + 3r) (x—rt) (x—7) x-7 x^ — iOx' f 'Six — 30 {x—6){x'—r)x + (i) x'—'yx + Q' Here we find at onee by inspection the factors of the numerator to be X — 5, X — 7 \ the last tenn (— 30) of the denominator shows that x—7 cannot be a factor of the denominator ; and x — 5 is .-. the H. C. F. Ex. 6= X* + a\r^ + a* X* + ax^ — a^x — a* (x^ + r/^ + nx) *// 4^/'-' — i\(th 6. 7. a" a;»+l a 3 /^» .t;'-7.r a-''-7.r + 12 9. -. 10. 11. a'-6»' (u'+l)''' (f—W' a-^-loVr + oO ' a-'^-y.r + '^o" «^a;— 20 a-"— 2.^—03 x'—1a,r—(^\)(i'' a-' + 7.r+13 icN^3a^— 35' .r^-I2.r + 27' ■i^l2(/r + 27(r ' "Tm-3)^ ' a;*— 1 a5'' + (r?4-6).'r + rt6 x'^—{n—b)x—nb a:'— 3a;'^ + 3a-— I ' .c«-f-(a+c)a; + ac ' a*"'' — (^/+ <•).« + r/c 21(a'4-4a/>-21^/^) (a^b f-o? ^ a?—b (a-bf 2S{a''~+2ab^^) ' {a+b + cf' ¥^i' {b-af (a' + i)'— (a"+o)\ (c—bUr—n) (a-^b)^ {x-af-{x^bf ' {^'bnii^nb^) ' a^-ab^W ' 2oa* — 'J<>- _ l+x + y + xy 5a*—8a^b+Jb^ ' T-a;'' "" * a^ + ab—ac ' {x + y)''—(x + 2f' Ha-''-2.r— 15 ' '(a-bf J-* + a-'//' + //* f a' + />" ) ( a' + ab -f/>^ U" - 1 ) REDrCTION TO LOWEST TEKM8— KXKKCISES. 15: 1. 2. x''-ir).r+nn (ft) .r' + .r- 13 4^'-25,r^ + 2().r + 35' iil.r» + ;.r-;{.r~ 15 ' :r»_5.r^ + 7^--8' 7.i;'- -',':{./•// + «// r, >.3 ./•' - :{./• + 2 na" +\{Uf *,(' + rui V (t'x + '2(1 \f'' + 2(10-* + .r* (ilw- 4- (('\vf/—(ih//^~()\) i^'-X} ••'-( i2,r'' + s.r -12 o4y-i_44.,.- 4- •■)( a* + 2n''b— 2(il,' ■''->/r .r + 40 12.r='-.r'-is.r-:{r, Kr-2()' 2H.i-^i-l\).r-\i2x-\r, !/ -h* 1 Tu-* — 72x''(r -f s h/< .,•=■ -f 1 1 ,/•' _ r)4 {(t + (^ 'i\- :?.< !T^/.r' + ;8^/'J 2x''+ lih f.r-4ir.> li(f'',v*—2(tx'- 4 + //' + v/.r4-J)— 210' 12ru'^ + 24.r4-l (X — // ) ■ 4- ( _y — ^ ) '^ 4- ( ^ _7,Tp (Hf,—n)'' + f,(c——(■]''+"{,■— iff I .i""— ;/.r4-w— 1 tix'"'—{N + r* .<•"'+ 1 iir — fry' +{/)'- ■ {(( — () '»■' + ('>-<■)' + -r"]-^+{c"—^,"-)'-\ j--—.ri/—C}ir~.r.r~ 1 li/z— 1 2.?'-' r- — C}Xi/ 4- !»//" — HX3 + 24y^4- 1 6^'^ -")• (X +!/)■• -.(•' — I/' ( ^r ' + (/> \(l* +{C~f/)/) i-t' + l/)*+x'+y* ' (/f-/;)'>4:76-_n*4-(c'-a)' 128. Addition and Subtraction of Fractions.— We have And conversely 20_12 8 '4 ~ 4 "^ 4 ' 12 8_124-8_ 4 + 4 - ~J ' - 20 CArt. 125, 5), l.'ig FRArTlONfl. So Also H r> + ;{ = , + , , etc 4 I a + f) — (' _ a h r I'lnd .•„ cniivcl'srlv, + f/ /^ f a f A > — (• I (I d o __ <>^> + ff b'^ i/'~ (r ■'■ i> ~ b' So If^^ ^t, Generallv, let , Jiiid ' bo any two fractions whoso sum is b d required, then Quotient x divisor = dividend ; (t that is, Multiply both sides by r/, a X b = (I. X bd = ad. Similarly, Multiply both sidt's by b, d X (f = c. d X hd = bo. Add (1) and (3), .'. , X bd -\- X ltd = ad + b<\ u d Divide both sides by bd, a c ad be ad + be b^ i bd^ bd bd (I) (3) the fmrti,)\uH r; thus :-- hose sum is (1) (2) \K\'\.v. FOK Ar)r)iTio>r. Ilcncc, to add fractions : - 169 1 . Find, if ncccssjiry. flic L. (". M. of the (Icuoniiniitora ; this will he ihv (oniuton (femnnhxtftir. 2". Multiply the nuincnitor of ciicli fnictioii hy llic <|ii<>ti<>i)t of till' tomimm (Iciioinin.ttor divided hy the dnioinJiiMioi- of tlir frac- iioii ; (his will j-ivc the iiuni('rat(»rs of the new fractions. :r. The sum of tjiese new numerators placed over the nnununi denominator will he the sum of the fractions. 1" . ^ '' (' ., '• ». + »~^" "«''••' ^''»' I- CM. <»f tho deiumunators i.s their |)r()duct Jcyz ; .-. xy2-\-x = yz IS tho multii)lier for the term.s of tlie first fraction ; is tlie multiplier for the terms of the second fraction ; ""f a + b' y y a + b 'Sia + b) rt{a — b) 9{a — b) a — b a + b EXERCISES IN ADDITION. 1G3 to shorten the ms by dividing h, c, a + b + c tions, and the roni a + b — c. udent will see to the follovv- o their lowest teed to mixed .) 7 ' 56 106' v-h 1 1 1 r/6 ,' b ' a'^ b'^ i'.' c " ab 0. -' + ,' > 1 (/6 or' ry/ ^" + •/ 4- -"■*'■ ^ y , ^ l ^ - 3 '/^ ■i.^ ./■// (I b y 7 J* 4- ^ + /' . •*'' 4- -^ i_ •*■ . " ^ ^ 1 1 1 1 X x—\ x+\ a — b rt + 6' x — n v X' X ub ab X a X 9. x-\-(i a—x ' r/ + 6 «' + 6"' a;- 1 a a^ — b'' a."" — 1 a* + i + 1 16. 1 X' — 5a; + « a ■' — ($.r + 8 « + X + r/ — a- + a" - I af + 1 >.3 a' +ax + X' a' — «a; f x" «« + c/'a;* + a 104 rUACTlONS. ('» Conibino the following : — Q _ J" + '^f' 86« a (f/ + h)" 2a + '.if> 2(i — iifj 4(1' — W (I'+ab ah + fr (rh — ah' 1 x — '.\ :{.<•- 2.r* J* — 2 1 2.r " a- - a '*' ~x' + a.^; + 1) "*" .i" - 2f ' .r'^'- 2^' + 4 "^ ;r + 2 "^ u-=' + H a* + 4 .r + ;{ .?• + 2 1 1 _ 1 _ -^ + '^ ■ 2'i + 1 "^ 3,r- 1 .■) "^ 4.?^^1 2 ' .r-1 ~ 2 (./•+!) 2 (.r^> 1 ) " 6. 3 3 3.?- a- + 2// 3// X — 1 .r + 4 ,v" + 3.r — 4 ' (.r + //)' ^''^ — f (•* y)^ , I 6 - ^ — + 1 + „ 1 _. • ,,'^ _ (/, _ rf ^ Ir -{(/ — rr "^ r' - (a - A)'' ' 1 2 4- 1 a-"' — 1 (,r + !)(.'■ + 3) (./• + 3) (./•— 1) !l ■»' 'I ''ill 7. if + + 3 1 _ 1 _ 18 _ i^— 9 .7^4-9 .*•' T hT ' 1 .v^ 4/r' a" — ^/ .<' + ^/ .f- + ir .r ah ar \-tl*' «l— (>){!)—<•) {(t—r){<-~fi) a + '.\h + (I + 2f> + f> II + (» \ {(( + h) {If + 2f)} {If 4- h) {If 4- 3/>) 4 in + 2/>) {a + 'M)) 10. + -..- 15 12 *■■' + ru- 4- <> x' + \).r 4- 14 .r^ + lOa- + 21 a + b — c — 1/ (t — f) — I- + (I (.r^ + l) .+ 1 - J^' ( 1 - .r)'^ "^ ( 1 .vy' II. IT). 10. (r + />'" ir - I," ir '• 3 3 + 1 a — X x^ + t/'' X y 1 -^.^ 1+^ 1 u; — f/ X — /y if) — (I)' v — a {.V — (()(.(• — f>) IS. + ;c- 1 :ir + 2 .r + 2 -iLr + JJ) ./•- + 5.r + ({ (<•) Siinplifv tlic followim (f- + (I h t ft {(I — (t >) tah ab- b' {(I + b) b (I- — i; + a' (i'~b'' // — (f 1 1 2x H.r ■I' !+.<■ 1 + *•'-■ 1 +.!■' 1 +, r-'^ X—b ;>+7 .f— f* l-(/ 1- + a — I- b- (I \—(i-^-u-—,r'' {a~b){x- ) ix—tn [b—ioib—x) J _ X— { .i — + X — .) ./• — It •/•-I .r + 2 7(^-8) 7(./' + 4]' 5. - + (a - 6) (rt ~ r-j (6 _ aj \b-c) "^ (c - a) (c - 6) us m Ml iii: , "'I' -J, 166 6. 7. 8. 9. 10. 11. 12. 13. FRACTIONS. a - + d (a — b){a — c) {b — a){b — c) (c — a) (c — b) a' + rr + (o — b)(a — f) {b — a)(b — c) (c — a) (n — b) W + (a — b) {a — c) (b — a){b — c) {a — c) (c — b) y + z z + x X + y (x -y){x- z) (y -z){y- x) {z -x){z- y) x'yz + y'zx z^xy {x -y){x- z) {y -z){y- x) {z - x) {z - y) 1 -t-« 1 + 6 1 +c (a — 6) (a — c) (6 — c) {b — a) (c — a) (c — b) p — a + — 2^"— + r — a iP -g){P- n {q - r) (q - p) {r - p) (r - q) a" + 7;;r (a" - 6") (a' - c') ' (i« - c») (6" - a") (c" - a") (c' - 6") 1 1 14. -„ _ :crT + (6 + c — 2a) (c + a — 26) (c + a — 26) (a + 6 — 2cj 1 (a + 6 — 2c) (6 + c — 2a) 15. — Z^' X (y + z) + y(z + x) + z(x-\-y) {x-y){z-x) (y-zjix — y) {z — x){y — z) 16. x' 1 '^ «■'• + a'b +6* „«• _ a*'b + a"b'' — a'b" + b* d^ — a'b + b' 18. a' ;«+l'f«_rt«-»f» ««+»-._ „,^»fi a' a' a*-'-' — a'-*''* a'+*+' + a'^-' 130. Multiplication of Fraction&— i. To multiply a fractiun by a whole number. u. To multiply a fractiun by a fraction. MULTIPLICATION — PROOF OF THE RULE. 167 tsi i. If — is to 1)0 niultipliod l)y 4, we have (Art. 125, 2), ^ Or, Po, also, Or, ?ix 4 = »•' = 12. 8 8 24 X 4 ^ 24 ^ ^^ 8 ~ 2 — = a*. b' That is, in boih the arithmetical and the algebraic examples we multiplied the numerator or divided the denominator. ii. Generally, if ^ is to be multiplied by -, this means that tne l)r()duct of the tivo dividends (a and c) is to be divided by the pro- duct of the two divisors. Thus, 36 _ 4x9 _ 4 9 "Cr'~3x3~2^3' 4 9 4x9 36 and, conversely, 3^3 = 2~x~3 ~ 6 ' " Tlie followinji; is a gemral i)roof :— '/ and ^, are any two fra(!tions, then 6 d Quotient x Divisor = Dividend. a Or, similarly. 6 c 7l X b =a (Art. 125, 1) ; X d = c. Multiply these equals. Divide both sides by bd, a c , , . , X - X bd = ac. b d b ^ d~ bd 108 FHACTIOXS. Case i is included in tiiis by putting <1 = 1. Therefore, To multiply two fractions : — Multiply the iiinnerators fur (I new niunrrntor ami the ilenom- inatorsfor <'" = '«• Dividing (1) by (2; hd cancels, (1) f2) n b c d ad br' 11 CI I CO. to divide one fraction by another: — Invert the dicisor and proceed as in ma/dph'mtion. \. H.— In nuilti|)lication and division of fractions the student slioiild jihvays tlrst indicate the operations to be performed, and thru iiDtCil id I factors common to numerator and denominator. K.\, 1. .sinipbtv . - . X ' t/^ — .r By rcsulviug into factors we have a' - u^ a^ + J-' (a + J-) ia" — ax- + ir») {a + .r){a —U-) ((<" + .r)(^/'^ — .r') .s> ' und this, by striking out the connnon factors. a — X <. 56 4 2a' 4 ^- 5 "" -V ' 9 a b a .3a' .36 ; , X ax \ - X ('X\ ~ nx\ , x — c a; b a 3a^ 26» ' 4a-' 2a — ;r- X 4a-» a' 3a;" a' 2^ 2a X y z XX • y z X FXRIK'ISKS IN MriTIPLIfATIOS ANT) DIVISION. 171 liJLE OF Signs 3f/' 3fi X — b a , y z X z X (I h r a h r 4. , H- X ; X -7- h c a o c (t (I 1 1 a ...I a' h ' (i^ ' y ' ' • 6' ' * 1 b-{-X 7. --- ' -^ (.r^ + 7.r+13) ; •\^ 4- — ' ; , 4- - ^ + 4 (.r-f//)' ^• + y .r— 1 1 (,ry—yz) -f- ;ry^ ; rx/'''"/^ -;- 'Mt^c. „.(>-■%;+■'■) „-.o;("+^ -":;')u,--6',; -X he c(l\ (a—b n+f\ ^ , ,. ( . + + -r I -5- (the ; ( ,4- , I -^ <(f'—b^) \'' a b I \(i+b (i—b/ .r+\ d-'-l 11. .r-1 (x + l)-' X + A ;r + 4 iX—ii) {X— 1 ) (X — ij (X nH-:>)' -T,\ X (j--\)U-'.))ix-4) i^+yf {x + yf (i^'b''*r^ "" -f a x-l x—h x + 'ii -7- af)c. .r-1 1 X (x—a)' x{x—'S) X + 'ii' x+\ ' x+l 1 (x—ny 13. 3- + 3 ^ (x—^)\ a X a- a n^-b^ a' «' + ax + X 3 • ah ' «'— j;' a{a+x) !72 FRACTIONS. ^, «'— 1 a;*— 1 a;+l « + 2 x + ^ in. -^ — - X ■— — 7 : X - - X — - ; «' + l ;c'-l a' + a x + % «+! {x—a)(x + h) a:'— r'" «;"-«» 16. ( x—'i d) (x—'ih) x—'ic {.r-\-a)'*{x—hf (x + b) {x—c)* {x^'^cf x—ia' {x+l)f~{x—n* *(.r + a)*(x— 6; «' + va + 6)a' + a6 , a; + (/ a^' + '^ar + O x' + lx^Vl 17. :; ;. -! ; : X x'-c' x-\-(i ' a; + 4 a-' + 6a; + y ' (-:)(:-) ■PfFfi. I •= tU (6) Simplify the following : — x*—y^ {x—yf + xi/ 14a \" 1 _£±^ j^ f-r ^ (;r-y)' + .ry . /i'f'X"^ /14a\ 2. ——..-.. X a + 26 a-'-l aj'-ia-r + liS a' + a-"' + ;{aa?(a + .r) a-'-a* (r/+a-)' / , a^ I , b\ a* + x'+ax \ bf \ a) /.»•'-■ + ;{.*• + 2 a- + 3\ Ma-V" /erw/'X- /a" //'X /a b\ vy'/ A a--' r W"^w^V' ■*""/■ _x^+12«.r- i;}r/- x^W^n ^b" {2ab)* (2a6)'-^a)* (j''(a-"' + i7«a— L{«'/) ^ x—a ' «V (867 '^ T*^f6)* ^ a;' + a-— 2 a-'' + 5,r + 4 ^•''— a;— 20 .r--4 Mn/ni'LlCATION AM) Dl VISION -KXKUdsLS. 173 (i''—(b-\-<')' (a—i^)'—b {nr-j-'u'j^ .r— (/' .r' + f/' '.<•-;{ au;'— ll./"+I" u.r'-l».r + 'J JO. 11. 1-J. i;j. •^••-l .//-l r- X .'/+l .f'H. (-,:,.) (.i*t''«)».(j*t'"-)5.(j'tJp,S (.*•''■.<•»•./•'•)" (.,.H.....-s- -,).(: J.r+l I- ^•' ) .r-()/ jji ) ft _ »•*»«. ..<■ ' a :■!•■_ ,i>,i ,aA ./-'•.r"u /■''+//V. A/ + /A' / 1 v" //-\-'r!r .(• + !/ ^ t^^^V /•'^"^'V l-'''^V a^-(b + ('y nhf ab + be + m a" + b'> + <■» - -.iab, „-'-b'+r-'-i Uh a -■ 4- b'' + <''—ab + be— CO 174 FRAf'TIONS. A? J In M 'I ' I' 'i ii .a 1 ,,T a;'+a {:-> 132. Complex Fractions. — A fmction which hw^ either iQm\ or btth trnns fnu'tiunul, is caUod a Complex Fraction. Thus, a X X I) n — , — , — , jire eoinpU X fnictioiis. Such fractions merely dcnoto u •*> sc b h divtsion, and arc simplified by the rules foi" division. Tims. a 1) (I f — , rjeans that , is to he divided bv ., and the result is, there- d fore, ; • This result mav he ol)taincd hv vc* (1) Jfultij)l>/fn(/ the tiro (.rtreme terms n and) of tlie tirodei>onin(itorst>a}i — a ducinj; th(> denominator to a mixed num- a a + b _ (fia + b) _ a Z ~ b {a + b) ~ b a + (> a -h 6 unnrcTioN ov complkx fractions— kxamples. 175 K.\. 3. ^-.y ^ + // •f''-ff' iiijitors cunccl, this i ; ai'.d since th: donom' •e'-y' _ (•f-H.y)' + (. i-// )- _ 3./- 4- 2 2.11/ '-,:]'-;.t-..i[ Kx. 3. .Simplify -^ . 1. The I..C. M. „r »// M, ihnomi„ators in tJ-.s cas*. is 0, lu-ncr nuiltiplvin^,' both terms by (J wi' luive, ' 6-3+l-.g _ 4-.r 6-2 + l-.c~ r)-.r' a mixed nuni- 1.3 A.)-'!/-' ix-ij .t' + i/ 4.*-//' Uv"— !/''}{. r+!/U -I,* ^•-r Hero we, in tlie first place, simplify tlie numerators an 1 +Vi 1 -X 1 +a a + X a ^- X a^ + x' X, X ^ £. _8 ^/ 1 1 \ n X I a + ./• .r-4- J' - T.r 1 1 1 + .. a + ./• - X' ft' — (u- + y' Aa 4 a? — 8 // + + — ./• If — X x + u—.i I _ [ a a — \ r* (r , X X'' 1 + + , (I a' 1 _ 1 f« — 1 f/ 9. 1 lO. - /" -J] (" +J'Y _ (" - f'Y (" + f>V ^ (a - by -i]^ 1.1 1 11, (" + ''4-^'-^V^^'-'V ll±ll- c a 1 Knliicc tl ■" r<.ll()\vin<,' t«) their .siinidest form :— (/.{.f, fi—h 1 1 X a— I) a + h :{./•—'.> :{./• + 2 a I -; 1 + l+ar-i- «- l-a; 178 FUACTIONS. jc .r w ;{ ~ 4 4 (t 2a r'— //— ^* — '2//V ' X w u- ./• ' '^(1 + ■{■'! + 5j ^ 4 4 r, ^ (/-8 a + 1 8. 1 — a ft ((i.r-ifhj/)" -\- {Kji—h.vf Ui—fi)- a \u' ijl \i/ .*•/ 'I M 7T7MT (a;-l)M.i-+l)H a;'' •r -I- .r + .r + 5. 1 III 4- // /«* + /''' VI— n m'-'—n 1 (I ■a? -j— f a b 0. 1 _ .r + a 1 1 .1' (I + .I- "'■ T of—n II— x d'-\-x* a a' + x' {(f + hf—4ah j_ 1 _ ■ 7?i "*" /.ii a > ah 4r + 1 + 4.r 2ur+i/) (i—X ^»\.r .. .. 7 .'/ .. ■'"-!/ 1 1 1 + 1 ^- b' •'• * '+l+i 8. + a-h 1 + V/A (I 1 — 'm^ — h) 1 +06 i- 1 - a - h 1 - uU 1«0 FItA< riONS. m IJ m UK tmtl tUe vhIiu' ol , + .,, wlicn w = • 11. Find tlic v.iliir of " "^ Y ^,— t', ^ v\i»en.7= -4.;m(I/>= - 3. 13. Kind the value (tf '/.r + /;//. wlicii .r = ^ — r--, and // = \'.\. Show that tlie value of ar 'iinr — 2h.c "iiib* — 2nj- h* . 1 , — - — , lii , when u 11. Prove that tht^ vahu' of C-.y^c:.)' is 11(11 — 1), when IT). Show t =vx.7'^ . ^ /.r + ay r + 2u + h is e» If show that '/ t- f- ■\- <' + ' . and f> = -5. l// = aq -, is IX 1 wlicn [11 — t), wIh'H .V - r) ah + (I = alM-d. 133. Ratios.— The tiacfidii ex 1 1 Messrs a n-laliou iM'twccn tlie () i\\(» (iiiaiititics a and A. wliicli is callrd tlirir nttio. When ///yj fraetions, or rali(.s, are e(|iial, they const it iite a />yv7/f>/7 /<;//. tl ins. a (• , ~ , means it : h :: t- : d. The student shonhl caretully attend to the foil deduced fn.ni the e«inality of two fractions, or ral owinj' resiwls lOH. If a _ r then ivi(h' both siih's of the sanie ith-ntify hy a/, and u _b Diviih' hoth si(h's of the ich-nlity l.v (tb, and we 1 lavu we nave lavo (> n A(hl 1 to hoth si(h's of b"~~ ,1 and we ^ct ivide (T.) by (4). and we I lave a— b _f' — (f a + b ~ r + 7l' (1) m ('h (4) (■-)) ^Oj 182 FnAfTroxs. "I' II Also a b d ma vr .•. ma — mbx, ac = mix\ ma + nc _ _ « _ '" mb + 11(1 ~ ~ b " -Id -%d **^ EXAMPLES IN RATIOS. 183 Divide both terms of the first fraction i)v /^ and the re restilts >. — 7 .n 5, -H 5-,-» .V- - 7r/ d liy pMttinK for its e(jijal a K.\. 2. If a h ~ (i: prove ff' + (ifr + />» r' +,.(/■'+ tr «•' ah' + h' r^—ar+fr hivide l)oth terras of first fraction by b\ an'd W(^ 1 lave r - .+ 1 ^Z'ciP'^ (P ('(/Halfofifrnr- )ved in any case r — (I J- ; for, on rms of the lirst on, leaviny ttro plication of tho Kx. ;}. If : -- h' + r« - a' r' + a' - ir (,'' + f/' - 6c ca (lb tli.ii each of these fractions is e«pial to 1. Take tiu; first and M( (.!id to^etijcr, then, by (8), each fraction /,-■ + r' - a-" + r' + a' - li 0/.3 2f? 2(1 be + ca <'{v svminetrv Ex. 4. If 2a + 2b 4- '2r a+'b + b + r + r + a = 1. find .»• in terms of a and b. Multiply numerator and (hMumiinat of right side by w and apply (H), and each fraction becomes or iT* + ft.i- — b = .r' — a.r + b 2ax = 2b X = 184 i |{A«TI()NS. L'!'' Kx. then If a + h h + r __ r + n a~^ h " '2 (h — r) ~ ',] (r — d) iMit cjicli (if tlio Kiv«'ti fractions = .r. so tliat ,1 ^ I, sz ,f{(l — h), ii \- r = .I'-'i (f) — r), , -\- (I ~ .f-'.\{i- — in. a h + + f> + — \{it — f>) \ {// — f) -f ('• — ")■ .r = t). Cli-arin;,' of fractions .-. hh 4- Uh + Tv = (>. Kx. i\. If (d + (> -f- c i- (/) ('/ — (> — r -{■ — r — f/j, hIiow that tl b d Hya I vision \v« luivi a -y h -^^ c + (1 _ n -ir h — <• — (1 It ~ h + (— (f a — f> ~ <• ■\- (f Now ai>|>lyinK (fl) wc j^jct 2(r/ + c) ill — /•) (f^ + c) 4- i" — '* 2a + '/) a (i — r/) {h + (!)-[■ (It — (I) 2h h _ ((( + <•) — (It — <•) _ 2c _ (' ~ [h +7/)— (/> — (/) ~ '2(/ '~ d' 134. The student should observe that. \)\ actually V >',r JiXKIKISEH I.N " VTIO.S. 18/ EXERCISE LX!!. a •I" - >V' ^ 3r - 2,/ 0„ ^ ;j/, ^ o^, ^ ,^^f 3f; f/^ f+f mh i' mtt — lir H /■ h — ,i—f ,„(,Z.,„f_ in (d Jf. (') — fi,' m(b +f) — niY IK' •tiially dividing'- lU'xact quotient. 'i. And that . = ' f>' -/•-■ {b +\iy _ (o — mr + //f)" • (6 — finf + ///)'' ^/^ -I- r» + t' nuP - 11/' b' + d' 4- /■■ n + b b + r. + a 'Ua — b) 4(6 — c) 5(t.- — 31^/ 4- 356 4- 27c' = 0. a; prove that rins as wo please N. If X y ij A-z a -{-X .V 4- // ' prove each ratio = 1 11. If a 6T ■\- a a + b' prove (I — b = it lf>. If = 6 d prove (^ J- r) {,P + r») ;6 4- ,/; {b' 4. ,/») t" - '•; la' - c'^j ~ y-(f)'{b' - d*) ' IMAGE EVALUATION TEST TARGET (MT-3) V // {/ y <" C^x .■^■' W!i .W I 1.0 I.I IIM ilM IIIIM ||||i22 m 12.0 1.8 1.25 1.4 1.6 * 6" ► v] <^ /a e. % e}. oj 'W/ '•>

!' II. .\ lltll; 13. Prove, also, . 6 oM-ft' i} - V7T7// 11. {a^^- f?)* _ (a* -j^ II ■ill :ill ::tl!: 14. If - = ^ = - , prove a 11. ct' + !/ _ y + z _ ^jf_^ a + 6 b + c~ c + a x' + y^ + z^ («;_+ y + ^)' a" + 6' + c* (a +"6' + cf a" '111 'I I If X y b + c— a c + a — b a+b — e^ (b — c)x + (c — a)y + (a — b) z = 0. prove '"tt" 16. If cy + bz az — ex bx — ay b-e c — a a — b then will {a ->t-b + n){x + y + z) = ax + by + cz. 17. If ay — bx bz ')Z — Off ex — az X a then will - = a I 18. If - = 'i^ == - , prove a xyz abc 11. X' \a + 6 + c/ + y^ + z^--^xyz _ x^ + //' f ^' a^ + 6^ + c' 3 ' a^ + 6' + c* ~ 3a6c a^ 19. If a' X' yz y' — zx z' — xy prove that a'^x + 6V + Cz = (a' + 6' + c") (a; + i/ + 2'). 20. If ?/ a (y — ?) b(z — x) c{x — y) then will ^ + f + f = a 6 r MISCELLA NEOUS EX EUriKES. 187 21. Obtain the quotients to five terms in each of the following cases : — 1. 6. 10. 1 1-x ax 2. a — X 7. 1 —^x a 3. 1 1 1 —hx \ ->r X \ +'dx 8, -"—. 9. 5. o- X — b 1 (a + xf l—x-{-x' \ ~ax + hx"^ a^ — (b — c)' 22. Show that ^t.: ~ + anal. + anal. = 1. 28. Show that + (a + by - V- a* + ¥ (a—b) (n—c) (a—d) (b—c) (b—d)\b—a) c* d* = a + b-i-c+d. + 24. Show that Z {c—d) (c—a) (e—b) {d—a) {d—b) (d~c) bed (a — b){a — c) (a — d) = 1. 25. Show that 2" — -^--"II^- =. ,?-^ {a — 6) {a — c) 26. If a + 6 + c = 0, show that ^ — = 1 2a' + be 27. \ix + y = m and x-y = v, sliow that ^±1-' = ^(^^^''1 x^-y" ti {n'' + Sin') 28. U X + y = a and x — y = b, .show that a* - ^ g-^y + 2xy^ - y* 1 iSab' 29. If xy = oh (a + 6) and a^' — xy + y- - a' + b' th(>n (M)(f-D=«- m. Prove ^^!-^- ^^ + ^' ^' ("^ - ^ ) + ^'*' ^i? - ft) _ , 31. Ux + y = 2a and x — y = 2b, show that «:♦ - "23a;y + y* (7a^ - 37^76'^^^») ' w !r*r Oil AFTER Xil. 4 'I, 31 ill" .iltl. ■Ii:ii ■fi, MISCELLANEOrS FRACTIONAL EQUATIONS. 135. ^\''' sliall now consider sinipl*' ('qniitiows involviiisj: frac- tions of pvator difficulty than those found in the (M|uations of Chapter VI. We saw that the general rule for cleaHnrj of fran- tions is : — Mn/ffj)!}/ both sides of the e(i>iation by tJiv L. (\ M. of all the (lenoininafors that orrxr in the e like numerators, senarat ing.the numerator as a factor, ~ n.i^^98 ~ :rirTi:)" ' ' U- Z^4 - ^--50) • .-. 21 y'^^^T^+j^^i ^ :i y'jz^- •'• + 94 ) ( U - 98) (.r + 44 ) f | ( o^- _ 94 j (T^l- 52) i" ' iliat is. 71 X 42 {X — 98) U; + 44) (x — 94) {x — 52)' where the numerators arc equal and, therefore, the dt;nominator,s also, that is or or (x - 9S) (,r + 44) = ix — 94) (x — 52), .^■' - 54^' — 98 X 44 = x- — 140,r 4- 94 x 52 .•. d'ix = 98 X 44 4- r)2 X 94, 2lix = 9S X 1 1 + 1 3 X 94 = 2300 .-. .r=100. (1) We might have got from 1, by division, ^ — 98 X — 52 /• Q« /»• ftoi A a i ,r-94 .c + 44 ' ^,r_y4)-(,/- + 44) - 138 ~ 3 *"• ^- ^+44 = 3 •■• ''-' - !•'»<> = .P + 44, and 2x = 200. Y"'9" WMkMMiiMMMIHII i. ::i!lli'^ 'i: ■lit. li 11 :;;:'' 111, '11 '"""I 190 Ex. 4. MISCELLANEOUS FRACTIOXAL EQUATIONS. ;r — 9 .r + 1 .)■ — 8 X + + x — 2 X — 7 X — 1 X — Horo ooniplcto the divisions, thus, 1 + .r — + 1 - O X - 7 o. or ic — 2 a; — 7~a;— 1 a; — 6' where 2 divides out. Coml)ine, x-l — (.r - 2) _ x-C ^-( x- 1) (a;— "2)7a; — 7) ~ (a: — Ij (J? — 6) or — 5 (a; — 2) (a- - 7) {X — -[) {X — ii) ' where the numerators are e(iiuil, and therefore the denom nai;ors are etjual, or (X - 2) (X - 7) = (X - 1 ) (x - 6) ; or x- — 9x + U = x"" — 7a' + ; .-. —9x + 7x = - 14 + «, and a; = 4. ^ r)x — 8 6a; — 44 10a; — 8 a; — 8 Ex. 5. + ^ ■-- = -• x — 2 X — 7 X— 1 X — 6 Here compU'te the divisions, thus 2 '> 5 + :, + - X v-7 — 10 — Simplify, and divide by 2, .;■- 1 + — X — a; _ 2 ■ a- — 6 a; — 7 a; - 1' m X? — Hx + 12 8a- + 7 ; .'. (8 — 8)a-+ 12 = 7, ?. e. 0-a-= -.5. which can be true only if x is infinitely great. If we combine the ^rftctiona differently we get EXAMPLES — EXERCISES. 191 1 1 1 1 or ,U1(1 x~2 ./ — 7 aj' — y.r + 14 o X X - 1 .r - (5 ' X' - 7X + «) ; -8, a; = 4. N. B.— The stiidont will learn from the abore examples that labour may be saved by combhiuiy frmMom in pai?-s, changiny f/iem to mixed numbers^ etc.. lenoiu naiiors EXERCISE LXIII. in.) Solve ^e combine the 8. X x-\ x — 2 5 _ 19 2 7j^— 1 • X X— 1 + "-' = X — 8 ■ 2 X— 1 X — 2 X — 4: 2 v-4 -2-*- 3 r-5 a; — 3 X r.-8 a; — 7 X a; + 1 _ a; 2. c-1 c-l- a:' + 3 ;(1 « + 2 10. II. 12. X 7x ;r + 8 X + a x-b~ X a x — b a-b x — b~ a + b X + 26 1 1 a + bx b + X X 13. "r + _:r_ = ax a a b ~a a + b 14. 15. 16. iia + X X - = .-5 + X + -r ic — a x — b X X X + d a-\-c a -\-b ■\-o = 0. >-T- If 1, : jit;.! ' w -. . f ^^ "iiiiiiu •■"!! .■..|[ Hill ilii''- :"ii«i!i: Iff- 7 ("••«» 'Ml ' :! Itffi [ill MISCELLANEOUS FRACTIONAL EQIATIONS. 17. 18. _a 6 j t(a^~ l'^) X — a 6 a- _ 6 a;'"' - 6" \x 8 ^ 2.C' — Tia- + _ .r" — 7j^ + 5 a;-l a;^ 19. : 1 « + 1 30. 2a;'' — 7.i' + 3 "" jc' — 9a- + 2 12a; + 10a 28a- +117^/ ,„ 3a; + a 2.r + 9a -^1 + 1 - =0. 4a;» - 1 4 2a; + 1 ax + b ex + d X — m 31. ^^:^— !-" + -"-^-" = rr + c. 80. 01 22. 23. 24. 25. 26. 27. 28. X X -\- \ 37 a-"' + .*; X — >A 33. '82,r+'+?i+.f = ,2. 3a; + 1 a; — 1 37 33. ?/> + // X ■{- a ' X— b 2a; + « 3a; — « 34. -r^ — + ^ , m -f II x — c =2\' a; + 2 "*" .r 4- 3 a-"'' + 5a; + 6 ' ' '3 (x — a) "^ 2 (a: + a) " 7 _ 6a; + 1 3 + 6 a;'' a;— l~a;+l a;" — I 6 4 (^ - 1) _ 3 a; — 2 a;' — 2a; a; a b _a—b X — a x — b x — c x — 2 X ~ 1 _5 2ir4^1 "*" 3~(a^T) "~ 6 " „^ 5a-^ + X - 3 7a;' — 3a; — 9 5x 37. X + 1 a: + 4 9 20 ■ X — a _x + a 2nx a — b~ a + b a* — b^ Gx_j- 7 _ 1 5a;- 5 '9x~+Q ~ 12 "^ 12a; +~8' 2a; + 5 _ 2a; + 1 _ 5^+ 3 5a; + 2 "~ ' 4 (a; + 2) 5a; +13 1 -a; 3 -3a; i-3a; 39. 6 4 (X - 2) 3 a; — 3 x'' — 4x + 3 x— .. a; + 3 2a- + 5 ^ ^a- + 3 ^a; + 1 2a; + 4 f a; + 1 1 Solve : — (6.) *a; + i3a; + 2 a;+l 23 EXERCISES. 19;j -bx + 'i' 7x+ 1 ~x—l 9 \x + 2/ + 28 — t 9 f 9a = 18 -. a + c. mx + rt + 6 _ WW + a +f! wa; — V — d ~ ttx — b — x 1 "*■ 2^+4 "^ aTTlt 2ar^ir4 " *« - 4 ~ x 'V 6. rt a — b b + ex a + ex c + ex „ a + b a + e b— c . . . 7. r X X 4- i (x — a)=z 0. a — c a — c o + c AT)V'A>rrET) RXRRCTSKS. 105 ' '- + + X -V a X ■\-b X — a + . A , = \x V — f, ./•'•' _ 1 (a' ^ f,i) I 13 — n.r rix + 2 H-x 9. .+-7 1 4 Hx + 20 12x + !}0 O.r + lii li 10. ^ \.r-l(.i- + r)l=: (t [^6]^"^■'■""*i 11. 12. 13. 1 ^- 1 + 1 + 1 Ax X + \ a; — 1 X ■{■ 8 .r — a (X — a) (a; — r) (>0 + b(p— m) x—n x—2) x — q _ a (w — q)+b(p — q) X — m 't^ IH R;!*' lip '<■ '1; \ :!l|L.: '■I Ml n ■ ,.j|r.,. '"•!'l .:";""l"'i| lOfi MIHCKLLANKOCS rUAcTIOXA r> KQt'ATION*S. 21. - m (a — h)+<' {in + ii) n (a — h)+r (m + n) X — a X — h ni(a — h) N m — f>) X — (a + 0) X — (h — r) oo ) + ) ~ (>' _ *«. "~ 1 — ' — X — (I X — 2! X— ((I + r) ./• — ift — r) \n — p — (// \n — o /> - 7/ X — 2a X ■— 2/» ,r — 'Hr, 24., + ,+ r ='^- + V — r train, >.) and V == tinu^ taken by slower 1 rain, and this is two hours more 3.) then the otluu", /. e. X . X X X 25 " 35 ■ ' 25 35 2 .-. 2.r = 350 and x = 175. Ex. 2. A number has two digits, of which that in the tens' place is 3 more than the oilier, and the number itself is 7 times the sum of its digits. Find Ihe number. L{^t X denote tens' digit, then x — 3 — units' digit ; and \{)x + X — 'd is the No. .'. Wx + x — '.i = 7 (x + x — 3), or lla^_3 = 14.i^ — 31. .-. 'dx = 18, and x = Q. .*. the number is 63. r:XAMi'r-Ks woukkf) out. 197 ^- faster train, () lioiirs inoro ' (liy-it ; and Ex. 3. 2850 acres are divided helwecii A, B, and (', so that A's share is to IVs as (i: 11, and (' lias ;50() acres more than A and H logellier; find eacli man's ^hare. Let O.r denote A's sliare ; tlien 1 l.r will (h'note I5's share, and ('.,/• -t- 1 1-r + :}(>0 = 17.*; + ;{()(» will denote (!'s share. .•. (i.r + 1 l.r + IT./' + :}()() — 2m) ; .-. .r = 75, ami A's share = x 75 := 150, etc. Ex. 4. Thoro are two lines of railway between the towns A and B, one ;}() miles l(»njj:er than the other. A train i)roceedin^ hy the longer route loses one-tenth of thc^ tinu? occui)ied by the joui'ney. Another train })rocee(lin^ by shorter route at a rate which is to that of tho tirst as 21 : 20, loses one-lifth of tlu; time ()ccui)ied by the journey. Both trains accomplish the distance in the same, time ; tind tho distance !)etween the towns. Let X = tho No. of miles in shorter t X = No. who vot I,-!' 'iiit; m!. !i 1 14 til '1 ■'X ■;■ f; ...III; .iii{ ■„,, .'Mf EXERCISE LXIV. PROBLEMS. («) 1. A farmer took a certain number of er of years ; for if of the time it ])rought 3f per cent, for I of the time 3|^ per cent, and for tht' rest of tlie time 4 per'cent. For how long was the capital invested ? 4. A student having swved scmie money attended a High School for a year ; lie ex])ended one-sixth of his savings in books, $247 for board and other expenses, and at the end of the year he had still oik^ per cent of his savings. What sum had he saved ? 5. A boy agreed to work a year for a dollars and a suit of clothes, but after m months he was discharged and received only b dollars and the suit of clothes. Find the value of tluj suit. 6. A person has only (f hours at his disposal ; how far may he ride at the rate of b miles an hour, and return on foot at the rat«; of c miles an hour ? EASY rivOBLtMS. 100 7. If the half of a cortain num1)er be subtracted from 468, and the remainder be taken from 185, 79 divided l)y the result will give a (luotient lf|^. Find the number. 8. There are 2 foremen, 19 mechanics, and 12 labourers engaged at a certain work. Each foreman gets $1.25 more than a mechanic, and each mechanic |1.25 more than a labourer; the total earnings are $111^ a day. Find the daily earnings of a mechanic. 9. A number has six digits, the last digit on the left hand being 1 ; if this digit be brought to the ri{j}d hand, the resulting nuni])er is three times the original number. Find the number. 10. A house cost 3 times as much for nu'terials as for labour ; had the materials cost 7^ per cent mori'. and the labour 5 per cent less, the cost of the house would have been $8350. What was its actual cost ? 11. A farm is let for $384 and the value of a certain number of bushels of wheat. When wheat is 1.18t^ a bushel the whole rent is 15 per cent lower then when it is $1.75 a bushel. Find the number of bushels of wheat that forms part of the rent. 12. Th(5 difference of the squares of two consecutive numbers is 181. Find the numbers. 13. The length of a room exceeds its breadth by 4 feet. If the length were 5 feet more, and tlie l)r('adth 5 feet less, the ar(>a of the room would be 85 feet less. Find the length and l)readth. 14. I invested part of $2000, and gained 10 i)er cent on the invest- ment ; l)ut I lost ()|^ per cent on the remainder, thereby reducing my gain (m the whoh' \\ per cent. Find the i)ar- ticular investnu'uts. 15. A number ccmsists of 6 digits, tlie right hand digit l)eing 2 ; if this be removed to the left hand, the resulting number is only \ of the original number. Wliat is tli«' numl)er ? 16. Three boys have each the same amount of i^ocket money ; the first would spend his in 76 days, the second his in 95 days, ■•p«* 200 MISCKLLANEOUS FRACTIONAL EQUATIONS. ,, ,11. » "■■ ■ . V ji .1111 1 1 It'll »!l ' .,.1.11 M. li ; ■'I'l 'tli^. '■I 11 »it|ili itH), ^ and the third spends 12 cents per day more than the second; their united spendings for 20 days would amount to the entire pocket money each possessed. Find this amount. 17. A man liought a number of eggs at a cent a piece, 6 times as many at 10 cents a dozen, and 4 times as nuiny at 6 cents a dozen, and sells them all at 88 cents a iTundred, then^by gaining 84 cents by the transaction. How many eggs did he buy ? 18. The fore-wheel of a carriaf,'e makes 04 revolutions more than the hind-wlieel in travelliii<; a mile. If tlie circumference of the fore wheel be ^ of the circumference of the hind wheel ; find the circumference of each wheel. 10. The sum of $17000 is divi(U'd aunrng A, B, C, I), and "E ; B is U> hav(! 1^ times as miu-h as A less $;}00 ; C is to have f of what A and Ji get, and $11:} besides ; 1) is to have ^ what A and C gei", less f of B's share ; and E gets I of what all the others receive, and $027 besides. What sum d cents left in my right. How much money have I in my right i)ocket ? if*) 1. A water-cistern was filled to a certain height ; -jV of the contents and 40 gallons were drawn off ; then a quantity 20 gallons less than ^^3 of the renuunder was pumped into the cistern, and of what it now contained j'r, all but 20 gallons, was drawn off ; the cistern now contained 700 gallons less than it contained at first. How uumy gallons were in it at first ? 2. A man had a number of iicres of nu^'idow land ; he exchanged ^ of it for a certain quantity of vineyard, } of it for wood- land, ;ind I of it for arable land, and he now finds himself possessed of 574 acres in all. If 5 acres of meadow were worth 3 acres of vineyard, and 6 acres of vineyard worth ADVAXrED PHORLEMS. 20 1 ^re are cents 10. 25 }ien>s of woodland, and -i acres of woodland worth 4 acres of arable land, how many acres of meadow had the man at first ? ' I have two eqnal sums of money to pay, on(^ in 9 months, the other in 15 months. 1 {)ay both debts immediately, one by $1208, the other by $1 100. Find the sum of money and the rate of discount (bank discount). If a man ^ains n per cent when he sells an artich^ at the price //, how mu(th i)er cent will he ^ain oi- lose when his sellinr gallon, gaining OOf cents. Of how many gallons was the cheating milkman cheated when he l)ought the milk :" A trader allows $500 for expenses, and augments that j)art of his capital which is not so e.\])ended, by .', of it ; at the end of three years, his stock is doubled. What had he at first i Two trains leave A for B at an interval of 2i houi-s, at 15) and 3S miles per hour, respectively ; on a cei'tain day the shtwer train was (h-layed 50 minutes on the way, and reached B only 10 minutes before flu; other was due. Find the dis- tance l)etvv(H!n A and B. A man sjM'uds the ///th of his income on l)oard and lodging, the /;th of it on clothes, they>tli of it on amusements, the qth of it in charities, and at the end of the year has $/• left. Find his income. -^aMMiM 202 MISCELLANEOUS FRACTIONAL EQUATIONS. \' t'Mh 1 ■. W-'ill. -»- .1. , 1 11!: .",;< , il^ nil. il' ; * S; '' -.rt, 11. A nuinher l)as tlirce (limits, tlu? hiiiidri'ds' iind the units' digit being each luilf of tlu; tens' digit ; if tlio tons' figure be interclumged witli the hundreds', the resulting numl)er will be greater by 360 than the original number. Find the number. 12. Find thr«^e consecutive numbers, such that a third of the greatest may be 3 less than the other two together. 13. A person bought a lot of land for !^5400, from which he cut off four-ninths for himself. At a cost of .f'2()() he made a road which took one-tenth of the remainder, and then sold the rest at the rate of 10 cents a square yard more than double what it cost him, thus clearing $400 ])y the transaction. How much land did he l)uy ? 14. Fifteen guineas should weigh 4 ounces ; but a certain parcel of light gold having been weighed and counted, was found to contain 9 more guineas than was supposed from the weight, and it appeared that 21 of these coins wcnghed as much as 20 true guineas. How many were there altogether? lo. A man rides one 7>th of the distance from A to B at the rate of J. miles an hour, and the n^st of the distance at tlu^ rate of & miles an hour. If he had maintained a uniform rate of b miles an hour, he could have made the distance from A to B and back, in the time occupied by the single journey 1 1 p Prove that (( ;% 10. A numlHT has four digits, of which the two middle ones are e^eh zero, and tlu^ right hand digit is half the left hand digit. Prove that the difference of the scpiares of the num- J)er, and of thc^ number formed by inverting tlu^ digits is 749,999:^ times the square of the left haiul digit. 17. At an army review the troops were drawn up in a solid mass 40 deep, when there were just ^ as many nu'U in fnnt as there were spectators ; had the depth been increased by 5 and the spectators drawn up with the army, the number of men in front would have been 100 fewur than before. Find the force of the army. ADYAXCED PROBLEMS. 203 18. A certain nnmbor is divided into a, and also into n + 1 equal j)arts; the produet of the it vqui\\ parts is // times the pro- duct of the n + 1 parts. Find the number. 19. A farmer buys m sheep for ^p each, and sells n of them at a Rain of 5 percent.; how must he sell the remainder that he may gain 10 per cent, on the whole ? •20. A passenger train which ought to run a certain distance in 2^ hours, is G minutes late at the end of 80 miles ; by in- creasing its speed to as many miles an hour as tliere were miles in half the whoh^ distance, it arrived at it.s destina- ti(m on time. Find the speed of the train at first, and the distance run. 21. A student was working with four innnbers in proporticm ; thinking to nuike the work easier, he subtracted a certain number from each term of the ])roportion, and .so derived the false proportior 41 : 93 :: 7 : 51. Find the true pro- l)()rtion. 22. A man has two kinds of tobacco, which cost, respectively, m cents and n cents a pound ; he wishes to take some of each kind, to make up a ruantity of a pounds, worth p cents a pound. How many pounds of each sort must he tuke? '2o. What value must ))e given to x so that the product {a- + ah + h-x) (a' - ab + b\r) may be equal to (t^ + 6V^ ? 24. A man wislu's to set a numlx'r of vines in a rc^ctangular field of which the length is to the; breadth as 7:5; by planting them a certain distance ai)art, he has 28:12 vines reiuiuning ; but if he plants them nearer together so that there will be 14 more on the longer side, and 10 more on the shorter side, there will be only 172 remaining. How many vines had he ? ^ ■> .III K ir 1", 1. .,•»..»..,. 1 1-.fi 1 t 1{ I::i:ii ' Aiiilj! R^ . . li ' .. ..11, ■1 1 .1 ,..., Ii, ! 1 1 ■ >. f 'i,;v:;r3i;ii| - <•>'}<, CHAPTER Xlii. SIMULTANEOUS J:QUATI()XS OF TIIK FIRST DEGREE. 137- In this chapter we shall consider thing unlimited. I N I) ET E U M I X A T E I» KG B L EMS. 305 Ex. 3. Twice A's age increased by three times B's age ia 50 years ; find their ages. HIEE. of simple ?:mknt. rst number econd. > problem is ' first num- d number." r, etc., etc., r, etc., etc. luaticm ARITHMF/nCAL STATEMENT. Twice A's age + tliree tunes B's age = oO. A's A's AUJEBUAIC STATEMENT. 2X + lit/ = 50, where x represents A's age, and y represents IVs age. It is clear from the arithmetical statement, that ani/ positive number may be taken to represent A's age, that 50 minus that number will represent B's age, and that, in every case, the sum of twice A's age and three times B's age is 50 ; so, various values may be given to x in the ecjuation 2x + ;3// = 50, each of which will give a c()rres[)onding value to ij. Thus, A's ag(! is 1 wlieu B's is 10, B's '' 15^, B's " 141, etc. This is exj)re.ssed algebraically l>y saying that the ecpiation 2.r + ;{// = 50 is satisfied by any one of th(! following i)airs of vahn^s : — x = 1, // r= 16. X = 3, IJ = 15^, X --- 3, y = 14f, eto, 138. From a study of the above problems wo learn :— 1°. That a siufjle equation inroIriiKj the first power of two unknown quantities expresses some one niatio)) between these (luantities. The ecpiation in Ex. 1 gives as a relation the sum of two quantities ; that in Ex. 2, the sum of twice one (juantity .'tml three times another. 2°. That one relation between two qutintities w not snffieient to determine definite mlues for these (luantities ; or. in other words., 3°. That there is an indefinite number of solutions for an equa- tion which expresses but one relation between twg unknowt} (pfqr^tities. mmm 20G SIMULTAXEOUS EQUATION'S. EXERCISE LXV. ^jfii* ■ii" ... .In, I, , "» lit, I*. I'll!' '1 11 MV, :itl "^J: •- '. 1. Writo equations which express thc^ rehitioiis ^ivcii in the following : — (I.) The difference l)etwoen John's age and William's age is 5 years. Let if and y represent tlieii- ages. (2.) Three times onc^ nnml)er togetiier with one-half another is 22. Let X and >/ re}) resent the nnmbers. (3.) If A's money were; increased by $3. 00 lu^ would hav(; three times as much as B. Let x represent A's money and y B'8. (4.) If A were to give $5.00 to B, they would have ecpial sums of money. Let x represent A's money and ij B's. (5.) Seven years ago th(^ age of a fating* was four times, tliat of his son. Let x and >/ represent tluiir ages tfofc. (6.) A certain fraction ))ecomes 2 wlien 7 is added to the numerator. Let x ^'present the numerator, y the denom- inator. 2. Express in words the relations givcui by tla; following equations : — (1.) 2x + ruj = 7. (2.) 3.r - 5// = fi. (3.) 7x - 8y = 5. (4.) 3a; H- 7 = 2y. (5.) S — dx=i/. (C.)? + ? = 10. Hi ■ 1 1 .r.A 11 ■: ^ :( 3. Find the value of x which will satisfy the following equations for the assigned values of y : — (1.) X + y = 10, when y = 1, 2, 3, or 1^. (2.) dx + 4y = 10, when i/ = (5^, 2, or 3^. (3.) ^x + iy = 6^, when y = 5, -,V- or 2^- , . , ir -1- 17 68 , (4.) — - — = X + , when w = 3. 4 • A' + y \o.) X — ^—j- = c, when y = h\l -\- nh — he -f- a m-'^. * ■ :, i ! INDEPENDENT EyiATIONS. •207 V(>n in the Ill's aj:;*; is nnothcr is monoy uiid (Hjual sums iiies that of 4 Find the ^'aluo of y which will satisfy tlie followin^^ (.<|natioTis lor tho assigned values of x :— (I.) x—"^ j^j^-— _ 20 , when .r = 21. }).ry_ no 151-1(1^ •'>•) T. . H :; -r- — 3a;, when x = J). By -4 4y-l (4. ) //«—•?=:_, when X— na + mh X X 139. Ex. a. The sum of two numbers is 9, and their (lifferenc is 1 ; tind the numbers. eneo ded to tlu^ ' the denom- e foUowinj; Ix — H// = T). %^ = 10. 2 5 ng equations ARITHMETK AL STATEMENT. 1st No. + 2d No. = 9, 1st No. -2d No. = 1. AUiEBRAir STATEMENT. X -h y = 9, •'• - /y = 1. It IS evident from the arithmetieal statem(>nt that, althou-h there are any number of pairs of numbers whose sum is 9 -ind any number of pairs whose difference is 1, there are onllj \u'o numbers, 5 and 4, whose sum is 9 and dilTerence 1 ; hence, although there are an 'indefinite number of pairs of values which will satisfy the equations, a- + y = 0, x-y= 1, when each ecpiation is considered by itself, yet there is only one })air of values, x^n, y = 4, which will satisfy the equations when the mme pair of values has to satisfy both equations. Equations that are to be satisfied by the same values of the unknown quantities involved are called Simultaneous Equatic's, r' ^" IIIP 208 SIMULTANEOUS KQUATIOXS. i S 4t : <• )» •'. ;^" 2S '1.1 l„ j..„,r ' 1 '■■ "* 'ul. < iljiiil; 1 ill U ' ' . 'i 1 '--I'lii; f Tf f 1',; 1 '■•il ""'■"TK 1'[ ■"•"""!.';; 140. Tt has b(Mni already noticcsd that each ('(juation involviiif» t\w first power of two unknown (Miantities expresses some oik^ relation between these quantities; notie(^ now tiuit the above e(iuations, x + y = 9, x-y= 1, express different relations between the same two quantities, the on(? expressing the sum^ the other, the difference of the (luantities represented by x and y. Equations that express different relations b<'tween the unknown quantities involved are called Independent Equations. The ecjuations 2x + 'iy = 8, %x + 2y = 4, are independent equations, because neither of the two given rela tions «in be inferred from the other. But the equations 3.t- + 2y=rlG, x+ y= 8, are not independent, because they express the same rehUion between the quantities involved. The second ciquation can be derived from the first by dividing both its members by 2. Hence, we infer that, in order that it may be possil)le to deter- mine definite values for the unknown quantities involved in simultaneous equations, thei'e must he as many independent equa: tions as there are unknown quantities. 141. Elimination. — The values of the unknown quantUies in- volved in simultaneous equations are determined by combining the equations in such a manner as to cause one or more of the unknown quantities to disappear, and thus to obtain a single equation containing but one unknown quantity. This process is called Elimination. There aro four general methods of elimination. I. By Addition or Subtraction (Cross Multiplication) II. By Substitution. III. By Comparison. Ty, By Arbitrary Multiplier^, KMMlf^ATroK — F'Ot^ll MKTHOnfl, m nvolvinn ,()iuo one 111! abovo titles, tlui (liiantitios unknown (/' mn rela ion can bt; e to detor- ivolvcd in itdent equa- antUies in- nbining the mre i>f the {\\\ a single s process is Hon) Wo sliall illiistra1(^ thewo methodft^ first in the ease of tiro unkiMwns, and then in the case of three or more iinknotrns. 142. L Addition or Subtraction.— Th«* following sohit ions will ilhistrat»i the method of eiunfnation by AdilUion or JSnbfrai'fion, Kx. 1. Solve x + 1/=\0 (1) a^-y= 6 (8) M) + (2) ^'ives 2x = 16. (8) Thus ail ('(juation has been obtained whi(!h does not eontain >/, i. e., y has l)een e/hninafeif bi^tween the equations (I) and (2). The value of .*; can be found by solvinj^ eciuation (8). 2x= 16; .-. a; = 8. The value of y can })e found by substituthnj this value for r in j'ither of the given eciuations. Thus a; + y = 10, or 8 + y = 10 ; .-. y = 3. The value of y could also be determined by subtriwthtn. Thus (1) — (2) gives 3y = 4 ; .-. y = 2. Ex. 2. Solve (1) (8) y=n- Ax + 'iy = 42 4a; - 7y = 30 ,1)- (2) gives r0y = 13; Substitute this value for y in (2) ; .-. 4a; — 7 X 1^ = 30, >v Ax = 38f ; .-. X = 9f . Notice that the equations are to be added to eliminate a quantity, when the coefficients of that quantity are alike in value nut different in sign ; and that they are to be subtracted^ when the coefficients are alike in value and in sign. EXERCISE LXVI. 1. Define Simultaneous Equations. Writ« simultaneous equa- tions which express the relations given in each of the following : — i. The sum of two numbers is 11 and their difference is 1. Let x represent one number and y the other. ; m )i\0 KlNfri/rAM'XH'S KQTATlONrt. I- S lb h ti« ... if,: 1 ui] - 1 •..., '« 1 1 '».„ fMit 1 ^ ■ .,' "" l:iiil 'tlj-- ;',• ^^■». •■H ii. The slim of 7 limes tlic jiro.'itcr of two nnmbcrs atul i\ times tho IcHS is 27, and the |»ro is 5. Ij<;t (C represi^iit the ^^reater and y the; less. ill. Tf to \hv. sum of th(^ ajjes of Iwo persons 18 he added, \]u) residt will l>e douhle the a^e of the elder ; hut if (> he taken from tlie difference of tlieir ajjes the reiuaindei" will he tlu! age of tln( younger. Let x represent the ag«! of the elder, y the ag(( of th(! younger. iv. If the numerator of a fraction l)e douhled and tlie denomina- tor in<*reased hy 3, its vahm will Ix; | ; ttut if the (h'uominator be douhled and the !iumerator increased ])y T), its vahu; will he ^. lict iV repres(ait the numerator, y the denominator. 3. What is meant hy Independent Equal iix — 5// z= 4, iv, 4x—7y= 6, 4x - 'Ay = 14. i^x + 4y=14. 2x + ;}// = 20. 2,r — !} • 5^ = 3. 3. DefiTK! Elimination. Eliminate x l)etween each of the follow ing pairs of simultaneous otpiations : — 1. X + y = 12, y~x = i. ii. Sx — 2y = 5, 3a; + 3y = 15. iii. 3y — 7^:! = 8, 7x + y = 12. iv. Gx + 3y = 27, 2y + dx = 26. 4. Eliminate y between each of the following : — i. 3a; — 4// =13, ii. 4x — 3y = 6, iii. 7x — 4y = — 1, 5. Solve : — i. X + y = 4, ii. 6x + '6y = 27, iii. 17x — l'6y = 2\, iv. Ibij -2x—4'6, 2x — 4y = 6. — nx + 3y = — 9. 4y — 2x = 6. x — y=l. 2y + ()X = 26. 2a; + 13^ = 17. 4a; + 15y = 49, ELIMIXATION HY ADIUTION. 211 (\ :i timcB uto r» Irt 5. »(l(l('(l, llm i; be tiikcn till) older, (Iciioinina- nominator valiH! will iioiniiiator. licli of tlio the follow 143. If IIh' cocnicinils of the (|iianlity to he clinjinatcd be not equal, tlicy can bi! made so by ciihcr niiiMiplyinj,' or dividing,' one or both ('(Illations l)y suitable tiiinil) (IcttTtnincd by inspection. Or, to h'tf I'se. rocjjiiucnts will give the required Kx. .'1 r^olve ^x + ni/ - 8, ;j.*; + 7// = 7. Multiply (r, by 8 aiul (2) by 2, and we obtain (8) _ (4) givjs 0^+ 9.y = 24 iix + 14 // --= 14 - Hy = 10 ; (1) (8) (4) • • if — '*'• Substitute th.i^ Value for y In (1) ; .•. 2u' - =- 8 ; .-. x=7. Ex. 4. Solvu 12a; + 15// = 39, 18.r-17y = 19. (1) (2) L. C. M. of 12 a:,d 18 is 36 ; 30 -=- 12 = 3, the multiplier for (1) ; and 36 -f- 18 = S, Iha multiplier lor (2). Then, (1) X 3 gives (?-) X 2 gives (3) _ (4) gives 36.r + 45.//= 117 36a; -34.//= 38 (3) (4) 79y= 79; Substitute this value for .// in dw .'. 18x-17=19; ti ii x = 2 212 filMULTANEOrs KQrAffOKhi. u m I .,. , «» -fJJ ■;. 1.111 144. Sometimes the given eijiiations luiiy be added and sub- tructeu in order to obtain easier forms. f T '' Ex. n. Solve 2x + 3y Sx + 2!/ = 12, = 13. « (1; + (2) gives 5x + 5// = 25 ; ¥ 4* ■* •^ or, (1) - (2) gives (3) - (4) gives x + y = r,, = -1, %ai = 0; :. .r = 3. (^} + {i) gives = 4; • ••• //• = 2. EXERCISE LXVII. Solve, by Addition or Subtraction : — \. Zx — y — 1, Zy — x — Z. 2. 2a; + 3^/ = 0, 8. 4a; — y = 1, 4. 3a; + 2^' = 23, 5. 4a; + 5^ = 1, 6. 9a; — 2y = — 24, a; — // = 1 0. 2x Jr'Ay — 11. 2.1; + 3y = 22. 5a; — 4y = 32. 2a; — 9?/ = — 31. (1) (2) (3) (4) 7. 32a; — 25// = — 18, 24a; + 2Qy = 76. 8. 15a; — 16y = - 3^, 12a'— 13y = l^. 9. 18a;— 19// = 9, 16.r — 17y = 7. JO. 196a; — 208// = 45 », 208a; — 196^ = 84f 145. n. Substitution. — The following solution will illustrate the method of eliiuination by substitution. Ex. 6. Solve 3a; — 4// = 2, 7a; - 9y = 7 (1) (2) SITBSTITUTIO.V— (OMPAllISOK. 213 and sub- (1) (3) (3) (4) From (1), 2 + 4y r. X = 3 Stir^stituto this value for x in (3); or 14 + 2S// _ 27// - 31, ••• // = 7. Substitute this value for y in (3); .-. .^^4^^ = 10. (3) EXERCISE LXVIII. illustrate m m SoJve, by Substitution : — 1. 11a; -9?/ =-5, 2. 15^: + 6// = 16, 8. 2\x V Sy = 47, 4. 7.t' + 0// = 5, 5. 13^- — 7t/ = 20, 6. 2u; — 18y=:21, 7. 9a;— IQi/— — o, 8. 2U- + 8// = 50, 9. 20a; + 12y = 44, 10. 9a; — 15^=23^, a; + 3y = 8. Tix — y = 10. 3a; + 5y = — 1. 6a; + 7y = - 5. 7a; + 3y = 40. r).f + 3y = 13. 6a; + ny = 28. 14a; - .5^ = 33. 13a; — 3()y = 108. 13a; - 7y = 36f 146. m Comparison.— The following solutions will illustrate the method of elimination by mmparison. Ex. 7. Solve 3a;-2y = 1, %y - 4a; = 1. (1) (2) 214 snirLTANEOUB EQUATIONS. m ♦ ' ¥ !i| » ijiii;. From (1), From (^), X = 8 4a; = 1 — 3i/ ; _3y-l a5 l3) (4) But the values of x obtained in (3) and (4) must be equal, that is, 1 + 2y 3// - 1 or 3 4 ' 9y-3 = 4 + 8y; .-. y rr.: 7. Substitute for y in (3), and 1+14 ^ "1 -KaiBi EXERCISE LXIX. Solve, by Comparison : — 1. 18a; — 5^ = 55, 2. 12a--lly = 23, 6x + lOy = 100. 3a; -15// = 18. 3. 133uf + 119y= — 93, lla;-13y = 61. 4. 19a- 4- 15y = 31, 5. 11a- 4- 9y= 19, 6. 4a; + 3y = 1 iX' 21a! + lHy = 15. 9a; + ly = 17. 3a; + 2y = 1. 7. 5a- + 4//=: 1«|, H. 108a; — 91y^ — 0, 9. 65a; + 54y = 33, .0. 2a; — 3y = 17, 7x 'W =-h 120.t- — 104y = — 24, 39a-45y = 132. 5a; + 7y = — 9. (3) AUniTRARY MtLTIPLIERS. 15 147. IV. Arbitrary Multipliera— In this nu'tliod we multiply one of the given e(iu;iti()iis by an arhitranj )i\ultij>lier {in, say), add the resulting (H|uation to the other, and then «>quate to zero the eoeffieient of y to find x, or tiie coefficient of x to find y. Ex. 8. Ix — 4// r= 20 3a- 4- o// = 22 (1) (2) (3) Multiply (2) by w, .-. 'imx + r^my = 22m (1) + (3) gives (7 + 3m).r + (nni — i)y = 20 + 22w. (4) Equation (4) is true for a/l values of in, and then;f(n-e for 5/n —4 = 0, or in = ^ ; in which case the coeflficituit of y is zero, and e/>) ^{nn — bin). Similarly, eliminating x, ])y putting a + Z//i = 0, we get y — iap — cm) -^ (an — bni). 148. Literal simultaneous equations may be solved by any one of the methods of elimination already exi)lained. Ex. 10. Solve ax + by = m, ex + dy = n. (t) m Mi I i. w I t m: It*;:: (1) X f? gives (2) X n gives (3) - (4) gives (1) X (I gives (2) X f) gives (5) — (0) gives Si M U LT A N' liOUS KQU ATI N S. acx + bey = cm acx + a fly = an (be — u(l)y — cm — (til cm — (in adx + })(hj ~ (fill hex + bdy = hn (nd — Ix-yx = dm — bn dm — bit ad — be X = (8) (4) (8) (fl) N. B. — The value of x might have been more easily determined by symmetry, by interehangiiig the eoefficieiits of x and y ; that is a into b, and 6 into a, c into d, and d into c. Thus. V = X = f^m — a7i be — ad dm — hn ad — 6c Ex. 11. Solve a(x^ y) ^bix-y) = \, (1) a{x-tj) +b{x-\- y) = \. (2) (1) + (2) gives 2ax + 'Ibx = 2, or {a + b)x =1 .•.x = \ ^(a -^b). (1) - (2) gives 2ay - 2% = 0, Of {a -b)y = (); :.y = 0. EXERCISE LXX. \'% Solve ;— 1. a; + y — a, 3. .-c + % = l, 3. mx — // // = ^>, 4. aa; — 6y = , '2h,v — (tjj = 2f/' — :ja' 7. aa-, + by = 2, b.v + ,i,j .-=. 2. 8. {m + n)x — {tn — ii)y = 4mn, (m — n)x + {m + n)y = 2 (ni + n) {m - n). 9. {a + b)x + {!' - b)y = m, (r- + d)x + {a - tT)y = m. 10. {a — b + c)x + (b — c + a)y — 2^/', (a + 6 + (',)x + {c + a — b)y =r ine(l ; but sometimes it IS convenient to coml)iii(' them bef )re simplifying. The student should always b«; on the watch tor any legitimate method of shortening his work. < f I Ex. 12. Solve («) X ;}() gives (2) X 12 " (8) -(4) " X 5 n^= : 18 X 2 4 21 6u + ryy = 540 Gu — %- 2r)2 ay = 288, and /. y = m; x = 00. Ex. 13. Solve I (x + y) + ^ (x - y) = D i(^ + I/) -iix-y)- 10 (1) X 2 gives {2) + (3j " (8) — (2) gives or. iU' + y) +l{x — y) = \{) ^ix + y) = 20 X 4- // -- 10 r~ y = {) ; and .-. x = y = 20. 4 Kx, 14. Solve — : : + z-z.^_:j^ _ •{ ^ ^ 4.^- — 10 +-3^"::^^-^+ r' (1) (3) (8) (4) (1) m (8) (4) (1) (2> f^ US STMl'LTAXKorM Kl^l' ATfON'S. Jl 1 •■:• ' ^V■^ *^ % «■> \ *• ,"' >* ? 2 '•;:: ' *■«.:: 4 =1 ' • Taking toifctlicr all the fiactioiis in ( 1 1 li.ivin.;- only luiiucricai deiKjniinatorsi we get, ?,X + ny _ 1!} + G.?- + H — C x — 9 4a-(r 4 * 3a; + 5// ;5. 4.t — (5 .-. 12.r — 18 = 8.r + %, or 9.r — 5^ = 18. rs) Taking together all the fractions in (2) having only numerical d(;nominator.s, we get, 6x - 3.y _ 40 + H i/ — 18 — 8// — 7 "2y-8 10 ~' 6a; — 3,y _ 3 " 2^-8 ~3* .-. 1 '2./' - ai/ = (],'/ — 24, or w — !/=: — 2. From (3) and (4) we get y=-M); .r = 7. EXERCISE LXXI. (a) Solve : — a; y_ a; y _ ^•2 "^3-^' 3 + 2-^- 2. i (21 - 6y) = Vj ^« + ^), ^ (^3 - ryx) =^(y + 6). 3// + 4 2a; + y a; + 13 (4J a- — 2 10 — a -_y — 10 • "5 ' a "" " 4 ' 3 4. It \-^!/=.ni 3.IM fy = 4|. . 1'.' - W + Ifi _3a- + y + 2 8 32 5t/ — Sx = 13. IJ !J 3 — 2a; 4 + 5// _ imcncju '3) umcrical (4) ±1^ 4 MISOELLAKEOUS EXERflSES. (^ _ i) _ m-i/ + 8) = , ~:^:+ - _ "»•'/ + « V- / 9 2 319 = 0. 4// 8 -"-^^ -^-2 3 • 4 10. ')a; + 4y = 38i+i (3^--//), a; = 5/, - i i^ U+>/)-^, (.r-y)}. 12. ^a-+ i/ = (ih a b a a io. a H V =r 1 w + ni n — ?;/ n — m n + m ^ 14. ^ + f=l, X — ^; // — b a 15 16. K ^-U 3 18. x^- y 5' —-=3, x+ij 2x — 'Si/ _ 8 3,c — T/ = 1. _^x — 2t/ _ 7 2~+ 2.in^4y~9' 2 + .f _ 5 2-i/~6' (b) 'i- + X + }/ _ 9 1 +x — y 1—x y 7 1—x + y —.1. 1 + ^ — 1 1 + y = -3. = 1. Hi I I ifl m vis SIMUl.TANEOrs KC^CATIONS. a; _ 3 2x-n 3. 2x — 1 4x — 5 y _ 4 2y - 1 3y + 4 Cy - H 4. u; 2y-x 23 —a; = 20 — 59 — 2x a + X— IH = 30 73 - % 3 "~ ■ ■ .5 (.J" ■ ^^i:^ oil iHiij; ::f:r m. r: ill ! + §(// + 5) - I (7x - (\) = 10 - -tV (iJa* - 10 + 7y), jl(12-.r) 1 5a; _ ^ (14 + y) 8 a^ — y + 1 a; — y — 1 = m:s woijkkd ott. mt 150. SiimiKaiM'oiis ('(iiialions contjiinluK fractions, the dcnoin- !nat()rs(tf wliich aivsinii>h' expressions containinjr Uic unknown <|nantifics, arc ^rmcrally more easily solved by eonibinin^' the e(iiialions Ix'fore elearing of fractions. IJ Kx. IT). .Solve 3 4 (1) V X 4, sriv es l,Vi K 3, giv( X — = (J. 13 10 =20 X y X y m (8) (4) I (!? ) — \\) (rives — o U or y^ Substitute tivu% calue for y in (1), 1 =1-- .) .r = - 1 . Ecjuations \\\\Un ^'-o rj -f- 7// =r 1 0.r//. ;}.t ,- ^;:;:■ ); ** * "i '/ i .'5 !«:-; li "■>;:: ' ' V J . is, i:r iff" -. it^,' % ). •'... ; i 1 :. ^i P \ . *t .3 [•«M . t » *< ?.'■»■,' ^-1.-..;; 1 -■'1 .1 EXERCISE LXXII. S()lv(i: — 1 ^ 1-1 ■ ic ,y ~~ - ' 2. ' + ~ = 4, X IJ o. — ~ ti. X y 4. x + >/ = X!/, 5. au; + 2// =: 4x1;, 1 J' 1 1 11 !5 ,^ ^ + = 25. •^'-'^ = 4. 0- !/ X — >/ = 2xf/. ;").*■ + 4// — n.j/,. 6. l«'l+I3.(/^_, 9.r,y 4x — 5// _ 14iry "" 7. - + = 1, X y h a (I y = 1. 8. hex + (ihy = xij , (tc-x + hey — ,ry. ^ rv 1 1 - 10.;. ^'^-1, ox ay 1 1 :^.'r '^// ff to; 6y 151. Three or more Unknowns.— Wi; proceed in a similar way to solve e(i[uations in three unknowns. Eliminate any one of tin; quantities between any two of tlu? equations, and the same X 2, gives (4) - (1), gives 2x + Ay -\- Az = 18, 3a; + % ■\-2z = 17, Sa; + 6y + 5g = 33. Qx + 6,y + iz = 34, 4a; + 2^ = 16. (1) (2) (3) (4) (5) I 'i'ii:.'i;i; r\K.N(»\v\ S— i: X A M F' L K S— E X K RC I S EH. A.'iiiii. I !) X 2 " 1 ■-)./• + l.')// + 10^ — 85, l'i)-(7j " 10;c + 1:2// + 10^ v: 04. Tu' + :}//- '^1. From (5) and (H), .,; is foimd to hv :;, mu\ //, 0. Suhstitutif these values for .r jnid // in ( i > ; .-. (I + H + 4.?= IS, ()r^= 1. 1 1 Kx. '2. Solve y 2' — C^j gives .) 4- (4) - 1 1 = 3, and .-./y — i. SuJ)stitute this value for y in (1) ; ■f- 1 = 3, ora;=: 1. x 223 {7} (1) (2) (3) (4) Siibstituto its value for y in (3) ; ••■ 1 -f- ^=3, or^ = -. ^ 2 N B -The student should he careful to combine such equations as shall require least multiplying, or dividing, in preparing them lor elimination. i i » EXERCISE LXXIII. Solve:- ^''^ l.X-y + z = i^ y-2^X = i^ Z-X + y = A. 2.2x + y-z = U, 2y + B~x = 4, 2z + x - y = - 4, 3. 2x + 'Sy-4z=4, Sx-2y + r^z^o, Sx + y-2=^Q, 4. 4x-5y = 2n, 3x + z=z2, 2y - 3^ = 6. I 1 iff I li 224 SIMULTANKOrS K(/t'AT10NS. I 5. X — 2,1/ + )]z- \ «, 2.1' + Wjj -\z= - M2, ;ir + 4// - (1^ = 1 3. 6. 1 Ox + 1 2!l— V.\z= —\ r>, !>.»• + iSu—lz = T,, . ir.^+ I/-Z _i/ + Z — U' z +.f - u •ry = m {ay + />;»), //^ = ;* (r^ + ) (0) From (5) and (6), x=^, y=\- then from (2) and (3), z = {), (I — 0. Ex. 2. 7a; - 3^ + 8^^ = ■n, 4^ -2z -^ t = 11, 5^. - 3a; — 2u = 8, Ay- - 3w + 3/ = 9, 3^ + 3w = 33. (1) (3;. (3j (4) (5) mi t 1> f 1 ■* " ■ jfSSSgSt 220 2(3j — (4) gives 3(1) + 7(3) " SIMULTANEOUS EQUATIONS. Ai/ — 4z + Ha = 13. 35*/ — 6^ — 5« = 107. (6) (7) r-i* From (5), (6), (7), y, z, u may be foimd : .r = 2, ,y = 4, ^ = 3, Ex. 3. X + y + z — a, (1) y + z + u =zh, (8) z + n + X = (', (3) u + X + y = d. (4) Add all the equations, and we get ^ {x + y + z + u) = .,^ •*•;;*, 1. 7.r-3// = 1, Wz - 7/' 3= 1, 4z — 7// = 1, 19a; - 3^/ = 1. 3. 5.r + 4y - 8^- = 29, 7.r - 3y = 26, 2z + 9?« = 38, ll.r_6?/= 31. 5. 7;/ - I3i' = 87, lOy — 3.r= 11, \\n + 14.r = 57, 2.r — \\z =50. 7. .r - 2y = 3, y - 22- = 4, z — 2u= 5, ?/ - 2/ =r 6, ^ - 2j: = — 8. EXERCISE LXXIV. 2. X + y = 3, y ■\- z = 5, / + y/ = 7, .r + ?/ = 5. 4. 3?/, — 2y = 2, 5a;- 7.? = 11, 2a; + 3// = 39, 4y + 3^ = 50. 6. X + 2// + 3^ = 4, 3a; — 2// — r = «, 3a; — 'iz — 3r = 0, // - 4r = 15 8. X- 3//+ 3^- 4?/ = -10, -5.r+ fiy- 7^+ 8;/ = 18, 9.r— 10//-ll^'4-12^/=4, -13.c + 14^-hl5^-16y/ = -4. PKOBLEMS WORKED OUT. 227 (6) = 4,^ = 3, (1) (3) (3) (4) (6) (5} l\ 4, 6, 0, 15. - 4?/ = -10. + 12^^=4, _16 subistitnte thi.s valu(! f added to 4 tiin«\s tlu^ result to make it ecjual to the original number. What is the number ? Let .r, y, and x be tlie hundreds', t 1 I0,r + // — I01,r + 10// + TA, and 4(100// + ll./'j + 5) — 101,/' + 10//. Jiy reducing, (1) gives ,/• — // — (I. or ./• — (5 + //, and Vl) gives i;jo// !«),/■ r= - (1) (4) III (4) substitute the value of ■/■ from (;{) ; ij =. \^ x — 1^ and the number is 717. Pin E.\. 3. There are three numbei's such that th<> i>roduct of the first and second ('(juals 15 times their sum ; the j)roduct of the first ;!nd third = M times their sum, and 7 times the prodnct of th(! second and third ecpials 9 times their sum. Find the numbers. Let x^ y, ^, represent the numbers ; then, x\j = 3 {X + IJ), or, dividing by ;Uy, + = . , (1) // X A nc(r. ist alike ; Is' (limits, uil to the Ls' (lijjrits, (1 niiils. and tens. (1) (2) (3) (4) 7, and the net of the of the lirst net of the iiiny^ers. (1) IMIO B LE M S~ K X K KC 18 ES. xz = 8 {x + ^), or, dividing by 8^-^, ^ + ^ (l)-C2) + (8) yives .'. // — J 44 ; ,?• — _L44 . ,. _ 144 1 8' 329 3_71 y/~72' EXERCISE LXXV. 1. Find the lumibei-s whose sum is Hr,7,142, and ditferenee 571,428. 'J. Find two numbers sucli that the first and half the second is 72|, and the second and iialf tlie first is 85. :;. Said Willie to C'har]i<> : (Jive me 5 of yonr mar})les, and T shall have three times as many as you : said Charlie : (Jive me 2 of yours, and I shall have five times as many as you. IIow many had each ? 4. If I divide the less of two )mml)ers In- the ^water, th(! quo- tient is. 21 and the remainder. ()41(>2. If I divide the greater by th(! less, the quotient is 4 and the remainder .742. Find the numbers. 5. The third of one num])er and the seventh of another make 58 • the third of the latter and the sevcMith of the former make 42. Find the numbers. C. A number of two di-its vxwv^U^ by ^, lo jj,,,,.^ ^,,,, ^^^^j^^, digit; if 9 be taken from the number, the result will be expressed by the digits taken in inverted order. :. If both terms of a ratio J)einereas,.d by r,, the iTsulting ratio IS 9:11 ; if })oth terms b.- diminished bv 5, the resultin.r ratio will b(, 2 : )\. Find the terms of the'ratio. S. Of a two-figure nunii)er the sum of the digits is 10 ; by invert- ing the digits a number is formed which is 36 less than the original number. Find it. rn y :1 '' ■ i i If r: iftm ■B m 230 SIMULTANEOUS EQUATIONS. hi. »7 1^," ■^^ r 1 i 9. A man has two full casks and an empty one ; to fill this he may take the whole of the first and a fifth of the second, or the whole of the second and a third of the first ; the three casks hold in all 1440 gallons. How much does each hold ? 10. Two investments, one at 5 per cent, the othei at 4^ j)er cent, yield !j>8o;}.20 in one year ; if the first were at 4^ per cent and the second at 5 per cent, tlu; interest for the same time would be $13.50 less. Find the amount of each investment ? 11. If the numerator and denominator of a fraction be each increased by 2, the new fraction is J ; if each be dimin- ished by 3, the new fraction is ^. Find tlie fraction. 12. If a two-digit number is times the sum of its digits, show that the number formed by interchanging the digits, is 5 times their sum. 18. There is a number of two digits ; a second number is formed by interchanging the digits ; the sum of the two numbers is 187 and their difference is 9. Find the original number. 14. A and B can together lift ooO lbs., B and C, 600 lbs., and C and A, OoO lbs.; what weight can each lift ? lo. A boy is sent to market with apples, which he is to sell at 8 cents a dozen ; having given some of them to a com- panion he was ol)liged to sell f of the renuundcr at 9 cents a dozen, and the rest at 7 cents a do/en, in oi'der to i:^vi the rcipiired amount of money ; had he .sold | of the remainder at 9 cents a dozen, and the rest at 7 '^nts, he would have realiz(Ml 2 cents more than the recpiinMl sum. IIow many apples had Ik; at first, and how many did he givt; away 2 10. A num invests $12000, }»art at 5 per cent, part at 4 per cent, and the ri^st at 3 per ci'ut, receiving annually $490 int(>rest ; the sum invested at 5^ is half as much as the other two parts. Find the three investni'^nts. 17. The number 232 is divided into three parts such that the first with half the sum of the other two, the second with one- third tlu' sum of the other two, and the third witii ont?- fourth the sum of tlu; other two make ecpial numbers. Find the three parts. MISCELLAXKOrs KXERCISES. 231 this he ond, or 10 three 11 hold? er cent, per cent uic time stnient 1 be each 3 dimin- II. its, show ;;its, is 5 s formed unbers is imber. nd C and to sell at () a c(mi- it 9 cents to get the ■eniaincU^r )uld have ow Tininy way 2 p(>r cent, interest ; other two t the first witli one- wit li one- iers. Find 18. A number is formed of three digits wlioso sum is 18 ; if the digits be reversed, tlie resulting numljer is 198 gi eater than the number ; and if the nunUn-r be divided by 13, the (juo- tient is 6| times the r(>maindei-. B'ind the number. 19. A banker has two kinds of money ; it takes it pieces of the first to make a dollar, and h pieces of the second to make a dollar ; he is otferiKl a dollar for c pieces. How many of each kind must he give ? 20. Find two nuni})ers whose difference is a times and product b times their sum. 21. The reciprocal of the difference of two numbers added to the reciprocal of their sum, is 8 ; the same, diminished by the reciprocal of their sum, is 1. Find the numbers. 23. A cistern can be emptied by two pipes A and B in 35 minutes, by B and C in 43 minutes, and by C and A in TO minutes ; in what tinu^ can (1) each sei)arately, (3) all together, empty the cistern ? 1. A man had a certain quantity of wine in each of two casks ; he poured as much from tlic first into the second as this contained ; then he poure fii'st ; he next pouri'd from the first into the second as much as was left in tiie second, and eacli cask was now found to contain IGO gallons. How many gallons were in each at first ? 2. In a four -figure number tlie sum of the first (units) and tliird di^'its is e<\.M t" make up the sum : hut if A's had been T.'A percent more and U'.; 13-A per cent less. C's suhscription would have Ir.'-ii "i^I 1. What sum did each subscribe ? 1^1 I 111 » J h'l I ■ 1 1; '^'^2 SIMlLTANHOrs KQIATION'S. (!.. m ^f'.: »' ,(h » * «-*t,''' 4. Two l)0(li('s ai-c r/ yjirds ajjiii't. When they move towards each otlu'i" with miifoi-in velocities, they meet in nt seconds; when one moves to overtake the other, tiiey come to<^ether in n seconds. Find tlie rate of eacli body i)er second. 5. Find two numbers whose sum, dill'erence, and jjroduct are as 5:1: l,s. 6. What A is wortl? ^,'^'V t ) I times what B and (' are worth is equal to /; ; what H i . worth added to m times what A and C! are worth is c'lual t») q ; w))at C is worth added to )>■ times what A and li are worth is efjual to /•. Find what each is worth. 7. What numl)er c(»nsi:- \- * i when its diijits are in.i '*' liiee di<^its is greater 1)V 99 than 'r. eater 1)V 1"^() than the su m of its dij^its, and grea;<'r i>y ,*r hau when its tens and units are transpose! ? 8. A train (A) starts from l' to' ,■>-, (• "^ :in liour afterwards (luur faster, starts anothe)' train (Bi, <;(iin;j: ten miles from il towards P, and tlu'y nu-et half way between P and ii ; had A started an hour aftei' B, its rate must hav(^ been iiuireased ::2M miles an hour in oi'der to meet B at the sanu; point. Find the length of the line from P to Q. 0. In a mile race A giv(>s B a start of 100 yards and beats him by IT) seconds ; in running tlu> same distances again. A gives B a start of 4.") seconds and is beat(>n hy 22 yards. Find the rate of eacli in miles per hour. 10. The sum of three numbers is (/> + 1) (// + t) n ; the sum of the two larger is e(|ual to p times the smallest, and the sum of the two smaller, to q times the largest. Find the numbers. tl. Find thre(> nunil)ers such that tlu' i)ro(UK't of the first and second is equal to the sum of the sam(> ; the product of the first and third is e(pial to twice the sum of the same; and the product of the second aiul third is ('(jual to three thnes the sum of the same. 12. I row a distance of c miles down stream in first :' 14. A pays to H and C as much as each of them has : B pays to A and C half as much as each of them has ; and C pays to \^ and A a third of what each lias ; A has then i^\AA. \\^'2A)H, and C 9(5 cents, llow mueli had each at lirsl i 15. A person walks fnmi A to B, a distance of 9^ miles, in 3 hours and 52 minutes, and returns in two hours and 44 minutes. His rate of walkinj>: up hill, down hill, and on the level is 3, 3f, and 3| miles, respectively. Find the length of level ground between A and B. 16. Two men, A and B, run at uniform paces from one station to another 4000 feet off. A starts 30 seconds after B, and arrives at the second station 10 seconds before B. Where does A pass B ? old ub 17. Seventeen ,ij;old coins, all of equal value, and as many suver coins, ai'c i)laeed in a row at raiulom ; A is to have on(>- half of tla; row, B th(M)ther half; A's share is found to includ(^ T gold coins, and the vahu; of it is $24 ; the value of B's share is $27. Find the value of each g(jld ami silver coin. 18. The road from A to 1) pas.ses through B and C successively. Th(( distance between A and B is miles greater than between C and I) ; the distance Ix'tween A and V is i\ of a mile short of being half as great again as that between B and I), aiul the ])oint half way from A to I) is l)etween B and C half a mile from B. Determine the distaiu-e l)et ween A and B, B and C, and C and D. 19. A laid out money in shares of two companies ; if he had sold out () months later he would hav(> lost $100 less than half his outlay in the shares of the first coi.ipany, $4t»| more than ^ of his outlay in those of the second, and on the whole would have received $40 less than | of his whole original outlay. Whim he did sell out he lost $40 less than ^ of his outlay in the first company, and gained $5(j less than r^ of his outlay in the second, so that his loss on thy 1 2U SIMULTANEOUS EQUATIONS. t'v :^?l: 1 !^ i::. 1 * • ■ ■• '^:-'> „. lib' 'II 'i'T, '* ,^ * '■' . ! ■ .,1 i^( ■'•-si'l i' : \ I *->., '•I • \ ^ whole, tof^ctlicr witli g's of his whole orif^inal outlay, made $376. Find tlu; sums laid out in the shares of each company. 20. To ('omi)l(;te a certain work A ref[uiros tn times as long a time as H and (' together ; B recjuii-es ;/ times as lon^ a time as A and C together ; and (' re(|uires /> times as long a time as A and B together. Show that _i _ ^ _ J - 1 lir+l n + \ j> + l~ ' 21. A stage-coacii left B for A two hoursbeforea locomotiv*" left A for H and they met in (J hours after the locomotive started ; had tlie speed of eacli been 1;' miles an hour greater, they would have met in oi hours ; had th«» speed of each been IJ miles an hour less and ha stage-coach loft two houi's later than the locomotive, they would have mot 7 hours and 5 minutes after the (h'i)arture of the loco- motive. Find the actual rate i)er hour of each, and tlu^ distance between A and B. 23. Separate the quoti(>nt -^ 27 + 34^ into the sum of two (3 + 4^) (H + 7^) fracti = 0, and ax- +li.r-\-c — ;ii'e adfected il'iadratics. ir an etpiation cmt.iins tl' • i'^U-iJ , ////-(roi' the unknown (juan- ii!\ it is caHcd a cubic equation, or an ('([uation of the f/iird "'■ ircc : if tiic fouiMli power is involved, the equation is of the ,/h/irfh de(jrei\ and so on. 155. Roots of Equations. — ^Ve have seen tliat a sini|)l(> e<|ua- tion can he reduced to the form ax + b — 0. and that the value of X which satisfies the equation, that is, which reduces ttx + b to icro, is the root of the equation ; thus, puttini; — foi- ,r, we find (I (-:) + ^ to l)e zei'o, so tliat the e([uation is reduced to = 0, /. e. , ft f's satisfied. In the equation 2.r — 6 == 0, the root is 3, for 2-3 — (i is z(»ro. Now, in tlie (Hpiation ^x- — Qx = 0, what value, or values, of x will mak(! 2x- — ().r become zero ? The expn'ssion has tn:o fm'tors, thus, x{2x — 6), and therefore vanishes (becomes zero) when either factor ranisfies, i. e., when either x = 0, or 2,r — (> = (), which jifives .r = 3: from this it appears that tlu^ ecpiation has fn'o roofs, viz. .{■ = 0, and X = 3. In the equation x'' — IQ = 0, we have, by factoring, (.r + 4) (X - 4) :=. ; and the equation i'\ satisfied if either a; + 4 = 0, /. e. if x = — 4, or a; — 4 = 0, i. e. if x = 4. I 'Via ](] i)\'\\)n\T\(' KQCATIONS. • i 1 : . idt-^' ' Till! ('(|iiation .r^ — Ki.r = 0, hccoiiics, l)y rMctorin;;, ./•(.»•■— H)) = 0, /. ^,, .r (./' + 4) (,r — 4) — 0, mikI if jiiiy one of tlir f/n-t'f factors vaiiislics, f li Illation is satisfied ; that is if ./• — 0, or ./■ + 4 — (>, or .r — 4 = ; so that 0, — 1, and 4 aro the roofs of the r(|iiation. T!n' ('(Illation .»••' — \.r — \7.c + ()() = 0, is satisticd for .v — :} = 0, i.e. for A- = :\[ dividin*; by .r — ',) \\'{) p-t the (|iiadrati(' factor .r'- — .*• — 20, wliicli has tlic factors ■*• + 4, .r — .") ; hence the ;,nven (Mination may he put in Mie form (.r— iV) (.r + 4) (.v — 5) = 0, and is satistied if any on(! of iho. three factors vanishes; i.e., if X — 8 = 0, or ./; + 4 = 0, or x — 5 = 0; so that the roots ww, ;5, — 4, and H. Tho ('(luation .r' — 1 ;},/•" + 'M\ = can bo i)ut in the form (x^ — 9) (.c' — 4) = 0, which is satisfied if a-'-O = 0, or a-"-4 = 0, i. e., if (.V + 8) (X — 8) = 0, or x = — 8, or + 8 ; and if (x + 3) (x — 2) = 0, or x = — 2, or + 2. So tliat tho four roots of the eciuation an^ + 8, — 8, + 2, — 2. ,F 156. From these examples we infer : — 1". Tliat a root of an equation of any do^jree in x is any valno of X which reduces thi^ ('(piation to X\\o ultimate form = 0. 2°. That if any factor, or factors, of the form, x — a can bo found (where rMnay Im» zero), then each of thos(! factors e(|uated to zero (/. r., .v — find that any expression, such as x — (i, etc.. when put = zero, reducios tho equation to the form = 0. then x -- t of tho mmbor of hmc root, \ttro " \ three " •m X — a, Iwo g( t ail 6°. That, (!(mvrrs('ly, in sof rf it (/ i\n ('(jnation wo are sookin^ (ho Unear (/. c, niie-dimension) fiictor.s into whieh it may bo ro.solvod. 157. Pure Quadratica — A iinadratic ('(piation may ho oon- sidrrod as the prodnct of tin> binomial factors (^(juatod to /oro. Tlins [.r + a) [x + b) = 0. If 6 = — r/. this boccmios. x"^ — {. and if x-^ 5 =: 0, ?, e. " a' = + o. Ex, 6. Form the equation whoso roots are ' and — ' we luive or ;{•'■■ - ^ = 0, or lG.r' — 9 = 0. 16 Ex. 7. What is the equation whose roots are ± (// + h) ? Wo have ja? + (a + 6)} ^a-- (a + 6)f =0, i, e. x" ^ (a + by = 0. Pl^RK t^rADUATlCK— KXKUClsKi •)'*( i[) re, we cx- "■); (1+'/')^'', , we liavc = 0. 6)? EXEF^CISE LXXVI. Solve thr .'(lUMtio.is iti iUv folNmin^r .-xamplrs : - - Il'.Vrrr 0. ;{. U- = 1 ; .," _ 1 .7- U 4 '''^•'■^-;5^'';4=;;.^^=18. 0. 4(.r + 8;-u.-4,^ ..^^(.,_/,,. . ,._(,_,„.^„. L-^) =.,^ !>. Form fhc (M|ii;iti(.ii wlios." roofs Mrc h :? i" ; ± i\ + If) : ± {(I —1,. ),(>()(• (). -+- J> ; ± ., ; ± ^^/> ; 3 ^ + 4-'' 5 + lv' = ^^ 7 +^^- = 1- 11. Form equations with roots ± \ (a ~ /,, ■ + " b ' ^\a~\)' ±C! + .!)= ±V^'= --t V'"^. .r-p:j-2' .r-.,+.rT7.^-^^' •%--'^,. '/ .r ^/ i-.7' 04V 9/ "hi i;j. '*'~J"- '/ — .r 4!) ./• + y// (« — .r) (.r^ />) __ 14. Form th(. equations wl.o.se roots an, ± (O,, _ y^^ . ±y,,y^/;. . ^•-10 10-.,' .,__i +^,:^ = >. ■I! ■Ill p» 240 QUADRATIC EQUATIONS. V' m '.', "I >•■. ■' II . >;.: ■ 5". i 'I . 'W. 16. ab cd a' - b'^x" c" - d'x* =30; a + X _x + b n — X _\ — bx a — X x — b'' \ — ax b — x ^^ a — X b — X a(a — h) . (h — rb — x — 18. 19. \ — ax 1 —bx^ X — a — b ' " ' " '" a + b — x X + m — 2w _n + 2m — 2x x + a + 2b _b — 2n + 2x X + m + 2n ~ 11 — 2m + 2x ' x + a — 2b~ b + 2a ^2x rta — 6b + x 3a — 56 + 3a; a + b — x 3 (a — b + x) a + X a + b + X ^ 'da — b — 'Sx a — ob + x 2 (m" + 1) fm 20 ^jL5* + ^ — ^ _ X — in X h m (1 + m) (1 — in) 158. Problems in Pure ftuadratica— A few examples are hero given of problems which are soloed by pure qaadratU^. Ex. 1. If I multiply by itself the number of dollars which I have, I shall have $133|. How much have I ? Let a; = No. of dollars ; then xx^ i. e. x^ = Vi2{ ; or «' = 4p; .-. a;=±V = ±lH, where + 11^ must be taken. Ex. 2. Find two numbers in the ratio of 11 : 13, whose product shall be 7007. Let \\x and 13j; be the numbers ; then 11a:- 13a; is their product. .-. 11. 13 -a;' = 7007 ; . . a" = 49 and a; = ± 7. 11 X (± 7) = ± 77, and 13 x (± 7j = ± 91 ; .-. the numbers are ± 77, and ± 91. Ex. 3, The length of a rectangle is to its brejulth as 3 : 4, and itvS area is 1200 square yards ; find its dimensions. Let 4a; and 3a; represent the length and breadth, respectively. Tlien 4a; X Sa- = ISa;" = 1200 ; .-. a;" = lOO'; and .t- = ± 10 ; ,-, the sides ure 40 and 30, : PROBLEMS IN PURE QUADRATICS. 241 - bx — X r c Vx) {■ X arc hero which 1 product product. ;j : 4, and tivcly. ■: ±10; Ex. 4, In a concert-room 800 persons are seated on benclies of equal h>ngth ; if there were 20 fewer benches, it would be neces- sary for two persons more to sit on each bencli ; find the numb<'r of benches. Suppose x—\ persons sit on each bench in the first case ; then x ■\- \ " " " second case. rru 800 . , . ^, „ , , 800 . ^. There are l)enches in the first case ; ana m the x—\ X + \ second case ; 800 800 -^ .^( \ 1 \ 1 ri =20, or 40 ( — r)== 1' x~\ X + \ or 80 .r»-l I; .-. .r' =81, and .r = ± 9. EXERCISE LXXVII. PROBLEMS. 1. If the number of dollars I have in my pocket were multiplied by itself, I should have 132|. How much have I ? 2. Find a number such that its fifth part multiplied by its seventh part is equal to 4285. 3. If I multiply 3f times a given number by 8.68 times tin; same number, the result is 5239. Find the number. 4. Find two numbers in the ratio of It : 13, of which the product is 9152. 5. I multiply the; third part of a certain numl)er by its fourth part, and the product by the fifth part of the number ; and find the result to be the sixth part of the number. Find the number. 6. Find three; numbers in the proportion of |, J, \, such that the sum of their scjuares shall be 10309. 7. A rectangular field which has a length of 3367 fe«'t, and a breadth of 37 yards, has the same numl)er of scjuare yards as a rectangular field of which the length and breadth aru as 18 : 7. Find the dimensions of the latter. 1 m i Ml il I, U2 QrADHATIC IvQl'ATIOXS. IT ••^vi : ili H. Find thron 0(iual iiinnhcrs whose sum is equal to tlioir product. 9. A uian l)ouf^ht a numlxT of pounds of raisins, twice as much coffee, and four times as much tea ; lie paid for every |)ound of each article 10 times as many cents as there wer<» jxtunds of the article, and his outlay was $8.40. How nuiiiy pounrtain numhei' is inen'ased by :», and also diminished ])y '.i ; the }j:reater sum is then divided by the less, and the less, by the >?reater ; the sum of the quotients is ;J|. Find the nundjer. 11. A man was instructed to buy 18 yards of cotton if it cost just 18 cents a yard ; but if it was cheaper or dearer, In; was to buy as many yards nio/r or /ess as the price was below or a})ove 18 cents ; he expended $^)A~}. How many yards did he buy i 13. What is the result in the last q(U'sti(m, if / price, I should have got $9.80 for them. How many eggs did <>acli take to market ? 17. A man sold 133 yards of cloth at a certain per cent profit; he invested the proceeds in goods on which he realized the same profit ; with the proceeds of the last sale he was able "~T^ ADFECTKI) QrADUATICS— EXAMPLES. 243 to tlu'ir ;is imu'li or every ere wcn^ o\v many led l)y >i ; the loss, Find tlu! cost jnst was to below or yards did or 18 and (jnare ; if 1 anothi'r re. Find tli«i i)arts I'a will be h reali/>e got ^ had had . have got :o market ? ^nt profit ; ?alized the e was able to buy IflH yards of eloth which cost 14 per eent more than .0 pau tor tlM> im yards ; what per cent pr.)ti( had he on the 188 >ardsi' 18. One side ,>f a right-angled triangle is 3? times the other, the liypothenuse being 1000 f,vt. Find the sides. is S2,i te(^t. Find tln! sides. 20. Two persons start fr,)m the same ,)]ace, the one travelling due east at the rate of miles an hour, tlu, other, due north at 4. miles an hour ; in how many liours will they bo 30 miles apart ? 159. Adfected auadratic8.-Tn many cases, the factors which form an a( f.^-ted rp.adratic may b,> found, and its roots at once d(>termmed, by some of the methods of factoring already given Ex. 1. .r" — ryr + = 0. This can at once be put in the form (x - H) (.V - 2) =z {) ; .-. u; — ;j = 0, or d—2 = 0, ^^- -■ .r' + .r - 20 -: 0. By factoring we have (.v + n) (j- _ 4) - q . .-. ./• + ;-) = 0, ,„• .,. _ 4 _ Q^ '"»'^ u- = - .-. or 4. Ex. 8. r»./-' — .,■_ 10 _ By factoring we have CU' — '^) ['.U' -f- 4)rr 0; •■ 2.r-8 = o, ,„. ;b-4.4 = 0, ^'"l ^=-*or|. ' 244 QUADRATIC EQUATIONS. fii* Ex. 4. x^ + 2x' — nr,x ^ 0. Here we have x(x + 7) (x — 5) = ; .-. a; = 0, or a; + 7 = 0, or a; — 5 = 0, and iB = 0, or — 7, or 5- - » * * • Ex. 5. «' 4- B^aa: — 2a* = 0, Wti have at once (a; — ^a) {x + 4a) = ; .•. X — ^a = 0, or a; 4- 4« — 0, and X = ^a, or — 4a. Ex. 6. a;' — 8a; + aa* — 3a - 0. By factoring we liave {x — 3) (x + a) ==0; .'. a; — 3 = 0, or x -f- a = 0, etc. Ex. 7. a;" — 4a; = 8a; + 16. By factoring we have a* (a;' — 4) = 8 (a; + 2) ; where a; + 2 is a fojctor of each side ; .-. a;4-2 = 0. Dividing by a; + 2, we get X (a; - 2) = 8 ; .-. x" — 2.r — H = ; from which we liavo {X — 4) (a; + 2) is; 0, Hence the given equation is satisfied if a; + 2 = 0, or a- — 4 = 0, or a- + 2 = ; • . ^ — 'w^ t, /w. Ex. 8. a;" + 7a- = 49(,r + 1). Here wo can arrange tlic terms thus : — x^ — 49a- r= — 7.r + 49, or a; (a;' — 49) = — 7 (a; - 7), where a; — 7 is a factor of each side ; .-. a; — 7 = 0. or .a; = 7. i-it.' FORMATrON OF Ql'ADUAT S. Dividing by x - 7, wo get ^■' + To; + 7 = 0, of which the solution may be found by the method of Art. IGi 24b 160. Conversely, a quadratic may he formed A roots'. or any given — 7 and — ;j. Ex. 1. Form tlie quadratic who.so roots sliall Ije We have at once {a; - (- 7) } |.r - ( _ y^ } = q ; .-. {£ + 7) (a; + '6) = 0, '^^ X' + n\v + 35 = 0, of which the roots arc; — 7 and — 3. Ex. 2 Form the equation which shall have roots -9 and + 8. ^Vehave (.r + 9) (a; - 8) = ; /. e. , x"" + ar — 73 = 0, whose roots an^ — 9 and + H. Ex. 3. Form the equation whose roots are - 4 and 5a. Wo I« Hf V i^ (x + 4) (a- - 5f/) = 0, or .r» + {\ - r^a) X — 2[)a = 0. Ex. 4. Form an ('((uation whose roots are a and -h \\> have ^ (.»• - rO (./■ + 6) = 0, x' - {a — b) X _ lib = 0. Off be^re'^' ^'''*"' """ «ination with roots a-l and « + l. As ra- - f/ _ 1) (^ ._ ,, + 1 ^ 3^ Q . t. c, x^ — 2a X + a- \ = o. K; :h : 24G grADHATIC KQIATIONH. a '. \v.» a Ex. f). What is the (Miuiitioii \vli()s(M'(M)ts iirc iind havo (■'-:)(-:;)-^ EXERCISE LXXVIII. Solve the following ('(lUiilions :— 1. {X - 3) (.r - 4) = ; (.r - 4) (x + 5) = ; x{x- 5) = 0. 2. (X - ;Vn (.f + 4^/) = ; ./• (.v-a) = (); U - a) {x - ?» = 0. 3. .2j: (.r — 7) = ; .<•-' — lo.r = ; r/.r- = 3fW ; x(x + h)= 0. 4. ;U-^ - \\\x = ; (2j' - 1 ) (.r - :5) = ; (2u- - 5) Clr + 4) = 0. 5. j:^ _ a.r + 2=0; x' - 7x + 12 = ; ^^ + 9^; + 20 = 0. 6. (.*; - (I ) (X - hr.) = ; x' - 8 + 15 = ; o-^ + Ho- - m = 0. 7. a-' - 1 7x + no = ; x' - 1 7x - 00 = ■ 4x' - 2x - 20 = 0. 8. a;" — {(f + h) x + tfb — ; ./•'— (1 —(i)x—a = ; .r'^-l If— 1 = 0. Form the ocjuations in th(> following cases :— 1. With roots 3 and 4 ; 4 and : - 3 and - 4 ; - 7 and 9 ; H and — 5 ; 18 and - 10. 2. With roots — a an ; 3 and 6. 4. With roots ^ and 3 ; i and - J ; "3// and A ; f/^ and - Ir. What inforonccs aro to br drawn from the following statements ? 1. .,•.// = 0. 2. ./••// -^ = 0. 3. 4j- = 0. 4. nx!/--=0. 5. a.ry^ = 0. (5. .*' Ui + h) = 0. 7. <' (x - a) = 0. 8. (.r - rn (6 + (') = 0. 9. x{\- a) (1 - 6) = 0. 10. {X — an (x — he) = 0. = 0. : 0. l)=0. 0. = 0. = 0. r -1 =0. ; H and nldiiij thv s(/ttuir. Tf all the Ici-nis containing./', when collected on one side of llic (;(lualion, form a coinplctc sciuarc, the ('((nation may be solved by (ixti'actin;^ the sqiian^ i-oot of both sides as in the case of a para (/tindnrffc. Tluis : — Ex. 1. .v^ — •-'.*• f 1 — 5; the left liand side is a eiMiiplete s(inare, .-. ./• — 1 = ± /y^r), and .)•= \ ± ^/r). Or, thns : (.r — \)'' — .") :l= o ; .-. (.r — 1 + y'.")) (.r — 1 — y^.')) = O; whence; x = \ — -y/r), or I + y^o. Ex. 2. .1^ — 'ia.v + (r'=h. Here the h^ft side is a s(iuaro, .-. .}• — (t — ^ ^//;. jnid ;*• — it ± 's/b. Tf tlio .r-t(>i-nis ilit not fortn a syifftn' we can, in every (MS(», com- plete the squitrc by addinj^ tlu; necessary (juantity to both sides of the (Hjnation. To deterinini^ this ((nantity, W(> have only to con- sider how the s product is th(! stpian^ of d (the second term of the liinomiot). 2°. The e(K>nici(!nt of x is ± 2 = 7, or 3. Or we may proceed thus :--- ;r'' — l().r + 21 =0 ; Add the pair of compl(;inoiitary terms 5'-' — 5*, /. d- — lOt; + •"»' — 5'^ + 21=0, or (u; — 5)'^ — 4 = 0, .-. {X — 5 + 2) {X — 5 — 2) = 0, or (x — 3) (X — 7) = 0, whence a; = 3, or 7. Kx. 2. 3.t- - 21a; - 450 = 6000. Divider l)y 3 and Iransjiose, .-.a-^- 7a' = 2150; Add l'\ to both sides, .-,.- -r,,. + (;)"= 2,50 + Q 864P 7 _ 93 , ■ ■ '^ 2 ~ 3 ' and 7 93 x = .^± .^ =50, or -43. • - ' • N. B.- -By factors : a;"* - 7a; — 2150 = 0, » 01 [x — 50) (X + 43) = 0, *' .-. X, as before, = 50, or - 43, 1«- • I of lul the T5XAMPLK.S— KXi:U(ISKS. Ud Ex. 3. ^-^ + "1^ ^ r^-zJ! x+2 x-2 u-\' First eoinplctc the divisi OILS. 1 1 ••• 1 + ---- + I _ .'. -J _ - 1 " X + 2 x— 2 •r - 1 ' .r-i' x~\' or a;' - 4j: = ; or ic(^ - 4) = .-. ^ = o, or 4. Ex. 4. x-' + 109 - ^''^'^ + ^^* _ -i Clear of fractioihs, .'. 35a;» + 4235 — 78(U" + Mx' = 18.' Transpose, «»- 8.50 ; .-. GU--' -. 780.^; = - 2)Mn, 780 61 ;C + P90y^^ _ 2375 /39(()y_ 7235 \61/ 61 '^\gW ~(6i?' and « . ^ 890 , 85 61 ^ 61 ' _ 390 ±85 ■ 475 61 61 Or. by factors, 61...^ - 780^ + 2375 = = (x - 5) (61:r - 475), etc. • • EXERCISE LXXIX. Complete tlie squares in the following' eases :^ 1. U-' -4x; x"-~ 10.r ; x' + 2ax ■ x' - 2.\; Solve I ilC ioli()v\ 111^ f 1 i: 'II- 1 . .!■' + ill- ./- — S./' = — ,/- + KM = — 21 2. .)'' + 3.r + 2 = ; ./•-' + 2///,/' = ii ; .r iii.r — II. 3. .*•" — lO.r — 24 = ; ,/•-' f 2(i.r + 120 = ; 2(i.r — 7- + 120 = 4. 22^ + 23-./' : .r A = 0; ^.v^\- r>. Tix"" - 17ir + 12 = ; 2 (x - f ) - 4 (.r + ^l" = 134. 6. 40 X— Ti X + 3 a; — 4 a; + 3 27 + = 13 X x-\ 1 + = 4 ,r X - 2 X - 4 IT » + a; — 3 a^ — = 3. + .T — 2 a; + 3 2a^ — 1 m X -f- !) _ X + 2 .r + 3 a- — 4 = m- 2a; -7 9. 10. 11. 9 64 id '*' 81 a- , it' T^ ^ 1 + 4 + a; a; — 3 a; -3 ' 3- -a; 5 9 — 2 \ 4 2a- + 1 f ~ 'Ax ' 2a- + 3 3a' + 8 13 3a- — 2 ^\\' 7_4-f.t_ X- 3~ 3a! ' X- -72 (X + 9) -4 a; + 5 ' +^-0:- a; a + a; _ 100 21 _1 2r).i- ~ 4 0. X + 2 oi rt + a; a; 12. a;" — (r/' + 6'') .« + (//^ — &^) «& = ; {m — n) a;' - nx = m. 13. r)57a; = 5,801^ + Sx"" x' 4mn {m + nf {in + /?)' ^x — (m— II f 14. {x — of — h (x — a — c) z= he ; ax^ — a" (.» + 6') = ah (x — ah] 15. a + /> + ^/ — & =: 2aa; a + h' a b + - = ;. - (a — b)+x X — a 3 1 rt 2a; X + (I %r 162. General Formula.— Tho gouoml form of a quadratic equa- tion is ax^ + bx + c=0, (1) or by dividing by the coeflacient of x, Hii OENEHAL FOinilLA. 851 120=:(.' = :}. 21 _1 V = m. - >n\ lb (x — at] Iratic equa- (1) h r a- + •!' + =: ; a (f (23 .11 which the .s(|iian. .nay Ix- comph.lrd hy a.I/lii.^r th<> s(|iiaivcf hiiUof '^, i.r., th.>s(iuari'()f^,'^, to l.<.th sides of th ,uafion. The s(jMarc may also hv comph'trd in (hr foUowin- niann(>r:— M.driply hoth si.lrs of (1) by 4u (/. r., by 4 tinu's the cocnicirnt of .i-') and add //-' (/.,'., th(^ siiuarc of fho eoeflidcnt of u) to b(.tb sides ; then 4' - 4r/c) , 2a ' or in one fonnula, as Ixfore, :r - rA±A/!''' - *^"*') 2 in — 7, and <; is ;{, wliciiee, upplyin^' tiic l'(»riiiulii, N. B.— By faetorinj;, \\v i^vi (;2.r — 1 ) (./• — :{) = ; wlirnec x is obtuinod, us by the formulfi. ^''''■- - 9 +4rT"«=^'+ 18-- Clearing of fractions, and transposing, 4x' + Vidx + 72x — 216.r = 114, or 4^'' -.V- 114 = 0. In applying the formula, W(? soo that o is 4, f> is —5, and c is — 114 X = '"> ± \/<~ ''» + 4-411 4^ 2-4 N. B. — By factoring wo get (4.r + 10) (.r-- 6) =0 ; whence we g(*t x^ as by the formula. 163. The artifices used in diminishing labour, by conihinini) fractious, etc., may be occasionally employed in solving (piad ratios, as in simple equations. ^ , 6j;— 12 2j'— 11 „ Ex. 1. + ^ = 7. — a; + 5 2a — o Complete the divisions. .-. - 6 + 18 ii — X + 1- 6 3 2.C - 5 1 .=7, or — o . 1^ y that is, 5 — if 2.C — 5 ix" — 23.'. + hu = i« 2, f) i« iiice X IS is -114 luuico we iig quad FRACTJOXAi, ;;qiations SIMPMKII:]). Apply tho foniHiIu, tlion ^._3;J± y'(2JJ»-4.4.;jO) «i/ ^ » — — •) ,,_ .1^ 8.4 — '"^ H- Ex. 3. ^'^ + ° ■ 4^-1 7./ + 1 2o;j 3u; — 7 ;t; — 2 a; — 8 on rv I . o + - 4-4-1- 7 _L 2a; — 7 or vvIkmico or that is svlit'iice Ex. B. a;-2 7 38 2'> 26 2a; — 7 ar^r2~^^r}j 40x- 101 33 (a- - 2) (2a; - 7) ~ ^^\ ' (a; - a; (_4()a; - 101) - 22 (.i- - 2) (2.r - 7) = 0, 4a;''-21a, + 5 = 0, (4a;-l)(.c-r)) = 0; 1 X = i) or 1 + 1 X + a + b X — a -f- b + X + a — b X — a — b = 0. Ttiko tho first and the last fractions together, and al middle ones, thus + Iso tho two )/ \x _ (a - 6) "^a; + (a - b)) ~ or u + {a + b) x—{a + b) x^i±b)_±^^{a^-_b)^ x+(a- b) + x ~ {a - b) x^ — {a + b) x^-{a- bf = which '-'.(Jiu" to 3a- + 2a; a;' - {a + 6)» x" - {a - b) r-„=0. Wheio 2a! is a factor, and gives x = o for one root. Divide by 2a;, x''~(a- by + x'-(a + by = 0, or X- » = o b\ and a; = ± >y/(a' + 6»). *■" !i 254 QUADRATIC EQUATIONS. 164. Equations Reducible to Quadratics.— Soinotiincs oqua- lions of higher degrees may l)o nf/nced to a qiiadnitic form, and solved as quadratics. Of this a few (;xaini)k's will now be given. I. We have already seen thai we can sonietinu's i/isroirr a fmtor, containing .c vvhicii will give us o/tf root, and will besides, enable us to reduce, by division, the ('(fuatioii to one of ji loirt^r dej/ree. Ex. 1. .<•=' + IW — 25,r + 'll = 0. We s<'e, on ins|)ection, that this is satisfied vvluni .c — 1, .*. .r — 3! is an exact divisor, and gives th(^ (piotient which is a common (juadratic. Ex. 2. ./■' — px = p — \. This may be put in the form x'' + 1 =.px + /), or ;r' + 1 =/>{.>• -I- 1), of which J? 4- 1 is a divisor, .-. .r + 1 = 0, or .r = - 1 ; and, by division, the e(iuation reduces to ;r'^ + X + 1= />, an ordinary (luadratic. Bi i Ex. 3. ,r* - Hj-* = 1 - .l.r ; or, .r* - 1 = :?.r' - :U: Which gives us (x^ + 1 ) (./•' — 1 ) := nx (./-' — t i, where x' — 1 is a factor in each side, .-. ./•» -1=0. or X = +. t , and the equation reduces to x^ + \ ='Sx^ a quadratic. EQIATIONS 01- IIIUIIKH Dl M KXSIONS. 255 n. !f thr imknown .luamiiy (xctirs in but two terms and //v ^ximneut in u„. of H,,,,, i.s t,ria thnl in the other, the fcmation may bo solved as a <|uadratic. * Kx. 1. x* — \u-- + 20 = 0. Ill this case we eonsich'r a- as the unknown (juantitv • put x^ = y, then y^ - 0// + :»(» = 0, .-.//.= r, or 4, ! are tlie other roots. Ex. 2. .r" - '.W — 2S = 0. Here we treat x^ as the unknown (piantity ; h-t .r' = //, then 2/' — % — 2S ur ; Ihi.jrivesus (y — 7)(y + 4) = (>, • ■ // = 7. or — 4 ; l)Ut x/ = \/,y ^ v^T, or ^{— 4) ./• = n Kx. ;{. .r -I- " =r know lis ^rives .r" - .-).r" + (J = 0, in whieh we take .»■» foi (} ii (jnantity. Let it = //, so tliat if — 'uf + (5 = (!/ — 'i) ill — '-2) =:(). H' un- and but y -_= ;{. or 2, E.\. 4. U-»-4)--H(.*--4 Hero we treat x" — 4 as the unk t + 1 ."> :^ 0. Jiown (jiiantily. Let it =zy. !f-Hy+ 15 .= whioh gives, by factoring, ?/ = 8, or li ic' - 4 = 3, or = 5, etc. f i : I? a56 QUADRATIC EQUATIONS. iBi: .*; t- I 1 1 ' '«;'' l^K M Ifi;- Mr:: j Ilsi iiiil "ii I'lr • •;•*■ ir 1 ' .,. . '. .-•::• ^.•i. ;"•» lii 1 . 1 ;v^s • }.^'*i • i i : '■ 'i, ..'" ai Ex.6. ^f-n'=L«/*_:'\_8. \3 x/ 3 \3 a-/ 3 Here we treat I - — j as the unknown quan^iv--. Let it = ^^ ; (y - 3) (y - 2) = 0, The first value, , gives the second value, 3, gives .-. y = ^, or 2, X 4 4 3-.. = 3'"''^- x== 6, or —2 ; a; = 3±-^(21). From these examples it appears that any equation that can In put in the form ax'"'' + 6af 4- c = 0, may be solved as a quadratic Ex. 6. X* + x' — 2x^ — x + l=0. Divide through by a-" ; then X X* ' or or a;' - 2 + 1 + a; - A = 0, aj' X h:)'-(^-:)-. in which we treat x — as the unknown quantity. r ■■■{-'h{->\-\' or 1 1 1 X — + = + - • which gives two quadratics for determining x. it = y ; hat can hi quadratic or irmifER KgrVTlONS-KXAMPLES. Ex. 7. x* + ax' + bx' + (u;+l=o. Divide by .T^ .-. a;' 4- oa: + 6 + « + 1 - 257 and or hence the equation becomes •^ + .-, = y, y" + a^^ + 6 _ 2 = 0, a common quadratic from which y may be found, and thence EXERCISE LXXX. («) ,ltowl!!lir"'"^ '"'™"^^ ^« «^*^i» the values of x i X. following equations : in the 2. 4^" — 9.tr = 28 ; a-» 14 2.C _ 3 ""^•'^ ~ ^• 3. 2^ -f 1 _ 26 "' - + __ = 3j 57.r-18jj» + 145 = 0. 6. oe* «-6' « "^ 2ii:~9 "= ^^- I - IT Mi 258 QUADRATIC EQUATIONS. It'. k: US',. :^> • '*>•: „ X + 2 x— 10 ^ , -^ , ^ , . 7. — -- + — -^ = ; (a + 6) (a^ — a) (a; — 6) = a6a;. a; + 3 a; + 8 (ft) Solve the following equations : — ' 1. a;* + 4a;'' — 12 = ; 4 = T^x' — x"- ; a* + 36 = 13a;'. 2. a*-y'-ir)n = 0; a-* — 14a;» + 40 = 0; (7a-»-4-9 (7.-») + 18 = 0. 3. a;" + 3a-=' - KH = ; (4a- — 1 )' — 15 (4r - 1 ) + ofi = 0. 4. a" - 35a-^ + 216 = 0; 3a'' - Ix' = 43076. 5. .r' — 4a- = 8a- +16; a-= -(.r^-4) : x'-.r (.*•=■- 1). 6. (a;» + 4)- - 5 (a?" + 4) = 44 ; (a-» + 5)* = 7 + a-^ 7. 2r)6a;* — 16a 2 O = ; .r" + 6a\i''' = 16a' ; a* - f/'-a'' + // = 0. 8. a;* - 7a-'' = 8 ; (4a-j* — 20 (4a!)'' + 99 = ; .*•'" + 31a-' = 32. 9. (a;» + 7a; + 5)' — x 7a;- 5 = 6; (^^^-•'■) + 60 u- = 6. 10. a-* + a;" + 1 = a (x' + x + 1) ; a-"-* - 7a-- + 14a; — 8 = 0. li. 4.r» + 6a-'''+a-— 1 =0; 2a-« + ./-'' - 1 la- — 10 = : 2a-"' — a-" = 1 . 1 2. a-=' + x' -12 = 0; x^ — x' + lix -27 = 0; (a-" + 1 ) ( a- + 2) =: 2, 13. x'-' —j)x" + px—\ = (» ; a'' — 2/}x' + 2/>a- = 1 — 1=0. 14. a;" 4- a;"'' - 4a; — 4 = ; x^ + hx'' = a' (a + h) ; 216a' + 19a-« = x. 15. a;' + ^ + a; + = 4 ; x* + U.r=' - 8^" + 1 Aa; + 1 = 0. x^ X '^ ^ (©) Solve the following oiiujitions : — a- + T) a- — 6 1. 1- 2a; + 1 « — X 2 J-^t^ 2 ('2a;— 1) 2(.r + W 7 -3a; 4-3** :) + 18 = 0. D. -(.»•=•- 1). '' + h* = 0. ' = 33. - ru- = 6. :0. ■' — x' = 1. a; + 3) = 3. -1=0. + 190^^* = iJC. 0. MISCELLAXEOrS EXEUCISES. 259 « - 5 0^ + 4 a; -Fr ' 4. ? _ ^^ + ^ _ '' ^■*'* — '>) + 4c« a X u{,/ +~3r) 5. (a; — 3)" + a; — 8 6 a; + 3 u- + ;{/-' 6. + ^- + 1 a; + 3 + -^-. + 24 7 '^ _•'>_ 3^;" _ 28 ■ 8 (a; + 1 j 8 ~ a;»T_ 1 - r^r- j^ -.. = 3. ba;'-' — 7^- + (J = ihu- ^ + 4 .f + 9 .r - 4 a; - 9 *" ■^■! , '1^+ »_.<■ — 4 a; _ 9 a^-4 ^--9 .r + 4 oTTt)' 10. ^'-±-^l-^±l + ^"'1+ «^" +ii> _ .'•••' + X + 3 11 '1^-jL^* — ^'^ — ' i bx + (I ~ ((x~a r ^ 1 = -^*^ 6 a:' + 6a: + 13 ** + ! ^- + 4 :r4-9. + ;^^-3 13. 18. X' + 1 a; 1 ^X"-\ 1 + O^; 4 a;' --1_.+ L ^ 1 + 1 la- - 8 ^ X' + 3.r - H + .I.;ri:YJj;^-^^^ = 0. 15. ^-^f^ _ ^:£l±if '• ^ ^'^ 1 l.r-^ .r^ - 3a- 38 33.r - . +-7 = 31 43 16. a?» + "_"L^ = 1^"'''Z1^^ '^' + -'') , i.V/ - 36 17. (6r,a-)« + (6.V.r)-^ + 1848^=0. 18. {X' - 8aj + 11/^ -u ,j,. _ 4,'j ^ 25^ I " -r. li::' 1; mi' 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. QUADUATIC EQUATIONS. «• + 27 = 28a-'. ie* — 97a:* + 1296 = 0. Ax* + Ax' + 8a; + I 2x^ f 2a; + 1 2a;'' + 2a; + 3 a b a; + 1 + = 2. X + a — c X + b — c 5 \a; — 9/ 9 V^ — -V 1 ^ Vr^49/ " 585 ' 1__ 1 1 {X + a]-' — />' ^ \x^ ly - «/■ ~ X' - (a + bf W lix* + r + // + 1 = 0. X* + 2x^ - .1 = 182. 1 (a--6)> a-" + n " _ 34 "is' s + if" _ «'• a;' — ff a;' + a 3a;'^ — 2a; + 7 _ .^ ^ .^ — ^ 6x' — 4x +T\. ~ 2i^+ 14a; -^ a;» + 7a; - 3 ox — t./: -t- II sSa;'"' + 14.C — 9 3a;' 4- 10 _ a;" 4- 2a; + 3 _ 2x^ + 2a; + 10 X ~~ar+ 2 ~ a? + 1 2j; + 1 _ a- + 2 _ 7a; + 8^ x~+'2 4«T^4 "~ 4a;T 13 ' Solvt! the foUowiiij^ (Mjuations : — 1. X* — \\y + 2a-' - Bj^ir +1=0. 2. a;' ± rta-" ± oa; + 1 = 0. 8. a-' -i 3^a'' + 3^a; + 1=0. 4. .*•» + '6.C- — Oa- — 8 = 0. a; + 3 a;+l 4a; + 9 12a; +17 5. a; + 4 a! + 2 2a; + 7 6a;+16 M ISC E L I. A N KO I'S KX K Kci li KH. 5»Gi 6. f^Jl''* + ? ±J? ^ 2j; + 5 _ a;' - 10 ' a; + 1 a; + 4 x + 2 a; + 3 + «. 7 Q-^ + a; - 3 _ 7j;^ - :ja; - 9 5a; — 4 a; — 8 7a; - 10 'dox" — 7»x + 40 8. a* - 4fr=' + r,^.!-^ _ 4lt + 1=0. x-l x + 2 a;-3~a;^+l;t_3'a; + 3 + ^+ " 10. x'-2x + 3 a-^ + 2a; + 4 _ / ii^' + 3a; + 4 "^ a;'^ - 2xT'S ~ ^*" ''• (l+agr^-^^- a; - 1 a- + 3 ^ a- + 3 "^ a- - 4 ~ 13. (r,x^ + u; + 10)>_ (4a;-^ + 5.r + 8,' = (3a;^-.f + r,/'_(.i.»H.7a;+l/^ 14. a;" +y;^« + (/> - l)a; + -L- (x + y> _ ij = q. 15. (x' _ 9a; + 18; (x' - 9.r + 20) =^ 24. 16. 1 -f a! + a"^ 62 1 - x* l—x + x'' 63 1 +a;' \a — x/ ^ b 18. ' +^.+ 1 3.r» 19. 20 ^-1 a;-2 a;-3 (a;- 1) (a;- 2) (a; - 3)' a;-l x a; — 3 {X -2){x- 3) (a; - 3) (x - 1 > ("J^fHa- 3~o , = 40 20 1** a;» + 2a--48 a;» + 9a; + 8 ^M^0.c "^ ;rM^5a^i::'56 + ^ = ^• 21. ^=il_?±l!^|_3/ _a;-lU 4a-/ ra--l )(a--2)\ 8 4 -21-^ 2-}^ A'-' -^- ) Ill M' ■ k: ' ^4 I, H 1 • 262 QUADRATIC EQUATIONS. 23. ^-.L+ 1 _ 1 1 1 + 1 +J- = 0. a; 1+aJ 3+a3 8 + a; 4+x 5+x ii+x 7+x „„a6 c c 6 a,. 23. - + + ^j + — - + r + p = 0. X X — I X — H X — ii X — 4 X — 5 1 oi 1 1 1 24. + 7 + + , X + a X + o X + c X + a + — c = 0. 25. X* + l>i — ^\x^ — 2n''x' + In- \x + \={}. 165. Solution of Problems.— We now give a few examples of problenis solved by q/Kuf/utics. Ex. 1. A man bought a number of sheep and paid for (jach as many dollars as there were sheep ; he sold f of them at $24 a head, and the rest at $30 a head, making $48 by the transaction. How many sheep did he buy ? Let X = No. bought, then }x at 24 + ^x at $20 is $48 more than cos/, which is x'\ That is, 5 5 Clearing and reducing .-. rw-— 11 2.« + 240 = ; or, (.r — 20)(ru;— 12) =0 ; .-. .r = 20. Ex. 2. Two couriers start at the same time on a journey of 80 miles. A travels two miles an hour faster than B and arrives two hours biifore h«m. Find their rates of travelling. Let X = A's rate per hour, then .^' — 2 = B's rate. Then 80 X = A's time {»nd ftO x-2 = B's time 80 80 =r:2. X PR0BLEM8 WORKED OUT. 2G3 X = 0. imples of ir each us at %2\ 11 insaction. iioro than )uriu'y of id arrives or or 80 whence and (z — :, -M = 2, or A()l-~\ = ,-. a'? — '2x — m = (), (.r— 10)(;C + H) =rO; a; = 10 = A's rate, 10 - 2 = 8 = iVs rate. Ex. 3. A man boii^Mit a number of (;ows for $540 ; if h(; had ^ot 3 less for the same sum, the i)riee of each would have been #6 more. How many did he buy ? Let X = the No., then - -^ = price of each, and S'^^ = urivi- of '*'' X — ii ' ftach had there been 3 less, and this is |6 more than the actual price. 540 540 X x — '6 -6. or or 540 !>0/-l ] = , whence we get and (.t'-18)(u;+ 15) = 0, •. x= 18, or — 15 ; of which only the former satisfies tlic eotiditions of the problem. The negative value nuiy be considered to indicjite a (limiinitii»i of stock, and is a solution of the following problem : A iiiiin sohl a ic giv(!n 3 cows mort' for the same number of cows for $540 ; had I sum he would have received !?(> ;i he:id / solved by the e(iuation V.v.v. This prol)lem is 540 X 5-10 ./• + 3 + 0. in which the values of x are 15 and - 1 ■I f ^. i ?59" M 20-i QUADRATIC EQUATIONS. M I'M* J N- EXERCISE (a) LXXXI. 1. If to the squaro of a certain number 13 times itself be added, the sura will be 364. What is the number ? 2. The area of a rectangle of which one side is 7 feet longer than the other is 494 feet. Find each side. 3. What number will give, when subtracted from its reciprocal, 6.09? 4. Divide a straight line a into two unequal parts so that one • part shall be a mean proportional between a and the other part. 5. A ganh^n plot 99 yards long and 66 broad is to be surrounded by a walk equal in area to the plot. Find the breadth of the walk. 6. A grocer bought $15 worth of coffee and paid an equal amount for raisins, receiving 10 lbs. more raisins than coffee ; he sold 20 lbs. of coffee and 30 lbs. of raisins each at a profit of 33^ per cent, for which he received in all $18. How many lbs. of each did he buy ? 7. By selling a horse for $432, a man gained a percentage P(puil to on(ir) 12. 13. added, . or tbau 14. iprocal, 15. liat one 16 e other rounded 17 jadth of amount ffi'e; be a profit i. How 18 re ecpial 19 >st biin. 20 number >r> more bad got Ul bave 21 • break- , and so iiy? A goes r before 23 The lengths in yards of the si(h's of a rig)it-angl(Ml triangle are represented by three consecutive i)uiul>ers. Find the length of each side. What are orar ges a dozen when two more for a dollar would lower the priee n cent a dozen ? By lowering the i)ri('e of ap|>les u penny a d(>zen an apple- woman can sell iW more for 5s. At what price did she sell thi*m at first ? Find two numbers whose difTeren'*e is 15, and of which the cube of the less is eipial to luilf their prodiwt. A company at a tavern had $;{5 to pay ; two of them having h'ft the room, th(i others had |2 apiece more to pay. How many were in the company ? Two persons start from different j^laces at the same time and travel towards each other; when they meet it is found that A h.as travelled 18 miles more than H, and that he could have done B's distance in 15J days, and that It would have been 28 days in doing A's distance. How far did each travel ? An arti(;le is sold for $9 .at a loss of as much per cent as it is worth in dollars. Find its cost. Find the quotient whose dividend is 2J less than the divisor, and which .added to its reciprocal gives 2J. A and B put the sum of |3,400 into business ; A's money w.is in 12 months, .and B's, 15 months ; on .settlement A received $2,070 as capital and gain, and B received $1,920. What capital did each invest ? Two pieces of cloth, ccmsisting of yards and 14 yards, respectively, are bought for $5;j ; and the buyer finds that for |15 he gets one yard more of the batter than of the* former. What is the price per yard of each piece ? The di.agonal of a rectangle, of which the ])re.'idth is d I'eet less than the length, measures h feet. F'ind tlu; length and bre.adth of the rectangle. Find the radius of a circle of which the area would be doubled by increasing its radius an inch. yoo yiADHATIC K(ii;ATI()NS. Hi;. H I II: (ft.) 1. A (;i.st(!ni ImH two pijM's ; to fill the cistern, ono takes two liours less time than tin- other; holh rufininj^ toj^ether cun (ill it in 1j( hours. How hm^ will it take each to fill thn cistern ? I'. Two bodies move towards each other from two jxtints 1800 yanls ajiurt, the lirst start inj^ "> seconds later than the so(!ond ; lait the first makes in each .>,«•. ...d city of each body per second. 3. A number has two dibits, the left-hand di^it bein^ (h)id)le of the rij^ht-hand dij^it ; if the di;;ils l)e revcr.sed, thci product of the number thus formed and the orij^inal number i.s 2268. Find the number. 4, A rectangular <^arden plot, of sides a and 6, is surriunded by a path of uniform width and ecpial in area to one wjth of tin? rectangle. Find the width of the path, 6. A company of men is formed into a liollow scpiare 8 deep, and afterwards into a hollow .ser of acres bought, and thc^ price por acre. 8. A and B start at tho same instant from two points, P and Q, 60 yards apart ; A, whose rate is 8 yards per second, runs at right angles to PQ ; B, whose rate is 10 yards per second, runs in such a direction that, without changing it, he may just catch A. TIow long did they run ? I'UOin-KMS. 2C7 cos twn hor (!iin nil tho Its IHOO lian \\n' (Is in(>r«5 ,«'('n tho oiiblo of product iinbor is i(l»'th bcin.ic ut when Ore than 1^ acres ff |7 per of acres and Q, )ud, runs jards per (changing 9. If H oarria^ie wJH'cl \i\}, feet in (•ircmiifcn'iicc took (uic second more to revolve, the rate of the cari'iaj^'e pel- hour would be I J miles less*. At what rate is the carria;,'e Iravellin;;^ If). Thei'e are two roads to a certain place ; one, over the liolds. is K miles, the main I'oad is II miles. \ man can i-ide a bicycle aloni; the main I'ojul 10 miles an lutiir faster than he can walk over the other road, and would arrive 5'J minntt's sooner than l»y walkini;. Find his i-ale of walkinjj;. 11. A man clindis a mountain, walkin<; | a mile an hour faster diirin^rthe firsf half of the distance than dnrin;; the second, and reaches the top in ."ii hours; he descends in IV^ hours, walkin;^ a mile an hour faster than when he be. and repays it, principal and interest, in two sums of $4410 at the end of the lirst and second years. Find the rate of interest. 10. A packet from Dover reaches Calais in 2 hours, but on tho return voyage proceeds at first (> miles an hour slower than 'H li f^ ^G8 QUADRATIC EQUATIONS. ili-' Hi 1:' ;* ! , ,i i a>- J it went ; but ufter half tho return trip was nmdo, th • wind changed, and the packet sailed 2 miles jyi hour faster, and reached Dover sooner than it would have done, if the wind had not changed, in th(» proportion of 6:7. Find thu xitinpl('s of tho solution of lu'i) (ytniUun.s, hi otit^ or both of witirh the unknown quantity is fomut in -f- 4. Substitute this in the lirst equation, and we have tir 25^" - 1 44.r + 1 T« = »), )r (2ru'-44)(.r-4) ^(K 44 .•..r = 4, or • Substituting these values of .t\ su<('essiv«'ly, in the first ecjua tion, we get 4'^ 4- ji"^ = 25, whenct^ y = -^ ; und Q -f-//^-ar,, - i^ = 4ii. From the secon*! eijuation, // J- -f- .»■!/ + (If = M). (3) L»'t = m, that is ij—m.v, and suhstitute tliis valin' of // in each equation ; thus ii'" + m.v" — Vinrw (5. Dividing (4) by (3), and nducin^;. we have (4> ;{()///- = 4 m = ± J. Bubstituting in (3) the.se values of m. we tjet .r = ±3 //= ±1. Ex. 4. ;r + // :^ 13. Assume x — my, and substitute in the equations ; thus J-* + m^if = «73. or .f» ( 1 4- ni^) ^ ♦>37, ;/• 4- my = 13, or vj(l 4- /w) = 13, (1) (2) (8) (4) F'MJMINATIOxN OF ABSOLITK TERM. 271 Divide the cube of (4) by (3), (1 + ill)' 1 + »»'' !:;'_ 169 637 ~ 4¥ ' •'• 1 -m'+7>i''~ 49 ' From this eipuitioii »i is nudily found to l)e | or |; and substi- iiitin^ Ihc value of m in (4), we get x = 5 or 8, whence t/ = H or 5. 169. Elimination. — Where ])olh the lioniogeneous e(|uations ;:!'(' of tile same dej^ree, they are, in jfelieral, most redd ilji sol ccd ill ('lhnhHitin li v«';- (:.) — (-'I jjives (»r r).r' + Tij-y - 30//' = 30, 4x' - 3t5y- = 0. .>■ = ± Sy, whence ihe values of w and y luay l)e found. Ex. «. .r*-3.r//- 10//' = l.i, j' + r}.ty + iUf = 'irio. Multiply the 1st ecjiuition by 17, .'. 17^' — 'Aj-y - 1 70//' = ',>r,.'i. Subtniet the 2d eijuation from :his result, .'. 10.r* — nan/ — 1 T(>//' = 0, or 2j-^ — l.ry - '2'iy'^ = ; i.e., {'2x- ny){J- + 2//) - 0; .-. .r = y //. or — 2y, etc (1) (2) i <'f I ■( 273 SIMULTANEOUS QUADRATICS. ft::' H *^5 170. S]rnun6trioal Equations. — Whim each of the equations is symmetrical in x and y, they may sometimes be solved by stich a substitution fxs> u + v for x and u — v for y. Ex. 7. a;' + y» = 18a-i/, (1) a- + y = 13. (3) Putting u f « for x and m — r fof y, ( 1) becomes (?'. + v)^ + (// — J')' = IH iu + r) (m — r), or «' + Sf/r" =^ 9 («" — t>») ; (3) and (3) becoines (w + '') + ('/ — '') = 12> which gives m = 6. {Substituting this value of u in (3), we got 210 4- 18<'» = 9(36 — v'), or 27tJ« = 108 ; /. V* = 4, ar.d tj = ± 3 ; hence a; = w + r — 6 t 3 = 8 or 4, and y = ;/ — r = « q: 3 = 4 or 8. EXKRCISE LXXXII. («.) Solve the following equations by the method of substitution . — 1. y = ti -\- X, x' = b. 2. X + y =a, x^ - b' = 0. 3. u-» + y» = 10, X ~ 2y = 1. 4. xy -1- // = 13, y-2x= -^ r>. a;' 4- 5// = 41, 5a; — 4y = 0. 8. «» - 4.cy + T)//" = 13, 2x - 7y = 0. 7. a- + y ■= 9 (.r — y), a-y r- 330. t*. -^ - y ^ !( (^ -»- //), iC* ^ xy =i 370. EXERCISES— HOMOOENEOUS EQtJATIONS. n:. (1) 9. a; + y = 40, xy = 300. 10. Sx^-2xy = ri, x~y = 2. 11. 4y = 5j-|-l, 2xy = 33 — ««. 13. a; + ^ = ^ + y 2xj^y 3 OP = * (4a; - y). 13. a:' + a-*/ + y' = 7, 2a; + 3y = 8. 14. .r' - y' = tV (a;» + y'), 5a; + 7y = 92. 15. a:' + 4//^ - 3a; + y = 67, a; - 2ir - 1 = 0. 10. 21a-* -8a-// = 53, 15a; + 2// = 7. 17. 5a;''-3Ar» + 118 = 0, nx' — UHxy ■\- 35y» = 0. 18. 9x* + 5y» = 161, 20a;* + 6y' - 23a-y = 0. 19. x-^(x- y) =, 4, X + Sy y- X + % = 1. 20. a; - y = 2, 15 (a:" - y») = nxy. 21. 3a;' + 2^/' = 165, 27a;' + ly'' = 66ary. 22. 3a;'' + V = 91, 2a;' + 5y' = 7ay. 23. ^x-Ay = 5, 3a;'-a-y-3y' = 21. 24. 5a- + y — 3 = 0, 2a;' - 3.ry + y' = 1. I (ft) Solvn the following honiojfenoous equations L 3a;' + lla;y-4y' = 126, a;' - 16y' = 9. 2. a;' -3a-//- 10//' = 15, a^' + 5a-// + «y^ = 255. 3. a-' + '^.r// + 3//' = 11, ;r' + 8a?y + 6y' = 22. 4. 14a^ - Irta-y + 5y' = 45, x* — 3a;y f y' = 9. 5. 2y' 4a-y + 3a;» = 17. y' - a;' = 16. 6. 20a;' + 7a-.y - 6y' = 21, 5a;' + 4a-y - 2y' = 7. 7. 2a;'-.r//- 15//' = 7, 6a;' + lla;y- lOy' = 35, 8. 6a-'-5a-y = 21, 2a:' -i- 3a;y ~ 9. I ' * ■ J 'if in SI M V i/rA N Koi's qv \ du atips. i< 11 la. i.r^ + v = -ifi, 15. .r' + .*•// + I//' = 0, :{./-^ I H//' :^ 14. 10. ./•' - ./•.// + .// = 21, !/' — '2.11/ \ 15 - 0. 11. \2Hj' -f IH.ny - V ■-= 14:{. 17. .r' f //' = 407, 2Tij' + T'2.r// - \iy' = 407. ./• + /y :^ 1 1. 12. ./•■' f .*•// r^ a, i;{. ./•■' - .77/ - «//' ^r 24, 14. .r f- .»•.// f 'V = 74, 2.r'' + 2.r// f //' :^^ 7;{. IM. .,•' - //' = 218, ./; — // = 2. 11 (// I \u") + ;{./•// -^ nm. 20. <».<•.// - 2./'' — If' - ai. S(»lv(' tilt' toMuwiii^ .symmrtricid ('(|iiiili(»ns : - 21. {X + .'/)(.f' + .'/■') = 175, ./•■'// -f ortf = 84. 23. X ^ .// r= 2;{, ./•=• 4. //' = :U7:i. x:'ii + .r// = IHO. 24. X' + .V' -- 272, X 4 .'/ ^ «. 25. .t* + //* = :W7, a; -^ = 1- 20. ./••• I //" ^- 275, X f // - 5. 27. x' - //* -- 22000, X + // == 22. 28. ./ * -//* = 2145, X -// = .{. Wk •'• + U = 4, ix' ^ U" )(./•» + tf) =: 280 80. jr + // . 5, (U.-' + .'/^ ) u-'' -f //•') = 455 171. Specip.l Artificas. Tlx' methods iilustritttMl in t\v' p inj5 urlif'lcs may In- ii|t|>lic(| f(» tin- classes of e<|iijitiotih f these ejasses, may he solvecl hy sfminl (irflfireM, of whieli a fv * txam['le< will now he Ljiveii. Ex. 1 X f y ^ 12 jsy = 85. m ., KXAMI'LKS OF SPKCIAL AUTIPfrKS. (Will (U (2) (1) sqiiurcd is i'-i) X 4 is V-i) — (1) j,'ivcs Now 4^7/ ^ 140, (4) (•'53 ^ow roinl„,H. fr.) uml (1,, aiMl ll„. val.ics of x mul y aro easily (1; 1)1 • •• •'• + // - \). I)ivi(l.! (2) by this, and we ^m1 .//y := -JO. Th.'s.. .■(lualions ,aii now l>., solved by substitution, or jw in Ivx. I., Kivin;<^=:r), 4; y = 4, 5. Ex. 3. Of icy (x- — //) = 2i({, xij {.r — j/)~ 0, (1) -f- (2) Kiv»;s x'jf' =. a«, «id)s»ilu(r in a>) ; ami // = 2. or - :{. Kx. 4. ,r - !, . 4, x' + !f' - \m subtnvft tlu! R(iuan! of tlu;y//-.v/ fn.in tin- swond ; tl lUS, ft ii III ^^mm V- I I.' .' n.i Tk.. ■ I JiQ •5 w^ are SIMl'LTANEOrS QrADRATirS. Add this to tlio second cqiiiilion and wo have (^4-//)'^= 196, or x + u= ±\^\ eomhinc this with tho Hrst lupiation, and x an«l y arc easily found. Ex. r,. (1) -i- (3) KJves (2> - (3) " (2) + (4) ♦* or (8) -(4) •♦ Ex. 6. a.'* + .cY + //«=.- 2188, y" -I- .r// 4- //-' =^ 70. a^' - ^// + If = 88, 2^-^ = 48, xy = 24, (a; + y)-' =z 100, hi: _..-,. 278 SIM U LTA xN KOL'S Qf A DRATICS. If^- 30. («.r 4- 4//) C^r — 2//) + '^J" -»• 4y = 44, CJr f 4//) (7.r - 2//) - 7.r -*- >i.y = 30. 81. Eliminato x and // from tho cnuations ax* + hxf/ + f//" = 0, a'x' + b'xy + c'y^ - 0. 5. a-' + xi/ = 35, 17. x*-y* = 8080, xy - y* = 6. x-y = 2. 6. a:'^ + U-y = 3, 18. X + y + xy = \\, y'^ + Axy = 3^. a-y' 4- x'y = - 30. 7. « + // = rt, 19. -ni 4-.'-//= 9, Axy = ii* — 4&'. u> - xy' = 30. 8. u;" - /y'^ _ 4 = 0. 20. a;' 4-y' 4-3(.r4-Z/) = 16?, xy — '/■' + h' — 0. .ry=13. 0. .r'^ - xy = a' 4 .V, 21. a?'4-A''' 4-3(.r 4-//) =44, ./•// — //" = tJf//' x + y = ^. m ' + • = 32. UJ'4-y^-4r4-4//=17, ^ 4- • - 1 ^ .*•//= 10. • 11. i_y + .,+ ,;- -^O- 33. ;r' 4. 4//"' 4- 6.'/ — 3.r = 393. (J.r -) =0, ••• i/ = 2, or 4. •y I)iittin^'f(,r //fl.r valuo3, .r + ^ =r 4 : O.J ' -d) gives 2 (a,-y + y^ 4- .,.^, ^ oa (•'">) - 3 (:}) which. gives eombi .ri- =r •} (1) (3) m 1 with (4). can hv solvrd by methods already given. .1' = il = 3. z = x are values for *•, y^ z ■ and thv. other val stituting the secoiiU value of •■■ft- 'J !/■ ues may be found bv suh *^*'^- ^'- a-^ + xy + .,z = m, f + .''y + liz = 48, ___£ ^ J-J' + t/z — HO. By adding the three ecpiations, wr get, (^- + i/ 4- ^)' = 144, **"* ^ + i/ + ^=±13. IMAGE EVALUATION TEST TARGET (MT-3) 1.0 I.I m I?; IIIIM IIIIIU illtt IIIIAO M 1.8 1.25 1.4 1.6 ^ 6" ► v] <^ /} 'c^l e: C)/, 'm -^1 /> '^y '/ M Photographic Sciences Corporation 23 WEST MAIN STREET WEBSTER, NY. 14580 (716) 872-4503 ^r .^. i f ? i^ 9: I!.: VI 'I m 280 SIMULTANEOUS QUADUATICS. The first equation is x(x + y -\- z) =M ; substitute in this the vahies of a; + y + -?, .-. X X 12 = 36, or a; = 3 ; the other value ot x + y + z (viz. — 12) will give x = — S; similarly, from the other equations, y and z may be found. x= ±d, y = ±^, z- ± 5. Ex.3. xy + yz + zx = a^ — x^, xy + yz + zx =-. b"" — y", ^y + y ^ + z^ = < ^^ — 2\ (1) is x^ + xy + yz + zx = a% OP x(x + z) + y(x + z) = a', or (x + y) {x + z)=i a?. And by symmetry, (2) is (3) is (y + z) (y + a-) = 6" iz + .*;) {z + THREE UNKxVOWN.S— EXEIICISES. 281 i-S (•« + U) + {2+x)- (y + z) = 3.r = "^' + "'"- _ '//"' . and tlie values of y and 2- may be determined hy symmetry. Ill (1< (3; (5) m EXERCISE Solve the follovvinj? eaeh ctions is ;he frac- lote the (1) (2.) Ex. 2. A train travels ir,0 miles ; luid it travelled at the sain rate for two hours, and the rest of the time at 40 miles an hou- it would have gone 210 miles. Find the time, and the rate per hour. Let .r = time in hours, y = rat<> per hour ; then ' . , .?^// = 150, ind 2y + (-^•-^) 40 ^210. From the s(!eond ('((nation y = H.-) — 20ir • .substitute this in the first e(juation, •. ir ( 14.5 — 20.r) = 150, 4a;' - 29^- + HO = 0, y = 25. or an''. whence Ex. 3. If $300 were allowed to remain at .simple interest for a eertani time it would amount to 1300 : if it were allowed to remain two years longer, at a rate of interest one \wx e(Mit higher It would amount to |405. Find the rat(? of interest and the Time.' Let X = No. of years, and y = rate per cent ; then 300 ..ZV/ or or = 60, 60. Also 100 ^xy _ .„, xy = 20. 300 (X + 2) (y + 1) -— = 105. (1) or or too • 3(x + 2) (y + l)=rl05, ix + 2)(y + \)=i)ry. By substituting the value of xy from (1), (2) gives .^3,i3_3^^ and this value substituted in (l) gives y(13-2^ = 2y, (3^ sill its 1; 1 284 iSIM ILTA \ KOrs Qr ADHATirs. h>V. that is, -Z/'- V.ii/ 4 20 = 0, Oi (2y_r,)(//-4) = 0; • •• >/ = -i, or '.>^ ; whence x —. ."), or 8. Ex. 4. Find throe numbers sncli that the sunj of the first and second added to their i)rodiict is 55, the sum of the second and thif•'», y + ^ 4 ^11 — '54, z + .<■ + zr = -in. Add 1 to both sides of (t) and factor the result; ilien (1 +.rUl +//) = 5G. Similarly with (2) and (8), (1 + //)(! + 2) = 35, (1 + z) (1 + .V) = 40 ; .-, (4).(5).(G)gives (I + x)-' ( t + i/Y ( 1 + ^)' = 40-35.56 ; /. {t +x)(l + y){\ +z)==2S0. (7) -i- (5) gives 1 + .r = 8 ; .-. X = 7. Similarly, 1+^ = 7; .-. y==Q, And 1+0 = 5; .:-. = 5; and the numbei's are 7, 6, i. (1) (2) (3) (4) (5) (6) n MISf'ELLAXKOrs PROBLEMS. 585 (1) (3) (4) (5) (6) EXERCISE LXXXV. PROBLEMS. CO ' 1. Find two nnmlxTs such tliat their sum. ))ro(ln<'t. and differ- ence of tlieir s(iuares shall be e(|iial to one another. 3. Find two consecutive ininihers such that the cube of the greater shall exceed the cutx' (A' the less by 188. 3. The product of two numbers is ")?(>, and their (juotient 2^; lind ihe iniinbers. 4. The product of two numbers is^^, and their (piotient is '^9 feet ; one of the smaller scpuires contains 41 sciuare feet more than the other. Find the sides of tla^ smaller s(iuares. 6. Th(> proiluct of the sum and difference of two num))ers is i i li tj i 28G S 1 M V LT A N i:u IS (^ U A I ) 1{ A TICS. lii: ill V s 5 II ll: rcmuiiidcr will ('(Hi;il U, liiiits tlir i»ro(Iiu'l of llic (ll miinlHT. 11. Divide 102 into tlirc*' i)ni'ts siu'li lluit llic |)r»)(liict of the first and third sIimU ('([iiid 102 tiiiifs tli.- scnuid, juid tli(> third shall \h'. li times the lirst. 12. A nuiiihcr ('(insists of two di<;its whose pi'oduct is IS ; lh(> dif- ference Ix'tween the s(|uare of tlie nunilx'r and the nunitier formed hy inverting; the diijits is 207:5. Find the nnm))er. i;{. Find two nnmbers whose sum multiplied by the sum of their S(iuares ('(juals 4SS, and whose dilVerenee multii)lied by the- ditterenee of their scjuares is 24. 14. The fore-wheel of a carriaj^'e tni-ns, in a mile, i;{2 times more than the hind-wheel ; if the eireumferenee of each were 3 feet greater, it would turn oidy SS times more. Find the circumference of ejich wheel. 15. Two l)oys run in opposite dir(>et ions ronnd a nM'tany;ular held of which th(^ area is one acre ; they start from on(^ corner and meet 18 yards from the opposite corner, and the rate of on(; is f that of the other. Find the dimensions of the field. 16. A ladder wliose foot rests in a given ])osition readies, on one side of a street, a window 'Mi feet high, and when turned about its foot just reaches a window 27 feet high, on tlie other side of ..he street ; if the two ])ositions of the ladder be at right angles, find the width of the street, and the length of the ladder. 17. From two stati(ms, A and B, 300 miles apart, two trains start at the same instant ; the train from A reaches H J) lumrs .ifter they met, and the train from B reaches A 4 honrs after they met. Find the rate of each train. 18. A person lends $1800 in two snms at different rates of inten^st, and receives ecjnal returns from both ; if the first portion . had been lent at the second rate of interest it would have produced $86 ; and if the second portion had been lent at tlie first rate of interest it would have produced $49. Find the rates of interest. ^ "-? MISCKLLANKOl 8 IMJOHLKMS. 287 ic (li,i;its. the first the third ; tlic (lif- > iuiiiiImt luimlu'r. 1 of tlu'ir 'd by tlio UK'S inovi' •ach \\vr{\ Find the >:uli»r Hold ;)iio c'oriU'i' (1 the rate oils of tliu >s, on Olio '11 turiu'd 1. on tlui u> liiddor . and the aiiis start ^ \) hours 4 hours interest, t portion luld liave n lent at ^9. Find O.) 2'.\. 19. Tlie si(h's of a rectan^nilar Held ai-c 1 19 yards and 19 yanls, respectively; how much must the h-ii^'lh he dimiuished and the l)rcadl!i increased in orth-r that tiie ai'ca nny l)e the same, and the i)erimeler I v '.'4 yards greater '' '20. The (lia;>:()nal of a rect a nude is "-.>(). t feet ; if tiie Iciiirtli of the rectan^h' l»e increased hy 14 feet, and thel)readtii he dimin- ished by 'i.4 feet, the diaiconal will be increased by 1'2.4 feet. Find the length and bi-eadth. 21. A person boil <;ht for $7')(»<) a number of !?H><> railway shares, <■ when they wer<' at a certain rate percent discount; and afterwards, when they were at the same rate percent i»re- niium, sold them, all l)ut (JO of them, for t^MHH) ; how many did he buy, and how much did he udve for each share i Tiu^ difTereiice of two numbers is :), and the dilTercnce of their s(iuares is «!9. Find the numbers. Two miMi s(>Il a (luantity of wlieat for ij59() ; ,\ sells 10 bushels more than B, and if he had sold the (piaiitity H sold, would have received $;}(> for it ; while H would have received $(54 for what A sold. How much did each sell, and at what rate? 24. The sum of three numbers is IS ; the sum of their .sipiares is 184, and the i)roduct of the second and third is 50. J'ind the numbers. 25. The sum of three numbers is 12 ; the product of the first and second added to the pnxhict of the second and third is 82, and the sum of the s([uares of the first and third is 84. Find the numbers. (ft) 1. 11ir(>e cubical blocks contain an agpcrcgato of 78 cubic feet ; the afi;^re)j;ate area of the faces is 12(5 s(i. ft., and the sum of their depths is 7 feet. Find the dimensions of each block. 5^. Th(>ro is a number (»f tliree digits, the left-hand diiijit being double the riss, which was invested Vi^ years Icmger than the tirst capital and at tlio same rate, was $2574. Find the first capital, and tht; time it was invested. 6. There are tlireo numbers such that the first multiplied l)y the sum of the other two is 68; the .second nuiltiplied by the sum of the other two is 98, and the third multiplied by the sum of the other two is 110. Find the numbers. 7. A, in running a race; with B to a post and l)ack, met him ten yards from the post ; to mak(( it a dead-heat, H must have increased his rate from this point 412 yards ])er minute ; and if, without chanjj^ing, he had turned back on meetin express<'d in fe«'t have th<^ sum of their cubes ecjufd to 52 times their sum, and th<' ditt'erenci* of their cul)es ecpial to 148 times their ditfi'rence. Find the .irea and the diagonal. 9. There are three numl)ers ; th<^ sum of the first and second adch'd to their pro(hrct is 89 ; the sum of the iirst and third ad(h'd to their product is 98 ; and the sum of tin; second and third added to their product is 109. Find the num))er. 10. Find a number greater than 100 and less than 1000 in which the middle digit is double of the right-hand digit, and if the first and last digits be interchanged, the new number is 99 less than the original number ; also the sum of the squares of the digits is 29. -?■ by tho uiin of ]\v I'lrst 111(1 r)i\. l.y the liict is I'd, tlio .! (li^it. iibcr of , wliicli and iit and till) by tlic by tho 1 liy tho him ton ist have minute ; 'tint:; A, low far sum of |tforon(ui hnd tho sooond lid third socond lumber. II which and if |m}>er is of the Mis( KIJ.ANKoi s Im;()MI,i;m«. to 14, Tho surfaco of a ro( taii;(ul;ir pjirallolojiipod is UMJ s(jiiaro foot ; tho loii^tli oxooods l»y ."» feet thi' sum of tho breadth and lioi<^ht, and the diaj^oiiid is i;{ foot. Find thi' leii^^th, breadth, and hei^dit. Find tlin'o numbers such tliat 5 times the mm of the first and soooiid added to their jiroduet is (5:5; ;") timsv t!io sum of tho so(;ond and third added to their product is 1'3JJ ; and it times tho sum of th(! first and third added to their product is y?. Four numbers form a proportion ; tho sum of tho moans is a, tho sum of the extremes />, and the sum of the sroadth of each of those parts. 18. A three-figure numl)er is ocpial to 43 times tho sum of the middle and loft hand digits, tho sum of tho digits is oipial to 9, and tho right hand digit is twice the sum of the other two. Find the number. in >1 w CHAPTER XVI. !0 ' V rh t •:'3 aj „j ~J C2I •:q: •-CI" li i i ■ * ■ lyili TIIKOUY OF Ql'ADUATIC KQUATIOXS. 174i /I quadrat hi vqaation cannot 7iaf>e more than two roots. For if i)()ssil)l(' let tho ('(illation x^ + px + ^ = hjivo thr(>(^ roots, a, /9, y. Then ;r — a, .r — /?, and a; — y are nil factors of iv''-\-p,v-\-(j\\vX. 150), wliicli is impossible, since x^-\-p.i + q is of only tile second deforce in x. Aithouj^h a ((uadratic cannot luive more than two roots, it may in etfect iiav(! only one, by reason of the two rcjots beinj^ e(iual to one another. Tljus, in ic* 4- px + root's. y are ail s, it may equal to ids us to tivofdc- itic. ')ots of ts of tho mt. fferent. KKLATIOX Ol- i{()(,Ts TO V{m'V\V\v.y{'X^, {\\) ir r—O, (I,,. r(,(,fsj||V()j,i„| _ XJOI (I (iii) If /.=.(), fl,,. roots Hivii/I^ ,-. ,.,.„., , .,, ., . > // ' •' ^^i"'^' '*^'^ Willi opoo- «!<<' si;,'ris. ^ I'lK tlijit the equation assumes tlio f(u-m vvliicli is Siilisficd l)y ;t' = — Honee when ., = (,, th. roofs nn> _^^ ,„i(l InMuify. (Vj If a = a„,i /. = 0, tho (>,i„ati(m assumes the form 0..r-' + o. Here, since tho eoeflficimts of a^' and + r. = 0. tities, only an indcfinjtclv I X are indefinitely small quan- tities O-.*;-' and O-.Hari^'eVnou.d »r^'e value for x could make tli wliol e expression a.r' + /,,• + ,. (o vanisi I <<> neutralize tlu^ e, and e (juan- caust; the 6 = 0, the roofs are hofh infj nife I. Hence when d =0 and (vi) If ^, = .,,1,1 '-' = (), t lie roof.s assume the f The value of h "orms X and th(i form y hero is s.-en lo l,o (), since tho e(iuation t Ox" + h.r + = 0. iikes H((nce when a = and c = 0, th(; roots ar<' z(>ro and infinity. (vii) If & = and e=o, hoih roots become --• H 6 = cind c = 0, both roots are zero, Examples of the disappi^aranee of terms from fre(iuontly mot with, ospeciallv wheiv tin a fractional form. Thus in the equation epco wlion equations are e o(piation is i>rosented in i !» '%>• lit, O l,,r. '■■■ U4 o '•■ r hi 1| I u :30;> THKOUY 01' QrADUATIC KQl'AtlONt^. 2x + T) _ 2x + 4 8a; + 2 8.r + 7 0, oil siini»lifyin + \^h- - 4ae) (- ?> - Jb' - 4ae) «/? = t-t; ~ b- - W + 4a( 4a" a which, by 00 satisfies PORMATfON OP QfADHATtOS, 393 We have ^A6-' + i^.+ ,,.^,,A,... ^'> ^. ^ r\ = ^' (-i- - a) (.?• - ^7), ohtained. /(lo,— a result previously "11 f^ hetwem ■iac — \ae) «|U„ti«n, the oquatiot ir ' '"' ~ ' ** ""^ ^"'^ °'' »» or Gi- - 2j Cc + 3) = 0, 178. From tlie results M -1 ^y __ ... «/? = '^r fif tlM equation l>e .,.^ + ^,,, ^ ^^ ^ ^^^ we may form, in terms f)f tiu. .„^- «• • ^ Thus. «' + ^y^^,« + ^,,._,^^^^^^^,_^^^ •.•' + ^= = (« + ^)(«._^^^^^^^,^^ % ^ = "^+1-1^ 11 1 . iiJi^t^nmBvnvi It . :C> '■a i; j ■!!::! ! ! ' 1 i ( 1 ;. M i4. ill 294 THEORY OF QUADRATIC EQUATIONS. Ex. 1. Form the equation whose roots are the reciprocals of those of x' + px + q = 0. If a, /? be the roots of this equation, the required equation will be = X a + S 1 = X' fix +- i. e., qx^ + px + 1=0. Ex, 3. Find the condition that one root of ax^ + bx + c = may be twice the other. Here x-' +lx+ ^ Hence and a — 2 9. -=«/?=2/?^ -'^'=i"0'^ and the required condition is 9ac — 26" = 0. Ex. 3. Find the condition that x'' + px + q and x'^ + p'x + q' ■. may have a common root. Let a be the common root. 0, a) m a) SVMAIETRICAL FUNCTIOXS OF ROOTS. Then (3) — (4) gives (3) X g' gives (4) X q gives a'+7>'« + , «/? = g, «(/?+ y) =q + s. O^ (/? + yr + (P + r) i^9 + y, + 2 (q + s) = 0, and /? + y is a root of <^' + ip + r)x + 2(q + s) = 0. II m I**'; ■ i! '; P '1 :;*h». I' -::> m m ! :'8 :i ?Q J ] M i I' i' ^ 39C THHOUY OP QL'ADRATir; flQL'ATION'S. EXERCISE LXXXVI. (a) 1. Form t]i(> (iiijidnitic whose roots ai'c 1 ± y^S. For wlijit valiif of p will tho o = () have t'(iiial roots ? Form th(! ('(luation wlioso roots arc — l«| and 5<)^. AVhat (£ua(lratic has the roots t'^'^/i, ) •*"(/ — IO7- + 277/' is 4/> — 5^ + ()/•. AVhat is the otlier root ? 11. Show tliat in (>verv cjuadratic of the form lueof 4b. Olio root -1234^. r + 27(j/- + a = 0. 1, show dac. t, prove 207 EXEurrsEs and problems. 1;"). Show that the roots of ar^ 4. A,- , . n of tho roots of a.^ + V. + 1 = o " "'' '^'' reciprocals 16. If ^•''=j,^^i,. + y^ show that where « + ,y = y>, «^:^^_ 1- If the roots of rnr^ + h. (ft) _ 7' +^'-^ + - = be equal, cu- + rr.r + 6 = q bo real, i1h>h o bet and those of ween a and 46'. IS intermediate in ma-Mitud le 3. Find the condition that th be respectively, tlie squares of th roots of a^r" + frw + r^- ^ 3. If the quadratic aa-- + L roots of (i.v--{h.v-^, in 0. 4. If two different values of values of x. fi^ /? be the roots of + (>.r + (' = 1 )e satisfied by •i', it will be satisfied more than hy all whose roots are H) «^,/\ (:2) U-- + p,r + q~ f, >nn the equations 1 1 «" 3^- 0^) rr+f, n^l U) ,,^. ^_ '">• If the e(iuati( /A «~,^. )n ^r f/ = ]() or ~\\ '<■>■ + '/ + 1.-) ~() I »;ive e(|nal roots, \\ len 0. If «, /? be tl le roots of .r^ +/>.r + y '>, .'lud n — '/ +P'J + 1=0. "// i-j ' th en Tf the roots of fh.^ e(iuation wiunnulti the n sy/ = (J,- pies of the roots of fl ■r- + />.r + 7 = ) "' respe('tiv(;Iy H' «'(iuatioi; x- + r.r + ,y _-_ () 8. Find the eondit be th ions that tlu* roofs of a.r + h reciprocals- of those of 9. In tlie equat a'.L- + h'.t + (/ :~ + h.r + r =: may ion .r^ _ a' + A.v + ?„i ^ ^i = have a real value, a cannot lie betwee 0, show that, if sn 1 and 2. X 3 3 i! :( \: 298 THEORY OF QUADRATIC EQUATIONS. 'l'^^ I 3.': .lat^ •.^•;o !|.l!|i' !r;-'<:j ■'^^' !JJ '' ■ ■ -J -J :,i8 lis t!:Cti '':-'!0 I i '■ \ ' 1 '■'■ I ■ ' it: 1! '■;; y 10. If the (lifTcrenoo betwoiMi Uic roots of ax'' + 2bx + c = be eqiiJil to tlu! (liftcrcneo between the roots of a'x' + 2b'x + c' = 0, then (j'^-ac _b"-a'c' a' a 11. If {fi — h) ic^ + (a + bf .V + (a'^ ~ b") (d + h) huva two equal root.s, tlien a = '6b, or '6a = b. 1. If a-x"^ + hi' + be + b'' = luive equal roots, then will Aa' b 2. Show that (a — x)(b — x) — <>rsoffluM.qnatio,i '^''' + «t ± h) .V ± ah = 0, 11. If iC^ + rt'.r + /; root, then '' = 0, and ,/-■ -f- ^/ •'■ + -'/ = 0, h , Have a eoirinion {' ~(i'(,)(a ~a') + if, 13. Find the condition that the '' ~ h'f- = 0. roots of tiK. o(iuation ff, *■' + 2l,.r + "lay be formed from tl 'lo roots of % adding th «'*•- + 2b X + r' 1'^. If the equations f^ same quantity to each root. ax solution, prove that + %=1, <-x:' ^dy-' = \ Jiave only one a^ !/■ + ^ = 1, and :r = a U (I 179. Maximum and Minimum Vainao «^ IJic results of Art. 17.5 enTd^,?^,, Th . ^^ *? Expression.- -^-^'-'- values of a quadratic Lp;;!^:;:""^ " """^"'^'^^ ^^ Ex. 1. Find the minimum value of x^ - 3 >■ + i .C — ,^x + ,) = 7/j,^ . or 2 ^ — Jjcf = )n — 5 • from this result we see thaf fn,. ^n , , B>.n,mum value of the given .y^^r " "'"''"''"•" ">» fl H 300 THEOKY OF QIADIIATIC KQCATIONR. c:;i li Ca IK. o !, o 1 o i::0 liQ I ^^ 'Ui If I' ij ■'. Ex. 3. Dotonninc tlK'iiiaximtmi and minimiiin Valiums of 4 +a;—a;'. Let 4 + .r — .*'■' = //* , or ;*■'■' — .r = 4 — III ; .-. •i- = ^ ± \/{-\ '/-> +|) = ^ ± \/< V- - ^'0- In this rcsull, (1) m may bo miuUi as siiw?l n.s ire please^ and then^ is tliorefore no niininiiiin valnc of the ex{)ression to which m is equal. (2) But /;/ cannot he ijredter tlian -y-, which is therefore the maxiniuni value of tlie given expression. 2.*' 7 Ex. 3. If a; be real, show that r-;r - . .can have no real value between 1 and ^^ Let 2.C"' — 2.f — 5 2.*;— 7 — ^ = III : 2x' — 2x - 5 .'. 'inix' — (2in + 2) x — (ryut — 7) = ; . _ ^ "^ + '-i± V \ (-'» + 2)- + Hin ( 5m — 7 ) i 4 III _ 2in + 2 ± 2^(tlm'^ — 12m + 1 ____ ; in which l\ii)'' — \'iin + t, /. e., (Win — 1) (in — 1) must be posi- tive if .r is real ; .•. 11/// — 1 and in — 1 must be positive, and .•. ui cannot lie between i\ and 1. EXERCISE LXXXVII. Find the maxinnim or minimum values of the following : — 1. x' + 4./- + 8. 4. ./•- — f.r + J. 7. ,v- + 8u; + 4. 2. a-- — 7,v + 8. • 5. rtilv' + 40x + 9. 8. — .r + 4a; —6. B. 1 + a; - x". 6. 72.*;- - G3a; + 64. 9. x -J- (1 + a-'). !0. Divide a line 12 inches long into two parts so that the rect- angle contained by them may be the greatest possible. 4-fiC— a;». therefore I'fore tlie 'eal value bo posi- , and .-. ■ + 4. 4a;— 6. + x'). e reet- B. MISCKJ.LANKOIH KXKKCISKS. 301 i*;' — 2^ + 21 if*) 1- Show that — _■ -' ,.„„ , 0^_14 - t.an have no real values between 2 and — 10 •'>. Show that (j- — a\ (h >-\ W a) {0 - ,,) can never ex(,,.(l | („ -, i)». 6. Determine the niaxinu.ni vahie of L-^ +JJ1(^ :z^) '■ ''•'^•'••""■'»<' ^I'o lin,i(s for the values of '-^^'1+ ^ n" + ft + \' - + -t = « /^ c • n Find the eondition that tho .lift- a^-'^ + 6.; + . = o ma Ith? '''^"''"""" '^f ^'"^ roots of roots of ma.^- + .l,"";'', !^\l^'' '^^''"^^ ^^'^ ^'•'' 'li^^vneo <,f the 12. The equation Spsr' + (Cyp _ lov,. + q a i determine // ^^ ^ ^ ^^•'^^ *^^^ ^'1"^I root« ; ■9 13. If a, ^ are th(> roots of .r- + ;>>• j. ^ n i , , a^'-n'a-4.o'~n i} \i^ +'7 = 0, and «', / the roots of ; OIIAPTKll XVII. C> <, fO «... ttU p ' ;;« ^'^1 ii ' Mi^ INDKX LAWS. — Sl'HDS. — SCil'AUKS AXD rmKS. 180. ^V<' shall now consider I lie ^ciicrfil rules which <,'overn liui Multiplication, Division, Involution, and Kvolntiot\ of roofs and jjouiers. Theso rules arc given in the following Index Laws. III. (((}>)•' = ffb". II. (a")" = urr = f'"". IV and the Law connecting the Index and the Sui'd Symbol is m V. r/" = ^y La\v I, (t. ^.J:=al'l^ '• '/a is one of flic ///mM'riii-il f.w * '*" • "" • a" . . in U factors = ,,-1 ' i • ,1 •....-. Wun. ^ n __ « It V"- Siinilarlv 10 ^ .•! '/<-r/l -ai.^ii f-+J - ^>i* a^ is one of tlic ./; ^^^^ fourth root of ^o, (jenerdUy^ 1 a d' '""'•<;t\ a root lif ti,., ^.. ... root of the (jmu>tifij uff^ted (>,, f/, .*' have ^/"-r/" = a"*" = r/" hut also f/" X 1 = r/" f/" = 1, which corresponds with Arts. MO, (M). Ex.1. Express ^UtHi') + ^\a*t,'-') - ^(aH)^), witli fractioiud i\i(li CCS. The expression = rpftS 4. rpftii —a^b^. Ex. 2. I{(Mnove the denominators from H6^ 4^'' 26< a'V^ + •i- \ahi The e.xpression =3 laHr"c-'- + -Ir'-V/---'/-' + '2f)ca~^ + yr'b~'c' 1 1! 1 1 ft a-* a^b» 2a'icf Ex. 3. Express with surd syml)ols — + bf 2c4 36* Tho expression = \ h Ex.4. Simplify (3a - 46)'»-(9rt.M- l«6T-('5« + 46)" By Law Til, the oxpressicm = I (3a — 46) . (9a'' + 106*) • (3a + 46) ["• = (Hla^ — 2r)«6*)" Ex. „. ,.. /3a'-'6V\'» / 9a^6V' V" Law IV, ^« expression =( 3a*6 3„4 !5r/Vy 5(i*ey' 9a*6\' '.\(ll,r By Lhw II, uohav,. t|„. ,.xj,n'.ssi(,„ •»//) _ j (N.^ — 0//) -i. (4./. _ ;{,^, J io„ _ ^„„„ 'M I HIK c press \ 1 =|.^-y •^^V^^-.-.V'V/^j 1^ = (.;?.yv.,}ii)f « = ^'V^^ EXERCISE LXXXVIII. '•Simplify :— iff) 2. 2^/" X iW X 4f«. 4. - X ''- X - . a" a" f '■lim,,., ,l„. f„]|„,vi„^, to q„a„ti,i,.., ,„„1,,. ,|,o ,,„,t ,,i,„ ._ s«5; ,v,>6i; T«te?.. ,„a6r „s,w. „iv,?,.v CImns. ,1,0 f,Mbwin« tn^titie., „i,h fra,,i„nal i.uli,.,.,:- «-V'M ^„-7,.; ^„.,.,.. ^„.«-^,.. ^,,,^^1, Change tho foIlowiiiL' so fJi-if m wi. i . a-='6- a-?6^r/ C'» (b) a Simplify: 3al.a8.a-».6„.^,3„-}.,J.„}; (^ij,-J^l, (xy~^)~i. i SHSni ..... tuJ Hi '-^<5 -J I O ;i a I! its ■ '5Z. i ^;l! i: ! I ,1 30C INDEX LAWS. Express with fractional indices : — 4. ^((rb') - r,^{(r'b') f ab^,\ab) - irP^Ui'b) ; ^'a--* ^ ^a" ; Express with positive* indices: — 6. a-Hr + b-Ur nb-\^ + (ibc Sit-V .1"!/-'' 1 (/-7-V-" 8/c-' a-y rixij-*- Express witli surd sij^ns : 3 2 3 23 1 ol l._- 3 1 4 8. ft 3fe 3 /> 3 h i (I a I „ 3 3J- ■-'■' i a i - -I U — — ; .l-an . (a-)4 • r; T -^ ^/ 3 ; an . «" .«". 9. Find the value of 82s ; 4"^ ; 12.^)-3 ; S^i ; pirh ■ 043-!. 10. 0'" = (/ : find 12"* in terms of ^/ ; is'' in terms of a. 11. Simplify: (2ab)'-{'d(ib)--(rui)* -^ {{:\hf-aaby'\ ; 13. Simplify : (49.r - lUJ/y'js h- (7.»' + 0^)^ ; ('ur + Hab—2Urf -=- (a + ;??>)'. 13. 5"= ir),r,3r), lindT)''^ ; lind the value of (//")" + ((/")" + ( 'V' 307 .-•/-I ]^/ «6-a^(rt6-i^^6-^^f| iut be icable whatev iclict A i <'\v ox ^iinplcs of this will now be '«/ Ex. 1. Mult: ply -a Arrai) ,'e botli fact •^ 1 _.i given. _i. 6« = + ^-6 '>i'>; in (leseeiKlin^r poM-ers of a :— 3«'> ^ 4r7.^ — ((~z 9.-e )(.«^ + 1 +.H). Here formula (B) applies ; the = \(x^ + x-^-^^^J^^^-^ ox])ression ,>.3 ==x^ -^ 2^-" + or-' \=x' + 1 + 1[= (xt + + x-\ x-iy ~ 1 , ii r 1 ~! u\ : m 308 INDEX LAWS. Ex. 3. ] {(ih)l + (ac)^- + (b(f)l + i(;(/)i\\ (a6)si-(ac)^-(6r/)a + {c + x'iy'^ — y. Ex. 5. (16.^; — y") -r- {Hx^- — y^). This is \ {:ixi)*— (yly } ^ (2.rT _ yh) = Sxi + 4x:^y^ + 2a;i^ + yk Ex. 6. Resolve x"* — Gx — 2. Complete the square by adding +9 — 9, and we have a;" _ Oa; + 9 - 3 - 9 = (x - zy - 11 = \{x-'S) + ^n\{{x-'i)- ^'\\\. 4 2 2 4 Ex. 7. Resolve x^ + x^ys + y^. o g 9 2 Add the eonipleuicntary terms xiy^ — x^y^^ and we have ,8 2. 2 a a 2 11 2 2 11 (,i;5 + ^5)'' _ x'-y^ = (.i'5 + y« + .t6^S) (A-r. + yi. _ .rSyS). 2 1 Ex. 8. Resolve 3«a-' — 'hi^x" — aso*. IS 1 The expression — a^x (3aaa-'^ — ^a'^^x — 1), whicli, using formula (I)), = a^x(a^x — 1) (jSa^x 4- 1). x^ I'- //"~^ \ -j- ;/ — ' Ex. 9. Reduce -^~ --r' This expression X + xir + 1 + zr = ?L[L- ^) - ^1 - •'/"'' = (t-/r')(-r^-i) ^ .J _ ^.. .^ _ jx ^(i + ir') +Ti + ir') (1 + y-') (^- + 1) ^ ^ ^ Sx. 10. Reduce l+J:iz:^l~ a-^ .,, J + (i^.^^ • ^l'« expression 3oy EXERCISE LXXXIX. Simplify :^ ("^ Mulfiply : — 3. (.r'+a-' + i)(J+^. + J, (.f 3 + ^^-^ (.^.f _ _j.i 4. (.c2 - .yi) (a- + ^.U ^^ + z/) ; (A-ir + yi) (x^ _ (•<• + 3 y'(ary) + y) (^ _ jj y^ + yh- xsy's + yjj_ C. lC2ro'' +/> H ..O i'^<'y'--{'la)U + i^ia)lb-h^. 7. (^_ 8. («i- O 1.— a .-) - '.U-" + 0.^«) . (o ^.n _ a 3 + c/ -i -- .- i{,y) (3.I- + ;j^). ^/ -') 0/i + ,,3 _ -.1. _3 f7 3_fj,-3) 9. Cc - 4yt) U + o.ri^/i + 4 i)r.r — '■2 A A r^y^ 4- 4y5). i 1 i-'i" + 1 ) («ir.C3 _ i) (^3^.j _^ ^^ ^^^|_3 _^ ^^^^ Divide :— il. a:3 — yS by .^•5 + yi ; ^^ _ ^3 })y x^ — wi y' 12. ir ■ fZ/bya-B + y^ -,(5^-3 13. 56t-66!r_ + C^a ' + na~' - 6 by 2a-\~ 1. 66!r - 46-t - ib-i _ 5 by 6^ _ 26"^. olO INDEX LAAVS. ll '%■■■■■ ■ (D ilU lii \ t -t :l' : 1. 1 11. ,*-- - //-" by x~^ — y"^f ; ^r-" — 04^' l)y a~^ + 26^ •-' 1 _i _■•? I."). Niuarc .<•!• — 2.i':> + ;5 — 2.i' :• + .<• «. . ,. , 4 a at 4 , _4 1(J. licsolve .ra + .f:'//"' + y* ; .fa + 4// ». 1 7. liosolvo X — iv^ — 50 ; dx^ + 3a;3ya _ 21/. 1 1 18. Kesolv(; .r' - 1 + .i-^' - .v^ ; 6 (u^ + //) — 13a;Sy^. ^,-1 + ^-'fi-s r/1 ('"-i^fe-' a + h + c- Sahsc^ 19. Keducc, , — - — r-;, ; —. , — ~\ , . — - 20. Find the square root of «'6-- + 2nlr' + n + 2a-'b + a-''b\ 21. Find the square root of a^ + iaij^ + \Oah/s 4, \2a^y^ + 9<,'*. (ft) 1. What does (vb'^ — a~''b~^ — a^ — 6^ + a~'^ + 6~* become wher ub — cr'b-' = ? 2. Show tluit the ex})ression in (1) = {ab — a-'b-') (a — «-') {b — 6-'). 2 r> 74 3. Find the square root of «3 _ ia^ .f 4a +/2a<5 — -if/T + a%. 6. Divide a" + 2(ib^ -i U' ])y a^ + 2ah^ + b. 7. Multiply tonio wher i-rt'^. f-1. EXKKCISRS I\ INDICES. 3|| 11. Reduce to lowest terms ~f^-'j^L:r "^ ~ ^ ■ , ar? ~ ax + a^x — al 13. Show that V)"" + ^,„V,.,! + ^|,,, ^ ^,„,,.„ ^ ,^^. ^ ,^5^3 14. Show that (/"_ ;tr) ^ a^-_jT) ^ ,,, ^ (,, ^ ^.:) 15. Divide a^- -,/-:// Uy ,,^ _ ,,-, 16. Simplify /^'-)'V ("i^^yy 1 7. Divide xi ~ ma J + maxi - J byxi-„i. 18. IVovo (^ + x-'y^y + ^.y^„.^_^_,^'^^^,^^,_^^,^^^ .«. Fin., the aap ot 4,,=.. + 9«?.. + ,„.._,„;,.- 4 and ■;2\ 19 20. Find the L. C. M. of aa;^ _ 1 ax^ + i (,,*>. , >. . ^ «^-«'-l, at.z;=' + i. 2t- Multiply 1 + ..-i. + .«-.,, ,y l~^a-i.4-i.-i.^-.V-V. So. 23. Reduce ^^!I±_Zl^i''f' - a^x ,, ^ ,, ; and + (f_+ r/a — r/2 — a3 0«i..'-„,,_i „t + o„.-+15 flnd tho value of ..-, +.. ,„„ „ „,t.r„; t^'!^ '' 24. simphfy ^,,„ + '.-,. v,„+,;:'.^„,^, ~-.^,„.^,;-. 184. Surd*— Wlion a renuircd m„t ,.f . found «„,;/„ it is /,,,«,.„r/ . , , ,r * " '1"""*".'' """""t he frmtional quauMy ' '''""""^ '" ''■''"'-■'' " ^^^ '"■ Thus, v'2, ^9, ^„., ^„- f„.6,!, „,„ ,„„^^ Sometimes a ra/- y>o//V7' of a (puuitity can he ne<;ative, no cnii root of (I iicf/atirc (/naiitfti/ can he found. Such roots may be indicated, e. /th poirer of (/', and «*" repre- p sents the nth root of this ;/th power, /. e., the quantity a' itself. 188. A Mixed Surd is one whidi has a rational factor ; this may be separated and written at» a coeflicient of the surd. Thus, y 18 = -y/(2 x 9) = ^2 x y'9 = W ^2 ; 3 ^(a'fe'O = 3 ^/a« x ^b' = '6a'' ^b\ OEXKFIA L PHINCIPLKS— i:x KUCISKS. dIS ' see thai :— ^ ' xarno vjihic t'en root ot a" li n)ot of f/'-'" e %;#. as tlir '- "iff/n as f/tt lien^fon' tli(, ive, no eroi >ots may he are eallcd ler Algebra t sign CArt. extracted. ' quadratic er. '(I'd of any generally, af" repre- f' itself. 'tor ; this Conversely, it is j.lain that a mixed surd may ho changed to an ^'iitn-o snrd by giving tlu^ rational factor a .v,7yv/-/;„vy« of tlu. same order us the. sv//7/./y/r^>/-. Thus, O. 3 ^7 =: ^2» X V7 = -^C^- X 7) = ^yrwj , rt ^b = ^V/" X ^6 = ^\a'b) ; 189. Similar or Like Surds are thos(> that hav(>, or can i)e made to have, the same order and the same snrd factor. Thus, 2 ^3, 4 y'3; .■5^7, ;5^/7, aro pairs of like surds ; so ;{y ;{ 2 ^'9, are like surds, for the latter = •> ^^9 n= 2 ^)\ 190. A surd is rednced to its simplest form, when its snrd taetor is rednced to tin- smallest i)ossibl(! integral number. Thus, ^12 = y'3 X ^4 = 2 ^\\ ; ^25(1 = ^'4 X ^U = 4 ^4 ; ^ma'h') = ^/c^ah) X Y(J)r/-Y/') = :\ah-' ^(•]ah). 191. Surds are Compared in Magnitude l)v n.lucing tluMn to entire surds, and then to the same snrd index. Thns, 3^5, 5y'3, give y'4o, ^75 respectively, y^n, ^11 ve 5^ lli, or 5i^, 11^, or 125^, 121^ so that ^^'is s<.en to l> give greater than ^11. be EXERCISE XC. 1. Change to the snrd form : — ^«i; Ix^ifl, nxly}, ^x^yt alb\ 2. Express with exixments whose numerators are each unity • ■ - 3. Express with exponents I and — 1 :— Reduce to entire surds :— '■'f%'fv,ty'' '^'" '^'- '^^^ iA^"^ a y^a ; 6 -y/(a^6). "^ 314 sriiDs. ! ^1 /h a /h n ■i/h ., /2ff 2ft ■w2h ia 9 / d / + xifyi ; ^{m*h - 24a='6- + 34a''6-' — ^ab'). 13. Show that 3 -y/^r), i V'147, ^^\, ^^\, ami 144-1 are simihir surds. Reduce the following to tlie same surd indices, so as to compare their magnitu(h's : — 13. 3, ^3; 3, ^'Q; 10, ^1000; ^/33, 3; 3^10, ^'50 ; rt^ y'a='; ^a, ^a- ^a\ ^a\ 14. Write in the order of magnitude: 3-^3, 3-^/3, 1-^/1. 192. If surds are similar, they can h{^ added or subtracted by adding or subtract ititj their rational coefflcients. If tliey an^ dissimilar, we merely connect them with their proper signs as in the case of ordinary nnlike quantities. Thus, 3 y'8 + ^^'m - y'98 = 3-3 ^2 + 5 ^2 — 1 ^2 = (11 - 7) V'3 =. 4 ^2. 193. To Multiply Surds we reduce tliem to the same surd index, and then multiply the product of the rational parts into the product of the sard factors. . f:lemf-:xtary hulks— examples. 3ir. 4a 3 / 9 3 r 4^' 2 • '\ V'ji 44-i are compare cted by lioy are IS as in c surd 'ts into Thus, .V3 X 3 V18 = .') X 2 y-.s X IS, = ,i ^ „ ^ ,,„ Also..^o,,^ .^^ ,j^^ ^,^^^ ^,^^ ^,^ = 4 ■ 4^ . yr = 4 . 83* = 4 ^;J2. 194. Polynomial Surds nw. nmlfipii,.,], t.-nn inf,, f,.nn ... .. th(. ease of ordinary Multiplication. '"' ''' '" ■'",1 ,l,v„la overy ton,, „f ;l,„ divi.K.nd by the ,livi„„. "■'"■"■*' 196. If the divisor is a Binomial Quadratic Surd of tlu> form Y« ± Vi, w, wnte the quotient in the fonn of a />-«.7/ ,/ tluMi multip V both terms hv .A/ 3- /a + J ^^ J iwuun m\{\ (divisor^ /Y//L> ,7 ■ ,/ ^ '^ ' ^*' "''''^■'' ^'"- ^^^^'^''^'^/./A)/- Caiv]M)r) /«//o//c./; tins is ealled mtionammj the denominator. Thus, (^5 -f ^2) - (y^5 - y'o) ^ V;M-_V3 ^ Gv/^J- v^" 8 :f^. B.-If the divisor is a surd of a Idglier decree th-in tl.o Ex. 1. Combine 5 ^80 + 3 ^370 + ^/640. We have, by simplify inij, 3-5^10 + 3.3^10 + 4^10^(10 + + 4^^10 = 30^10 Ex. 3. 3 ^Ul - I y^. _ ^^y We have 7-3 ^3 - M ^, _ i y, ^ ^oj _ «^ ^3 ^ ,,,^ ^^ Applying formula (B), we have «»(1 _ I) _ j^ (4 _ ^.^ ^ ^,... _ ^. 310 srijDs. : * 5:: I i P lit. ;•. 1 l¥H Ex. 4. Multiply togcthoi- a y h, 2 /y/0, and :{ y^oJ. Tli(! pnjduet of Iho WMjfTieiciits is ;j . 2 • 53 = 18. Tho product of tlu; surd factors is = 3.2M3-2)i's =z: 12'-^<{. Now iriultijUy this by 18, tlu^ product of Uic cocHicicnts, and wo havo IH X 12'^ nr=2H;'^<}. This (\\ainplc shows that soincliincs labour may bo saved ])y fac- torintj the (plant itics to be multiplied. Ex. T). Divide 8 — T) y'2 by :$ — 2 y/2. Ex.0. Simpliiv ■ .*•- ^i-V' - 1) .T+ y'l.r'^- 1) Tho expression = --^-^--^^^^ = -^ =Ax^U- -1). Ex. 7. Simplify ^Vi'!^/'!:. he — c ^(ab) f-\ \/(nh) — a \ x/a i\/h — Vo) /a Expression = ; — ^ _ = -7— — ;^ — \ < ~ 'V / ' Ex. 8. Simplify (;\b- ^{(th) \ ^h ( y'h - yr/) 1 The expression ^/2 + Y^3 — ys i V^ + i Vi + ? V" = i \/'^ + T \/- + tV V^O, ^^U^MKNTAHV HCLKs-KXICUr-lSKS. 317 3) A s, and wo J by/at'- /(^'-i V^O, »> ;{. 6. EXEliCISE XCI. Multiply: ;{ v'H f'V "iH/fi- ') /,^ , < Find t\w continued product of 4 + 2 ., ^ _ ,, and 1 - ^/r,. ^ ' - V-^ 1 + V'). I>ividc : o ^, + , ^/o + V30 by 3 W, • 7. Divide : ;{ I \/7 ~ y'i S. RationiUizo the donominatoivs of '^; :5 + V^ by ;} _ yr V-^ O i) V^ . 4 y 7J- 3 y V 0. 10. n. s -I- ^5 ' 4 + a y o ' ,,5 ^] Find tlio continued i)rod + 2 V' - ^(a' ~ 1,' a Sinij)Iir l>''^>'l"^*t of 2 ^24, 8^18, and 4.^2. A/.? a; — r/ ^■f' + 2^(jr + 1). .*• -I- r/ fi" + ah + 6' 3-y 3 + V'^ 12. 13. 14. If U:?) = 0, tl Prove 3-1^^ + 2 + vY.r^_4,i ^ len ^7/ + ncrz + .Vyy?). + ^/(. >=^ - 4). X + VV^-4)i Find the value of 7 yT) -^ ( ^/\ placeti 1 + V'S) to four decimal IF 318 sriiDs, 15. Find llio value of (:{ + 2 ■ + y (.{•■' — 1) 1 I 1 4 (1 + y.n n 1 - y.D 37r+ ./•)■ IS. Simplifv ^ +— _ '^ 1. 19. Fiiul the value of -— — ^ , when ,v = -.-, -v/(i+.n-y(i-r) ^/^ + i 20. Simplify ^(a + a') — ^ /(a — x) y^Ca' — ^'') ^ a; '/ + ■'• + yC' — a;') y^" + y.*" " — \/( are ratinnaJ, and y/y and y.// are irrafioiui/, then = //; /. ^., the rational i)arts are ecpial, and, also, the irrational parts. For, we have (I — .v + ^h ■=. y.y, and, squaring, {a — x)" + 2 (V/ — ,r) y^y + h = t/^ or 2 (f/, — .V) ^h — jj — h — ((t — x)'\ in whicli the rofjjicienl of ^/J) niHsf nntlsh, /. ('. , (I — .r = 0, or (I = :v ; otherwise we sln^uld liave an irratioual quantity (^qual to a ratwnal one, which is impossibh;. %. M^ 198. Tills proposition enables ns to find in a simple form, where possil)le, the square root of a binomial surd, one of whose terms in rational and the ot/ter a (juadratic surd. 'u■^ <•■ /f*) to tliivo ^ +'\/{a—,c) a' + 1 ft'onal, and ; i.e., the i. For, we jual to a rm, where iose terms KUOTS OF niNOMlAI, SI'IM )S. 310 Tims. It. (iiid (|„. ,s(|ii,iiv ruoi o| (/ -) ^/{„ -|. /,, _ y/^', llssiiiMc 'licii, siiiiariii;,' l)i)tli sid \ ■>' + \/// cs. ■''<'"<•<' I'y the lasl Art., .r + (t + y'/, -^ .,. + y .,_ o y,.,,^ '/. and or 6 - V •'•//. SoIviii,i( (1) and (i> 4.r// = /, ) VVC <'(•( as = i 1" + \/("' — ft (I (:i y = h Vi" + ^/f>) = ^i \a + ./('/■-• - A V<' M; ^) It is I.Iain, however, that unl »l + yi ]a-^(a'~h): ri^-ht side (.f this identity is t'ss r/- — /, '^ is a itcrlVct s((i lii.'irc. the more eoiiiplcx than the jcff, and tl I'X. !. Find the stjnarc root (.f 7 + oyi Assume y'C + '2 y'l 7 + 2y 0) in ■v + // + 2 y.ry ( 1 )' — 4 (2) s?i ves •• ■-*• 4- // == 7. •'77- 10. 49-40 = {,,'- //)■' or (') + (;}) .trives (1) -(.*{) .-. the ;{ = X = 5. y = 2. (1) (8) iv(ini-red root is ^/^^ + y'.^ = ^o ^ V 199. When the root can he cxtraeted times, by / nspetiioii, by remend ' + !J = (i, and ,r}j = h. Tl it can b(( foand, JCTin-;- that, in the ^'ciieral sum is 7 and product 10; tlie\ Ills. Ill Hx. 1 ; fi,„| t some case are, at once, 5 and '' \V() nunihcrs win )St> Ex. 2. ^a + 2 ^3). Here we find 4 and product 3 ; they are 3 and I ; h two numbers whose enee tht; recjuired root .'inm IS = v/1 + \/^ = 1 + -^3. 320 SQUARE ROOT. CI' Si:: a Ex.3. Findthesqnarerootof ll+6y'3. Expression = 11 +2-^/18; th(> two numbers whoso sum is 1 1 and product 18 are 9 and 2; lienco the r('(jnin>d root is -y/O + ^2 = 3 + -y/2. Ex. 4. Find tlie square root of 3 ^G + 2 /y/12. The expression and the square root of this = V^6 X ^(3 + 2^2) = ^G (1 + a/S) = ^6 + -^24. Ex. 5. Find the square root of 2a + 2 /y/(a^ — 6'-). The two quantities whose sum is 2a and product a'^—b^, are a + b, a—b\ hence <^he required root is ^((i-\-b) + /y/(a— ?;). EXERCISE XCII. Find the square {a) roots of 1. r, + 2>v/G. 6. 7 + 2^10. 11. 10-2^^21. 2. 13 + 2^22. 7. 7 + 2 Y^G. 12. 13-Gy'2. 3. 16-4>y/15. 8. + 4^5. 13. 29-12^5. 4. G + 4y'2. 9. 57 + 12^15. 14. 140-G-Y/4r)l. 5. lG + 2y'55. 10. y'17r)-y'147. in. 9-2y'l4. Find the square (6) roots of 1. y27 + 2v'6. 5. 100—2^2499. 9. 117 + 3G^10. 2: 3^r, + 2()0. 8. r)y'2-2^12. 12. 2+iv/l«- 13. Y^27 + 2^G. 14. ab-'inby'i h-b^). ir,. l + y'(t_.r'). IG. 9^+25/? — 3G^#??«. Find the fourth roots of 17. nr + 2ii^{nr-7i''). 18. 2,r + 2 ^(.r -!/'). 19. .v + !/ + ^ + 2^/^{.r + l/). 20. 2 + 2(l-.iOV(l+~'^-^'')- 21. r -|-12y^2. 23. r)G-24y^r). 25. 49-20y'6. 22. 248 + 32^^00. 24. 48y'^ + lliy'15. 26. 6^/5 + 14. 11+2^18; 1(1 2 ; hoiice expression 24. The two + 6, a-h] '21. o /451. 4. v/10. /(.6). 0. 6. EQUATIONS INV()I,\IX(; srUDS. 3-41 200. Equations Involving Surds.— We sliall now give a few examples, illnstratiuj^ some (^f the artifiees employed in solvinj^ the simpler forms of equations whieh inmh'e surds. 2 Ex. 1 . a/.I" - V^(^ - H) =: Clear of fractions .-. ^(x" - %x) - (X - 8) = 2. Transpose, etc., .*. -y/la;' — 8.r) = x — 0. ISquare both sides, .-. x^ — ^x = x' — V2x + 'iQ, vr Ax = 86, and .•. x = 9. Ex. 2. X -\-h— lv l)v '-3 and s(iuare, .■..,,,, Hj^' — 1 9 .••:id ch'arinj,' of fractions, etc. 6:U'' — Wlx + 100 = ; ^. e., (21iB — 50) (3;/; - 2) = , ,. — 8 ~'. N. B. — Tli(^ student sliould note that in solvinj; surd ecjuations, •values of ,r may sometimes \w found whidi will not satisfij the i'i|iiation. Thus, on solving th(> e<}uation • 34 4^/(1 + .v) - ;? ^( 1 - .V) = r,, u-(. get .r = ± ^^-, 'oT wlisch oHtj/ \ho posilive value will satisf.v the ecjiiation. The reasi>i> is tliat in c/airhti/ of radicals, we reall.v introduce n r I'i maiizhifi fador iacolritKj .r, /. f., we coml)ine a new ecjuation with the given one ; thus, — ff will not satisfy the given ecjuation, hut irill satisfy the conjoint (Mjuation, 4^(1 +.r) +3y'(l -a-) =:5. Tt sometimes luippens that somc^ of the oon,}oint equations thus introduced have no solution. Let the student solve, e, + x)\=0. 2. /> + X - y(2/>r + x') = a. (i. v^(r + ls)+ VC''-10)-14 = (). ;?. ^/(r-l)r^y';./_^,o+,,,,j_ 7. r - (i ^(.r + ;}) + lo ^ 0. . 2y^.r - 1 W.r _ o 9. 10. 1 -./■ 4- 4- V •'■ + 1 V •'■ - 1 Stes v<- 2) + 1 V /(.r - 1. H. /y/.*' + ^(4f/ 4- X) = '2^{h + X). U4: SQIAUE HOOT. ;c;> '9 it-'i d O IP 1} ' !'l4 1 M 12. //(1 2./; — ;-)) + y/c.lv - \) = y^^{27.f — 2). i;}. ur — ;j:{ + <> v'(u'- + 7j == o. li. «■' — 2x + <) -y/l./-- — -Iv + 5) — 11 =0. 15. (5 ^^{x' — r)x + 10) - (./•• - ThC + 18) =z 0. 16. <^(«* + x'} + Y^(a'^ - x') = h. 1. 13 V'(a;* - 10a; + 40) = x" - \Qx + 72. 3 + a- . 2 — a; . 3. i_i_r + _.___r: ^/2 4- ^(2 + ^) V'^ - V^'^ ~ *> 3. a + .f = ^|a^ + a: ^(W + a'^)f. 4. 6. ) — ^/(x + a — b) _a — h ^/ix -(I + 6) + ^{x + a — l>) (f +b ti Ir^Vn - Va - a^U ^ ^ " 1 + VI 1 - vn~ ■■>[ 1 + aa; r \1 — bx/ ■ V'(2.7a,') + 1 2 14 a/(^/'^--<-'^)- V(?>'^+a;') ^c_ • '^^a'-x-") + ^{b-' + x')~~d' 15. V(l + •^■)' - V^l - -t;') - 2 V(l - ^)' --= 0. a s HOOTS OF I'OfA'yoMlALS. 325 -b — c. 1(5. ^(a + .r) + ^/{a _ .,.) ::,, ,._ 18. (1 +.r)^ +YV cun us.nliv fiml by inspection the s<,nare root of,, polynonnal. But a Z^al Ex.1. a" + 2r/6 + 6" I a + 6 ±2ab + b-. ./^^.s^ tenn, 2 . 7, me the //V.v^ U^rm into the seroHd\m] ^" fi .vy//a^-(' of the seeo?ul tern.. Henc(>, ' ' ^^'^^ 1°. AVe arrange in descending (oi- asc(.nding) powers of a. ' . iz^:::::^:^''' ^^™ ^^ ^"^ --^-^- ^^y ^"•^- ^^ .-ting 4°. We add + b to 2a, and multiply the sum by b ^^V.../^..v..forthesu<.cessivermI:;;her:r '' '^ "'"'" ^"^ = )^/+ [2 la .{-/,) +,.<.,. (a + 6 + c + r/r ^ l(a + b + n+>n'^ i" + b + rr + 2ia + b + n,f + ,r where the /a,r shown in Ex. ] plainly holds. Ex.2. 0.^-^4.v-i2.y + ia.v + iH.,^ + 4,-{a^^_4.,^-,,^ ea;" - 4xjf l-24a-'i/-'-\ 2x'if + 1 (U-^y* -24^'V +16^V' -Ili^^-!± ^ ^'^'^' + 4/. ■ 1 p I: [Hi 4- id I.- iP ?Vtl \, i ! fti. 32G CUBE UOOT. 202. Cube Root of Polynomials.— Similarly, a (/eneral rule for tixtracting the citbe root of a quantity, may be derived from Fonniila G. (Art. 91) for the cube of a binomial. Ex. 1. (t'^ + Wd-b + \lab- + />'•' ,(( + b a' ya- + '^ab + b'' b + '6a-b + ;{a// + 6'' = r^a:' + :{^//> + b") x b = + iia'^b + Dab" + b\ 1°. Ari' • ;;^o in deseending powers of a. 2 . Exti'Juit th(^ cube root (a) of i\\G first term, for thofi'rst term of the root, and subtract it.s cube from the given expression. 3°. Divide thafi'rst term of the remainder by 3 times the square of a, for the second term (b) of thc^ root. 4°. A'' ' ■'., i''v trial-divisor (3rt^), 6'- and 3 times the product of a and b, and n" . .'"jVly the whole by 6 ; this gives the complete subtrahend. If ciicco ire more than two terms in the root, the ])ro(" same ; t! • • a -r b + c, is (a + b) + c, where a + b is tne first ten.', t,t' K !J. {a + bi-rf = {(a+ b) + e ;- ' = {(I. + b) ' + 3 ( a + bfc + 3 ((7 + 6) c- + c\ =:a' + {Da- + Dab + b"-)b + \ D(a + bf + 3(r? + ?')f' + c'* !- <\ {a -{-b + c + d)'' = (a + b + cf + D{a + b + (■)'d + 3(// + b + (■)(P + (P = f /' + ( Da- + Dab + //-)('; + \ D (a + />)" + 3 ( a + b)r + c- [ c + \D(a + b + <>)" -\-D(a-^b + r)d ->rd''\d. etc., etc., where the law exemj)litied in Ex. I., plainly oi)erates. F.x. '2. 2.»- 1 a"' + (kC' ( 15 (" + 20.r:' + 15,(- + {\r + 1|J'^+ 2j' + 1 .1" + ()»•" + l.-jj*' + 20,1 •' + 33!* + laa-' + 15j''J + to + 1 + 3f< + 12''' + l:")/-'' + fu- + 1 N. B. — If a foarth i-oot is nnpiired. we take the square root of 1 1 the square root, /. e. (a^) 5 — ^/f ; etc. So a cabe root of a s(puire 1 t root gives a sixth root, /. c. ( = a^ ; etc COMPLKTE SQrARi:S AND ClUKS. m 203. Complete Squares and Cubes.— W(> .rivo a few examples of questions that may 1„. solved 1)n tlir methods exphiim-d in the last two artieles. Ex. 1. Find the eondition tliat a,r + hx + r may be a perf(>ct .s(iuare. Applyino; th(> nietliod of extraetin^r .s.jua.V root, w,. ^^et tlie root aix + h - 2a^^ aiid remainder .' - ^^ ■ now if th.' quan- titrv^^is to ])e a perfeet square this remainder must vanish, i.e., ^ ~ ^,1 ~ *^' ^^'' ^' — ^<"' = 0, th{! requinnl eondition. Ex. 2 What value of x will make x'- 2ax'-{- (a- + 2?» x' - '.iabx + 2fr, a eomplete s(iuare^ Proceeding as in last example the root IS found to bo x'~ax + b, and the reniaimler is -abx + h\ which must vanish ; i.e., ~abx + b" == o ; .-. x = ^-. ^Ex. ;K Find the relation between b and e in order that x^ + ',iax- + bx + n may be a eomplete eul)o for all values of x. Extracting' the cube root, wo j?et x + a, with remainder {b-',in') x + c-a\ which must vanish for all values of .i- ; .-. b — '.]((- = 0, or a" = b^ -r- 27, and (. _ ,,3 _ (^^ ^j, ^^r, _ ^.'2^ or 27c' the required relation. N. B.— Ex. 3 and similar ones may be treated thus :— We see that the lirst term of the en])e root is .r, and the second term must b(i a, therefore assume ,f^ + 'Sax- + bx + r=ur + (()' = x' + Sx'a + :]x,t' + f,' -. wljirh is t me for all ralars of .r. and, tln'refon-, the coefficients of the like powers must be equal, /. r., b = ;}r/-, or (('■ =z b-' -^ 27 ; r; = tr, or « ft' BE nooT. i I O o ! t i 1 , : l' ; 1 ' ,■ ;■ 1 ( i ■ • ] ; ; \: ;■ '. { : r ! A'. : ii- 1 1 i;L ILik^ :5. r.l.r" — 192.r\v + 240.r''//'' - lfi(b-='//=' + fi0.r-//« - \2j'!/'' + y\ 4. ./•" — 4 J'"// + H.r\//'^ — 10.r\(/' + H.i'"//* — 4,r//" + //"• 5. 4 - Vlx + r).r + Uj'" — 11.^" — 4,r" + 4.r". 0. .r" — Axh/ + 4,rY + B^^'V' — 14.*'*// + 4.r'//''+ Wv^if—Md-if' + if. 7. FinV + qx + qr + (f bo a jx'rfect square, show that J--'' = l. 13. If aV — *>)a%x' + '^%ih\v" — 51^/ be a perfect cube, then 14. If ax^ + 6a;' + ex ■'r d be a complete cube, then will ac^ = dlr^, and }f = Sac. o ■> answp:rs. 1. 2. 5. 7. EXERCISE XIV [b]. (Page 32.) ex' - 12xV + iJ'ff - '^ff - !r + 2,v>/\ — 8 (m + /i) + -■) {(I + b). ;}. 0. 4. 8 (m + w)' — y. lOx -^ y — VSm -i- n. W 1. 2. 3. 4. 7. 8. 9. 1. 2. 3. 5. C. 8. 9. EXERCISE XV. (3 + 66 + 7a) a: + (- 2 - 4) // + m + n, (a + m + 1 ) .i; + (1 + ^/ _ (I) y. (6a - 36 - 2) a; + (1 + |6 + |o) y. (2d - 2/) a- + (Se - M) y + (4/ + 4^^) z. {a + h-A)x+ {^(v + (■:" - la' - 6) y. 6. Zax - Sby. {a -(3)x + (5m + 5) ^y+(h—\) y _ ;} y'a-. (a - <•) x'' + {b- a) y"- + (e - 6) 2' + ax + by + cz. (1 - 9a — 26 - oj .C + (1 + 76 + 10a - 3a6e) y». EXERCISE XVI [ft]. (Page 34.) - 9.»:^ + \2x^y + 6.ry - 18.r,y^ + 21//^ + 20. />^ _ I7,y2 _ 22/-^ + \lpq - Hyz' + 99. - 'J U - >J) + 34 (x - z). 4. 19 {a - 6) - 8 (a + 6) + 14a + 6 i2'^'-ir'/_8^-H''-oy. y .^; .1' 6 z 2.r^ - ^r.v - y\ 7. - |a^ - fa'-'.*; - - 2.^=" + \\xy"- - y' - Ux' + 2xy - 10/ + 2. a"b' — 'da-be - Sab"(; - (rr' ;laa;^ aUr — byr. 10. a* — 2a'''6' 4- 6* 11. (a' - b') X' + (b' + <"■) y-^ + (c^ _ a') z\ 12. 19a»-176»-2(*+ 10(/'. 13. (a -P) x^ + (^ _ 6) a;^ + (1 - r) x-^rX. 'm HINTS AND ANSWERS. ..... Hi O 3 id ■il t H N •1: 14. a' + a'b + i\ah' - 2h' + nb". \n. ^ - \,»rr - ^x. 10. — ? \,xyz — bx + cjjy — 4^8^, (^ — // + ''■'). 17. Sltm.\A\^':l(t--\^■lh-\\\<■ — lltl — 2Hh'\ the several "e- niaiiiders an; ;{.">jf/ — \)^]J> — 1 l^je — l^'/ — ~4;|^t.' ; ;{;{ J ^,, - 1 1 lb - ^^r - .'.f/ - k)\ \i' ; ;n ^r/ - viy, - w^^c - ^d — 2^(' ; 29^/ - H^h 4- ;}^r + 1 JJ -\le\ 2~W' - i}/> - ^<- - 4H'' - i~f« ; i«f^/ + '^i^' — \\\<- — 1 iV' — "i« ; O^rt + H^/> + 4i'<- + <}:i' + c- + d^- 2. f^ - Sb' + r\ 3. 2/?i — M + 6. 4. — 2a! — 3y — 2z. 5. l^a; — 4|// + V^z. EXERCISE XIX. (Page 48.) 1. — 2^/ + 3,r + 36. 2. a + b -^ e. 3. 'iab + ib\ 4. — 3.r — !/ + 4z. 5. r> — 4a;. 6. 2a — 36 — 8c + 4d. 7. — 4«. 8. — .^•-10y^-2^. 9. — 2.i; + 2//. 10. 2a; — 6// — w?/ 4- 4flf6 — .5. 11. 3rt — .56 — c. 12. 0. 13. —y. 14. i^^a-2b. 1.5. {ra;. 16. 9. 1. i EXERCISE XX. (Page 44.) a; — (a 4- 6) ; x— (a + 36 — 2t/). IV (a + 36 - 2«/), ! - (36 - 2c — f — 26) ; a? — (6 — a — c — w + w). c - 12); x—\{a + b) + (p + q) + {m 1. a; — {a -{■ o) ; x — [a + no — •41/). ii. X — (2m — 2w) ; a; — (36 — 2c — 5rf). iii. X — (2m + 3a — 26) ; a? — (6 — a — c — a; — (a + 6 — c — 12); a; — j(a + 6) + ( -n)}. ELK mi: NTS OF ALfJKRHA. 5 2. i. (2n - 4h - .'V') .r - (6// + \\r) ,j ^ ^4f, _ ,,,., ^, ii, ((t — h + r,.r—{(f+l,-r)t/~{ff — f,-(}^. iii. (1:3^/ - l.V) ,r _ ( i^^, + 4/; ^ ,},,, ^ _ , ,0^, _^ .5^.^ _ 3. i. 2 + (7 - 2r) ,/• + (.V, _ ;}, y^ ^. ,j,^, _ ;,,.:. ii. (2r - r,-', .r- + (a _ ;{/„ .,4 + , j _. ^^,, ,.n + ,4,. _ .{,,/^^ ^, iii. (1 - ,().,■■' + ,!_/, + ,., ,,.:. + (/,_!, y-' + ,,, _ ;^ _,. ^ ., 4. i. — ( ;{r''^ _ r,a ),/•-( r//>r - 7 , .,•» _ (^..^ _ 7^ .^.r._ ii. 1 - (./ - 1 , .,. _ ( 1 _ /,, .,.. _ (,, _ ,. ^ 1^ ^.:. _ ^^,_^^ __ ^^ ^,,^ - (^ _ ;{//-■) .,.4 _ , , _ ,,^ .,.n _ ( J ^ 3^.,^ ^., _ ^/^ _^ ^.^ ^. (// - <' + 1) .<■=• - (./ + 2/, + 1 , ,r^ 4. (/, ^ ,., ,,. _,. .3_ ii. ('K/ + 4r) .r^' + ( Tr - (;i + 3^/) x^ + (2(f - 7h) x. iii. (// -b + <■) y- -2{a + f> + ,') .1' + ah - hr - ra. «• i. «; (5. ii. -17; -9. 7. {,.4 y. (0// + 1) .^'^^ _ (^ + o^j ^.4 _ ^2_^ ^ jj^ EXERCISE XXI [6J. (Pagk 48, ) 1, r,0; -48 •"); 9; - KJN; - ISO. 2. i. /;/V/y^; r/tcai^" ; — L>4r/W. ii. ;}6a"//<^ iii. — I4a-b'.v'- 1H.V 3, ,2-4 a'h'a\t'i/'2'. 3, ,3-2 //V ; - i].)''i/V 3. i. 40 c:5 ; - 2 ; - ;i7. ii. 130; —880; 0. 111. s 1 'v t7* EXERCISE XXII [b]. (Page 49.) 1. „^h-('.~nh* + ah<-^ ; _ ix" + U-11 + -y )ab.vij + Cufc.m + iru tax ; 9«V' - V2.v'\l/'^■ + 1.5.r !/^ - ^-'^'"l/ - lO.ry + sru-'i/' - nw"-//' ; 3^^ + 2a'b - aHr 4. 3.?-y^ - 3.rV^ + 3,rV'^' - ^3.i•^y*^=' ; t,,2^ _ — 'In'W^ -L 7//4 ,.4 1 fi rr".jr^ + l.f>,,« 3^4, * // + if?- /y ix^ys- - f .1-*//='^- + ^r'^y V - ^ij 3^3 . 5 .,4 >.2 ^r,r' - i|r,'.r' -f- a'x 3.V.* i. - 2 (a + 6)^ + 3 (a + ^»^ ; - 3 (a - hf - 2 (a - 6)» 8. (m' - ;?)^ + (m^ __ „)!* . 3 ^,f, _^ /,,«.. _^ ., (^^ _^ ^^^,^.4^ 9. (a + 6)"+' + (« + 6;"+' ; (a _ 6)"+' - (a - 6)»+^ HINTS AND A NSW K US. i V i •;::► uJ no ! f U'V; EXERCISE XXIII. (Paoe no.) .<•" - \)(r.v. 4. — 10// — 1 :»///•' -f 14(///- + 'i\(iV>. ■• + //•'. «. fr' - /'•'. 7. r///^ - lb\ H. //" — D// 10. (^/■^-ft'^)./;"H. f/m + (itn — bill), I' + {(I/) l.l. f/m + {(Ni — bill ) ,1' + {(I/) — bn) a'^ — b/u'''. 16. fi — (//' — b) X + ru-'-' — «ic — //■').*••' + brx\ n. J''' - ;{./•' - Wx' + iSx' + 4.*'' - 5.^'=' + «.*■' - 12 J- + 0, 18. a^ + /,n _ ^.3 j^ ^^^,jj,. ,5) ,.3 ^ y3 ^ g^py _ , 16 \i 18 20 21 22 23 24 26 28, 30 31 84 12. 1 - u:" a^ + /,n _ ^.3 ^. .J^,^,, , 5) ,.3 ^ y3 ^ >^j^,j _ , iKi-'' +3T.f' + ;.«•" + ;U''" — 2^'^ + ^r^x"" + Uru;'' + \\)x + 6. U- + yf - (^ 4- fO'- 16«' + 24 + W - ^r'' - Acd - (P. 16a'^ - 2A(tb + 9/>- - Ar" + 4r•" ' ^ — />.*•' — (ix^ x""" + x""'ij"' — .t'""//" — //•' ic'^ + x*'-' - \h\ I - -x\ + 0. + \%nhc.. -1. -1. 15. nkx*' + {al + \)k) .r* + (///// + U + rU) x' + {an -f bm + <•/) x* {Ini + cm)x + en. '-3. 0. a. ,/4_-j.yj^n) r, 709.,." _ n 7(545) 7. 0, put ^'; ii,i-\ (i. —2(1"-'/)" '; — )"-". EXERCISE XXVI. (Paoe 57.) 1. X — 3// ; — X- + if ; uH) — a. 3. 1 — Wax — 4rr'.r' ; — 1 -;- .^' + %ih r, 8. - rf + 6 + r ; -a Jrh\ (r ; i-i'/y - |. 4. — awj:;'*-" + 2ain^ — fa^yj.t-^^". 5. (r/ + h) ; 4 (^/ — 6)''' ; a"-'' + r/"-" 'J 3 8 2 5 !J + 4. 7. - w"l 1 «••' i q m Q 0. i 1 ii i . . — i i t 1 > 1 ~ • 'V. ' ..' ,■ *■ 1 ' 1 r- 1. 5. 9. 13. 17. 20. 23. 2G. 28. 30. 33. 35. 87. 1. 3. 4. EXERCISE XXVII. (Page 60.) 2. (I - 0. iV + i 'Ax +1. 6. ;),<• — 7. 'ix — -Zij. 10. x-1. X- - tf. 14. 9.r + 4//*. ^x' - X. 18. .r- + ;{,r + 1 ifx- + f/^f +1. 21. -Idh. x' + 2.r, 24.1'-' — -lax — WT^e %x' ,2 3. 3./- 2. 4. r/-24. 7. Wx + 2. S. 4.*- + 3. 11. 3.r + 4. 12. ~ur—\. 15. 8,<' + 3//. 10. .r' + 14a;. 19. a- + r/- 1. .. ^ . .... , .. ,.. , 22. ^/* - 3?M^ + 26'rt. a;' + 2.r// + 2//-. 24. a;- - 5r + 0. 25. .r-2,r4-2 ; — lOOa^. ''"^•'-' " *"-■'' 27. .r' + .r//" + 7a.r. — 5fi'- + \hd — Scf. 29. 3^/- — 5^'- + 3c-. X + 2. 31. a — h — r. ]P(j 4- Apq- + 2f/\ 34. a;'-' — ///,r + iir — n ; {in" — v/j') if. «' + //; //'" ' ' + 2.r//'". 36. x-" — lx"i/" + if. ax" — bx" — a'-x + abx + a' — a"b. (Read a^"^ for li^b" in text.) EXERCISE XXVIII. (Page 61.) a"^ + 6* 4- •-■ - (f = 0. /> + (^ =r 0, etc. Or. multiply out.' ((■' — (lb) -i- {a + b — 2c) ; eijuation is ix + a) -4- (.r + 6) — (2x + n + r) -=- (2.r + 6 + <•); eoniplete the divisions, scpiare and transpose ; .-. (.c ^ 6) -4- (.1- + b) = {a - 6) 4- ( 2.r + b + c), etc. 19. 18. 9. 19. 9. 72 ; remove l)ra('kets and eonibint! nunierieal quantitie.s 4i 22. 4. 8 ; eip.anon is ,\x + ^\x + p- - \p- ^ ,s^ + ^ ^. :i^ ^ . . ,„. IU-^"-ff'' = ^"">l ^<5<5 X 91 ; /. ('. m-)\x 4- 77 X 284 = 80.-) 1 -^ OO x 91, etc. (2!ib'' — rut) -^ (2(1 — 2b + :)). 2'). H. EXERCISE XXXI. (Pagk 72.) 1. 240. 2. 5. IKl + lib 4- 7. 12 mile 8. S4 niil,.s. 0. nut i(t — b) -r- { 7. Price = |( 22.r - 2 Iz) ,j -r- 20x (a- - z). 4. 8 men. ntn — /// — 11). 9. One-third. 10. 188 oz.; gV (•^- + 5«) = tV ( H. noj^rjii. ;x: = vv.t. of lump. ■^j(x — .-jO), where m iMi 10 11. 1.-). 19. 20. 28. 29. 01. HINTS AND ANSWERS. 30 eggs. 12. 4. 13. 40. 14. |78|. 16200,23000. 10.63. 17. 30 gal. 18. 1080 -=- 251 miles. B ill ''-!^u4 . 5929a'^'' — 7744//"". 4.r^ 10. a' + b' + 2a/> — <•"' ; x"" — 2x1/ + if — ^^ 4^'^ _ />2 ^ ,.^,, _ 9^,2 . y2 _ 4^.. ^ ^2x2 — 9^^ ( >r + .//)'- - (X + zf ; (,v + tf - (u + ry. 13. (a + '' + 4pk — 127m' — 2mk. EXERCISE XXXIII [a]. (Paok 80.) 6. 1 + 2x + nx"" + 2x' + x*i 1 - 2.r + \ix" - 2x^ + x* ; 1 + 4.-r 4- 6.r» + ■ix'' + x* ; 1 - 4.r + 6.<-- - 4.r* + .r^ 7. 16 + .}•'- + 4//'^ + Hx — 16// - 4.r//; 25 + //- + 9^•--10//— 30^ + 6//^; 1 Or — r'^ •1 >'»»/^ -L Oi.sy^ 1 oyj >.M ^' + 2.*;=' + X' ; .<•- + y' + ^' + 2u'''y^ + 2yV' +2/^i • $781 T- 2ol miles. H. 24. 23r,. -r + 2)=3a i*b'a\ '■' + 2/V. ELEMENTS OF ALGERRA, II /O. «. 1 + 2a;* + Gx" + a* + Qx' + O.c" ; 1 - 2x'' + dx' + X* — Cyx' + [).)•« ; 4 - 4i, + 9y^ - 4y-' + 4f/ ■ 4x^ + y^^i + 4^.y _,y_ ^^, 9. 1 - 2x + 5^--' - Ax' + Ax* ■ 1 + 2x - .u-' - Qx' + y^. . 4a* - ^a-^ + 4 - 4«» + 4r, ; 1 + 2r.-^ + 2«^ + 2«^ + J«. 1 + .v" + 6-//- + 2x + 2/;// + ',V;.r// ; 1 f ^/-^•■^ + I'^y^ ^ 2ax + 2hi/ + 2(ikn/ ; 1 + a\i^ + frf - 2ax - 2% + 2af,x,j ] 1 - 2ax' + 2bx' + a'x' - 2ahx' + b^x". 1 + 2x + ;i^••■ + Ax'' + :}./•" + :>.r' + x' ; 1 - (j.f + iru-^ _ 2()x' + iryx* - (lr\ x" ; 1 - 2x — x' + Sx* + 2x'- + x'\ 1 — 4ax + lOrt'V — 12a''.r'' + 9,/V ; a-« - Qx" + nx* - Ux' + mr - Ax + 1 • T" - Ax^' + 1 Ox* - Ax' - 7.r-' + 2Ax + 16. ' 13. Aa' + b' + Ac'' - Aab + Sr/r - Abr ; "' + i//-' +^--a6 + r/f.-lk•• .i 12 ^a« + |6'^ 4- //2 a^ + 1. 10. 2a%'' + 2^>»r-^ + 2,,'v " -a*-b*- r* 'U- r*\ SCO Ex. 3, p. 128. EXERCISE XXXIV [b]. (Pag 4. (3x + 4//)- - 2r,/- ; (2a + Acf - 9/r. .'). \(x' + 2,r + 4) - 3a-!- X \u' + 2.i'-^ + 4) - .-,.,•;. = x'^ + 4j'^ - Ax* - Hu-« + 31.r - \\2x + 1«; • E 84.) \{x + z + (r) + ^yi \(x + ^ + a-i + 3^;. = {x + z -\- frf + Ai/{x + 2 + ,r) + 3/y 0. .f'' + !b-' + 2<».r + 24 ; y' + }>.r' + 23.r+ 1, X' + 14.r- + iio.f + 42 .*"' - l).*'^ + 2()./- - 24 ; ,/••' - 1 4 r' f 5o,/- .r'~y.r- + 23,r- 1/ M. .t-' -^ 3.r^- l().<-24 ; u" -f- .t'^- 17.f + lo. 42 .r^* - 1 2.r' + 29.r + 42 •ir? ii'iilij 1 i 1 % t t - n HINTS AND ANSWERS. 9. ba;' + 12a"- + 22.v + 6 ; S.r» - 12.t'^ + 22.<- - (5 ; Sx' — 4j'- — U)x + G. 10. X* + x''(!j + 2 + ir + /.•)+ .i;' (>/•// + u-2 + (fk + yz + yU + zk) + X (y2tr + yzk + zwk + ykw) + y^z/Vr ; X* — (a + h ->r (' + <^/) a.^^ + (a6 4- «c + oc/ + he + 6,/ + cc/j x* ~ (abr -f r''r + 2Air-r + ^i2frr' + \Qr* ; iv^ — 'Sw"",- + ^tvr — ;•' ; ir* — 4ir''r + dir-r" — 4u'r* + r\ 13. A-' + loA-^v + {)()/»• V + 2T0A-V + 40ryks* + 24;iv" ; (f — \2(i'h + ()(k/V/^ — lf)(V/^// + 240a'''6^ - 192^/?/ + 646« ; 8a' — M'lv + §a//'- — \w' ; |a' + i^'''^' + 6rtW'' + 8m-'' ; 18. 183()r/'"6=' ; - 22(m)tiHi' ; - 2f'/^^;^ 14. l48r)a-7/'^' + r).ia/>-"' + h"-' ; 214r).t-^,y"-' + Ofi.r//" + /" ; — (UJ.loa' + 13h/ — 1. 15. :Aa'b'\ r)40a=Y>'; 1680• ior .v + r/, ;// for x + />, /. product = A"* + k'-nr + m* = a-t"* + 6,j''' (a+b) + .r (Tr/- + 4(ib + T/r) + .<*(4rr' 4- 2trb + 3^6- 4- 4h^) + {((* + a'-7r + b^). 3. ^--Yy^ + r-V/- - f/V - //-V/-. 4. 4- 6'- ; -t- /i*. 9. I 4 2x 4- ''^x' ; 8f^- 4- 3f/ + :5. 13. .r 4- 4 ; 2.r - 8/>. 13. a 4- 86 ; 2a — 76. 1 1. 1 4- a\ lo. x" — 20- 4- 1. 3rt a 3a' 26 -2 8/M [^•] 4- y-- + zk) /•/•=' + 16;-«; .H + ,.4. f 646" ; 875. •nr + wi* •r + 1). /> + h\ ELEMENTS OF AUJEHRA. 13 2. fr' + i^- - 1 ; .r' 4- .<■ - A : .r'"" + ^Iv + 1. 8. U- + 8// - o^ ; ■iur — X + 1. 4. "^'^ - 4 + •'' ; x — 'i- ^ X y X ^ 3a 1 2h b 5 3r; 0. 4(/ — ob ; ~ — 3^. 7. 1 - 2,r + :}./•-■ ; f <• - 1. H. ,, + lV> - r. j). % l + ''^; ■'■ + 2. 10. Cube both sides by formula G ('i.), p. 8."). 3 X 3 EXERCISE XXXVII. 1. a^b + Z^'^^' ; 'Mr/ + h)"- + />(/> -f '/)■' : (th (h — c) + /><•('' — (I) + <-(l [(I — b) ; trbc + //'r' + ^ + (^ ■ a r + {x~r}{(i — by (I- (be ■+■ b■(/) + b" ((I ■ + (Kf + (■ + ^/ + r + r/ lib + r/r -f (Ul -\- hr + hd + rr/ ; a- {(I — />) + 6- (b — (■) + '■-' (r — f/) + ^/- (,l — (,) (n — bf + id — cf + {(I — (/)■' + {I, + {b-(h'' + ir-fh' 13. r/. A. 1."). (/ fi (I. -(. ■>, (■ (I. - b. 14. (t. 1 (1 /> '/, b ; (/. — 6, b. Hi. '/. />, '■ ; (i.r and // '■ : (I, b. //. .r and //. (. (I and It ; (/, o, r 20. a, b, r. 23. .r\ ./•-'//, .r//^. 26. .r,y'\ xyz. 21. (rb. n». '/. A: (t. b. (i\ (t 'b. . f/Ar ; .r\ .r"//, ,j' 2i»U' ,.3, a .//, x"DZ, x'lfz, x'z 4, ,3 .'/'. ■'■7/ 11. 11"" + b* + (•' + iP - 3 {ilbc + «6r/ + bed + Cf/«J. 1^ 14 HINTS AND ANSWERS. 32. n + h — n \ a — h + n ; — (( + I) + <• ; a — h — c. 33. i\(a- hf + (h + rf + ('• + '/)'!■; i ](« + '')''' + {b + vY + ('•-(')'']; \\{a + hf + (b-cf + ((■■^af\ ; ^\(a + hf-\-{b-cf-ir{c + af\\ tlio three expressions are derived from the, ffrsf by, respect- ively, substituting — c for (\ — h for h, — r + vd). 5. 2 (.*:'■' 4- //' + Z' — H-i' — //.? — 2'^'). 6. 14 ('/'•' + //■' + '•'"') — 14 (ab + be + rr/). 7. 4 (a'' + 6- + c- + (/-). 8. 4 u/V + 6^' + ^•'^')- 9. 2 (a" + 6« + c') + 6 («'6 + etc.) - 12a6c. 10. a' + 6^ + c' + rf'. 11. 3 (a' + ft'' + c' + 'P) + 2 (a6 + etc.). 12. 0. 13. 6a6c. 14. a6c(a + & + c). 15. 4 (x' -f y' + ^') + 24 (a-6^ + 6'c' + c'^a'). 16. Notp:. — The first term in each of the l)inomiaI fadorfi should liave index 2 ; /. c, d" for r ])airs by symmetry ; re.sult is 4 (<(}> + («■ + <^/(/ + /"' + bil + cd). 20. Type terms are (t\ )l(f'Ub + r), a'-lr, and both expressiotis reduce to sam<' form. (h\ use identity, Kx. 7, \). 105, put- tin*' a — b for a, b — <■ for 6, and .•. a — c f or u + b. EXERCISE XL. (P.\(iE 98.) 1. {a — b){x + 2y). 3. id + xjiti — b). 5. ini 4- ") <'t''"' — ^0. 7. 0/ + b) (3.r + y). 9. id — //) (r + y). 11. (3tt — 6j (.<• — //). 2. (-7 + /')(2.*- — 3//). 4. {r — + ). (.«' — ^/'H-*''-' + (f.r + ((-}. i + (/)(l -r/ + 6j. 11. 12. ••"f slioiild 1 y identify, \ 1 )' ; rr.siilt 1 '« W'.ssioiis 1 19. 105, put- 1 ^^^- 1 22. 1 23. EXERCISE XLI. (Page 100) (a by \(a' + «6 + ¥) - (a^ ^ ah + 6»jp; i Z''" (J-OT 10. (;v,.r + o/^y ^ ^,^y^ (■V + U + Zf ; (;) - ry + /.)^ (^/ - 26 + 3^)'-' ; (!_.,. + _,^,._ (3rt + 26 + r)*-' ; (2rr'-:w + 4,i. (2a^ _ 36 + 4r-)^ (,r _ ^^ _ ,..,._ ±fi^/.y; ±U)a'fj; -±12.rV/-': ±:.V'6";\r'. ^''/''/' ^= ^= ^^- ^1- i^^ ±46.y; ±0. ^o.^. 81-10; .r^ + 4; ;r^ + i:}: -.+ 1^., EXERCISE XLII [/.j. ,Pa.. 103.) t. ^^4-6±.; 2u. + ^)±,.,i,^^^,^ 2±,. + i, ^- ^^^^-^•>;'' + ^ + ^^±.-;(8 + ..,ao-.,;6-.±Ua;^ 1 w IC HINTS A\D ANSWERS. S ia ■, III 7. 8. 9. 10. 11. 12. 18. 14. 15. IG. 17. 18. :J 1 2 (a' — he) ± (b- - ac) { ; a- r^b ± 1 ; i ± (x - ij + 2) ; (a* + b*) {a' + W) (a + b){(i — b); {a — '.)(•) ui + Ab + iic). (— (t + b — 4cj (3a — 5/; + 4c} ; {\ — a + b) {1 — (t — b) (1 + 2f/ — a' + b'') ; (12a;— l)(2.f + 7). {x — z± !/) {X + 2 ±y)\ 4 (.c + 2) (>J -f '0 ; U'±(!/ + 2)\\.l'±(U-2)\. {x — 2)± (II — u) ; a ±{x— ij)\ X ± (11 4- z). X ± ( // — 2) ; X ± (// + ^) ; X\ 2 ±l]\ .*•■-■ ± (.<• — 1). {X -{■ + (■) ± «i + d) ; (a + d) ± (2b — :J?:). ;5r + r/ ± (a — 26) ; (a — 8r) ± (26 — (7). ]a + r/± {b — r)\ \b + r-± (a-r/)}. (0,'- + I -=- //-') (.r + 1 ^ //) (,r - 1 -^ ?/) ; x^ ± -,V., I'to. ; a;' (.f" — 2r)) - y (.}•' — ^.-)j = (x* — 25) (^-^^ — ^), etc. ; (.!"• — 10) (y-* + 1), etc. EXERCISE 1. 3.t- + if ± xy. 3. 3a'' + 6'^ ± Tuib. 5. x' + 1 ± X. 7. .<.•■-' + 25 ± 5.C. 9. .r- — //■- ± 3.r//. 11. a- — //-' ± 2(11/. 13. 9a'- + 6' ± 3a6. 15. 5//- — 4(/ ± pq. XLIII. (Page 105.) . 2. 4a'-' — 6'- ± 3(/6. 4. nm^ + 4///* ± linn, 0. .<■■ + 4 ± 2a;. 8. .i- i + ^ ± t^. 10. X- 4- l; m' II* ± Ainn. 14. 4a'- + 6'- ± Ga6. 1(5. 9,r'- - //- ± Axy. 1 n o J'" 2.r^ - 1 ± 2x. 18. ■'•■ + .y' ± xy. 19. ;c' + 2ttY' ± 2aj:y 20. 2a'' + y' ± lay. 16 + 8c). ELEMKXTS OF AI/JKHRA. 21. ^ + V' ± x'\i\ ,>((' + 1 ± ^•' 'f* + 1 ± itx' 23. (ft +h r + 2// ± 2.vy. r + c" ± We ((I + (,) ; I 4. :.^- ± 2.r. 24. 4u.- + 2 (// - 25. 1 + 0^,4 ^ o,,j 26. 2(1 +,, +,ry 27. ^)' ± TW (y _ ^-) ; 14. ;-,^^4 ^ jj^, X' + 1 H- //■-■ ± ,r ^ //. X' + 1 - J/y' ± .^• -^ // (// + /»■-' 4 (,,_/, ^r + '>r ± ( + 1). 20. U'+17)(.r + 23). 22. U + 83) {X + 27). 24. (a + \Ht,)\ 28. (x~\\))\ 31. U-='-r,j(.t.='_o5-)^ 33. (u;-18//)V 3o. (a - 21h) {a - 2b). 37. {26 ~ (,b) {rt - ab). 20. «(.*■- 4)1 29. U~2{)f 27. r.r-ir,)^ 30. (,('—-)())■' 39. .*•' + 1 ± J 41. iix(x-2)(.v-8). 43. .^- + 00± 17.r. 45. f« + 6-4)(^/ + ?>_8). 47. (l-8./;V)(l_r,l.ry). 49. A"" (« — ^rtbx) (a — rAv). 51. {p — 21q)\ 1- (a' + l)(a»-o^. 3. {x - 8) (.*• + 2). 5. (.r + 12j {X - 7). 32. ()n ~\i;u)(ni — rui). 34. (.^;■- _ ny^) u-'^ _ 4y^^. 36. (4 - .^■) (8 - .r). 38. (r/-25)(r/- 15). 40. (.*•«_ 27) (;f=-,S). 42. a(x~r)){x~i\). 44. U-" _ 7^ (^.n _ 3;^_ 46. (18-«A-)(ll_f,^). 48. (^/ - 276)-. 50. {m-\%f. r^-> 52. \(x i')"-33} lOr- 2/)" 11|. [b.] (Page 109.) 2. ' <( + 8) (r/ — 2). 4. U'-16)U + 3). 6. {y + 12) (y - 5). i ', f ; ■ (M ;■' f &, fl ML 18 7. 9. 11. i;{. 15. 17. 19. 21. 25. or 29 31. 34. 35. 37. 39. HINTS AND AXKWKlJ^i. ((I + 20) (i' — '^). (.r^-4S)(.r- +H). (./• + !i- \\)){.r f ,// + 18). {.r-» 4- 4) (.7"" - ;{). (13 — ah) (5 + f//>). 3//(^/ + 14//.r) U( — -ifu-). {lU- + 7) {•'U- + 5). {7(1 — Hh)-. a-*{H!j^ - U)2)\ 32. (^z^- H. (^/ + 25/>) (). 10. (r — 10)(.<' 4- 2). 12. {a/) — 4) (^//> 4- 1). {II* — 20) (r/' 4- 5). (,,•7/- — 30) U/'-V/' 4- 3j. (.r"— Ifj) (,r" 4- 3). la—:\{){h-\-r)\ \(t-{-V2{f> + r)\. 22. (20 + in{\\) — in. 24. (12 — //n (17 f /;/)■ 20. (2.r f 7)(2.r 4- 5). 25. (2.r'7/ — 7/-') (2.r-.// 4- <>/■'). 30. .<• ( 2/' — //) (2// — 5.//). 40//-) {ir 4- 5//-). 33. ( 1 \u' - 13//)- 14. 10. 18. 20. 3 (,v^ 4- //'-') (3.I- — 4//") ; wlicro ,r — a — A and // == '' {Sx'' — 2b"){Hx" + 0"). (fa 4- 76) i^a - 86). 30. 38. 40. {U-- -f 7)(^.i'' — 0). (5^'' 4-21j(5a''''-31). 21. {l\m. + 2{)){2iii— 19). 23. (3.*- 4- 7//) (4./- — 5//). 25. (5.r-— l)(4.r- 4-1). 27. (12.^•-7)(2d; + 3). 29. (Ha 4- 6) (3« — 46). EXERCISE XLV. (l*A(iE 112.) (4.r 4- 1) (.<• + 3). (3./' 4- 2)(2.r +1). (2./- 4- 7//) {U- 4- 3^). (1 4- /^/) (7 4- 'ini). (./■ 4- 7)(3.r + 2). (4.r 4- 3)(3.r — 2). (4.,. _ 7) ('J.,' 4- 5). (5,r — 1) (2^- — 3). (.r — 7) {7.1'— 1). (^/-■— 19) (r/- 4- 17). (2// + 20) (3a— 19). (3- 12.r)(5 4- 11.*-). 1. (2.1- 4- l)(2.r 4-3). 3. {ilr 4- 3)(5.r 4-4). 4 5. (2.i- 4- 5)(3.r 4-4). 7. (4a 4- 9) (a 4- 1 ). 8 9. (.r 4- 5) (4.r 4- 3). 10 11. (4.r-3)(3.r + 2). 12 13. (4.r 4- 7) (3./-- 5). 14 15. {:li- 4- 2) (2.r— 1). 10 17. (3.*- 4- 4)(5.<' — 2). 18 19. (5./- 4- 2//) (3.r — 5//). 20 >>o 24 20 28 30 (15a — 1) (a + 15). (0 — //)(3 — 5//). (8 - 9y) (3 4- 8y). KI.KMKNTS OK AI-OFOHKA. Id V2{h + r)\. 31. (2H^-' - 25) (u;' + 5). :{;{. 4 ilx —')}/) {'2j' — !j). ;{r,. (Mr/ — 56) (7(/ — 4/>). :J7. (H// + r.^)(»y — 8^). 'M. ma' + 41/') in' — nh'). 41. {V.\x+ 12//)(;U"-4//j. 4;J. (39./;— 2(;)(.r +1). 45. (1 - i;{.r-')(l + ll.r'^). 47. (:J.r='-21)(l.r' f 11). 32. 4(14.r + 5//) {.V — y). 34. 4 (1 l.r — 5//) (./• + //). 30. 2(2H// + 1) (/y — 10). 38. (»// + 3^0 (4// — 5^/), 40. {r>{\(t — nh) Ui — ih). 42. V.ix-\-y){UU-- 11//). 44. (12^ + 13//) (5,/ — H//). 40. (^/"— 13/>'')(r/" + 116"). 48. (17^;— l)(.r+ 17). '). EXERCISE XLVI. (Paok 114.) 1. (.r — (n (■>■" — 2.r — 1 ). 2. {.v — m (.r — ju- -f 7). 3. (y^/// — //) {((y" + hij — '•). 4. (26 — >■) (./•'■ — 26./' + h). 5. ()ix — a) (.r — .r — 1 ). 0. (6.1; — ^roup, etc. 7. {y — ^)(U — *<)'. H. (.f — 6) (a; — <0 C-*'' + -/>). 9. {x-\-p + q) {.!■ + (/— /)} (.v—2(j). 10. (X — r/) (.f + h) (x + 3). 11. (x + b) \x(x—\)—(f(x~\)\. 12. (2.f — ^0(2^!'^ 4- 4u- — 3). 13. (;>// +q) iff -y + \). 14. (//j.» — n) ( ;ac"' + qx — r). 15. (mj; — w) (^w;'^ — fu* — 6). 10. (px — q)('.\x'' — (;x — b). 17. (.r'-y>.c + 7 + 7*). 21 . {(ip — 67) (3//^ — ])q — 27"). 22. (ax + 6) (ca-' + cfx + c). 23. (r/.r + 6) (2i-x' — r/.r — 3r'). 24. (3// — ah) (3// — 6r') (3y + 5). EXERCISE XLVII. (Paok 118.) 1. (6.r - // + 1) ix - (\y - 1). 2. Oix + 2// + 1) (2x — i\>j - 1 ). 3. (4<^ -f 56 + 4) (3a — 46 — 5). 4. (x — y + 32) (x + 2y — z). 5. (3.r + y + 3)(.f-3y + 9). 6. (2a — 56 + 6c) (3a + 46— 80). 7. (3a — 6 — 7) (4a — 36 + 8). 8. (Ix - y - I) (x — y + 3). 9. (a +3//)(a — 4y — 5). 10. (2x - 5// - 7z) {2x + 3y + 3^). 1 1 . (3.r + y — 4z) (3.k — 3//-2^). 1 2. {^x - 2y + 3^) (2.I-— 3y + 2?). 13. (5.r — 3// + 25')(.r .?). 14. (a - 26 + 3c) (U^r - 6 - c). 15. (2« - 6 — 3c) (4a — 36 - c). KJ. ( 1 — 3,r + 4//) (1 + 7x — 5.y). HINTS AND A NSW Kits. i a. c:i IP; c:> HW*| (D BiJ (I. o )^ uJ -J -J ;o i,1i^- Vf io EXERCISE XLVm. (Paok 122.) 7. 8. H. -Ha\ 1). -'2i)ry. 10. 1. 11. (1^ + /)(!' + (/a + r. 12. ~ ;m;. la. 1055. 11. ./■ + 2, .*• - a. 15. (.»• + 1 ) cte -f. 2) (2.r — 1). 10. I.fist term should be r)2(i\ 18. (a; — 2) (.i- — 5) (a; + 7). 11). - 5:{.-). 20. -mi), 21. 101. 22. IIT). 2;{. — (/'' — jm'^ — 7'' — 4^//> id + h). 2». -1. :{2. 2. 84. 2(f/ + />)•'. :{■). <). 39. 0. 40. 0. 41. yes; i)Ut Ifor .r + //. 42. 0. 44. a' + pa + (/, ir + />V/ + v'. EXERCISE XLIX. (P.voe 12«.) ( /> — 1 )■- - ( /> - 1 ) (7 + 1 ) + (7 + 1 )' ; a" - f>\ ^•'" _ ,ir') + Ul — h)- — (it — 6)1 ^;'^ _ 1 + 1 -f- ,r'^ ; .r" - 2.r" + ;{.r' - 2.r- + 1 . ./•' + .'/■' ± .'7/; ("■' + 4//-') {(1 + 26)(<; " 26) ; 2.n.f- + 12//'). {a + 6)((/- + 6' ± r/6j ; .r'J — "iff and ic^ 4- 2.r''7/'- + Ax^y' + H.r-'.//" + 1(1//". ((/- + />r') (//' + 7/V-r'^ - 4//''6r) ; (x + 1/ (^: - l)"* ; (.r+ l)(j:-l)(a;'' + \)(x' + 1). a" — {26-')=' =\a — 2/y-\ \a'' + 2r///' + 46'" [ ; (^/ - 6) ab. E.xi)n'ssi(m = (4a*-96'^)(.r''-8a'),<'t('. ; In - ^^ )("'— 1 + yX E.xprcssion = {x" — //-) {x — //)" = 128AOEl2 '■ + //) ('/ + 2) (.? + X). a. (./ _ />^ (/, _ ,,) ^^,, _ ^,j^ •J ^« - i^) (y ~2)(z- X). T). 7. 8. y. 10. 13. 1!{. 10. 17. f** + y) (Z/ + ^) {z 4- ^-i. - (a - b) {b - r) (r _ ,,, ^„ ^f,_^ ^^_ ) (^; _ ,., ,,, . -11). It. lUf^'-^yX/," )(^ <■){<■-.„){„•' ^b' + ,.■' ^ ah + i ■)('■'- a). 0' + b + r)ur + fi^ + ,.-^^. 14. (u' + ')(■ + >■!/). (iab( y + z)\ ir,. (,,-^h^ rf o 4. 0. 7. 8. (r^ - />) (^, _ ,,) (,. _ ,,, (,,j^ ^ j^^, _^ ^^^^^ [*.] (Paok 130.) 2. By syniinctry ; or fonniila (II % symiiicti'v ; or \ ) (4), p. 8,'). .) (X + //) f/y + ^) (3- + X) (cc--' + //■• + fansposo ;W6r, then r/ is a factor, oto. (a + i — r) (((■' + i" + (— ^/ + I'y + (■) ( //• r/7> + ba + ca) z' 4- .r// + y.? -f. 2.j.)_ a b + r, (^/« + // .^ + />■' + '•- + ab — be + ca) (•• — 12f ; 2(1 term should be l;U'^ ah + ftr- + ca). Use synthetic division ; 4//i-12 = m-^3 "■v..i expn-ssion = (x - S) (x + 2) (x - r^u whiel Gi a/l values of ?. e. , a 0. 12. l."). 1(5. "^■v of .«, .-. co(>mei(>nt of like powers of = -6, 6=1, r.= 30. 1 is true for X are ('(pial ; ?> = — 6. ry. 2a + 5 ; a + 5. fl" + ah + h' ; a + h. x" (3a' + 2). 3if + 4rt ; unity. 21. 3a' + 1 ; 5.r - 1. 23. x + 3 ; U' — 1 )". 25. 2.r + 1 ; •'■" + y. 27. a;- + 2/y- — 2.ry. 29. 4 ; 1 + ^ a rt* + 1_ a' 31. 2a; - 1. 2a + 3& — «7. /»a' + m — X. X 4- 2ah \ omit r/ in ax^. 3 (2a — 7). (2aa' — y) . {X - 1)\ EXERCISE LIII {({]. 2(.<; + \)\ 2. ./■- 5. -la'' + 3a - I. 5. y' + Hy — 2. a;' — 2a' - 3. 8. .r' - 3. 3a''' - 2a-y + y\ 11. x- 1. a^ {2x^ + 2a'y — ;y=;. 14. x - {X - 1) (a; + 1) or a:" - 1. 33. 5 {X + 2y). 35. aj> — hq. 37 (a — h) (X + a). (Pagk 141.) 3. 2a''' — 3a; — 1. 6. y-=-3y-5 9. 5.C'' - 1. 12. (X + 2)". 2. [ft.] 1. a;' _ 3;J' + 2. 3. a* - 8. 5. 12.t' + 5. 2. a'' — 13a; + 5. 4. W - 3). 6, 2a;'' - 4a:'' + x -1, 5 ELKMEXTS OF ALGEHKA. 2S 7. 9. II. 1:5. 15. a;' + 3.r + 5. a' - a^ -a-1, •*"' — 7.r — '.I 8. (^ + .1) (X' + 1). 10. 2y« _ 7. 13. «=• + ;v^ _ 5.,. ^ o 14. 2a;' (2a; + y). 2. 3. 4. 5. 6. . EXERCISE LIV M. (P.oE 14,) 42a b ; 4/y-..,-; ,y.io.,-y; ;V-.7rr7.V- y^V; 2.:irrL^; 7a'. W. 2, ab-r 7. 3x {a ~ ,v) ; iia'h (a + ^ 8. Aa^x (a + x) ; 21 ( i"/-." f/ (»■/ - ^>') ; abc (/?' - r») 9. V + //) (,/ -f. l) a(a + h)ih^a). x{x+l)- .,.-'( rA {p + ry) (y, _ ,.,. 10. <76 (.f + a) uv + b) 11. {x-2)(x~\) 12. (.r + 1) (X + 2 :0: (.i;-l ; — ) ; ('/ + br i"' + b'). )- ia + bf 19. -d _ //)=•; (}(r/ + 1 )•-•(>,_ 1,. !-^')(l + 2a'); (y^_ 20. -Ga-i-c^^-i^; a;^-^^. (^.v J .'/-')'•': a%-{„- \)ib ^ \ l)(a'- + 1) or X* — 1. !1. ab(Aa'-\)- (S.r(^^^ [*.] (Page 144.) X {X — 2 oo '-i'^. ix + l)(.r_ I,'-' ) (•'■ + 2). (•'•- l)Oi' + 2)(r- (a- — I ) (,r _ o, ( .. _ :{). 4; 23. (a- + :{ ) (a- + 4l(.r + ;"i) (a i4. («- !)(„_ — 1 )' fa' + 2). !» (a + 2) ; (X' - y^y-' (,r (.r _ yf ,^. ^ i/J' i ;i f K V-" i; ; i'l'M^ i '^'U j i ■ ^ ) ■ ^1 : ,3;,: \4 o it—I 24 HINTS AND.AXSWEHS. 25. (X + 2) (x — 4) (.r - 10) U + 1'2) ; (.r + :{) {.r — 3) (\r — 12). 26. (X -2} (.1- — 4) U — Ir. (./' + 1) (.r -f- ;{) (.r — 4). 27. (X + r/) (j;"' - l/') ; (1 — .r) (i + .i-)\ 28. (J- — a) (.r + a) (x — h) ; ( — r-) (r/ + /> + r)'. 29. {X - 2)- (u^ + 2)'^ ; (.*; + :3) (.r - ;}) c^.r - 1 ). 30. (// + 2) (// - ;J) (3// + 1) ; (2.r + 3) {2.v - 3) (3.^ - 2). 31. {2x + 3.V) {2x — 3y) (3.r - 2y); 3 (.r — 1 ) CTr — 1) (3^ + 2). 32. 20r/7/ (4(1 — \) (.V/ + 1) (3fA + 1) ; (4r/ — 1) (4^/ + 1) (Tut + 1). 33. (.r + 2) (,r - 2) (3,r - 7) ; .i'^" — //^". 34. (./■ — 2) (./■ — 3) (,r — 4) (.r — 5) ; 2aif (4.r— 1) (.r + 3) (3.r— 2). 35. U + y) ix — !/) (.r + //-■) : 12 (.r- — 1)-. 36. (x — a)'' (X + It}- or (.»•- — (rr : — 12.-f ix — \) (x + l)\ 37. Ixiia'' -Ir)- xUx-~ 1 )■-■(./• + 1)'. 38. (X + I/) {x- + xij + if) {X — ij) or (x + y) (x'-f) ; 24 {\-x*). 39. ^/^r" - h*i/' ; f/-* — h\ 40. 12a='(rr — //-■), (./•=* + «"). • ^ 41. 1 - .*•" ; ( 1 - 2xf ( 1 + 2x + 4.r-) ; (.r + //)=' (.r' + //'). 42. (x + 1) u-'- — .r + 1 ) (.r- + .,• + i ) „r (./■ + i) (.i;" + x' + 1) ; (.i; - y) {X' + .r--'//'' + i/*). 43. (a;" - «/») (.r* + .r>' + //') or .'" - // ; ix -\- Af (./•^_4,r4-16) ; {.r" — 1) (if- + 1 ). 44. 1 + J-' + ic' ; {'V- — y") (x" + x-y- 4- //*) or (.r' - //=') (.r" + y^j or .r" — y*. 45. - {ti -b){b- c) {(• - «) ; - (y-' - ^■-) (.r^ - y') (.<•- - ^'^j. EXERCISE LV. (Page 145.) 1. {x + \)(.v + 2) {X + 3) (x + 5) {X — 5) fx - 6). 2. (X + 1) (j:'' + 1) (6^-=' + ryv' + 2x - 1). 3. {X + 2)ix + 'S) {X + 4) (x + 5). 4. (X + 3^ (x 4- 4^" (X 4- 51 mi ELEMENTS OF AUJHBUA. 25 I) ix - 12). 10. • 11. 12 . 13 14 -2). 15 )(''\x + 2). 1) (oa + 1). 1 f!{)(;u-3). i) 8 r + l)^ • j» ; 24(l-.i•^). a{a + 5) (7^/ + ^h) {A(r + ;{'/ + 0). (a- - 1) C,V + :j) (:ir - 2) (.r - ,,• + 1). {X -!)(./- :{) (.*• + 3) (.i--^ - 3) {.r - i.v + 3). (3.r-^ + 2.r + l)(3.r=' - 2.r'' + 2./' — 1)(2.J"' + 3.r- - 2j' + 1). {x+ l)(.r + 2)(.r + 3)(a,--M). (^- + \){x - 3)(3.r* - 4^ + (>)(3.r'' - (!r -f »)• EXERCISE LVl [r/J. (Page 146.) — 30. >) o 0. ;5. „ =f>= 12. r/ - ;0 and 6 = - I. O. - 114. 4. 10. 7. 13. '^'3! r = 'i'4,, a fy"). + u-'^ + 1) ; 1. y-_4.r + 16); 6. f){.r^ + //=") 8i :.*- - 2'), . 1. 2. 3. 4. 6. 1 6. = 48. 9. \—(i. to. /> = 2. 11. c = 6''(l-«). Divisor = {x — n>j){x — (>z), and dividend vanish(\s for ('a(!h of these factors; /. f., for x = (ty, x = 1)2, sul)stitute and sn})tract. 15. Substitute x = —a in each expression ; subtract and a = ;;— 1 ; substitute this in ri' — qa -|- 1 := 0. . »l«>rriainders on divi<]inff by x + '• is zero. First quotient = (^a; + a — r), which niuUi|)h'd into x^ + a'x + 6', ^ives required expression. Unity. 7. 3.r- + 2.r + 1. 2x{\x' + 1)(.U-^ - ])-(ru,-^ + .r + I). EXERCISE LVIl [a]. (Page 1.53.) a + /v ?> _ r^ .J- +1 X — \ 3j' + 2 xy + 1 .r'// — 1 a b X X X X x" -\- X + 1 c' — fi — 1 it^ + a' + 1 0^=* + 4£ _ ^^; ; —ff ; ^]i ; "~ "2;^ 2x + 1 2.i.'' + r — 2 3.r=' — ^"^ 4- 2 3f/''/>' - 1 a; ah X ?,x- 5 u;'(4a-=' + 3) .f(,r^ + I) X —\' a- — 1 ' .r^* + 1 ' .t ■' 4- 1 (g + ^ + f) (ff + 6 ^ r*) (/> -I- c — g) (a — /y + r)^ £^1^ + 1 iab ^' ' 2rt^ ' X - \ ' x{x + n)(x + 5) «* ... f/'^ -f- .»•// -I- r/*/ 4- 1 . 3.r-" -|- 2 »^f ] X + 3 ' X + a ' X + 4 w Ih.i r ■•.■ If m «... O s .J I "JO ■1 •1 t i. L.f 4 1 i^ ^■|: 26 8. 9. 10. HINT.S AXD ANSWERP 5a — 66 ' a+ X 2x* + x'\f + y\ x"* + xy + y"* x'^ — xy + y'^ ' X + a ' x — I iix* — 34.^" + 3a\ (g - 26) (a + 26) (a'^ + 46" ) 8a;' - 27 a' + )lax~+ X* ' (^/ — 86)~(a + 36) ' llF— T ' a^ — IP x^ + if x^ — '/'' x^ a + h' X — y ' X + y ' 2.r( l-iiaf" + 'ix- U x* + \i. i' + 27.y"- 27.g^) i"+~3i [6.] 1. a + a 1 w , 6'^ a" ,. a a X (V b^ X + a 3.7.2. //'^ -.7,2, y' ' + — >= '+,^4' '+ir-5' '+:?f2p' .i; - 2' 1 - a» + r r; ^/ - 1 - 1 a + 1 3. u a" f 4- .f + 1 + — ^v; « + II ; x — ii-^ .-»! — I .r, — n. a' 1 +x + id x-1 V.3 X - a' 1 + iC _ 4a' 2_ . g '^ X + 2a' "^ .«' - 3.f - 1' 2.r'^ - 3.r + 1' a;'' — aa; + a"* X + a 3 , 2 ,. TT + '+.-- 1 _ 4.1' + 3a; — 7 a;" + 4a-' -5 -J <5 1 1 -— 4- 1 + r; 1 + — -7 + 1 + »• J. 1 »• _ 1 a; — 4 X — 5' a + b — r — (I b — r 2 nix + c + d ' mx — 6 — f/' aa; + ?><^ — 1' 11 3a; + 1 — 11 dx — 9 1 + 2 a.r + ni — 2 a; — 8 + 1 + „-; aa; -f- »/ — 2 1 a; + 2 + + a; + 2 + ^• - — 8 a; — 6 ELl-MEXTS OI' ALOKHUA. 27 8. 12a; — 25 + ^l^^.J=iO ^)x'^ + i)x — 2 .<■■■' — .1-1/ + ,f a - 66 - .. _ 1 + 1^_^1 166» + y/>r- + b a + 2b + EXERCISE LVIII [.,]. ( Pag k 166.) 1. 2. 3. 4. 5. 6. «^. c. a^ I X ab x^ 6' rf' 6»' c 2^ «^ Zbx' ^'' oa' 3a V' 3a6c' «" . r . a;^^. ff-6 a n_3i a a-- + y 2« — 36 a y-x 2« + 3c' a--y' 3^7^ro6' r7~=" g + 6 ff — 6 1 1 1 x-^-l a + b 1/ (« + xy y' a' _ 6 - -» + 1 a' + a6 + 6' «' + tf6 + 6'^' x-' + 2a; + 1 iK_+ji, «_+_7. X + 7f/. a; + 4 a; a + 6 a* — ^•-8' a- X + 7 a; -3' a; - 3«' X + 3 8. 9. 10. ^■'' - 2a; + 1 X + <;' X — a 4(« + 56)' . I_ 7.1,' oo; — by 2x — y — z 4 ^ ? . (-g + l )(3a; — 7) ' 2a;— i' («— l)(7a! + 3)' a — 6 (2£_— 3a)' a;" — 2a''' + 12a; — 18 J , a;* 3 (a; — 3a) {x — 4a) a{x + 8«) 2 (a; -TSa) (a; + ia)' x {x +'7 a) — 2a;' +"x~+ i Soa;' + 1 c --; irredueible. 8 (a; '' 4a V + 2aa;' — i' 1 a« + 6« a; _ 5 . 8. ; •-; ' a X + T) - a' + i' + f/- — ab — be — ca x + y + z a' + 6' + ?'+ 2aF+~26f' +'2m' ^2 in 1 ,, , _ (a- - 1) (a^' + .g"-" + .... + t - » '"• "(rt + 6 + 6')' ^P^''"~(a;-l)l;^-(a;-^+af-''+-.. Vl)! = (^' - 1 ) (-g^' — 1 + af-' — 1 +. ... + a ; — 1) ^ ~ (a; — 1) (af — af-' + af — i^^ + i-" — 1 ) ~~ (x— \y |af~' + 2af-'' + 3af-* + .... + (^*j::_l)i x^ ly I af^' + ^' (a; "+T) + .r"-=' (a-' V a* + 1 ) + • . .7-|-\V ar-'' +J2x^-^ + 3a;"-^ + . . . + (// — 1) "~ «^'~+^"^"(i'T 1) +~^" (a;' +~ar+'l) + ....' "+1 _ last numerator ~" iijr-' + (// — iT-r^^' + (w— 2) af-MTT 77+ 1 * (a + b) {b + e) {n + a) x + 2y + Zz 1 'a; — 3/y — \z fixy (x + y) a' + &' + fi' + aft + 6f' + m 2 (^TTry ~+ 1?) ' 5 (a' +"6' + r' —~iib — be — caj ' EXERCISE LIX [a]. (Page 162.) 3 1 a x + y 12 • a;' a;' a;' a + 6' y 11. 13. 1. 4JajJ.). 2(a-6) _ 26 _ _ ., J» + X ^}l -a + 1 i://K\f i:\Ts OF MAii: nuA. 29 a + b b — '-^ (tb n . a{a-\-x) a(a^ ab. h r>x-i7 X + \: 1: U( 7X 12 J- 2 :vj^b^-a) ab + i '2(rb — 1 SB ab ><• + ni (ff;^ _ !1+ &jf' '. a;* + y/ + aba ry^ '"'>^' ' a be ' 2\ ^^ + ^,,.y + abz «» + 18a; - a6c 27 12a; -^L^+Jll^J/z), a+bx + M' 8. — "*-*l. 26 «y^ 2a;* abx x'-r a'~b^' ^n 9. 11 a' + a;» c — a X- +3^- -28' (a + 6y(6T7-)5 „ 10. H^l+i_). 3(.:^+«0 2a.- -82 X + 4 ia a;" + Tu- -f. 6' (rt - 1) a' - b' a-^-1 11. ?i^2a'^ + ^* ^- — a') 'i\ ' r + i27 *•'- ira+ 72 • 24a- }i(2x + 7) 4x'~i) 3 (1 -f_«a') (w" - 1) (1 --J^) 1 +a -M^- + 4)' a; + a 12. i;j. 14. 15. Sa;" .'C* + x' + 1' Sax x' + a 'Ja )ia ..-j' u'-u:'' a{u~xy u' a Sax a *> (1 - a)- 2a;'^ — 2xu- X' -t X •in + 2. 2a; - 2 (a;" — ax + a") a;=' - rt» 4.r// « + 6 (a-'-' - {x~2)Kx-S)(x'Zri^^ !/■)■ a- — ab + b J (a + X) (/* Vax+^ a ~h a» a' -{■ ab + b at t: 1 -^ X^ X 1 - 4a;''' '^^^^ [&■] 4a« + 6' 18 2a;» «'-27' a;" -8 2. -"-'l-X^ 4. *«•*'' +J^lni04 a; + 3 13 (a- -5) (a; + 5)' ^^"-71' , t ■ :*, ...LIM. ! 30 HINTS AX I) AXSWKIH. .2;: ;c:> flIJ iS ■ . ?i M' II '■ a a; 4- H ^ 2^" — Tu-^v + lO.ry' + 5y» a + 6 + 6' e (a + h — rj(a + r — huh + r — ((V (.r — 1 )\.v +1) (.i -{- lij - 1 ai)io 0. 11. 13. 14. 15. 16, 17. X + '.iy' .r — ()5()1 1 . . 1 «. - Ha'' a' a + b 1 ; 0. 10. 12. x" — a"' {ii — (>) (a ~ 'J 4 {itb — c"'' //"-'^ (("b + a'c + rt^^ c^ S,/'- + 4.r — 3 o^^ ' (.i- — 1) (X + l7T2iTT) r«(ft* + 2ax +iix^) 2x 4 (a* — a;*) ' i*' + ,y 2 _ ,v.3 + X' - X 2{x' + l)(.r' + 1) • 2 \H l'--l 1 (./• + i) {X + 2) {X + '6) ^ 4a'—2(fh ,..,,. , , b- {(/■' — ir + b')' 1. — ,--, no ror trb in muii. iind mcuoiu. ; .. — ,,. • (i^ — b' a — ()'■ m 1 —./■'"' (.*• — a) (X — b) ., (I — X) ( 1 + 2fr + "') • p (>' — <■) {.>' — by -4* y ib—c) {x—di 4. - (1 — a) {\ + (t){\ + -; rr f'-^"; ~; fr~ (r a" ; h + 6. (a + A) -f- (,/_/,) . 4,,-^ ,,-, a' — 9 '•'•"; ^r>; 1. u--^ + ir {X + 4) y t ■ . 1 : - '; + rv/. 9. 2 + ;.'.,•■'; 4)^' .r' e .-<•■ + <'/ + if /^t-' —7. .' — .'i - 1 ; {.r - ay ~ a' ("-•'•)('7+.Ty-' ' a- — (ff, + // ■'• + 1/ ^ -b" + h 'in- \ {^-J)U+^)^ ,. U'^b)(.v~ + a- (a + /,)(a - hf ' (i + l7u\ -^^ 16. 17. X* + .1' + 1 X — ab (.r-ry ^-2c (x + a)(,v-b)(x + by .r + b X Jf~C ' X* w '^■^tXs i 3' V ■Ci, O lU It. 8 i2 ijo (J 1 ' .11 11 '-■Si 'Ni: -I 32 III NTH AND ANSWERS. [&.] 1. 3. 5. 6. r»rt&»\* 1 /r»rt&»Y ix-yf' \ y) X + (I (I x — a' ft 2 1. O^'-tx^ + i) • • (^_f)(,._^5)»• 4. a — 6 2^///' c(/ ^'<^-^>- (mxur- lV''^-i ~ — - , i)iit a; for a m first term of numerator and a b*v(x — 'ia) ^ 8646' for (r in second term ; , -,„• 7. 9. 10. 12. 14. 16. IH. a + h — c ; 1. a;* /nbc^fPeV 6 — — a (j; + a) { x* 4- (» ') 1 ix' + ay (X'' + ax'+a'') ' {x — 4) (Hx — 3) i-y a? ; 1. \x-i/r {a + by ^' 8fl6 ax' ix^ + a") 11. 13. ;e« _ (\x — 8 ; 1. X - 12 (.e-^/)''(a; +3a) {na'^'^y (y - X) ix' + y'^j 67r)a^ (x-(-y{x + M)' 7" 15. 2; 0. 17. ab< {a — i' + 6) {ah + 6^; + ca) pV 1. 2. 2)* + p'q^ + ^^ EXERCISE LXI [«]. (Page 176.) 4a — 3 8a + 1 x — 3 a^ — 6o «' , "l6~ ' ~28" ' ~6~ ■' ~h'~' ' 6^ ' 2 (a — 6) 16 + ^;. 6.x- — 20 "Ix — 5 1 2x — 8 V ' 16-^;' 127+5' aT— 10 ' 'l2.t; + 9' X 1 a+6 . 6+ a 3- — ., ; :: — E ; 1 ; 6' a-b b — a' 2 +x 6^' i2a + 86' i^' + l"' i:li:ments ok alokbka. 33 5. 6. 7. 8. 9. 10. 11. a — ajr—l 1 2ax 2x — 85 x' - 2^' — 8 1 x'Oi-J'HH+V)' x' ''"^'^'• a» + ^« - r?.» 12 o^ + x'-x'' {x - \) (i'lTo] ; a X (a^br a 2ab ; (a + ly ; a''~ax+ x\ _2a6 . 4a6^aM^«)^ (a' + b') (a* + \4a'b' + f,*) a' + 6"^ ' a* + 6a^b^ + b* ' ~2iIbJ^^F:^-m^irr^b(a«-a;»)' «» 2 4-3;C (a» - by ' • 1. 1 ; a + &. . »— y 3 — 3a? 4, ; . » + y 4 + 4* 3. 16a; + lly. 5. 1. 8. 5a! + liy. 2 6 4; I ,(1' 1 i u til NTS AM) ANSWKHS. 8 "^ . » ''^ 7 (!i-'^\ ■ {II '+ fj)' ^ "' 2jr "' a -+ h 10. 2. 11. - j^ or -g»g. I'i. r. 16. Tuku tlio fractions in pairs, thus : (_L_ + _i_\ + / _ 1 _ i\ ^ 1 ,_ g y.v — a s — bf \.» — c .v/ {.s — a) {n — b) s{^N—()' by sul>stitutin^' for !2.v, etc. 17. f/. *-30. Multiply ^ivrn relations out and transpose, a + 6 4- '' + ^/ = abc + abd + bed + tied = abed I +('t('. J t'tu EXERCISE LXII. (Paok 185.) 21. 1. 1 +U- + J''' +.c' +u-*+ L\ 1 + •.U- + U.i' + 27.1'' + H\.t* + .i. 1 — .*• ■]■ .r' — .<■=' + j' - 4. 1 — ;{.<• + t)u'^ — au-' + H U-* — ^ «' a'b u'W (I'b^ a'b* o. h - 7 + -3+ , + - 5- + « iC* a?' x^ .r ;r' .1* a-" ) 10. d; = (f 11. x = ' M - b 12. x = l. •' k + Si^s-r)' •so. 'tc. j vt + <'tc. ll'tc I:LKMK\TS (.!• Al.dKMUA. aa 13. r^'^r''). b {a -f f>) 10. ^ = ^ii!!.±_£). 14. ./• = 17. aJ = 8a — 6 a' + rt!>-rh 1 •"). .*' =: Srtft 1». .r = - 22. u- ^ 1. 2.5. a- = ah 20. 3- = 3. 23. .. ^ _ 1^. '/ + h — r.' 20. .r = J8. .r .-= _ a I ff- 21). .r = _ 31. x = '-ii"l-'" "*" " !!' 'L'^ ''"' " h + ritl ~ • 31. .f=3f/: riju'lil mom. .^li. ho 21 .K. ./• = 40. u' = - 0. 1. .r = - 4. J' = 3. 3i. ./• z= or - H- [b.] j: = — 107. 3. .r r= ^L±^+ 'i±l^ •). X = _ i(r/_^4.,.) /* -- m a 10. ;r = IT). 13. ./;=:-2|. 16. w = — . 2 1 H. x~ ] =0 and 4 20. x=z2. 8. X = — T II. .r==;{. 11. .r=_6. 17. .r=r ± 3 or oo r + rtX + 3 rrr 0. 31. x = a. 0. X = 9. .j=r f//> fn 12. ..z.1^. 1."). .^—7. 19. x = a + h. oo V, 76, or 0. 12. a; = r. X Cix + 51=0; .-. x = {) or — 2^. U- = + r. 15. .f = ^/'■' + />■•' + r'. 05 = a*. { First numerator on eight hand should \ni x— 1.) Take in pairs the fractions with lik(^ numerators ; ?ip {ti — (1) + n.p [a — 6) + mil {h — /•) x = 18. (—7^5 + 49) III (a — c) + //. (b — o) + J) (/• — b) 1 1 ./■" + x — 2 x^ + X ^..[ = X ^ I or X li). Complete the divisions, cancel and transpose X — \ X ~ ) X -'.. = .) >> KOi h: i V\' \ & - - ■ 1 i - •• ll 1 >. li \m or a? — 4 3 X ~ li X — -i 20. X = whence {x—H){x — 4 ) = ; .-. x = 8. Tho value 4 is not admissible. _ bn (q — p) {m — p ) + ap U/ — //) (m — 7i) f^{Q—p) ('« — p) + « (y — n) {/n — n) 21 . X— \m (b ~ (') — 11 (a + c) \ -• ; :n — n) {11^ X + b'\c — (I'-h — ah' — h'^r. 4- ifc) X 00 \ 1 \{x — a){x — h 1 ) {X — a — r)(X —b ■\- r) X~ |f/' (h — <') + h' (r + f,)f -f- (^/"^ + //). f = 0. 2;{. 24. ('/ -'■'( n — (t !> — (j h 71 — 0, x = ~~ p — q X — 2^/ /> 4- c — a 1 + aiiMl. + anal. = 0, whence {X — a — h — r) X = a + b + c. (b-^c- a + anal. + anal.)- =0 ELEMENTS OF ALGEBRA. 37 25. o — aj 1 a' — 6c a + b + c + &Q. + «fee. = 0, or ab + be + m — (a + b + <-)x ^ , , , _ — - -; — ; \- anal. + anal. { + '*). 0; 1. 4. 10. i:{. 10, 'g dozen. ^:S00. 14. 15. EXERCISE LXIV [a]. (Paok 108.) PROBLEMS. 2. |;{<>ooo. ^ ilUI — Vib 5. ! '^ — m 8.. if>{.75. 11. lOOiY, buslK'18. K(ji:ation n>s to (4 — 4) .r + -lO = ; j*=x; /. r.. concli- lons of i»rol>leins are inconsislmt. In fact, urea will abraijs be 45 ft. less, under the j^iven eondition.s ; for \\A\n\f 45 tor H5, the result inj^ e in first lino; 4 times and ets. in seeimd line 18. i:{^ feet and Ifi^ feet. 19. A. !j;2800; B |3900 ; (\ |51:{8 ; T), !j;219(} ; E, |29({G. 20. 1 14. 2. 420 acres. _y(100 + //)^100/> «. I12H0, 7J^. accord ing as 1. 900 j;all(ms. 4. Gain or loss i > 100/> ft -^ _ ' < 100 + 11 r>. B makes 1«40 yds. in 4 m. lUsee. ; (' makes 1700 yds. in 4 m. 82 see. Lot;r=tiniein min. from startlnj; at which A overtakes B, then ' • 17no = 20 + "^ ,1740, /^IJiUmin., I?. 4.^ J "'^^ distance 775^^, yd-\ from start. Similarlv A is found to pass (.> in ;J8| in.; distance 1450^§ yds. Cfom start. ^T vlfc." 38 HINTS AM) AVSWKKS. 6. 12 miles. 7. 5 ^ral. 8. !J7 mileA 10. nutpqr -j- (jn/ipq — //'7>7 — ///^7 — /////7 — iniij)). 11. 4H4. 12. 1, '2, and 53. 13. 12000 s(22m — 21/<) 20 {m — h) 20. Krj^ulai' ra(<' 40 miles, diininishcd rate ;{8^, miles; KiO miles. »)1 001 001 . 07' 2:51. •).) ap — f/// /M — n 23. i. 24. 14172. EXERCISE LXVII. (Paok 212.) 1. .r = 3 3. X = 1 •) !• — r = (5 ; // == - 4. 4. .t = y = 4. .r = 4 ; // rrr — 3. 6. .r = — 2 ; y = 3. !/ 9. .!•= 10 — o = 9. H. ./• = — 20 fl 1 //=-toi. 10. ./• 3tr:t 1 .'/ = '1^5 1212- H *'-5l 3. ,/• = i>. .*• EXERCISE LXVIII. (Paok 213.) 3. •'•-H; y- 1 * — o 4: // = 4. 3 ; // = 2. 4. a; := ." 6. a: = Vr 8. a; = 2 ; .'/ = - •»• .'/-I. 9. ./• :=^ 4 ; // ^ - 3. 10. x = :^; y = \. 3. .rn= r). a* = ;> 7. a'=: EXEF^CIbE LXIX. (PMiK 214.) X — ;> ; // // ^ - 3. .'/ - - 4. 2. a; = 1 4. x= — ^ 0. ./• ^ jf ; //= — !. 4 ' .M «. J'=li{f U ' _ I r. fl fio- H. ./• 10. .<• iro r mile» 99 100 m lies. 24. 14172. m- 7 .'5 12- elkmf:nts of ai/jk liUA. 30 EXERCISE LXX. (P J. x = a + f) (I'AOE 316.) !/ = a~b 1 - // 2. a; = — 3. a^ = ^-^"/. „_"/> a — 6 i^ = c/ - 6 /^r — // •i ' // = -V- my 4. ^^ m' — n' 0. .r 4//- - :{r/- i' = _ a (W - 4//^) x = „ + f,. y = _i. .'i«'-' — -I//' 7. « = // = (t +b a. ^-z^ "* + " ; ^ = /M - n. 9. .C=:V = m 1(1. a? — — - ) 2(2r/6 + 6r— f/r._ca; y EXERCISE LXXI 1. X r,<}; y= lo_ 2 (2^/6 + U-—ai^(7) f"]. (Paop: 218.) 3. .r = 4 ; w = .f =-. ^ ; y = 10. 7. .y-:.S 9. a- = »-=r U. x = 8^ y==12. // =-i. 4. x= |;{, 6. ar = 4 5 //=-41I I »• 2 ' .y^ - 12. 8. .J;=r3; /y=7. 10. X = 4; y = r,. •ii-i X = y = u* — Wt* 14. a- (t~l, l/ = b~ a ."■). .r = 4 A'- 1. •^' -= - 2 ; y = 4. /'^ -f 0///-'/r' -\. ,n* 10. .t = f5; //:zr IH. x^ l/ = -4. ^- ^ = ^sV ; .'y -= 1 1 0, X = ;j : V = 7. »!• f*. 2. ./ := 14 14, 4. u; = 21 ; y = 30. ill » i *:: n.:i a It. o., -J O It , .1 II H llH II •' I ij 40 6. x = — — HINTS AM) ANSWKHS. ab — i a — h (I -(t){\ — 6) ' ^ n — fod /» 7. a; 8. X 10. ar 12. X 13. .c 14. a; 16. X 17. a; i. y= (tl, + 4/> — 2c ^ = ± 9 ; y ^ ± 3. = 9; y = 2. = 8; y = 2. _h -\ f — a — d _ 4]7;c -^(/7/7~ ' ^ ~ 2 (hr^^ad) = -2f; //=-6i. = 6|; 2/ = 8. a" 4- 2f/ + ac — ah + h — c 7i(j — l> + 'id 9. x = (t \ !/ ~ hi .. a h 11. x= — , ; tj— — -.• c + (7 — a — b he — ad) lo. x=t/ = ^. _ b'c — be' _ ac' — a'n ~ ab'—~a'l ' ^ ~ af7— a'b X—. y = 70 , and tho »>qiintinns are inoonfihtent ; thus, put a b a c ,— —/>•, and .•. (/ = ka', b = hb', and suljstituting these values of a and h in ^r.*' + % = <\ wo ^et r/'.r + h'tj = k' 2, which is incoiisistcnt with tiie second given e<[uation. x = if=i%^ i.e. tile ('(juations are mtt ttidependent ; thus, put —=.- = -■— m. Then a = nta', b — mb', c = mc', and a b c' substituting in (1), we get ma'x + inb'f/= ;;,c', which is a multiple of the second given equation. EXERCISE LXXII. (Page 222.) ili 1. .^=3; .y-0. 3. x = ^[ !/= 1. T). a^--|: !/ = l 7. X = y = a + b. 9. .,. = -^\: y = T-Jij. 2. .r-^; y=l. 4. x:= -2: y = I 0. x = ; .'/ = ! b(a'^—(M-) (-((r—bc) 8. .r = -— ; y = . n—b a—c 10. x=y a' +- b' ah a + b ELEMENTS OF ALGKHHA. EXERCISE LXXIII [«]. (Paoe 223.) 7. a; = 4; .y = r); ^- = 6. 9. a; = 1 ; y/ = 3 ; ^ =, ;j 11. a^ = 3 41 8. .r=_ 1; y=_o. 10. ^ z:z _ i»j. , Z/ = 21; ^= -a. 12. u=8. 13. X = Q; y=S; Z=Ul 14. u'=ii; „=_.. , ' ./ », ^-H. 16. a-^-r,; ^ = 9; ^ 17. a.-=_4i, y=-4f, ^ = _ :}^^. =3 -2f. =: -4. 7. z = 5. = -3. = -8. ?/; thus, pivt ituting these ,, e -« + ^U = ,- , uutioii. ndent ; thus, c = .v + mna — y^y;r . _^. _ P»m{arb + bf/r) 15. (.r + y + ^)('^ + ft ^.,.) ^ _ «' + &» + ">"J>'f + fipff-p/nfb t(f i- b + '■)' ^[tJ^o + t- - ia + b) ib + mr + o) + r + // ■ z = (,^.fj^, a' + 6^ + •^a^, (//y — 6f — ca rar^ l! ! .4- 42 1. X 2. X 8. X 4. ;i; 5. « 0. X 7. a; 8. X HINTS AM) AXSWKKS. EXERCISE LXXIV [«]. (Page 328.) = 4; // = J»; z =z n\\ u = 25. = :{ ; // = (); z = ^\ fi-2. = 5 ; // = ;J ; 2 = 1 ; H .— 4. = - •"> 1^ ; // =■ ifiA ; ^ = - •^>t', ; " = i i-i^r. = ;{ ; !/ = 2; z = - A ; tf = 5. = 0; !/ 1; ^ — «) /' = - 4. = — 1 ; // = — 2 ; ^ = — a ; // = — 4 ; / = — 5. = 1 ; y = 2 ; ^ = ;{ ; *< = 4. 1. X = II + h + r ; y — a + b — c; z -= a—h + t-j t — b+e—a 2. x = « ; // = — t ; -?=:{; ;< = 2. y. ./• = jf(a — 2b + r + f/) ; //, ^, and /f by symmetry. 4. X — ^{ + '^ + r — ^^ J (//y — he ._ 9 . :i 2;{ vf-iU' rf—de 25. 16, 20, 42. 24. «.*■ - 7 ' .).r - 4 ' 2j; — 1 EXERCISE LXXVI. (Paok 2:59.) 6. -if, -10; ±(v/3, 0. 11. 15 or 21. (Pa(je 241.) 3. ± 13. 6. 78, 52, 39. 9. 2, 4, 8. 12. a± y^U/" - b). 16. 140, 120. 19. 285, 152. 14. 8, 12, and -40,60. 15. 12 ft. 17. 20 per cent. 18. 960, 280. 20. 4hrs. EIJ:>fi;\TS OF AI.OKItFtA. 40 1. 2. 8. ■4. «. 8. 10. 12. 18. 15. 1. 2. 4. 6. 7. '• -h --iiV.; :{j. -21. 2f/, (f. EXERCISE LXXIX. (PA(iK 250.) - 7. 1 : (;, 2 ;-:,_:{. - 2, - 1 ; - m ± ^{„r + n)- ^ \ni ± y'(m' + 4w)|. 12, -2 : -(>, -20; :?(». —4. ~=»'-1: il--i; f-1. 5. 2f,l; -|±iV-^-<3 ,5 ,, 51±y^lT21 IS' 0; - y- ; -r)±y'2. 4, ^ 2, iV; 2S, -;]. {). 3^. _,, 4 fMr/ — />). (t{a + h) ; — I, //, ^ (,/, _ ,^,_ 5'51, I2« ; (/// + /o^ -- (m - /m'"'. 14. ./ + b, ,i ; a + f», 0. (I ±b; ~ 2f/, 2/> ; :{, (r -^ (0 - a). EXERCISE LXXX [a]. (Paok 257.) •*- Si; -|. 1 -1.4: -2;.. 2^. m), -4; (',±^-04; V. - k- 5. «, 6 ; 6, a. 2, -:U: ,,4-6, "^' 9(f + h [h.] (Page 258.) ^im:. — I)na!/iN(in/ roofs (jeneniUy omitted, 1. ± V-; ± 1. ±2; :i 2, ± '.\. 2. ± V^ia ; ± 2, ± y 10 ; _ •!, _ 6. 3. 2,-^11; 2,21. 4. :5.2'; ±.11. .5. ..0,4. o^ 1. 6- ±y'] + ^(-8±y^>01)[; .r-' + 5 = 2. or-1. 7. (±V2)-4; -2r,, ,/^2; .r' =: A .;./'^ ± ^(„^ _ 4//)}. «• -1, 3; ±^, ±1^11 ; 1. _o. 9- i!-7± v'41},i!-7± ^211: _4,8.3±21. 10. ^{1 ± ^(4,/-8)f; 1,0,4. II. -1; _i _o o,. 1 12. .r='-H+,,"_4 = 0. .*— 2; ;j ; 0, _ 1, _ 1. 13. 1, ^|y>-l±v/(//-'-2/>-a,J; l,iJ2y>-l±^,4y/^-4/>-3)}; 1, ^•'+^-+ 1 =0. ^'' f w ^;: ] 3 ■V V p lU lu O * if -J IB O r V|i 4f) HINTS AND A NSW l: US. 14. -I, ±2; ff, -,,, -?,; 0, J. -I ••"■>• I. 1. i ( - :i ± y.-,) ; ',>, A. _ o ^ ^.,. equation is - 0, ((!• . 'I a 1 7 _ '» 4. a + ',>,., - :. :{. - I. r/ (n + b) Vkv Art, l;{;5. ^- •■"•' - (IciKHiiiiiatorKK 14. Coiuplctc the divisions. traiisiM.sc and divide by ;}. jind U!'' — 1 - U.r — a y+ ;j ~~ = •'• - »i- '4'- = 25, X = 10. 15. 4, 0. 16. (4rt - 5^) -i- (hib, id -2b)-^ Sab. 17. ±i!*A/-l- ±ii1 V-1- IH. I'si' (irsf (|n;inlity for the Hnkiioiru: 7, 1, 4, 4. 19. 1, 8. 20. ±5V±1. ±','V±1. 21. 2. • 22. r. r-i„", +/„.' 2:{. (V.njpicir the divisions, n^dit nirnihrr (•ancd.s, and 1^13 1 _ 1 ^ 1 _ 1 24. Scparat*' flio fat'lor .r + a -\- b .V = , ^. oj) 4 _ , ;j,) „^ _^ \d.] (P.voK 2(i0.) 1- 15, -1. ± ^/— 1 (.s»'<' second <'(iuafion in \b\ 15). 2. Kijuation is [.v + 1 ) u'' - x -\- \ ± a.r) ; if = - 1. i , 1 — « ± y(^/' _ 3f/ - ;{)[. \- t:in) ; H2, etc. to. H. - f - ^. 5. - ({, 4. «■ ;». .!. id .-b V-!<». 1». T;.k..|„;rHluT pairs of lik,. imim.ntt(.rsaiHl ,M,iiali.m iv,|„,,.s 10. —I, -',>, .'{ :t ^/7; ohsrrvr timt on.. Iracti.,!. is ivipr..,,,! of the olhcr; ptit il (•(piai //. then y + ' - '>A ,.(,. .'/ 11. L'sc formula (1). p. |n1, and fa«lor l.y Kx. 7; ,,. lor, ; I'-'- - i ± Vn- l:{. '2 ± ^/e. and r^ - ($.,• + « = fl. H- 1 -/>, ('(Illation reduces to (.«■ +/,-]) L- ^ j- + ^ \ _ ,)_ !•). 7, '^^ ; if ,*■•-• - U.r 4 |,s = //, ecpiation is //-' + 2// = 21. etc. 10. Divide l»y left nieinber and clear, .-. «:{ ( I +.<•) = ({2 { I _.,). c|„. 17. «^1 ±2|/^^'| orO; t'omplt'te the division in left meniher. square, and 1 cancels. IH. \L 10. Sij,Mi before hist fraction in h-ft member shoidd be -; ('(ination n-duces to 7.r — M.t + =r o. 20. Write - for -+. before r>0 in denominator ; combine first pair fractions and .second pair ; factor (hnoniinators and cancel ; tlicn takin;^: .*•' - ."».<• = //. y- + isy — 21 _ o, etc. ^1- ill '^'2. Combine first and last fractions. s(>cond and last I'ut OIK', etc., and 2.1- + 7 is found to be a factor ; then put a' + 7.V =z I/, and rosnltin;^ (-(ination is //- + isy + «?, ^ &*- & t/j f/x 1.0 I.I *"IIIIIM IM 1116 2,2 11112.0 1.8 1.25 1.4 1.6 -* 6" — ► V] <^ /a '<^. e". ei, CM jjt o 7 / M Photographic Sciences Corporation 23 WEST MAIN STREET ^ WEBSTER, NY. M580 (716) 872-4503 f>C iV ^9) V ^ :\ \ %^ %_<^'' K ^ O^ ■'^ ■!\ S il i. ■ # •«:' < o D Hi St.!' m- 5 -4.- -J O; J' i ''' I .1 t( f '< ' \:l.-' 48 HINTS AND ANSWERS. EXERCISE LXXXI [a]. (Page 264.) 1. 11 or— 34, 2. 26 and 19. 3. /y or — 6|. 4. 1 (yr)_t)^/, ^(a-y5)a. 6. 50 coffee, GO raisins. 8. 100. 9. 12. 11. A, 11 miles; B, 10. 13. 25 cts. 16. 7. 18. $90 or $10. 31. $3. 23. 2.414 inches. 14. 4cZ. a dozen. 5. 16^. 7. $240. 10. 11 vases. 12. 3, 4, and 5. 15. 3 and 18. 17. A, 72 miles; B, 54 miles. 19. ^-J-- 20. A, $1800; B^ $1600. 22. ^ { ^(2Ji' - (P) +d\,i{ ^CZh' - d') - dl. lb.] (Page 266.) 1. 3 hours and 5 hours. 2. 36 and 30. 3. 63. 4. - i (a + b)± |/(£^ 4- -xV (« + 6)")- 5. 961. 6. 4200 ; read 780 in question. 7. 14 acres at $75. 8. 10 seconds. 9. 5f miles. 10. 5 miles an hour. 11. 15 miles. *' 12. If a; be cost and « selling price, then a? = ,9 + ; on solving it is seen that 4* cannot be greater than 100 ; see Art. 175 (i). 13. $333.1,, |G66|. 14. 72, 12, 8; Let ic'' = number remaining in .smaller bag after handful is taken ; then a-" is left in larfjrr bag, and .r^ = number in handful, and :v* is number in larger after second lot is taken out; then x* + x^ =^{x^ -\- x^), and aj = 2, etc. . 15. If X represents per cent, then 620 = 82a; + (3790 + 82a-) '* X = 5. 16, Let 2a; = distance, then X X _6 2x 100 X = 22. f -6f les. nd 5. 18. ELEMENTS OF ALGEBRA. 49 17. Let X = rate backwards, 4x = rate forwards, tl.rn dr + ^. ' *• ^' Tff« - 4^^ + ^^,2. 1 ' ^ '"'!•' '>" 'ionr. 1 4a; a; 18. 90. 19. 4900 ; x^ being number „f lin.s, oquati.m is '^'('^- 10)^^ = 3 ^•^«- 4^ ,.,_lO,t, or 00L.^-20u;.2234 ^-100.49.43; x=70. 20. A, half-past 4 o'clock; B, 5 o'clock. \i, 11600. 63. 901. <> t <}. Iiour. on solving bag after , and x^ = rger after + x^), and MOO ; X = 22. 1. X 3. X 5. a; 7. X 9. a; 11. a; 13. X 15. a; 17. X 19. a; 21. a; 23. X EXERCISE LXXXII [a]. (Paok 272.) '^' — i y = i, -If 4, -10^, y = n, -12<- 4. 6. 8. 10. = 1 .1 - ± 20, y = ± 16. = 30, 10, y = lo, 30. = =^' -H, y = 4, -8^. 12. l.f // = 2, 2f 14. -4f, y = 3, -2|. 16. ±7, ±5, y=±ll, ±9. 18. 2, 5, y = 6, 3. 20. 7,1,2^ = 3,9. 22. 3, y = l. 24. [6.J ,f = ± 6, y = a ^ h. *-=3, -2^,// = 8, _8. .*■ = ± 7, y = ± 2. i» = ± 15, y = ± 3. •^■=1. -'^'^ y = -l, -7. .-r = 2, - ^, y = :}, |. a- =10, 11.-,, y = 6, _69. *' = !' -M- .y = -4, S^f. X = 2, 3, y = .5. 4. a; = 5, f. y = 3, -IJ. a- = 2. 5 ■8) y=-7, -I 1. a;= ± 5, y= ± 1. 3 3- ^^ = 0, ±2, y=±^M, ±1. 4" 5. a;=±3, ±iy=^:^^ ^,,^ ^ 7. ^=±2iy=±^. ^^ x= ± 11, y = ± 2. *• = 0, + 3, ^ ^ ^ .5 ^ 9 a— 0, ±x,y = ^i, ±1. 9. ^-±2^, ±l,y=±i, :p3. 10. ^ = ±7, ±3,y=±2, ±6. 11. a;=±l, ±1^, y=±.5, :fi|. 12. a;=±a^y^(,i; + i^^y^^^^^^^^^_^^^^^ 13. a.= ±6,y=±2. 14. a;=±3, ±8,y=±5. 15- ^=±2, ± Vf y=±^, T2v/f 16. a;=: ±4, ±3V3, y=±5, ± ^/3. 17. a; = 7, 4, y = 4, 7. 18. a;= 7, - 5, y 5, - 7. w mr I < V '■ o lU o ., lb UJ ;o o o ' JO r i lU, HI 1 1 t i L 50 19. 21. 2li. 2-}. 27. 29. 1. 3. 5. 7. 8. 9. 10. 12. 14. 15. 10. 18. 19. 20. 21. HINTS A XT) ANSWERS. ^ - ± •'>, ± :^, >/= ± 2, 7. .V. = 4, // = ',\. .(• — 5, 4, // = 4, n. X — 4. // = :}. ./— 1;}, J), /y^j), ly. .*• ^ ;j, 1 , // = 1 , :5. 20. .V 00 't 24. .1 26. A 2H. m. ± r,, ± 4, // := ± 3. 14, 10, i/= H), 14. 4, 2, // = 2, 4. ;j, 2. .// = 2, 3. 7, 4, // = 4, 7. ;j, 2. >/ = 2, 3. iV X EXERCISE LXXXIII [a]. (Pagk 277.) 2, 1, // = - 1. - 2. 2. .r = 4, - 8, y = - 3, ±h T-h >/ = l 0. 4. .r = ± 1, ;/ = ± 7. 10. x= ±.y, i/= ± 2. x = i{n±2b), i/^^(((-f 2h). X = ± (d 4- 6), !]■= ± ((, — h). /= ±?y, ± 7. x= ±4, ±3, )/= ± 3, ± 4. xr^2, -U i/ = \,-~2. 17. a; ==11, 2/ = 9. a- = - 2, - 3, 3 ± ^ y^56, y r= _ 3, - 2, 3 T J V"'^- X = 5, - 1, ^ (± y'41 + 5), y = 1, - 5, ^ ( ± y'41 - 5). Troiit X + !/ as tin; unknown ; x = | \a ± y'lr/'' — 48)^ y = i ]'' =F V^"' - •*''^)h whoro « =. 1 (- ;j ± y'853). « = ^ (9 ± y'- 47, // = ^ (9 T V- 4")- « = 5, - 2, - i ( 1 ± y'41 ), y = 2, - 5. ■» = 17, -(!, ± y(llS)-4, !/ = 'S, -S^,2±i^nS. X = 5, (), 7, 8, y = 8, 7, 6, 5. ^ = ±^^WvW >/=±7x^s*-i's- iiO. a; = 4, 3, // = 2, 4. .f=l, 10. //=10, 1. 28. ;i=3, 2, //^2, 3. a- = 8, 4, /y = 4, 8. ' 30. .r ^ 1, 1 ,V, y = 2, -^. (a'c - ac'f + iab' - a'b) (b'c - be') = 0. ELEMENTS OF A LO KHIIA. 51 J>, 14. k t. 5, •7.) - 3, - 10. = ±5, ± 9. - 5). 153). 118. 4. 1 3 5 6, 7. 8. 9. 10. 12. 14. 15. 16. 17. 18. 19. 30. 21. 32. 33. 34. 35. 26. 1. 3. 5. 7. 9. 11. EXERCISE LXXXIV. (Pace 381.) ^ = 8,3, y = 4, ^=2,8. 2. .r :. ± 4, ± 9, .y = ± 6. etc. x-7, y = ry, ^ = 5. 4. x=l, >/ = ^2, ^ = 3. ^=±tf ^/^i, !J=± ^3, z=±^i y;}. a- =r 4, - 7, // = 3, - 8, ^ =: 6, 2,s, _ 22 i„ text. *= 1, 9, y=. ±A, z = 2, -6. * = 2, - 14, y ^ 3, - 15, ^ = 4, - 16. «=], i/ = -2, 2 = 4. 11. .i-==4, //=-5, B = 7. a; = 2, 7, y = 3, ^ = 7, 2. 13. ■r = i/ = 2=±i^^2. X = 3rt6c -=- («/• 4- ic - r/6), //, 2- syniniotrical. .'i' = ± cr -V- ^(./'^ + h"- + c'^), y, ^ i,y syninu'try. •^- = ± a/K« + 6 - e) (^? + c' - 6) -^ 2 (6 + c - «)} ,y, ^ by syninietry. ' X = (fbe -=- {ab + ae - 6c), y, 5- by symmetry. * = Vl (^^' + a-h) (a -rb-c.)-^(h^c-a)\. Add /i'^ to eacli equation and factor ; x=-n±{n + n) {e + n) ~ {b + u), //, ^ by symmetry. a^=±a^\{b + c--a)-^\{a + b-r)i^a-b^i;)\\ y, ^ by symmetry. ^ = (/^c' + ca + f/6) - «, ; y, ^ by symmetry. .i-==l, 2, 4, y = 2, 4, 1, 2- = 4, 1, 2. a,-^ ± (_ ,M + n + y» -^ ^2im + ;. + y>, ; ^, ^ ;,,. symmetry. X - ± (- ^'- + «/ + ab) -^ ^i2abr) ; y, ^ })y symmetry. ^' - ^.' 01" ± 1 -^ (/' - <0 ; y, z by symmetry. a; = .{/(6V ^ a) ; .y, ^ by symmetry. EXERCISE LXXXV M. (Pag k 385.) HI + V5), 1 (3 + ^5). 2. ^ ± . ^2193, -^:t « ^2193. 36, 16, or -36, -16. 4. ± V'( ^ ±\/( /> -^ J). ^' "1' •^•'- 8. 343, 64. T^' ~^' 10. 36. 34, 17, 51, or - 204, 612, - 306. 13. gn, H. \'l mo C o 5 iD ■ ■ n^f 52 HINTS AND ANSWKUS. 14. 8 ft., 10 ft. IG. «;} ft., 45 ft. 18. ii%, 1% 20. 18, 9-6. 22. 13, 10. 24. 3, 5, 10. Add and subtract the ('(juations. m. 88 ;ds., ."io yds. 17. 20 m., 30 m. 19. 102 from l(>n<,'th, 114 to width. 31. 100 at $T5 each. 23. A 40 at $1.20, B 30 at %\m. 25. 3, 4, 5. [6] 1. Edges (;c, y, 2") are 1, 2, 4 ; a; 4- y 4- ^ = 7, a-' + U' + ^' = 21, a;" + y" + 0' = 73. Cube tirst equation ijy formula H (3), p. 85, and substitute from tiiird, second, and square of tirst. 2. 864. 3. 2, 5, 8. See Ex. LXXXV, 16. Add first two equations and subtract third, then symmetry. 4. 76 ; tiie one digit remainder is of course 9. 5. $2145, ^ years. 6. 4, 7, 10. 7. 290 yds. 8. 48, 10. 9. 8, 9, 10; see Ex. 4, p. 284. 10. 342 ; in last line of problem read 29. 11. 12, 4, 3. 12. 3, 6, 9. See LXXXIV, 19. 13. \\h±^(c-(V)\:^\a±^(e-l?^\ v. )^\a^ ^{c-h'')\ 14. a; = -y/ j ( 1 + fO ( 1 + &) -j- (1 + c) f , y '"^nd z by symmetry. See 4, p. 284. 15. 28 workmen, each 45 lbs., or 36 workmen and each 77 lbs.; X = number of workmen, y lbs. carried each load, 2 number loads in one hour ; then 8,vi/z is whole weight moved, and 7 (a; + 8) (y - 5) z = S.vyz = d(x-S)(y+n) z. 16. — — ^ — = — — ^- , where w = product of ex- a - y^(a* ~Ap) 6 + ^(U' - 4/>) tremes (or means), and = (a^ + 6* — c) -4- 3 (a + h). 17. Of the first, 135, 62; of the second, 182, 57, (yds.). 18. 126. i'j'-^^ ELK.MENTS OF ALfJKimA. 5;i quations. 114 to width. 30 at 11.60. = 73. (1 .substitute Id iir.st two letry. . ich 77 lbs. ; , 2" number noved, and duct of ex- IH. 136. EXEF^CISE LXXXVI [a]. (Page 296.) 1. «'■' — 2.r — 3 = 0. o 2r, 3. 'i2x^-Ul2.v-2lV2(m = 0. 4. x' ~ a'b = 0. r>. jr -q- p ( y/^ _ ',i,^j . , ^;.. _ 2,j^ ^ ,y. . ,,,.,, ^^j.j ^.^ 6. x'-(4u-nh).r + {),r-H)af, + Sb'=:0. 7. 3.141.-,9. 8. Positive for all values of a-, expression = {,v-2^)\ 9. 7Mii|o 10. y>-lV/ + 3/-. 11. See Ex. 1, p. 294. 16. Assume a- = A.;- + By, th.-n, sine.> «, /? are values of x, a" = A.i; + By, and ^ = A^ + By, whence A and B. ' 2. 6»-r7c = 0. 4. i. a^f'^ — (h' ~ 2ae) x + c'' = 0; 2. rA'^-(/y^-2y).T+lr:zO;' S. x^ - ( p' - (J) X + q ( y/-' _ o^^) _ Q . 8. cc'-aa' = 0, 6'c + a'6 z= 0. 4. 579 and IS.I are the roots of the first equation, .579 and - 135 those of the second. 12. 4a6^ + «'c-rmV = 0; let roots of first equation be «, ,? of .second a + m, (i + m • form equations from relations of roots and coefficients and eliminate la. 13. (Right side of first equation should be 1.) Substitute^ for .// in second equation, and apply condition of equal raids to resulting equation in x. V- 3. Max. |. EXERCISE LXXXVII \a\ (Page 300.) !• Min. 4. 2. Min. 4. Min. tV- 5. Min. 7. Min. |. 8 Mill '> n at- T ,r 10. Max. 36 area, Le. line i.s bisected. 1 Min. M"' ^^^SSl^^B III ^m O o '9. iij' ' UJ 18 if K ISlll i! lil 1 1 -"'■ n 1 j:-J - I ! ■'; t." j ..'; .' k - -') ' U] M 54 HINTS AND ANSVVF':HS. [*.] 3. 81. 3. Min. ^(i\ i. c. \'\\w is l)is('et('fl. 4. ^a /y/3, tlie sides are ('(iiial. (5. (d + f>f -r- 4rt6. 7. ^1// miin))('rs bctwoen .', and :{. m. ^ y^i + ^, 11. (I)' — 4«c) H- rt" = {n' — 4tnr) -r- m\ 13. ^> = 6, or «. EXERCISE LXXXVIII [rjj. (Paok 305.) 1. 1. 3. 346\ 3. 5. 4. 1. Cy^a^b; ^a^b^r; ^a' ^b" ^c^\ 7. i. r/^t-"; a-'6-V'; «2?>-^ ; 7(<^6i ; «-'^6-^ r?^6^ 1 1 6 1 )c-' «■ ■h^ %■>' ab(r^' '^^' ^-Vf' «^^^' «-^«'-^' 3. la'V'; 1. 1. (a' - 6')" ; (.i; + .y)*^' • (.r — //)". 3. .r"3 — 4a;2 ; a'ib — a^b" + '^a"b'' ; 1 4 4 oil rt , 4 ao3 4- rt^ — a bi — r/sfts". 1.4 ,- . « 4i^ 14,1(1 _1 1 , _2 4 3 4. a^ft*^ — -Trtfis + o 1 C a ?> 1 • (' ftnc^ 1 «'^6=' 6=* • a'(;' *> J ^«l^6 yy* ^3 q,,a a^b^ 6V'^ ahf if''' O^u 3 y/^" 4 ^^' ELEMKNTS OF AUJKBKA. m •'• ^' 3'.; .',; -^i tI,; 1. 13. 344140(520; 3. 10. 1034^/; n^)Vl. 13. (T.r-O^):}; (ru,_7A)' 14. 0. [c] (Page 306.) Off 34 81a;* 8 16a« 3. a'*h- 5. ^-J; a-6- orY; a^^^'.h^. 7 /«'^'n. .^•'' 493^37 \ J/' "z/«' "^"^"^ • 6. tt''6V; a^6V. EXERCISE LXXXIX [a], (Page 309.) m3 2- 2"'; I; (a- . ^/-'i'jp?. 3. ^U^^ + .;l + 3.r3 + ^U.r^ + ^t + ^. + l. ^^ J a 3 ' ' .7- G. 4r/'^-6'. 4. «%-y^; .-i;^ + y2 7. 2a;--43-9.^- + 6.- + lU-; 4^^_9y^ 8. «=• + «-^ - 2 - at - «-i + o« t._ 9. a;^ + Aa^ij^ _ 4^2^ _ i(. ,,^t ^ ^ ^^_^5 _ ^^^^ 10. 3+2a-U3.c?+.i-» + .,.n. «|.4_^,._ 11. .i;^-^i; x + y+a^ifi^a^fjh, 13. a;l - .rM + ^rfyt _ ^^yl ^ y^ . g^_, _^ ^^_. ^ ^^ 13. nfti + 46^ + 36-ff + Sft-i. "3 il ^ 1.;: < V D o lU o , TO : JO 4t i ' HINTS AM) A\S\VKF{S. 14. u; » + ;c «// !• + .jr'// a + x «//-' + .r »// » + //« ; 15. xt — 4x + lO.ta - \{\x^ + 19 - 1(U~« + KU"^ — 4.t-' + x~^. 16. (^3 — .iM//:t + ^3) (.fS + ;*•"//« + >p) ; 1 _i 17. (J - H) {J + 7) ; (ar" - //f) (:{.<;" + 2//5). 1 8. (,r — 1 ) (.<• — .ra + 1 ) ; (;}.t;3 _ 2yh {2a-^- - '\fA )■ (a\l(r>i + 1) ) ' s a 2 11 11 11 rt» 4- ^^ + '•* — M^6s — b^fi — ('Sai a^ + b^ + ca 30. nb-^ + 1 + «~'6. 21. «5 + 2c/s^^ + 3^^^. [&.] (Page 310.) 1. ; oxpr(\ssion — (ab — «-'6-') {ab + a-^b-') — aft (db-^ + ) + ar'b-' {(r'b + ub-') = ((ib—a-^b-')iab + a-'b-^)-'(ab-' + (r'b) X {ah—(rHr')=(ab — fr'b-') {ab + «-'&-' — wft-' — a-'b)=0. 2. See last (lueslion. 3. For last term read a^, Ans. a^ — 3rt2 + ««. 4. (.f^V")i 5. 2^- (3 + a'?)^{\ ■\- x)\ 6. as — 2ab^ + 3a§6 — 3a'36§ + 6^ in divisor read 2aafti 7. a;* + 2x^ — 8a;'' — 6a; — 1. 1 18 8. aa;' — a*a; = a^x{asx'^ — 1), and second factor of this is con- tained in the product of the other two quantities, .-. L C. M. i=«^a;(aV- 1).- 9. at-' + 1 — ^a-^b] in second term in text read b"^ for 6~'. 10. ^'j ; in first term read a^. , . (a; - ai) (a- + a) 11. ; 1 x — a in second term of numerator read a. 12. '^{x'>f) -r ^{a-^b% ELK.MENTS OF ALliKHKA, o7 i — iab-'+d-'b) 6-' _ rt-'6)=0. d 2a37>i of this is con- ties, b' for b-\ read «. 13. Expression = a" (a' + ^?)i + b' (a? + hi)i = etx). U. Niunerutor = x' (J^" ~ j-'^'")^ neiiorninafor = same faetor x (w" + .z^), 16. a6.r/.; <,ivi,or = r/"' (say) = ./■' x cl% apply Uw IV. 17. xi + (1 _ y„) ^fi,.| ^ „ 18. Expression =. .^ - ,^ _ ,^. _ .^.^ ^ ,,^ ^ __ ^^^ ~ ^'ir' ur' - ^-'y), etc. 19. a^o^H-a; -second remainder is - 5 (««;' + 7,^. ^. ,0) = -0 (,,if + 2) (ai^- + r,), and H. C. F. is «i,,- + o 20. (m-^ _ 1) (« = + 1^ ^^,^„ _ j^ ^ ^,^,„ _ ^^^^^^ _ ^^^^ ^ • 21. 1 - ^^fl-V. 33. . . \x - 1) H- (3«5^ _ 1 . read x^ in first term of numerator • denominator = {M>>x + 1 ) (2mx - 1) ; denominator = J {a + ai) (na + J), 2S. x-,J + ^=^\ 1 ^ 4. 1 / Expression = :^l^LrL?^Z_ll^JZ_^ _ 1 (« - 6; (6 - c-j (a _ e> ~^- 24. (a + 6)^'^-l±-'J^-^*. 2)1171 EXERCISE XC. (Page 313.) 1- ^«-^ 7V(.r^3^; r,^(..y,; «^,,.y^. ^^^3^.^. 2. 27i; 512^ H; (f)!; g^. 58 HINTS AND A NSW Kits. ; ! 4.' o CI U. I:» ,0 i ■ !IK. 1 P ' ; ' ■ ' :. iJ^KM •'>\/''>; '^a/''*; «\/«; i«v^3; ?\)/i2; 7i. 8. 8^2; (5/^'48; 3 ^^o ; 3 ^:i ; 10 y'JJ ; 2; 3^18; 13; ab ^b. »• a\/3' Av'3; A^i«; ^'//^ ; r^v^^r; «'^^ ^au--' - i), 1 (I. •! \/l 50 ; /^;}7r) ; a» {x + 5) ; (.r f //) ^(^- _ //). 11. (.« - r/) ^\ (X + a) (x^ - «»)[ ; x^ (x + //) ; 3 (^/ - 6) ^(rz6). 13. 10 ^'.i, i V^, A \/=^ * V'^' i \/3- 13. 4*, 321 ; si, OS ; 10,000^, lOOOT ; 33^, 33b ; 80», 50^ ; 4n.4n 4n2R 14. 3 \/'i = 34t ; 3 V^ = 18? ; 34a = 57Gi ; 18^ = 5833^ ; and g ^1 = (344,V)i EXERCISE XCI. (Pagk 317.) 1. 2^/2^ 8^5. 3. -13^v'3; llf ^9. 3. (50 ^l] • 80 ^3 ; 24. 4. 6 - 5 y/O ; 6 y'3 + 3 -y/30. lilll. 'Hi J'.! 1^- ^ i ' 1t 7. IV' + a/'^)5 i(7 + 3v^5). H. A ( 17 - 3 yT) ; ^ (16 - 13 >y/3) ; ■j-V O \/U - 13) ; 2u'' - 1 + 3fl .^(a" - 1). 9. 288 *^73 ; sco 4, p. 316. 10. X + !/ + 2 + 2 A^{xij) - 3 ^{xz) - 2 ^{ijz) ; 13^-^ + 4 + \2x ^{x^ + 1). 11. ^x — y/a ; a;^ — (xa)^ — a^ \ a— /y/(rt6) + 6 ; ^(35 -6,^/3) (3 -^3). ELKMKNTS OF A l.(i KHRA. 50 f('.:")> ^12; 71 ? ^{ax' - 1). a - b) ^inh). = 5832^ ; /;{ + ;5 y';j(). 120. 13 Square and transp„s,. radicals, s(,„an. aKain thm {ax + % + rzf = 4 (r//>..y + („•!,. + ^,.,^, IJJ. Ftati.mdizr. 14. jjj,,,,;,. ,> .^ j^,,j 17. ix ^{x' - 1 ) ; 1 -^ , I _ .,-■,. ,,^ ^,^., _^ ^^, 10. o\ rationalize and siil)sf if uir. nator of first fraction and rationali/c, rcsulfin.^ numerator caneeLs denominator of last fraction, etc. so. io|. EXERCISE XCII [a]. (Page 330.) 1. V^+V^- 2. V11 + V3. 3. V10-V6 4. 3 + V2. 5. VH + V.^. 6. V.^,-f Ve '• '^' + ^- «• 2 + V^. 9. 2V^J + ,V5. V7 n. V7-V3. 10. -Y(a/7-V-^. 12. 3 — //3 ; change 13 to 11. 13. 2Vr>-3. 14. 3Vn-V4T. 15. V7-V^2. [k] 1- ^'^ n + V2). 2. ^.5^(1 + V2). 3. ^o ,y, _ ^,). •*• •>+V«. 5. V;^l-7. 6. V1T+ v/t9. 7. \^« (1 + VS). 8. ^2 (V3 ^ V3). 9. 3 ^5 + (5 ^2. to. Y Va - Vh 11. V30 - Vf 12. ^2 + ^yi 13. ^lui + v^). 14. V(«& - «6'o - -v/c^^n 17. ^rm'-/?,^) +,?,. 19. V.i;Tl/ + V^. 16. iW'n-n^n. 18. Y^(a; + //) + y'(,j. _ ^)_ 20. 1 —X + \/r+~27tr^^~^ i- ■! ' V, 1'.: < o MM*' D f0 'J ;.0.:: ;o'' \ ii JO iii ' pv 60 HINTS AND ANSWERS. 21. 1 + .y/2. 23. ^2 (^5 + ^3) ; in text GO under root sign. 23. ^5 - 1. 24. y/2. 20. ^ (^10 + ^2). EXERCISE XCIII [a]. (Page 323.) 3. ^5. 4. 21. 7. rr. 8. 10. 11. 25, TfV? ''^'•'^^l "H ''1 text. ' 14. 4 or — 142. 15. 71 or 4. 18. 2. 19. 2. 22. 3|or 10. 23. -243 or 32. . 20. 1. «' -J- 2. 8. 5. 9. 0. 2. 9. - 1. 10. a. 12. 3. 13. 9. 10. 303. 17. 9. 20. 5. 21. 0. 24. a + h a — b 25. 9. t r. 1. 5. 9. 10. 12. 14. 1. o 1 2. {a - bf [6.] 3. -1. 4. 16. h - 2 ' 2a - in + h). 0. 46. 7. 0. 8. f I. Ill text remove parentlieses from denominator. 27. 11. <« ::!-'"'. 2f? — o 13. ± 3 ; use ^{.r- + 7) as the unknown. 15. —1,2.3,0. 10. ^\{-i(r(/ -b*). I 5i 3. 4. [c] (Page 324.) -2, 4, 0. 12. 0, ± y'3 : simplifies to x \ y'(2 + x) + ^/(2 —x)\ is a factor, etc, 0, (6'" - 4a-) -=- 4a. a (6 — c) -f- 2 ^{bc) ; see Ex. 5, p. 322. ELEMENTS OF ALGEBUA. (H r root sign. •1. • 0. 1 }Xt. I or 4. • - 243 or 32. ')• ^(2 - X) ^- a\^-n-z:-i)- 6. 0, ± y-j. '^- ^- 8. 81 -a. 9. 3a* s- (1 + ««) 10 , ■. ,o n. (1 + 4i'> - U)b' + 4f/ + //)-- (1 + f,,\ •square and use formula (6), p. i8l. !•}• 30. 14. J^^liiZL^-_^'i'' + ^/)'^|^ ( 2{//^ + f/Y j' ■ i<] j<'lz_^*^;l+_2rav). I 2rc= ~ f ' ^'"^J^' ^y formula G (2), p. 85. 18- - tVbV' - i^fi ; divide by right member, and ir,. I /l±^l 3 /l -J., V-xJ lG\l+a'/ 3 1 ^ =1. or^ + _. l,etc. Z/ EXERCISE XCIV [«]. (Page 327.) 1. 4.r'-3.r'^ + 2^--l. 3. 8.f^-12.r*// + 6.^■//-y^ 5. 2 - 3.r - .r^ + o^.3_ 7. 1 + |r, - Irt^ + -jV/^ - ^f^aV 2. 8.c^_l2.i-^ + 6.r~l. 4. .r'' - 2,r7/ + 2 6. X* — 2x^ o^.yi .r-'y + 3^-y^ - y\ [^.] (Page 328) 2. 2-4.r + .i:^ 1. .^;*_.r+ 1. 3. 1 + :}.r _ .i.» n. x'-x'y + 7. 1 +^^-_l^^+ 5^.3 6. a* - »«-> + !,/-■: 1. If. ST-' O 1 1 [C] 5. (a* - c/) -J- (c - 2a'). 3. 10. 6. 6-i-(a). 4. -|. G2 HINTS AND ANSWERS. 7. 8. 8. 6. 9. -i. 10. ± 3a. 11. (m — //)'+ a' = 0. 13. Condition for S(iuaro is