m 
 
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 EACHER'S HAND-BOOK 
 
LIBRARY 
 
 OF THE 
 
 University of California. 
 
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M. 3 Cage's ^atljematital Scries. 
 
 THE .TEACHER'S 
 
 Hand-Book of Algebra ; 
 
 CONTAINING 
 
 / 
 
 METHODS, SOLUTIONS AND EXERCISES 
 
 ILLUSTRATING 
 
 THE liATEST AND BEST TREATMENT OF THE ELEMENTS 
 
 OF ALGEBRA. 
 
 BY 
 
 J. A. McLELLAN, M.A., LL.l)., 
 
 M 
 HIGH SCHOOL INSPECTOE FOR ONTABIO. 
 
 The object of pirre Mafliemati'r<>.ivhicli is mwther vanipfor Alfiehra,istheunfoldivo 
 of tlie laws of the human intelligence.'' — SyiiViiHTEB. 
 
 FIFTH EDITION-REV ^SEO AND ENLARGED. 
 
 W. -T. GAGE & COMPANY, 
 
 TOKONTO AND WINNIPEG. 
 
Entered according to Act of Pt^rliament of Canada in the year 
 1880 by W. J. Gage & Coitpany. in the office or tne iviinisier 
 of Agriculture. 
 
PREFACE. 
 
 This boot — embodyincr the substance of Lectures at Teachers' 
 A.ssociations — has been prepared at the almost unanimous request 
 o^ the teachers of Ontario, who have long felt the need of a work 
 to supplement the elementary text-books in common use. The 
 following are some of its special features : 
 
 It gives a large number of solutions in illustration of the best 
 methods of algebraic resolution and reduction, some of which are 
 not found in any text-book. 
 
 It gives, classified under proper heads and preceded by type- 
 solutions, a exeat number of exercises, many of them illustrating 
 methods and principles which ai'e unaccountably ignored in 
 elementary Algebras. 
 
 It presents these solutions and exercises in such a way that 
 the student not only sees how Algebraic transformations are 
 effected, but also perceives how to form for himself as many 
 additional examples as he may desire. 
 
 It shows the student how simple principles with which he is 
 quite familiar, may be applied to the solution of questions which 
 he has thought beyond their reach. 
 
 It gives complete explanations and illustrations of important 
 topics which are strangely omitted or barely touched upon in the 
 ordinary books, such as the Principle of Symmetry, Theory of 
 Divisors, Factoring, Applications of Horner's Division, &c. 
 
 A few of the exercises are chiefly supplementary to those pro- 
 posed in the text-books, but the intelligent student will find that 
 even these examples have not been selected in the usual appar- 
 ently aimless fashion; he. will recognise that they are really 
 expressions of certain laws ; they are in fact proposed with a view 
 
 l(w220 
 
IV PREFACE. 
 
 to lead liim fr, investigate these laws for himself as soon as he 
 has sufficiently advanced in his course. Nos. 8, 9, 10 and 11 
 afford instances of such exercises. 
 
 Others of the questions proposed are preparatory or interpreta- 
 tion exercises. These might well have been omitted, were it not 
 that they are generally omitted from the text-books and too often 
 neglected by teachers. Practice in the interpretation of a new 
 notation and in expression bv means of it, should always precede 
 Nits use as a symbolism itself subject to operations. Nos. 23 to 
 36 of Ex. iii., and nearly the whole of Ex. xv. may serve for 
 instances. 
 
 By far the greater number of the exercises, are intended for 
 practice in the methods exhibited in the solved examples. As 
 many, as possible of these have been selected for their intrinsic 
 value. They have been gathered from the works of the great 
 masters of analysis, and the student who proceeds to the higher 
 branches of mathematics will meet again with these examples 
 and exercises, and he will find his progress aided by his familiar- 
 ity with them, and will not have to inten-upt his advanced 
 studies to learn processes properly belonging to elementary 
 Algebra. In making this selection, it has been found that the 
 most widely useful transformations are, at the same time, those 
 that best exhibit the methods of reduction here explained, so that 
 they have thus a double advantage. A great part of the exercises 
 have, of necessity, been prepared specially for this work. 
 
 Articles and exercises have been prepared on the theory of 
 substitutions, on Elimination, &c., but it has finally been decided 
 to hold these over for Pt. ii,, which will probably appear if the 
 prSsent work be favorably received. 
 
CONTENTS. 
 
 Chapter I. — Sobstitution, Horner's Division. &c. 
 
 PAGE. 
 
 Sect. 1. — Nnmeiical and Literal Substitution 1 
 
 Sect 2.— Fundamental Formulas and their Applications 10 
 
 Sect. 3. — Horner's Methods of Multiplication and Division, and their 
 
 Applications 21 
 
 Chapter II. — Principle of Symmetry, &c. 
 
 Sect. 1.— The Principle of Symmetry and its Applications 33 
 
 Sect. 2. — The Theory of Divisors and its AppUeations..- 39 
 
 Chapter in. — Factoring. 
 
 Sect. 1. — Direct Application of the Fundamental Formulas 62 
 
 Sect. 2. — Extended Application of the Formulas 71 
 
 Sect. 3.— Factoring by Parts 79 
 
 Sect. 4. — Application of the Theory of Divisors 83 
 
 Sbct. 0. — Factoring a Polynome by Trial Divisors 90 
 
 Chapter IV. — Measures and Multiples, &c. 
 
 Sect. 1. — Division, Measures and Multiples 101 
 
 Sect. 2. — Fractions IO9 
 
 Sect. d. — Ratios 122 
 
 Sect. 4. — Complete Squares, &o 130 
 
 Chapter V. 
 Simple Equations op One Unknown Quantity 138 
 
 Preliminary Equations. Resolution by Factors. Fractional Equa- 
 tions. Application of Ratios. Equations involving Suids, 
 Higher Equations, Sec. 
 
 Chapter VI. 
 Simultaneous Equations I70 
 
 Equations of Two Unknown Quantities. Systems of Equations. 
 Application of Symmetry. Equations of Three Unknowns. 
 Systems of Equations. 
 
 Chapter VQ. 
 Examination Papers 207 
 
:ir.R^R>t^ 
 
 CHAPTER I. 
 Bection I. — Substitution. 
 
 Exercise i. 
 1. If a = 1, 6 = 2, c = 3, rf = 4, a; = 9, ?/ = 8, find the 
 value of the following expressions : — 
 
 l_a_(l-l-a;)}. 
 a-{x-y)-{b-c){d-a)-{y-b){x + c). 
 x - !j^y - {y — a)\d + c(h-- c)v']■ 
 {x+d){y+b+c)+{x-d){a—b-(^)+(y+d](a-x -dy 
 {d-x)^ + {c + v^* 
 
 la-b){c^ -b-'x) -{c-d){b^ -a'-'x) + {d-b-c){d^ -f'*'^ 
 d — ad + c (\d + b 
 d + a d—G d — b 
 
 2. If a = 3, 6 = - 4, c = - 9, and 2s' = a + b + c, find the 
 value of the following expressions : — 
 
 s{s — a){s — b){s-c). 
 ««H-(5-a)2+(s-5)^+(s-c)2. 
 «« - (s-a){s — h) - {s — b){s -c) — (s — c)(s— a). 
 2{s — a)[s — b){s — c)-j-a{s-b){s-c) -\-b{s-c){s~a)+c{s- a){s-b). 
 
 8. If ffi = 2, 6 = - 3, c = 1, a; = 4^, find the value of the 
 following expressions : — 
 
 a2_62 „2+/,2 (a-^,)2 (a-h)^ * 
 
 c^^Tb^' ~a^b^' (a + by' (a + by' 
 
 a^^ab + b^ a^-b^ a; (2x-3 3«-l ).x-l 
 
 a'^-ab^b^ a^-b^ 2 
 {a + b)\{a + by-c^\ ^ 
 
 Ab^c^-{a^-b^-c^)^' 
 a'{b - g) Vb''{c-a)^ ",^{a-b) ^ 
 - Ja-b)(b-G){c-a) 
 
2 SUBSTITUTION. 
 
 \ 
 
 4. If a = G. ^ = 5, e == - 4, rf = — 3, find tlie value of tin 
 following expressions : — 
 
 y{b-' +ac)+ y(c2— 2ac), yib^+ac+ ^(c^— 2«c)}. 
 
 5. If a; = 3, i/ = 4, 2; = 0, find the value of: — 
 
 {3a;-v/(a;3+?/2)}2{2a;+v/fa:2+2/2+2)}. 
 
 G. Calculate the values of {x+y+z)^-^{x^+y^ + z^, ^^^^ 
 
 xyz 
 
 (a) 9^=1, y = 2, z = H. 
 
 (b) x = %y = 3,z = 4:. 
 
 (c) a; = 3, 2/ = 4, 2 = 5. 
 (J) a: =10, .v = ll, 2=12. 
 
 7. Givcu x=S, y = 4:, z= —5, calculate the values of 
 
 {x+y+x}^ -S{x+y+z) (xy+yz+zx). 
 x^{y+z)-{-y^{z+x)+z^{x+y) + 2xyz. 
 x'^[y-z)-{:ij^z-x}+z^x-y). 
 (5«-4z)2+9(4a;-2)2-(18a;-5«)2. 
 
 {Sx + Ay + 5z)^ + {Ax + 8y + 12z)Z-(5x+5y + lSz)*, 
 
 8. If s = a+ 5 +c, find the value of 
 
 (2s_a)2 + (2s-ft)2-(2s+c)2, given 
 (1) a = S, 6 = 4, c = o, (2) a = 21, 6 = 20, c = 29, 
 
 (3) a=119, 6 = 120, c = 169, (4) a = S, b= -4, c = 5, 
 (5) ^=5, 6 = 12, c=-13. 
 
 9. If a = l, 6 = 3, c = 5, f? = 7, f = 9,,/"=ll, prove that 
 
 <i + 6 + c + (i + e+/= 
 
 2 
 
 1+1+1+1+1=1(1-1 
 
 ab be cd cle ef 2 \ a / 
 
 A+ V + l_+±=l/l-l 
 
 abc bed cd^ def 4 i fl6 et 
 
 J_+J^+_l.=l(l- M. 
 
 o6';i 6cde cdef 6 \a6p t/(// 
 
SUBSTITUTION. 
 
 a2^h2.\.c2_ab — bc — ca = h^+c^+fl^ — bc — ed — db = 
 t.2 4. ,f2 _|_ ^2 _ cd - de -ec = d^+e^ -\-P -de- ef-fd. 
 10. lia=l,h = % 6 = 3, d = A, e=o,f=&,(j = l, prove that 
 a+b + c = U-d, a-{-b + c-\-d = ^de, 
 a^b+c+cl+e= W, a + b+c+d-\-e+f= ^fg, 
 
 aa^fc3+c.3= ^4l+^), a2_{.b3 4.^2+^3=: 
 
 de{d±^)^ «3+^,3 +,3+^3 +,3 = ^0 
 «3 +i2 +,2 +,/3 ^.,2 + /^ = M+'/J. 
 
 a3_|_i3 4.^3 4-^3 = (« + J4.(.+ rf)8, 
 
 a^-\.b^4.rS + d^+e^-{-p = {a+b+c + d-\-t+f)» 
 A , lA . A. cd(c+d)(c^d—l) 
 6c(o+c) 
 
 a4 4-64 4.^4 4_J4 = 
 
 6ie((i+e)(crfe— 1) 
 
 ^44.,44.,4+J4 + ,4 = €Ai-^i^), 
 
 ,4 + ,4+,44.d4+,44./4= ^(Z+^) (^^^tI). 
 c2+t^2=e3^ C3+J3 4.,3=y3 
 
 11. Assume any numerical values for x, y, and z, and calculate 
 the values of the following expressions : — 
 
 (a;5 - 10:c3 4. 5a;j 2 + (5a;4 - 10a;2 + 1)3 - (a;3 + 1) 5 . 
 
 (a,4.i)3_2(x+5)3-(a;+0)34-2(a;+ll)3 + (x+12)3-(a;+16)3. 
 
 (^2 _^3;3 4.(2x^)2 _ ^^2 4.^2)2 
 (a;3_3x^3)34.(^);c32/-^3)3_(a;3 4.^2)3. 
 
 (3x3 +4a;i/+2/2)3 + (4a;3 +2a;(/)2 - (5a;S+4a;.'/+i/2)2 
 (a;_2/)34-(v-5)34-(5-a;)3-3ia;-2/)(2/-5;(«-aj). 
 Art. I. If X = any number, aS, for example, 3, then x- 
 (which = x.a;) - 3*, x^ (which = x.x'^) = ?..(;2, a;* (which = x.x^) = 
 Bx-3, &c. Oi- 3 = x, 3j; = a;2, 3a;3 =0;^, '6x'^ = x^, &c. Hence prob- 
 
4 SUBSTITUTION. 
 
 lems like the follomng may be solved like ordinary arithmetical 
 problems in " Keduction Descending." 
 
 Examples. 
 
 1. Find the value oi x^ — '2.x — 9 when x^H. 
 
 x^-2x-9 
 5 
 
 5x 
 
 -2x 
 
 3» 
 5 
 
 15 Explanation, 
 
 -9 x^^6x, 
 
 .-. a?2-2a; = 8a;=15, and 
 
 6 .-. a;2-2a;-9=:15-9 = 6. 
 
 2. Find the value of a;* — *^ — ^x"^ —dx—b when x~'^ 
 
 x'^—x^ — 4.x'^—Sx — 5 
 3 
 
 p^ 3x3 
 
 Tj 2a;3 
 
 3 
 
 Pa ^*^ 
 
 -4a;3 
 
 r„ 2ar2 
 
 3 .-. x4-a;3-4j;«-3a:-6 = 4 
 
 — if ic = 3. 
 
 7>s ^-^ 
 
 -3a; 
 
 Tj 3a! 
 
 3 
 
 P4 9 
 
 -5 
 
 % *• 
 
SUBSTITUTION, 
 
 Explanation. 
 
 :. x^—x'^ = 2x^ = 6x^ , 
 ,•. x^-x^ — ix'^ —2x^ — 6x, 
 :. x'^-x'^-ix^-Sx = 3x^-9, 
 :. cc^ — x^ — 4a;2 — 3a; — 5 = 4. 
 8. Find the value oi 2x^ + V2x^ 4-6a;3 -12x+ 10, 
 Using coefficients only, we have 
 
 2+12+6-12+10 
 -5 
 
 Pi •• 
 
 . -10 
 
 + 12 
 
 r, 
 
 + 2 
 
 - 5 
 
 Va ... 
 
 -10 
 
 ^2 ■•• 
 
 + 6 
 
 Vm. ... 
 
 - 4 
 
 '9 ••' 
 
 - 5 
 
 1t>, ... 
 
 20 
 
 r3 
 
 -12 
 
 fg ... 
 
 8 
 
 
 -6 
 
 Pa ... 
 
 -40 
 
 r4 
 
 +10 
 
 r. ... 
 
 -30 
 
 the quantity = —30 if ar= -5. 
 Art. II. If the coefficients, and aiso the values of x are small 
 numbers, much of the above may be done mentally, and the work 
 will then be very compact. Thus, performing mentally the mul- 
 tiplications and additions (or subtractions) of the coefficients, 
 and merely recording the partial reductions r^, r^, r^, and the, 
 result r^, the last example would appear as follows : — 
 
6 
 
 SUBSTITUTION. 
 
 -5)2 4-12 +6 -12 +10 
 2 
 -4 
 8 
 
 Art. III. In the above examples, the coefficients are "brought 
 iown" and written below the wodncts p^, p^, p^, p^, and are 
 added or subtracted, as the case may require, to get the partial 
 reductions r^, r^, r^, and the result r^. Instead of thus " bring- 
 ing down " the coefficients, we may " carry up " the products j9^, 
 P2' Ps' Pv writing tliem beneath their corresponding coefficients, 
 and thus get r^, r^, r^, r^ in a third (horizontal) line. Arranged 
 in this way Ex. 2 will appear 
 
 1 -1 _4 _3 _5 
 + 3 +6 +6 +9 
 
 1 +2 +2 +3; 4; 
 and Ex. 3 will appear 
 
 2 +12 +6 -12 +10 
 -10 -10 +20 -40 
 
 -5 
 
 2 +2 -4 +8; -30 
 Comparing these arrangements with those first given (Ex. 2 
 and 3), it will be seen that they are figure for figure the same, 
 except that the multiplier is not repeated. 
 
 Art. IV. When there are several figures in the value of a;, 
 they may be arranged in a column, and each figure used sepa- 
 rately, as in common multiplication. Where only approximate 
 values are required, *• contracted multipHcation " may be used. 
 
 4. Find the value of 3a;5-lG0a;4 + 344a;3_^700a;3-1910a;+ 
 1200, given a;= 51. 
 
 3 -160 +844 +700 -1910 +1200 
 
 1 3-7-18 37 -23 
 
 50 150 -350 -650 1850 -1150 
 
 •7 -13 +37 
 result is 27. 
 
 -23; +27 
 
SUBSTITUTION. 
 
 5. G'ven »;= 1-1S3, find the value of CAx*-lUx+4:5 correct to 
 three decimal places. 
 
 64 -144 +45 
 
 64 75-712 89-5673 -38-0419 
 
 6-4 7-5712 8-9567 -3-8042 
 
 5-12 6-0570 7-1654 -3.0434 
 
 •192 -2271 -2687 --1141 
 
 1 
 1 
 
 8 
 8 
 
 64, 75-712, 89-5673, -88-0419, 
 .'. result is —-004. 
 
 •0036. 
 
 1. 
 
 2. 
 
 3. 
 
 4. 
 5. 
 
 Exercise ii. 
 Find the value of 
 
 xi-Ux^-Ux^-lSx+n, fora;=12. 
 xi + r)0x^-lGx^-16x-61, for ic= -17.' 
 2x4 + 249a;3-125a;2-|.l00, fora:= -125. 
 2.^3 _478a;3_ 234a;- 711, for a; = 200. 
 x'^—iix'^-8, for a; = 4. 
 
 6. a;6-515a;S-3127a;4+525a;3-2090a;2+315Ga;- 15792, for a: 
 = 521. 
 
 7. 2a;5+401a;4-199a;3 + 390a;3_602a:+211, for «= -201. 
 
 8. 1000x4 - 81a;, for a; = •!. 
 
 9. 99a;4 + 117a;3-257a;2-325a;-50,- fora;=lf. 
 
 10. 5a;^+497a;4 + 200a;3 + 19Sa;3- 218a;- 2000, fora;=-99. 
 
 11. 5a;5-620a;4-1030a;3 + 1045a;2-4120a:+9000, fora; = 205. 
 Calculate, correct to three places of decimals, — 
 
 12. a;3 4-3a;2-18a;-38 for a; = 3-58443, for a; =- 3-77931, and 
 for a; = -2-80512. 
 
 13. 7/4- 142/2 +1/+ 38 for t/ = 3-13131, for y= -1-84813, and 
 for ?/= -3-28319. 
 
 Exercise iii. 
 What do the following expressions become (1) when x = a, (2) 
 
 when x= -a? 
 
 1. a;4 -4ra3 + 6fl2a;3_4fl3a;-l-rt,4, 
 
 2. y\x^-ax+a^). 3. y(x^ + 2ax+a^). 
 4. (a;2-|-aa; + rt3)3-(a;2-ax- + r/2)3. 
 
 If a; = 1/ = 3 = «,^nd the value of the following expressions r 
 
8 SUBSTITUTION. 
 
 5. {x-y) (y-z) (z-x). 
 
 6. (x-i-y)^ {y + z-a) (x+z-a). 
 
 7. x{y + z) (y2 +z2 -x^) +^ ^^, + x] {z^ +x^ -y^-)+z{x ^y) (x^ -^ 
 
 8. -^ + JL + ^_. 
 y+z x+z x+y 
 
 Find the value of 
 
 n ■^ , X 1 abc 
 
 9. — + — .wtienjc= 
 
 a b a-{-b 
 
 10. + — + — -, when x= — (a-b+c), 
 
 a[b — x) b[c — x) a{x — c) a 
 
 11. ^+ -^-, wheua.= ^(Ll«)_. 
 
 a b — a b[b+a) 
 
 12. (a + x) {b+x)-n{b + c)+x^, when a; = —. 
 
 b 
 
 13. bx-\-cy-\-az, when x = b-\-c — a, y = c + a — b, z = 'i-^-b -c. 
 
 14. <l±^l±b^ - __1^ _, when a:= -a. 
 
 a(l+6) —bx a-2bx 
 
 15. — — - —! —, when a; =*(/;- ^0- 
 
 \x+bj x — a — 2,b 
 
 16. (p — q) {x+2r) + {r — x) (p+q), when a; = ^ ^ ^^ ~ ^ ■ ' ■. 
 
 17. ffl3(6-c)-f62(c-a) + c2(a-6), when a- 6 = 0. 
 
 18. (a-\-b + c) (6c+m+rt6) - (rt+6) (6-f-c) (e+a), when rt= — 6. 
 
 19. (a+6 + c)3-(a3+63_^c3), whena+6 = 0. 
 
 20. {x+y +z)'^ - (x+y)'^ - {y+z^ - {z+x}'^ -\-x'^ +y* +z*, when 
 x-i-y-r-z = 0. 
 
 21. a3(c-62) + 53(a_c2)_}.c3(j_a2)+a6c(a&c-l), when6-a» 
 = 0. 
 
 22. aW«_!+!iV + 6^/^^l±lT. when a^ +6^=0. 
 
 23. Express in words the fact that 
 
 (a-&)2=a2_2rt6+62. 
 
 24. Express algebraically the fact " that the snra of two quan- 
 tities multiplied by their difference is equal to tjie difference of 
 the squares of the numbers." 
 
SUBSTITUTIOSI. 9 
 
 • 25. The firea of the walls of a room is equal to the height mul- 
 tiplied by twice the sum of the length and breadth : what are the 
 areas of the walls in the following cases : 
 
 (1) leuglh I, height h, breadth h. 
 
 (2) height x, length b feet more than the height, and breadth 
 6 feet less than the height. 
 
 26. Express in words the statement that 
 
 {x-^-a) {x+h)=x^-^{a+b)z+ab. 
 
 27. Express in symbols the statement that " the square of the 
 sum of two quantities exceeds the sum of their squares by twice 
 their product." 
 
 28. Express in words the algebraic statement, 
 
 (x+y)^ =x^ +y^ + Sxij{x-\-y). 
 
 29. Express algebraically the fact that "the cube of the differ- 
 ence of two quantities is equal to the difference of the cubes of 
 the quantities diminished by three times the product of the 
 quantities multiplied by their difference," 
 
 30. If the sum of the cubes of two quantities be divided by 
 the sum of the quantities, the quotient is equal to tl: ^ -square of 
 their difference increased by their product ; express this algebrai- 
 cally. 
 
 81. Express in words the following algebraic statement: 
 
 ""Lzyl^ix+yy-xy. 
 x-y 
 
 32. The square on the diagonal of a cube is equal to three 
 times the square on the edge ; express this in symbols, using 
 I for length of the edge, and d for length of the diagonal. 
 
 83. Express in symbols that " the length of the edge of the 
 greatest cube that can be cut from a sphere is equal to the square 
 root of one-third the square of the diameter." 
 
 34. Express in symbols that any "rectangle is half the rectan- 
 gle eoutained by the diagonals of the squares upon two adjacent 
 sides." [The square on the diagonal of a square is double the 
 square on a side.] 
 
 85. The area of a ckcle is equal to x multiplied into the square 
 
10 SUBSTITUTION. 
 
 of the radius ; express this in symbols. Also express in symbols 
 the area of the ring between two concentric circles. 
 
 36. The volume of a cylinder is equal to product of its height 
 into the area of the base, that of a cone is one-third of this, and 
 that of a sphere is two-thiids of the volume of the cii-cumscribing 
 cylinder ; express these facts in symbols, using h for the height 
 of the cylinder, and r for the radius of its base. 
 
 Exercise iv. 
 Perform the additions in the following cases : 
 
 1. {b-a)x+{c-b)y, and {a+b)z+{h+c)y. 
 
 2. ax-lnj, {a — b)x-{a+b)y, and {a + b)x-{b—a)y, 
 
 3. (y~z)a^+iz-x)ab + {x-y)b^, and {x-y)a^-{z -y)ab-{x 
 
 4. ax+by + rz, bx+cy+nz, an."c.>t4-.''.'y + &z. 
 
 5. (a+b)x^+{b+c)y^+{a+c)z^, {b + c^x^ +{a+c)y^ + {a + b)z9, 
 {a+c)x^ + {a + b)y^ + {b+c)^^, and-(a+/;+c) {x^+y^+z^). 
 
 6. x(a-b)2 +y{b-c)2-\.z{c-a)*, y(^a-b)^ +z{b-c)^+z{c- 
 ^)^, and z{a - b)^+x{b - c)3 +y(c-a)^, 
 
 7. {a-b)x^+{b-c)y^+{c-a)z^,{b-c)x^ + {c-a)y^+{a-b)z^, 
 and [c-a)x^ + in-b)y^+(b-c)z^. 
 
 8. {a-^b)x + {b+c)y-{c + a)z, {b + r,)z + {G + a)x-{n + b)y, and 
 (a+c)y+(a + b)z-{b + c)x. 
 
 9. rt3-3rt6-^^/;2, 263-363+C3. ab-^,b^+b^, and 2a&-^?;». 
 
 10. ax^-nbx'', -Qaaf+lbaf, and - 8bx" + I0ax'' . 
 
 11. What will {ax-by + cz)-\-{bx + cy-((z) -{cx + mj+bz) be- 
 come when X - y - z = l ? 
 
 Section II. — Funovmental Formulas and theib Appcication. 
 
 4. By Multiph cation we get 
 
 {x + r) (x + s)=x^ + {r + s) x + rs A. 
 
 (x-hr){x-ts){x + t) = x'' + {r + s + t)x'' + {rs-tst + tr)x + rst B, 
 
 From A we immediately get 
 (a; -}-j/) 2=^2 +2x^+2/2 [1] 
 
FUNDAMENTAL FORMULAS. 11 
 
 {x+7j + z)^=x^+2xy + 2xz + y^ + 2yz t z^ [2] 
 
 (2«)^ = 2rt2 + 2 nab [3] 
 
 {x+y){x — y)^x'-^—y^ [4] 
 
 From B we derive 
 
 {x±y)^ ^x^±8x^y + Sxy^±ir [6] 
 
 = x^±/j^±3xij {x±y) [6] 
 
 {x + y4 »)^=a;2 +y'-^+z^ +'Sx-{y+z) + 8y^{z i- x) i-Sz^{x + y) 
 
 + 6xyz • [7J 
 
 = x^+y^ + z^ + 3 {x + y) (y^-z) (z + x) [8] 
 
 = x^ +y^+z^ +8 {x+y + z) {xy -\- yz + zx) — Sxyz... [9] 
 
 (2a)3^2a3 + 8^a^b -\- Q^ahc [10] 
 
 [The symbol £ means the sum of all such terms as] 
 Formula [1] . — Examples. 
 
 1. We have at once {x -\- y)'^ + (^ — y)'^ = 2(^2 _j_ ^2^^ a^^j 
 {x + yY —{x — yY=4.xy. 
 
 2. (a + 6 + c + d) '^ +{a — h — c 4- d) '^ may be written 
 
 {{n + d) + (6 + c)}-^ -h {{a + d) — (6 + c)}2, which (Ex. 1; == 
 . 2{('< + J)2i-(i + f)^} ; similarly 
 
 l^a — h -f c— (/)2+ [a + b—c — d)'^ = {{a — d) — {b-c)}-^-^ 
 
 .-. (a+ 6 + C+ tf)2 + [a — h — G -ir-dY -f {o. — h + c — d^Jr 
 (a + 6-c-(/)3 = 2{(a + J)-^ + (6+c)s+(a-c/)2 + (/, _c)a} ^ 
 (again by Ex. 1) 4(«3+63+,.2_i.,i2). 
 
 3. Simplify (« + ?j-fc)s - 2{a+b+c)c + c^ ; 
 
 This is the square of a binomial of which the first term is 
 ^aJrb-^c) and the second -c; the given quantity .•. = 
 {(«+6+c)-c[- = (a 
 
12 FUNDAMENTAL FORMULAS. 
 
 4. Simplify {a+b)* -^a'-i + b^) [a-i-b)^ + 2{a* + h^). 
 
 By Ex. 1. 2[aA + b't)=:{ct2^b^)^ + {a» -b^)^ ; :. given quan- 
 
 tity = (« + 6)4- 2[U^ + //2) (« + 6)2 4- („2 +02)2 + („2 _ ^>2^a = 
 {{a + 6)2 _^a3+62j}2 + ^,i2_63)2=rt4+2«2y3^ /;i = (a2 +62j2. 
 
 Exercise v. 
 
 1. (a;+3//2)2 + (a; -32/3)3, |i,,,3 + 3J2)2 _ ^,,2 _ 37,2)2. 
 
 2. Shew that {mx-{-n;j)2 + {nx-my)^ = {m^-^n^) [x^ + i/^). 
 B. " " {mx — ny)'^—{Hx — iny)^ = \^m''^—n^){x^-i/^). 
 4. Simplify ;./ + 3ij2_^2(«H-36) ((i-^jH-(a-&)3) {a-b\^. 
 
 6. " (a;+ 3)3^ (x -1-4)2 -(a;+ 6)3, and (4^3-2^2^2 - 
 
 (i(/2+2a;3)2. 
 
 6. Simplify (a + 6+c)2 + (6 + (;)»-2(6+c) {a + b+c) 
 
 7. Shew that ['ix + by)^ + {cx-\-dyY-^{ay - bxY -\- [cy - dx)^ = 
 (rt2 + i2 + c3+(Z3j (^3.34.^3^. 
 
 8." Simplify (a;-3y2)3^.(3^2 _^)3 _ 2(3a;3-^) (a:-3v/2). 
 
 9. " (a;3-i-a;^-^2)3_(^3_a.^„^3^3^and(l + 2a;+4a;3)2 
 -|-(l-2a;+4x2)«. 
 
 10. If « + />= - Jc, shew that (2(t-6)2 + (26 -c)3 + (2c-a)2 + 
 2(2«-6) (26-c) + 2(2Z>-c) (2c-«)-|-2(2c-a) (2«-?>) = /^jc2. 
 
 11. Simplify- 2 (ff- 6) 2 -(a -26) 2; (a^+^ah-^b^)^ -{a^-\-b^y. 
 
 12. " ((/ + M2-(6-H6-)3 + (c + t/)2-(^+«)2. 
 
 13. " (^a;-2/)3+ai/-«)2+a^-a;)2+2(ia:-2/) (Az-cr^ 
 
 + 2(A2/-^)(i-^-^) + 2Q^-2/)(i2/-=)- 
 
 14. Prove that [x- yf + {y -zf +{z-xY = '2{x-y) {z- y) + 
 
 ^{y-x) {z-x) + ''l{z-y) {z-x). 
 
 15. Simplify (l + o;)* -2(1 +a;2) (1+^)3 + 2(1 +x4). 
 
 16. " (a;-l-.(/+,^)2-(.c+?/-z)3-(^+2-a;)2-(z+a;-y)2. 
 
 17. " (a;-2y+3z)3 + (3z-27/)3+2(a;-2?/ + 3z)i2?/-32). 
 
 18. " (r/2 4.62-,,. 3)3 _f.(c2_ 62)2 +2(/;2_^.2)(rt2_(.62_c2), 
 
 19. " {x+yy + {x-y)^-\x-y}-[x + y)». 
 
FUNDAMENTAL FORMTJIiAS. 18 
 
 20. " {5a+3b)^ + 16{da+by^-{lSa + 5b)9. 
 
 21. Shew that (3a-ft)2+(3fe-6)34-(3c-a)3-2(i-3fl)(36-c) 
 + 2{'db-c){Sc-a)-2{a-Si.){3u-b)-i{a + b + c)2=0. 
 
 22. If z2= 2x7/, prove that (2x3 -2/2)2 + (22 -2^ 3)2 +(a;3-2z2)3 
 
 -2(2x3-2/2)(22-2^2)^_2(x2-2z2)(23_2j/3)_ 
 2(x3 - 222) (2x2 -1/2) = (x+2/)*. 
 
 23. Simplify (l+x+x3+x3)2 + (l-x-x2+x3)2 + 
 
 (1-X + X3-X3)2 + (1+X-X2-X3)2. 
 
 24. Simplify {ax-{-by)^-2{a^x^ + b^y'^) (ax+by)^ + 
 
 2{a^x^ + bhj^). 
 
 Formulas [2} and [3] . — Examples. 
 
 1. (l-2x + 3x3;2 = l_4x+6x2 
 
 + 4x3-12x3 
 
 +9x* 
 
 = l-4x+l(ix-- 12x3 +9x4. 
 
 2. (ah + bc + ra)^=a^h^-\-2ab^e + 2a^bc-{-b^c^-{-^abc^ + c^a^ = 
 a2b^+b^c^+c^a^+2abc{a + b + c). 
 
 3. \{x + y)^+x^ + y^}^^{x+ij)^ + 2(xi-y)^{x2+l/^)-\-x^ + 2x^ 
 
 y2^^4 = (a. + j^)4 + (^|.y)3|(a;-H^)2 ^ (x-y)^} t X-^ + 2x^7/3 + z/4 
 = 2,X + ^)4+(x2 - 7/'3, 3 +^4+2x2^/3 +yi = ^{ix+y)^ + X* + 7/4}. 
 
 4. (x3+X?/ + /y3j3=x*+i^.X-'^ + 2x2^/3 +x3^2 -f 2x7/3 + ^/^ = 
 (x+7/)2x2+a;'//3+7/2(x + 7/)3. 
 
 5. In Ex. 3, substitute 5 -c for x, f -a forj/, and consequently 
 b — a for x+7/, then since (b — a)^ = {a — b)^, Ex. 3 gives 
 
 |(rt_/,)2 + (6_c)2+(c-a)2}2 = 2{(a-Z>}4 + (5-c)4 + (c-fl)4}. 
 
 6. Making the same substitutions in Ex. 4, we have 
 (a2+63^c2 -ah-bc -caY = {a —b)^{b - c)2 + (6-c)2(c- a)2 + 
 (c — a)2(a — fi)2, or, multiplying both sides by 4, 
 
 {(a_6)3^(5_c)3 + (c-a)3}2=4(rt_6)2(J_c)2 + 4(fe_c)2 X 
 
 (c-a)2+4(c-rt)2(a-6)2. and .-. from Ex. 5, (a-i)4 + (^6-o)4 + 
 (c-a)4 = 2(«-6)2(i>-c)2+2(fe-c)8(c-a)2 + 2(c-a)3>(a-6)3. 
 
14 FUNDAMENTAL PORMULAS. 
 
 Exercise vi. 
 
 1. (l-2.'c+3a;3 -4x3)3, {l-x-tx^-x^)». 
 
 2. {l~2x+2x^-3x^-x^)^, (l + 3a;+3a;3+a;3)2. 
 
 3. (2a-6-c2_l)2, (l_;^ + y + 2)3, (la;- 1?/ + 03) 2. 
 
 4. (:c3-x2^+a;^3_^3)3, („a;+te2+cx3 + (/a;4)3. 
 
 o. Shew that (a^ ^b'^ +c^) (.c^ i- y2 ^z2 ^ _ (^^x + bij + •z)^ = 
 {a/j - 6a;) 3 + (ex - flz) 3 + (65 - cy ) 3 . 
 
 6. Prove that {a + b)x + (6 + c)y + (c + ^/)3 multiplied by {a — b)x 
 -^[b — c)y + (c — a)z, is equal to the difference of the squares of 
 two trinomials. . 
 
 7. Shew that (a-b) (a-c) + {b-c) (b-a) + {c-a) (c-b) - 
 i-{(a-6)2 + (6-c)34-(c-a)3}=0. 
 
 8. Simplify {a-(6-c)}3 +{6-(c- a)}» + {c-(a-6)}». 
 
 9. Shew that {a^+b^ -x^- )^ +{ai + bl-x^)^ + 2{aa^ +bb,y 
 
 ^(,,2 ^ ,t2_a;3)2+(t2_l_i,2_^2)2+2(a6 + ai6i)3. 
 
 10. ^rovethsii{{a-b){b-c) + {b-c){c-a)-]-{c-a){a-b)}^ = 
 (a-bY(b-GY + {b-cY (c-a)3-h(c-rt)3 (a-6)2. 
 
 1 1 . Square 2(« — \bx —^cx-\-2dx. 
 
 12. If a; + ?/ + 2 = 0, shew that x^ + //* + 2* = (^2 -7/2)3 + 
 
 (^3_22)3+(22_^2)2. 
 
 13. Provethata3(6 + c)2-}.//2(c + rt)3-fc3(rt + 6)3 + 2a6c(a + 6 + c) 
 = 2(rt6 + 6c+crt)2. 
 
 Art. V. To apply formula [4] to obtain the product of two 
 factors which differ only in the signs of some of their terms : — 
 group together all the terms whose signs are the same in one fac- 
 tor as they are in the other, and then form into a second group 
 all the other terms. 
 
 Examples. 
 
 1. Multiply a + 6 — c-f t^ by a-6 — c — d; here the first group is 
 a — c, the second b-\-Ll\ :. we have 
 
 {(a -c) + (6 + <0} {{a-c)-{b+d)}={a-cy-{b-\-d)^. 
 
PUNDAMENTAIi FORMULAS. 15 
 
 2. (1 J- Sx-^Sx^ + x^) (1 - nx + Bx^ - x^.) = {(1.+ ?>x"} + 
 {Sx-^x-)} {(1+3x3) - ('3a;+«3)}={l+3a;3)2 - (3a;+a-)3 = 1- 
 3x^ +8x4 -a;«. 
 
 3. Find the continued product of a +/'+c. h+c — a, c+n —b and 
 n + h — c. 
 
 The first pair of factors gives {{b + c)-\-a} {{h-\-c) — <-i\ ={6+c)^ 
 -o8 = 62+26c+c2-«2. 
 
 The second pair gives {a — {h — c)] {a + {h — <:)] =a^ — h^-\-1hc 
 — c2 ; the only term whose sign is the same in both these results 
 is 26c ; hence, grouping the other terms, we have 
 
 {26c + (/y2+22_a2)} |2k-(63+c3 - a^)] = 
 (263)2 -(63+c2-a2)3 = 2a262 + 2^2,•3+2c3»2 _„4_ft4_c4. 
 
 4. Prove (^i.^-^ah^h)^ -aU^ = {a'^+ahY + {ah + b^)^ . 
 
 The expression ^{>t^-\-h^) {n^ + 2ah+h^) = {a'^ +b^) (^+6)2 = 
 a2(a+6)2+63(« + 6)2 = (a2_f.a5)2_}-(a6+63)2. 
 
 Exercise vii. 
 
 1. (a2+2a6 + 62) („2_2a6+62). 
 
 2. {:L;c^-xy+y^){hx^-\-y^+xy). 
 
 8. (r<2_n6+262) (o3^.«?,+262); (.'c4+4.T2/) (a:4_4a;y). 
 
 4. {(x + y)x-y{x-y)] {{x-y) x-y{y-x)). 
 
 5. Simplify: (.c+3) (x-8) + (a; + 4) (a;- 4)-(a; + 5) (a- 5). 
 
 6. " (l+a-)'i+(i-a;)4-2(l-a;2)2. 
 
 7. (a;2+j/^)2-(2^?/)3-(a;3_y2)3. 
 
 8. (2a2-362+4c2) (2«3+3J2_4c2). 
 
 9. (2a+6-3c) (6 + 3c-2a); (2a— 6-3c)(/^-3o— 2a). 
 
 10. (x4+2/4) (a;2+i/2) (x+?/) (x-7/). 
 
 11. (a;2+a;2/+?/2) (aj3 _a;?/+?/3) (x* -a:3?/2+y4). 
 
 12. (a+A-aZ;-l) (a+6+a6+l). 
 
 13. Prove (a2 +62 +6-2)(6'-i+(-'- _ as^^^s+as _fe2) ^^2 _j./,2_c2j 
 = 46*0* when a4 = 64 _j_c4. 
 
 14. (a!2+t/2-|a;i/) (a;2 + //2+|xy). 
 
 15. (a-*— 2x3^3x2— 2a; +1) (a;* +2x3+3x2 +2a;+l). 
 
16 FUNDAMENTAL FORMULAS. 
 
 16. Multiply (2:k— 2/)a2 - {x-\-2j)ax +x^ by (2x-y)a'' + 
 {x+y)ax-x^. 
 
 Prove the following : 
 
 17. {a^A-h^+c^+ab + bc+m)^-(ah + bc + ca)- = {a + h +c)3 
 x(a3+62-|-r.2). 
 
 18. (a3-|-/>2 +c3 +a6-j.5c+ca)3 - {a^ +ah + ca-hcf = 
 {{a+h)[b^c)]^ + {(b+c) (c + «)}2. 
 
 19. 4.{ab+cdY-{a^+h^-c^-d^)^ = 
 <a->rb-\-c-d) {a + b-c + d) {c + d->ra~h) {c-\.d — n-{-h). 
 
 20. Find tlje product of a;2+//2 +2'- 2x// + 2xz-2?/0 and a;= + 
 .,3+22 ._ 2xy - 2x3+2^/2. 
 
 21. (a;2 +;/2 +.r//|/2) [x^ -xy^2+y^) (a;* -//"). 
 
 22. (l-6a+9ri2)(i + 2a+3a2). 
 
 23. {(m+») + (/?+r^)} (m-9+p-nj. 
 
 24. Obtain the product of 1+x+x^, x^ +x — l, x" — x-f-l, and 
 H-a;-a:3. 
 
 25. («-ft2)2 (a+i8)3 (rtS4-64)3 (a4^.ft8)3. 
 
 26 Shew that {xP- + a;y + ^/^^s (^,2 _ xij -f ;y2)2 _ {x^y^y = 
 
 Formula A. — Examples. 
 
 1. Multiply x' —x+5 by x^ —x — 1 : here tKe common term is 
 a;2 _3.^ ^}ie other terms +5, and — 7, hence the product = (x^ —xY 
 4_(_7+5) (a;?-a;)+(-7x5) = (ic^-a;)2-2(a;2 - x)- %o=x^- 
 2x^-x^+2x-S5. 
 
 2. {x—a) (x—3a) (a;+4a) {x+6a): taking the first and third 
 factors together, and the .second and fourth, we have the product 
 = {x''+Sax - 4a2)(a;2 + 3aa;-18a2)=n(a;2+3aa;;2 _ (ia^ + lSa^) 
 
 x(a;2+3aa;)-72a* = &c. 
 
 Exercise viii. 
 
 1. (a;2+2a;+3) (a;2 + 2a;-4); (x-y + Zz) (x-y-^^z). 
 
 2. (x+1) (a;+5) (x-l-2)(a;+4); {x^ Jra-b) {x^+^b-aX 
 
 3. (a2-3)(a2_l)(a2 + 5) (a2 + 7); {x^ + x' +l){x*+x'' -2). 
 
 4. {(a;+2/)^-4a;i/)} {(x+y)^ +5a;//}. 
 
FUNDAMENTAL FORMULAS, 17 
 
 6. (?«a;+?/+3) (nic+y+7). 
 
 7. (x+a-y) (x+a+dy). 
 
 8. (x-" +a;" -a) (x^" +a;'' -*). 
 
 9. (ia;4-2/-' + 2)(|aj4-2/2-4). 
 
 10. (-+--;5 -+- + 2i) . 
 \x ' y 2j \x ' y ' 'I 
 
 11. Multiply together a- 2 -f 1/2, a;- 2 + 1/3, a;-2--i/2, and 
 K-2- v/8. 
 
 12. (x+a + b) (x+h-c) {x~a + b) [x + b + c). 
 
 13. (a + b+c) {a + b-{-l)-\-{a+c-\-d) {b-\-c+d) - {a+b+c+cl)K 
 
 14. Prove that 
 
 (2aH-26-c)(2//+2c-a)-H(2c + 2a-6)(2a4-2i-o)+(2// + 2c-a) 
 (2c + 2a -b) = 9{ab+bc+cu,). 
 
 Formulas [5] and [6] . — Examples. 
 
 1. We get at once 
 
 (a;+^)3-(a;-2/)3 = 22/(3a;^+2/2). 
 
 2. Simplify (rt+6+tf)3-3(« + i+6')2c4.3(fl+fc + c)c2-r3. 
 This plainly comes under formula [5] , the first term being a+5 
 
 +c, the second —c; hence the expression is {(a + i+c) — c}' = 
 {a+b)K 
 
 3. Shew thsit {x-" +xii+y-)^ + {xij -x- -y^)^ - 
 6xy{x* +a;2y3 +?/*) = 8x^y^. 
 
 This comes under formula [G] , the first term being 
 {x^+X7/+y'^), and the second- (a;- —xy+y^) ; we have therefore 
 {{x^+xy+y-)- {x^-xy+y^)}^ = {2xyy = 8x-y». 
 
 Exercise ix. 
 
 Simplify 
 
 1. (l-a;')3 4.(i+a;2)3^ (x^ +xy^)^ -(x^ -xy^)^. 
 
 2. (a + 2fc)»-(rt-6)3, (3a_6)3_(3«,_2i)3. 
 
18 FUNDAMENTAL FORMULAS. 
 
 4. (a-fc)3+(a + &)3+6«(a2_62). 
 
 5. (x-i/)^+{x+!/)^ + d{x-yy- {x+y)^3{7j-x){x+y)\ 
 
 6. (l+a;+x^)s-(l-a;+a;2)3_6a;(l+a;-+a;4). 
 
 8. (3a;-4?/-r02)3-(52 - 4?/)3 + 3(52 - 4^)2(3a:-4^ + 5z}~ 
 ■S(3x-4.y + 5zy{5z-iy). 
 
 9. (l+^+x-)3 + 3(l-x3)(2+a;-) + (l-:c)3. 
 
 10. Shewtbat «(a-26)3-^6-2«)3 = (rt-^/y)(a+c/)3. 
 
 11. Shewthata3(rt3_263)3+i,3(2a3-i3)3 = (rt3_i3)(«3^i3)3 
 
 12. (a;3+u:?/+i/'-*)3 + 6(a;-+i/-) {x^+xy+y^)-\-{x^ -xy+y^)^. 
 
 13. Shew that aS^^s + 2^3)3 + b3{2a- + 63)3 4- (8a-62)3 =r 
 
 14. Simplify {aa!;-\-l>y)^+0'^y^ +b^x^ ~Sabxy{ax-i-by). 
 
 15. What will a^+b^+c^ —3abc become when a+b+c = ? 
 
 16. Find the value of x^ -y'^+z'^ + 'dx-y^z'^ when x- -y- -fz^ 
 = 0. 
 
 Formulas [7] , [8] and [9] . — Examples, 
 
 1. Simplify (2x-3?/)3+(4^_ 5a;)^4-(3-c-y)3- 
 .■i(2x-3^)(4^-5a;)(3a;-^). 
 
 By [8] this is seen to be {(2a;- 3jr)+(4^ — 5a;) + (3a; - ^) } 3 = 
 (0)3 = 0. 
 
 2. Prove that («-/;)3 + (6-t)3 + (c-a)3 = 3(a-6) ib-c){c-a). 
 
 In [8] substitute a — 6 for x, b — c for y, and c — aforz; for 
 these values x+y+z=-0, and the identity appears at once. 
 
 3. Prove (a + 6 + c)3 -(6 + 6--a)3_(a + c-i)3-(a-h6-c)3 = 
 24Labc. 
 
 In [7] let a; = 6+c — a, y = c+a — b, z = a-\-b—c, and therefore at 
 + 7/ = 2c, y-{-z = 2a, z-\-x = ''Ib, and this identity at once appears. 
 
- FUNDAMENTAL FORMULAS. 19 
 
 Exercise x. 
 
 1. Cube the following: 1 -x-\-x^ . a-b-c 1 - 2:r+3a;2_4a;3. 
 
 2. SimpUfy (a?2 ^ 9a:_l)3 + (2a;-l)(a:^ + 2a;- 2) — 
 (a;3 + 3a;2-l)3. 
 
 3. *Prove that. {x+y){y+z)(z+x) +xf/z = {z + y-\-z){.r]/ -\-i/z+zx). 
 
 4. Prove that {ax — b?/)^ + a^i/'^ — b^x^ •^'dahxy{ax — bij) = 
 
 5. Simplify (a;-22/)3 + (?/-2z)3 + (z-2a:)3 + 3(2'-?/-22) x 
 (y-z -2a;) (2-a:-2?/)+(a;+?/+z)3. 
 
 6. Simplify (2a;-' - Sy^ +422)3^(2^2 _ g^s + 43^2)34- 
 (222-3x2+4|/-)3. 
 
 7. Simplify (2fl;B-fe?/)34-(2?>7/-c5)3 + (2rz— ffa;)3 + 
 8f2fla;+6i/ - rz) {^by+cz — o^a;) (2r2+rta; - by). 
 
 8. Prove (x^ + Sx'^y-y^y + \'^xy{x +jj)]^ = \(x- y)^+^x^y] 
 x{a;'^+a^+|/%'-3. 
 
 9. ' Prove 9{x^ +y^+z^) - (x-l-y + z)^ = (4a; -f 4y-!- z) {x - tj)^ + 
 {4y + -iz+x) (y-z)-+{iz+4:x+y) (z-x)^. 
 
 10. U x+y-\-z = 0, shew that a;3 4-7/3 _|_23 :=3a^2_ 
 
 11. If a;= 27/+3z shew that a;^ -8y3 _27z3 - 18a;^2 = 0. 
 
 12. Shew that (x^+xy + y^)^ + {x^ -xy+y'')^ + 8^^- 
 ^z'' (a;4 +a;22/2 +?/4) = 0, if x^ +?/2 +2^ = 0. 
 
 13. Prove that 8{a + b + r)3 - (a + 6)3 - (6 + c)3 - (c+ a)3 = 
 3(2a+6 + c) (a + 2i4-c) (a + 6+2c). 
 
 Prove the following : 
 14 (ax — by)^ +0^;/^ =a^i;^ -\-?>abxy{by — ax). 
 
 . . . ^ 
 
 *Note that the right-hand member is formed from the left-band one by changing 
 additions into multiplications, and miMipHcations into additions ; hence in (x+y+ 
 s).(x.y-i-y.z+z.xi the signs + and . maybe interchanged throughout without alter- 
 ing the value of the expression. 
 
20 FUNDAMENTAL FORMULAS. 
 
 15. a^+b^+c^-3nbc=i{{a-b)* + {b-c)^ + (c-a)^} X 
 {a+b + c). 
 
 16. {a + b + c) {{a + b-r) {b +c-a} + {b + c-a) {c+a-b) + 
 {c+-a-b) {a + b — c)} = {a + b — c) {b + c — a) (c+a — b)-+8abc. 
 
 17. a^ + b^+c^+2iabc = {a + b + c)^-S{a{b-cy+b{c-a)^-\- 
 c{a-b)^}. 
 
 18. (fl. + />+7c)(a-i)3 + (5 + c+7a)(6-c)« + (c+a+76)(c-fl)a 
 = 2(a + fe + f)3 -54rt/>c. 
 
 19. (a + b + c) {{2a-b) {2b-c) + {2b-c) {2c-a) + {2c-a)x 
 {2a-b)} = i2a-b) {2b-c) {2c-a) + {2a + b -c) {'Ib + c-a)x 
 {2c + a-b). 
 
 20. li x^{y-'rz) = a^, y^{z+x) = b^, z^(x + y) = c^, and xyz = abc, 
 shew that a^ -{-b^ + c^ +-2abc = {x+y) {y-\-z) (z+x) 
 
 Expansion of Binomials. • 
 
 We have from formula [5] 
 
 {a+b)^ —n^ +Sa^b + 3ab^ +b^ ; multiplying by a + b we get 
 la+b)'^=a^ + 4rt.36 + Sa^b'^^ +iab^+-b^ ; multiplying this by 
 o+i we get 
 
 [a-\-b)^=a^ + 5a^b+10a3b^ + 10a^b^ + 5ab^ + b*. 
 
 From tiiese examples we derive the following law for the form- 
 ation of the terms in the expansion of a+6 to any requu-ed 
 power : — 
 
 (1). The index of a, in the ^rsf term, is that of the given power, 
 and decreases by unity in each succeeding term ; the index of b 
 begins with unity in the second term and increases by unity in 
 each succeeding term. 
 
 (2). The coefficient of the first term is unity, and the coefficient 
 of any other term is found by multiplying the coefficient of the 
 immediately preceding term by the index of a in that term, and 
 dividing the product by the number of that preceding term. It 
 will be observed that the coefficients equally distant from the 
 extremes of the expansion, are equal. 
 
MULTIPLICATION AND DIVISION, 21 
 
 Exercise xi. 
 
 1. Expand (ic+i/)«, (a; + 2/)^ («+?/)% {x+ijY*. 
 
 2. What will be the law of signs if — y be -written for y in (\) ^ 
 
 3. Expand (a-Z>)^ (a-26)4, (26-a)4. 
 
 4. Expand (l + m)6, (w^-fl)^ (2m+l)e. 
 
 5. What is the coefficient of the 4th term in {a—by° ? 
 
 6. Expand (a;2- 2/) 4, (a-2''.^)^ (a3-2/>3)G. 
 
 7. In the expansion of [a — bY^, the thii-d term is %Qa'^*^lt^, iind 
 the 5th and 6th terms, 
 
 8. Shew that {x-{-yY -x^ -y^ —^xyix+y) {x^ +x>i-{-y'^). 
 
 9. From (8) shew that 2 {(a - by -^r {h - cY + [^c - ay] =-. 
 5(a-6) (6-c) {c-a) {{a-hy + {b-cY + {c- aY). 
 
 Section III. — Hoknek's Methods of Multiplication and 
 
 Division. 
 
 Examples. 
 
 1. Find the product of kx^-\-ix'^ -'rvix-\-n and ax'^-\-hx-^r. 
 Write the multiplier in a column to the left ot the muitipiicaud, 
 placing each term in the same horizontal line with the partial 
 product it gives • 
 
 kx^ . +ix'^ +mx -fw Q 
 
 ax^ 
 
 •^bx 
 + c 
 
 akx^ +'.'?j;* +a)iix^ -\-a}ix^ Pj 
 
 -\-hhx'^ -\-'ilx^ -\-bmx~-\-bnx ■■■'P^ 
 
 -\-rkx-^ ~\-(:tx^ ■\-cinx-'rcn p^ 
 
 akx^ -f (rt/ + 6/.-)a;* + ( '.nn + bl + ck)x^ -f- {an + bm + cl)x' + 
 (b7i+cm)x-{-cn P. 
 
 Art, VI. The above example has been given in full, the pow- 
 ers of X being inserted ; in the following example detached coeffi- 
 cients are»used. It is evident that if the coefficient of the first 
 term of the multiplier be unity, the coefficients of the multiplicand 
 will be the same as .those of the first partial product, and may be 
 used for them, thus saving the repetition of a line. 
 
22 
 
 MULTIPLICATION AND DIVISION. 
 
 2. Multiply 3^4 -2.7-3- 2a: + 3 bya;3+3a;-2. 
 
 1 
 +3 
 
 -2 
 
 3 -2 +0 -2 +3 
 
 +9 -G +0 -G +9 
 
 -6 +4 -0 +4 -6 
 
 3x^ + 7x^-12x^+2x^-Sx^ + rdx-Q. 
 3. Find the product, of (^-3) >- + 4) (a: -2) (a; -5). 
 
 +^ 
 
 1 
 
 -3 
 
 +4 -12 
 
 + 21 
 
 
 -2 
 
 1 
 
 + 1 -12 
 
 -2-2 
 
 
 -6 
 
 1 
 
 -1 -14 
 -5 +5 
 
 + 24 
 
 + 70 
 
 -120 
 
 
 a:4 
 
 -6x3_9a;3 
 
 + 94a; 
 
 -liO. 
 
 4. Multiply a;3 - 4a;3 + 2x - 3 by 2a;3 - 3 
 
 
 1 
 
 -4 +2 
 
 -3 
 
 
 2 
 
 
 
 
 -3 
 
 2 
 
 -8 +4 
 
 
 
 
 
 -6 
 
 
 
 
 
 -8 +12 
 
 
 -6 + 9 
 
 
 2.'c« 
 
 -8a;-^ + 4.r4. 
 
 -.9a;3+12a;2 
 
 -63; +9 
 
 [x^ X Z^ =X*] 
 
 In this example the missing terms of the multiplier are suppliecl 
 by zeros ; but instead of writing the zeros as in the example, we 
 may, as in ordinary arithmetical multiplication, " skip a line " 
 for every missing term. 
 
 5. Multiply a;4 - 2a;3 + 1 by a;4 _ a;2 + 3. 
 
 1 
 
 -1 
 3 
 
 1 +0 -2 +0 +1 [x*xa'*'-xn 
 
 _1 _o +2 -0 -1 L« xa- -X J 
 
 + 3 +0 -6 +0 +3 
 
 x° 
 
 -3aj« 
 
 + 6a?* 
 
 •7a,-2 +3 
 
MULTIPLICATION AND MYISION, 
 
 ss 
 
 G. Find the value of {x+ 2) (a;+3)(a;+4) {x + 5)- 9(x + 2)(a;-i- 3} 
 X^^^-4) + 3(x•+2)(.^■^-3) + 77(a; + 2)-85. 
 
 1 +5 
 -0 
 
 + 4 
 
 + 3 
 
 + 2 
 
 1 -4 
 +4 
 
 -IG 
 + 3 
 
 -39 
 
 +77 
 
 
 1 +0 
 +3 
 
 -13 
 + 
 
 
 1 +3 
 
 + 2 
 
 -13 
 
 + 6 
 
 +38 
 -26 
 
 + 79 
 -85 
 
 + 5.^3- 7.f3 + 12a; - 9 
 
 7. Find the coeflficieut of a;* in the product of x —ax^+bx^~ 
 cx + d and x^+px-\-q. 
 
 1 —a +6 —c 
 — ap 
 
 + ? 
 
 -{-d 
 
 + {b-ap-{-q) 
 
 Exercise xii. 
 Find the product of 
 
 1. (l+x+x3+a;3+a;'ij(l ~x+x^ -x^ +x^ -x'^^+x^^). 
 
 2. {l'+x^){l -x^+x'^){l+x+x-^+X'i+x^). 
 
 3. (x-5) (x+Q) {x-7){x+8); (2^--a;3 + i) (^4_^+2). 
 
 4. (aj3 + 5a;3 - 16a; - 1) (a;3- 5^-3- l(3.c+l). 
 
 6. {6x^-x'^-j--2x^-2x^ + 2x^ + Wx+6) (3a;2+4a;+l). 
 Obtain the coefficients of x'^ and lower powers in 
 
 6. {l + ix-ix^-\-^\x^-^§^xi) (1 - .^x-ix^-^^^x-^-^^^x*). 
 
 7. Multiply 2x'' -x^+Sx -4:hj 3z^ -2x^ -x-l. 
 
24 MULTIPLICATION AND ERISION. 
 
 Simplify the following : 
 
 8. (x+l) {x-\-2) (a;+3) + 3(a;4-l) (a;+2) - 10(a;+l)+9. 
 
 9. x{x+l) (£c + 2) {x-h'd)--dx{x+l) [x+2)-2x{x-hl) + 2x. 
 
 10. x{x-l){x-2){x-3} + dx{x-l){x-2)-2x{x-l)-2z, 
 
 11. {x-1) (x+l) {x+3) {x + 5)-U{x-l) [x-tl)-rl. 
 
 VI. Given that the sum of the four following factors is — 1, find 
 (1) the product of the first pair ; (2) the product of the second 
 pair ; and (3) the product of the sum of the first pair- by tiie sum 
 of the second pair. 
 
 (1) 
 
 X 
 
 +^^ 
 
 +xi3 
 
 + X^* 
 
 (2) 
 
 x^ 
 
 + X8 
 
 + ^9 
 
 4. .-IS 
 
 (3) 
 
 x^ 
 
 +^-» 
 
 +a;i3 
 
 +x^^ 
 
 (4) 
 
 x^ 
 
 + a;' 
 
 H-r'o 
 
 +a;ii 
 
 13. Given that the sum of the three following factors is equal 
 to — 1, find their product. 
 
 (1) X 4a;' 4a;8 4a;»8 
 
 (2) ic2 -ha;3 +xio 4a;* ' 
 
 (3) x^ +x*^ +x^ +x^. 
 
 Art. VII. Were it required to divide the product P in the 
 first of the above examples by ax'^ +bx+c, it is evident that could 
 we find and subtract from P the partial products ^g, ^3, (or what 
 would give the same result, could we add them with the sign of 
 each term changed), there would remam the partial product ;>, , 
 which, divided by the monomial ax^, would give the quotient Q, 
 This is what Horner's method does, the change of sign being 
 secured by changing the signs of b and c, which are factors m 
 eacli term of _pg, p^, respectively. 
 
BITJLTIP LIGATION AND DIVISION. 
 
 25 
 
 
 + 
 
 H « 3 
 
 + ' 
 
 
 + 
 
 ^^ ^ 
 
 •^ 5 "1^ 
 
 + 
 
 + 
 
 + I 
 + 
 
 "5 
 
 * 
 
 - O 
 
 
 a 
 .2 
 
 *-i3 
 
 r-l 
 
 03 
 
 Co 
 
 a-i 
 
 CO 
 
 o 
 O 
 
 
 <D 
 
 CD 
 
 a 
 
 * 1— H 
 
 o 
 
 f-H 
 
 
 
 
 ■ ^ 
 
 • ^H 
 
 p— < 
 
 
 
 
 O 
 
 r-4 
 
 > 
 
 
 
 CD 
 
 
 
 
 
 "^ 
 
 -<-i» 
 
 S 
 
 a 
 
 > 
 
 P^ 
 
 > 
 
 c3 
 
 o 
 
 r---l 
 -4-3 
 
 a 
 
 
 
 
 
 
 fcD 
 
 J? 
 
 'Hi 
 
 •8 
 
 a 
 
 • <— 1 
 
 r-i 
 
 
 
 o 
 
 
 
 
 
 O 
 
 n3 
 
 
 •"3 
 
 IN 
 
 
 1— ~i 
 
 •1-H 
 
 «J-I 
 O 
 
 CO 
 
 -1-3 
 
 a 
 
 a 
 
 • »-4 
 
 CQ 
 
 
 
 
 P* 
 
 CO 
 
 
 
 o 
 
 
 r^ 
 
 "« 
 
 
 }-\ 
 
 .2 
 
 a" 
 
 
 
 
 <s 
 
 Pi 
 
 • * 
 
 
 Sh 
 
 'Ph 
 
 
 S» 
 
 '^ 
 
 Q) 
 
 'o 
 
 o 
 
 
 
 
 o 
 S3 
 
 
 aj 
 
 
 Ph 
 
 _a 
 
 
 • 1— ( 
 
 c3 
 
 a 
 
 
 
 
 o 
 
 'S 
 
 ■4J 
 
 CD 
 
 'to 
 
 O 
 
 CO 
 
 a 
 
 • 1— t 
 
 g 
 
 CD 
 O 
 O 
 
 a 
 
 
 
 
 '3 
 
 -»3 
 >—< 
 
 « 
 
 &. 
 
 fl 
 
 r^ 
 
 g 
 
 
 
 CD 
 
 ■—1 
 
 -4^ 
 
 
 ■ 
 
 
 2 
 
 OQ 
 
 ■ 1— 1 
 1 — I 
 
 
 * 1-^ 
 
 CO 
 ■4^ 
 
 o 
 
 -1-3 
 
 1 
 
 c 
 
 a 
 
 CD 
 
 ^-4 
 
 -4J3 
 
 ^ 
 
 
 
 
 3 
 
 O) 
 
 o 
 
 "*^ 
 
 O 
 
 
 TS 
 
 1— H 
 -4-3 
 
 C^"* 
 
 
 
 -f 
 
 - f 
 
 a 
 
 o 
 
 CD 
 
 a 
 
 1— ■ 
 
 o 
 
 3 
 o 
 
 i 
 
 o 
 
 -k3 
 
 o 
 
 
 o 
 
 P^ 
 
 a 
 
 o 
 
 (—1 
 
 a 
 
 O 
 
 o 
 CO 
 00 
 
 • 1— t 
 
 -1-3 
 
 to 
 
 ■73 
 
 a 
 
 1 
 
 et-c 
 O 
 
 g 
 
 ■73 
 
 a 
 
 rH 
 CD 
 
 • f-H 
 -4-3 
 O 
 
 a 
 
 <D 
 
 O 
 
 CO 
 
 •rH 
 
 -4J 
 
 o 
 
 a 
 
 ■-3 
 
 CD 
 CD 
 
 a 
 
 
 CO 
 
 1 
 
 -1-3 
 
 c3 
 
 O 
 
 -< 
 so 
 
 en 
 
 -4-3 
 
 a 
 
 o 
 
 g 
 
 _a 
 
 > 
 
 P 
 
 
 ■73 
 
 a 
 o 
 
 CO 
 
 cr" 
 
 CD 
 
 -1-3 
 
 0) 
 
 P-. 
 
 <D 
 
 f^ 
 
 r— * 
 -4^ 
 
 ■4-4 
 O 
 
 CD 
 
 a 
 
 • r-( 
 1 1 
 
 
 o 
 
 ^ 
 
 4 
 
 ' 1 
 
 - 4 
 
 "73 
 
 ■73 
 
 a 
 
 c3 
 
 CD 
 
 ■ I— 1 
 
 -4J 
 
 (0 
 
 'q3 
 1 -^ 
 
 -1-3 
 00 
 
 
 'Hi 
 •■— ( 
 
 cu 
 
 3 
 
 g 
 
 o 
 o 
 
 g 
 
 r-4 
 -1-3 
 
 CO 
 
 o 
 
 +3 
 
 ■^ 
 OS 
 
 
 
 
 c5 
 
 
 Jj 
 
 
 o 
 
 o 
 
 <D 
 
 tp 
 
 
 -*-3 
 
 -4J 
 
 ca 
 
 
 
 
 
 0) 
 
 
 ^ 
 
 ?H 
 
 
 *^ 
 
 
 fcB 
 
 .^ 
 
 
 ^ 
 
 
 
 
 o 
 
 
 
 o 
 
 CO 
 •i-H 
 
 > 
 •r-( 
 
 a-t 
 
 CO 
 
 3 
 
 > 
 
 00 
 
 
 a 
 •p— 1 
 
 Pi 
 
 -4-3 
 
 -1-3 
 ■M-l 
 
 O 
 
 
 
 i e; 
 
 
 > 
 
 n3 
 
 1 
 
 o 
 
 o 
 
 
 a 
 o 
 
 ■73 
 
 CO 
 
 o 
 
 -4-3 
 
 a 
 
 CD 
 CO 
 O 
 
 
 4 
 
 - 4 
 
 
 CO 
 -1-3 
 
 O 
 
 r-» 
 
 -a 
 
 -t.3 
 
 O 
 
 
 -1^ 
 
 Ol 
 
 p—H 
 -1-3 
 
 O 
 
 
 CO 
 
 -1-3 
 
 60 
 
 _a 
 
 -4-> 
 
 o3 
 
 (D 
 
 "3 
 
 o 
 o 
 
 -4J 
 
 a) 
 
 
 
 « 
 
 
 -1-3 
 
 CD 
 O 
 >4 
 
 o 
 
 g 
 
 CD 
 
 a 
 
 -4-3 
 -t-3 
 
 1 
 
 'cS 
 CO 
 
 g 
 
 a 
 
 
 
 Pi 
 
 CD 
 
 Ph 
 
 ■73 
 
 g 
 
 eg 
 
 CO 
 
 • 
 
 i-H 
 
 -4-3 
 
 ^ 
 
 
 ^ 
 
 Ol 
 
 
 tfl 
 
 so 
 
 CD 
 
 rid 
 
 'o 
 
 CO 
 
 g 
 
 • • 
 
 r— 1 
 
 a 
 
 -1-3 
 
 O 
 
 -a 
 
 c* 
 
 
 <x> 
 
 
 02 
 
 o 
 
 ^ 
 
 c3 
 
 ■73 n3 
 
 o 
 
 
 
 
 
 
 o 
 
 • f-l 
 > 
 
 
 
 g" 
 
 g 
 
 2 
 
 a 
 o 
 
 o 
 
 CQ 
 
 CO 
 0) 
 
 -^ 
 
 ri 
 ■ 1— ( 
 
 a a 
 
 ce eg 
 
 o o 
 
 m 
 
 •f-H 
 
 CO 
 
 o 
 
26 
 
 MULTIPLICATION AND DIVISION. 
 
 2. Divide Sx" +1x^ - rix*-i-2x^ - 3a;3 + 18x-- 6 by z'+Zx-2. 
 3 +7 -12 +2 -3 +13 -6 
 
 -3 
 
 + 2 
 
 _9 +6-0+6-9 
 
 + 6-4+0-4+6 
 
 {x''^X'^X'>'} 
 
 8a;4_2a;3+ -2x + 3 
 
 Compare this example with the secoEd example of Homer's 
 Multiplication, performing a step in multiplicatio.u, then the cor- 
 responding step in division ; then another step iu multiplication 
 and the second, (corresponding) step in division, and so on. 
 
 3. Divide x'' - Sx*^ + ia;-^ + 18x-^ - 7x.+ 12 bv x^ - Sx^ + 'dx - 1. 
 
 -3 
 + 1 
 
 -3 +0 -4 +18 +0 -7 +12 
 
 + 3 +0 -y -36 -27 
 
 -3 -U +9 +36+ 27 [x" - x^ =.'f4] , 
 
 + 1 +0 -3 -12 -9 
 
 > x-i +U -3a;3-12x- 9; (lr-'+ 8x +3 
 
 The quotient is therefore x'^ —ox'^ - IJx — 9, and the remainder 
 6x2+8a;+3. 
 
 4: Divide x^ - Zx' -ox'' +2a;4 + 5x3 + Ix- + 1 by xS 4. 2a;-l. 
 The zero coefficient in the divisor may be inserted, or it may be 
 omitted and allowance made for it in the 2x — iiue. See examples 
 4 and. 5 in multinlication. 
 
 -2 
 ■+1 
 
 1 _3 +u _5 +'2 +.J +4 +0 +1 
 -2 +6 +4 -4 -f; +2 
 
 1-3-2 +2 +3 -1 
 
 I 1 _3 -2 +2 +3 1; 0+5 +0 
 
 \_x^ ~- x^ =x^] . The quotient is therefore x^ — 3x^ — 'Ix^ 4- 2a;' 
 + 3a; — 1, and the remainder 5x. 
 
 5. Divide lOx^ -lla;»-3a;4+20x3 + iOa;3 + 2 by bx'^-3x^-i- 
 
OF 
 
 MUI.TIPIiTCTI TON ANT) r)I\aSlON. 
 
 27 
 
 Arranging as in the ordinary metliod, we have 
 
 +3 
 
 -2 
 +2 
 
 10 -11 -3 +20 
 6-3-6 
 
 -4 + 2 
 + 4 
 
 + 10 +0 +2 
 
 + 12 
 
 + 4-8 
 
 -2-4+8 
 
 2 _i _2 + 4 
 Quoiieat = 2x^-x^ - 2x+4 + 
 
 24-12+10 
 
 24a;2....i2a;+10 
 
 5x^-Sx^+2x—2 
 
 We first draw a vertical h'ne with as many vertical cohimns to 
 the right as are less bj' unity than tlie number of terms in the 
 divisor. This will mark the point at v,'hich the remainder begins 
 to be formed. We then divide 5 into 10, and thus obtain the 
 first coefficient of the dividend. We next multiply the remaining 
 terms of the divisor by the 2 thus obtained. Adding the second 
 vertical column and dividing by 5, we obtain — 1 ; we multiply 
 by the — 1, add tlie next column and divide the sum by 5, and so 
 on for the others. 
 
 This method is not, however, always convenient. If the first 
 term of the dividend be not divisible by the first term of the divi- 
 sor, the work would be embarrassed with fractions. We may 
 then proceed as in the following exaTQples : 
 
 6. Dividea;5-3a;4+a;3+3a;2-a: + 3 hy 2x^+x^ -Bx+l. 
 
 Let 2x = y, or x = — . 
 ^ 2 
 
 Substitute -^ for a; in the dividend and divisor, and wo have 
 2 
 
 y' 
 
 3,y4 y3 Sy^ 
 
 V 
 
 2^ 
 
 24 -r 23 ' 22 " 
 
 2 
 
 + 3 - 
 
 %^ 
 
 + 
 
 y 
 
 _ y^ 
 
 23 ' 23 
 
 -2 X 82/^1 +22^/3 + 23 X 3^3 - 24?/+2s x 3 
 
 37/ 
 
 — + 1 
 2 ^ 
 
 25 
 
 y 3+y3-2x3y + 23 
 
 — p '■ 
 
 2/ 6 - &ij^^ + 4y3 + 242/3 -16y +96 
 
 ^2/3+,/2_6iy + 4. 
 
 .A. 
 
28 
 
 MULTIPLICATION AND mviSION. 
 
 Dividing ?/» -6^/* +4»/3 +247/2 - IQy+dQ by y^+y^ - 6?/+4 by 
 the ordinary method, and the quotient by 2' we have 
 
 y2- 7y + 17 
 23 
 
 1_ 39 y2- 1147/ -28 
 23* ^3_f.,/2_6^_|_4 • 
 
 Substituting for ?/ its value 2x, and simplifying we get 
 a;2 7iB . 17 1 39a;2-57a;-7 
 
 -T + 
 
 8 
 
 8" 2a;3+:c3-3a;+l 
 
 B. 
 
 By comparing the dividend of A with the original question, we 
 find that we have multiplied the successive coefficients of the divi- 
 dend by 2°, 2^^, 22, &c., and. omitting the first term, we have 
 multiplied the successive coefficients of the divisor by tlie same 
 numbers. Dividing then by Horner's division we get tlie coeffi- 
 cients 1, —7, 17, and for coefficients of remainder, -39, 114, 
 and 28. The first of these divided by 2, 22, 2^ are the coeffi- 
 cients of a;2 &c. ; and, -39, &c., are divided by 1, 2, 22. Hence 
 the work will stand as follows : — 
 
 a.5 _.;:)^4 +a;3 + 8.c2- x+ 3-2a;3-fa;2-3«+l 
 1248 16 82 124 
 
 
 1-6 +4 
 
 + 24 
 
 - 16 + 96 
 
 -1 
 
 -1 -f? 
 
 -17 
 
 
 +'•> 
 
 + 6 
 
 -42 
 
 + 102 
 
 4 
 
 
 - 4 
 
 + 28-68 
 
 
 .1-7 +17 
 
 -39 
 
 + 114-f28 
 
 1 -6 +4 
 
 "Quotient = 
 
 x' 
 
 X' 
 
 2 
 
 7x 17 
 4 "'" 8 
 
 1 
 §■ 
 
 7x 17 
 4 8 
 
 1 
 
 8 
 
 S9x^ 
 
 lUx 
 
 2^ 
 
 28 
 4 
 
 2x3 +3:3 -'Sx + 1 
 39x2 _ 57a; - 7 
 
 2x^+x2-Sx+V 
 
 *lt will, in general, be as convenient to multiply the dividend by such a, num- 
 ber aa will make its first ti'vin exactly divisible by the ftrst torm of the divisor, and 
 afterwards divide the quotient by this multiplier. 
 
MTJLTTPTJrjATION AND DIVISION. 
 
 7. Divide 5x^+2 by Sz^-2x+B. 
 
 29 
 
 5x-' 
 1 
 
 +2 -f- Sx^-2x+S 
 
 3 9 27 81 243 13 
 
 
 5 
 
 
 
 
 
 + 2 
 
 
 10 +20 
 
 -50 
 
 -9 
 
 
 -45 
 
 -90 
 
 5 -|.io -25-140 
 
 + 486 
 —280 
 +225 +12G0 
 
 -2 +9 
 
 - 55 +1746. 
 10 25 140 
 
 Coeffs. of Quotient = ^. - - 
 
 3 32 o^ 
 
 3* 
 
 Quotient = 
 
 5x^ 
 
 + 
 
 lOiB^ 
 
 25^ 
 
 140 
 
 J^ 55 -H^ 
 34" 3_2 + 3- 
 
 55a;- 582 
 
 9 27 81 
 
 Exercise xiii. 
 
 81 3a;2-2a;+3' 
 
 1. Divide 6ar-^ + 5^4 - 17x3 - 6x^ + 10.r- 2 by 2x- +?.x - 1. 
 
 2. (5a;«+6a;5+l)-f (a;-^+2a- + l). 
 
 4. (x5-4a;-//3 _ 8x273. „ i7a;,y4_ 127/5) ^(cc3 _2^v/_3^2). 
 
 Divide 
 
 6. 4a;'i + 8x3-3a;+lby a;3-2a-+3. 
 
 7. 10x5+ox-4-00a;3 _4i^.2+i0a;+l by a;2-9. 
 
 8. x^ —x^y-{-x^ii^—x'^ll^^xy'^-y-' hyx^—y^. 
 
 9. Multiply a;4-4a;3a + Ca;3«2 -4a;«3 4- ^4 ^y a;2 4. 2x(< + a", 
 ftnd divide the product by x* — 2ic3a+2a;a3 —a*. 
 
 Divide 
 
 10. x^ - ax^ + I'X^ - bx^ +r,x - 1 by a; - 1. 
 
 11. 6a;^+7.T-i+7a;3+6x3+0a;+5 by 2a;2+^ + l. 
 
 12. m{x^-\-y^)-\-S)lxy{x'^-y'-^) by 12a;3 - 13a;// + 5.y*. 
 
•jO multiplication and division. 
 
 13. 6ae-481a;'^ + 79a;4 + 81a;3_81a:2-f86x-481 l.y«-80. 
 
 15. a(« + 2//)3_/;(2rt4-6)3 by («- A)^. 
 
 16. (x+uy^-{-3{xi-ijyz-{--d{x + y)z"-+z^hy{z+y)^ ^^ 
 
 — 17. 10a;"J + ina;C + 10a:3-200 by a;' +x3 -x + l. 
 
 18. bjnx^ + {bji-}-cin)x^ +ciix~ +abx + ac hy bx+c. 
 
 19. Multiply 1+ 2^a;-18a;3 by l-L3a;3 + 3a;3 aud divide the 
 product by l+Va; — 8x2. 
 
 Find the remainders in the following cases : 
 -r 20. {xJ-\-nx^ + -i.x + 5)-{x~2). 
 
 ^ 21. (x^~Sx^-]-x-S)~{x-l). 
 
 22. (a:^+4.«3 + Ga; + 8)-(a;-i-2) 
 
 23. (27a:i-?y*)^(8a;-2y). 
 
 24. {3x'' +5x^ -3x^-\-7x^ -r)x + 8)-^{x" -2x). 
 
 25. (5a;4-f90a;3 4.80a;2-100a; + 500)-^(a;+17). 
 
 Art. viii. The following are examjiles of au important use of 
 Horner's Division : 
 
 1. Arrange x^ —6x^-\-lx— 5 in powers of a; — 2. 
 
 11 -6 7—5 
 
 2 2 _8 -2 
 
 2 
 
 1 . 
 
 -4 
 2 
 
 -1 
 -4 
 
 -7 
 
 — 2 
 -1-2 
 
 -5 
 
 I 1; 
 
 Hence, x^ -Gx^ +lx-5 = {x—2)^ -5{x-2)-7, or as it is gen- 
 erally expressed, x^ - Gx^ -\-7x— 5 = y^ — 5y — 7 if y = X'-2. 
 
MULTIPLICATION AND DIVISION. 
 
 81 
 
 a. Express x^+l2x^+'i7x^+G()x+28 in powers of x+6. 
 
 -3 
 
 
 12 
 -3 
 
 47 
 
 -27 
 
 60 
 -GO 
 
 2S 
 -18 
 
 -3 
 
 
 9 
 -3 
 
 20 
 -18 
 
 6; 
 -6 
 
 10 
 
 -3 
 
 
 6 
 -3 
 
 2- 
 -9 
 
 
 
 
 -3 
 
 
 3; 
 -3 
 
 -7 
 
 
 
 -*- ) 
 
 0. 
 
 
 Hence a;4 +12x3+47x2 +66a;+28 = //'i -7y/- +10 if y=a5+3. 
 
 After a few solutions have been written out iu full, as in the 
 above examples, the writing may be lessened by omitting the 
 lines opposite the mcrements (—2 in Ex. 1, and 3 in Ex. 2), the 
 multiplication and addition being performed mentally. The last 
 example written iu this way would appear as follows : 
 
 1 12 47 66 28 
 
 1 9 20 6 (10) 
 
 1 6 2 (0) 
 
 1 3 (-7) 
 
 1 (0) 
 
 Exercise xiv. 
 
 1 Express x^ — 5x^ + 3a; - 8 in powers of x — 1. 
 x''-^Sx-+Gx+0 " x+1. 
 
 ;c4 _ 8x3 + 2 Ix'-' - 32x + 97 in powers of x - 2. < 
 
 -3 
 
 2. 
 3. 
 
 4. 
 
 5. 
 
 6. 
 
 7. 
 
 8 
 
 9. 
 
 x-' + 12x3 + 5x3-7 
 3x'-x3 + 4x-+5x-8 
 x^ -7x3 + 11x3 -7x + 10 
 x3-2x--4x+9 
 
 x3-9x3// + (JX^2_8^3 
 
 X* —bx^ij-k-^xy^--y^ 
 
 (< 
 
 x+2. 
 
 <4 
 
 x-2. 
 
 (< 
 
 x-n 
 
 (( 
 
 x-h 
 
 ({ 
 
 x-2.y 
 
 <f 
 
 x-y. 
 
32 SYMMETRY. 
 
 10. " 8a;S+12x2y + 10«.'/2+8?/3 •• 2z+y. 
 
 11. «' ,r3-|a;2+§a;-f^ " ia^-rV 
 
 12. *' a;'^ + 8a;3-15x-10 ♦* x+2. 
 
 CHAPTEB II 
 
 Section I. — The Principle ov Sym^metky. 
 
 Art. ix. An expression is said to be symmetrical with respect 
 to two of its letters when these can be interchanged without 
 altering the expression : 
 
 Thus if in a^+a^x + ax^ + .v^, we write x for a, and a for x, we 
 get x^+x-a+xa^+a^, which is identical with the given- expres- 
 sion. So, in a;2+i3^_f.^^_|_rt23. if ^e interchange a and 6, there 
 results a;- +«-iB+ai'+62a; which is identical with the given ex- 
 pression ; but it will be seen that the expression is not symmetrical 
 with respect to x and b, or x and a. 
 
 An expression is symmetrical with respect to three of its letters 
 a, b, c, when a can be changed into b, b into c, and c into a, without 
 altering the expression. 
 
 Thus a3 -^i,s^sS—Sabo remains unaltered by changing a into b, 
 h into c, and c into a, and is therefore symmetrical with respefct 
 to these letters. So, aH-\-b^a+a^c+c^a-irb^c-\-hc^ , and {a-h)'^ 
 _|. (h — c)^ + (c — «)', are each symmetrical with respect to a, b, c. 
 
 Again (x-a) {a-by + {z-b) (b-c)^ + (x-c) {c-ay is sym- 
 metrical with respect to a, b, c, but not with respect to x and any 
 of the other letters. 
 
 Generally, an expression is symmetrical with respect to any 
 number of its letters a, b, c, . . . h, k, when a can be changed 
 
 into h, b into c, c into d h into Jc, and k into a, without 
 
 altering the expression. 
 
BYMMETRY. 83 
 
 A symmetric function of several letters is frequently represented 
 by writiDg each type-term once, preceded by the letter 2 ; thus for 
 a-\-b-\-c-\- ... . +/. we write 2a, and for rt^+rtc-f atZ+ . . . . 
 + 6c + k/+ . . . [i. e. the «um of the products of every pair of 
 the letters considered) we write "Zab. 
 
 Exercise xv. 
 Write the following in full : 
 
 1. 2a^h, 2(a-i)3, y_a{b—c), ^ab{x-c), ^aH^c, ^{a-\-h) 
 X{c-a){c—b), 2 {(a+c)--63}, and va(i-fc)3, each for a, h, c. 
 
 2. y.abe, y.aH, ^a-bc, 2 (a- i), and 2a 3 (a _jj^ each with 
 respect to a, b, c, d. 
 
 She"w that the following are symmetrical : 
 
 3. {x-\-a) (a +6) {b->rx)-'rabx, with respect to a and b. 
 
 4. (a+6)^+(a. — 6)3 with respect to a and b, and also with 
 respect to a and — b. 
 
 5. {ah -xyY^ -{a -^-h-x-y) {ab{x-{-y)—xy{a+b)} Mniih. respect 
 to a and b, and also with respect to x and y. 
 
 6. a'^ {b ~c) — b^{a — c) — c^ {b - a) with respect to a, b, c. 
 
 7. {ac+bd)-+{bc-arl)- with respect to a' and i-, and also 
 with respect to c- and (P . 
 
 8. x^ +y^ +?>x>/{x^ +xy + y^) with respect to x and y. 
 
 9. •{«^-?/3+8x^(2a;-f2/)}3 + {2/3-a;3+3a;?/(2|/ + a?)}3 with res- 
 pect to x and y. 
 
 10. a(a+26)3+6(6+2a)3 with respect' to a and 6, and also 
 with respect to a and — b. 
 
 11. ab[{{a + c) (b+c) -f 2<7(a + ft)}2 _ (a_e)2 (j_c)«] with 
 respect to a, b, c. 
 
 12. a'^b- -{■b'^c'^ -\^c^a- -^'2>abc{a-\-b-\-c) with respect to afc, 6c, ca 
 With respect to what letters are the following symmetrical ? 
 
 13. xyz-\'5xy-\-'l{x--\-y'^). 
 
 14. 2(a3a;-' + h-h/) - 2ab{xy + //z -\-zx). 
 
 15. {P - /t--^)-2 + 4//2( z'+Zi r-' 4- ( 2/7^ - 2.7-)». 
 
34 SYMMETRY. 
 
 16. (x+y) (x—z) {7j — z) — xyz. 
 
 17. «,252^Z»2c3+.c2a2-2fl6c(a + ^-c). 
 
 18. x^ -y^ +z^ -^x^ -y-^){y^ -z^) (z^ + a;'). 
 
 19. (a+ft)2+(a + c)2 + (//-(-)*- 
 
 20. (a + i)4 + (,i_^)4^(/,+c)4 + (« + f)4. 
 
 Select the type-terms in : 
 
 22. ,,2a.2ai-f/*2+2k+?^+2ca 
 
 23. a(b^ -t"~)+h[c^ -a^)+r{(r- -b"-)-\-{a + h) (h-\-c) {c-k-a). 
 
 24. «(6 + '0'+^'(''-l-«)^ + ^'(« + ^)^-l'^«/;c. 
 Write down the type-terms in : 
 
 25. (a;+y)5, (a;-?/)^ (x + y)^ -a^s - ^/S. 
 
 26. (x+yy + ix-yy, {x+!/y-{x-yy. 
 
 27. (a;+7/-f2)4, (a;-y-^)4. 
 
 28. (a+b + c + dy, {a-+h^-+c^+d^y. 
 
 29. (a + &)3+(z, + c)3 + (c4.«,)3. 
 
 Art. X. In reducing an algebraic expression from one form 
 to another, advantage may be taken of the principle of symmetry : 
 for, it will be necessary to calculate only the type-terms, and the 
 others may be written down from these. 
 
 Examples. 
 
 1. Find the expansion of {a + b-\-c + d-^e-{-&c.y 
 
 TJiis expression is symmetrical with respect to a, b, c, &c. ; 
 hence the expansion also must be symmetrical, and as it is a pro- 
 duct oi' iM;o factors, it can contain only the squares a^, b^, c^, &c., 
 and the products in pairs, ab, ac, ad . , , , be, bd, &c. ; so that 
 rt^ and ab are type-terms. 
 
 Now {a-^by —a"^ -{-2ab-^b^ ; and the addition of terms involv- 
 ing a, b, c, SiC, will not alter the terms a^ -f 2fli>, but will merely 
 give additional terms of the' same type. Hence from symmetry 
 we get 
 
SYMMETRY. 35 
 
 (a + !) + (■-{- I + e + &Q.y-' ^ a'-' +2ab-^2ac + 2ad + 2ae+ 
 
 + b- +2bc+2bd+2be + 
 
 + 6-2 -{-2cd+2ce-\- 
 
 + fZ2 +2de + 
 
 This may be compactly written 
 
 (2a)3=Xa2 + 2Safe. 
 
 2. Expand (a + h)'^. 
 
 This has been found by actual multiplication — see formula [5] 
 —but we may also proceed as follows : 
 
 (1) The expression is of three dimensions, and is symmetrical 
 with respect to a and b. 
 
 (2) The type-terms are a^, a-b. 
 
 ■ Hence [a-^-b)^ = a^ +h^ -\-niu'^b-}-b-a), where n is numerical. 
 
 To find the value of n, puta = ft = l, and we have (1 + 1)3 = 
 1 + 1 + »(1 + 1); .-. n = S. 
 
 3. Expand (a;+^ +2)3. 
 
 This is of three dimensions, and is symmetrical with respect 
 to X, 7j, z. We have 
 
 {x-^U+z)^ = {{x+y)+z]^ = (x+^)3 +&c. 
 
 = x^ + ^x'^y-\-kQ., which are type-terms, the only other possible 
 type-term beiug xyz. 
 
 Now, since the expression contains ^x-y, it must also contaia 
 ^x'-z, that is, it must contain Qx-{y-\-z). Hence 
 
 (x+y+z)^ = x^-\-dx''{y-{-z) 
 +2/^+3y-(z+a;) 
 + z^ + -iz^{y^x) 
 
 + n{xyz), where n is numerical, and 
 may be found by puttinjf x-y = z = l in the last result, giving 
 (l + ]+l)3 = i + i^.i+3(i4.iA4.3(i_^l)_j.3(l_^l)_^,,. 
 .•. n = 6, 
 
36 SYMMETRY. 
 
 4. Similarly we may shew that 
 
 {a + b-\-c + U)^= ai-\-nn-{b+c + d) + (ibcd 
 + hs+3b2{c + d+a) + Gcda 
 + c^ + 3c^d-+a+b)-\-Gdab 
 -f d^ +'Bd^(<i + b-i-c)-\-6abc. 
 
 5. Expand (a+6+c+&c.)3. 
 
 Tho type- terms are a^, a^b, abc. 
 
 Expanding (a + b-\-c)^, we get a^ + Ba^h-\-6abc-{-&0, 
 
 Hence by symmetry we have 
 
 6. Simplify (a + & - 2c) 2 + ( fc + c - 2(y) f + (c + « - 2b) 2 . 
 
 This expression is symmetrical, involving terms of the types 
 a^ and ab. Now a^ occurs with 1 as a coefficient in the first 
 square, with 4 as a coefficient in the second square, and with 1 as 
 a coefficient in the third square, and hence 6a^ is one type-term 
 of the result : ab occurs with 2 as a coefficient in the first square, 
 with —4 as a coefficient in the second square, and with —4 as a 
 coefficient in the third square, and hence — 6ah is the second 
 type-term in the result: hence the total result is 6 {a^+b^+c^ 
 •—ab — bc—ca). 
 
 7. Simplify {x-\-y+z)^-\-{x-y-z)^+{ij-~z-xy -\- (z — x-y)^. 
 This is symmetrical with respect to x, y, z; and the type-terms 
 
 are x^, Sx^y, (ixyz : 
 
 (1) ic* occurs in each of the first two cubes, and — os^ in each 
 of the second two cubes, .". there are no terms of the type x^ in 
 the result. 
 
 (2) 3x^y occurs in the first and third cubes, and — Bx^y in the 
 second and fourth, .•. there are no terms of this type in tlie 
 result. 
 
 (3) 6xyz occurs in each of the four cubes, .'. 24a;^z is the total 
 result. 
 
 8. Prove (a^ + b^+c^+d^) (w^ +x^ -^y- +z'')- 
 
 {aw -r bx-\'Cy -f- dz) - = (ax — bw) ^ +{ay — \cw) ^ + (az — dw) 2 -f 
 (by-cx)^ + {l)z-dxY + (cz-dy)^. 
 
SYMMETRY. 87 
 
 The left hand member (considered as given) is symmetrical 
 with respect to the pairs of letters, a and iv, b and x, c and y, 
 d and t, that is, any two pairs may be interchanged without 
 affecting the expression. As the expression is only of the second 
 degree in these pairs, no term can involve three pairs as factors ; 
 hence the type-terms may be obtained by considering all the 
 terms involving a, b, w, x', these are a^ic'^, a-x^, b^w^, b^x^, 
 — a^w^, —b^x^, — 2«fettic, and are the terms of [ax — bw)'^ -which is 
 consequently a type-term. From (ax—bw)^ we derive the five 
 other terms of the second member by merely changing the 
 letters. 
 
 9. Prove that 
 
 (x^-yz)^-j-{y^-zxy -\-{z^ -X!jy-B{x^ -yz) (y^-zx) {z^-xy)k 
 a complete square. 
 
 The expression will remain symmetrical if (x^—yz) (y^—zx)- 
 (z^—icy), instead of being multiplied by —3, be subtracted £>-om 
 each of the preceding terms, thus giving 
 
 (Kg - yz) { (a;3 - yz) ^-(y^-xz) {z^ —xy) } 
 -f (?/2 -zx) {(y^ -zx)^ — {z^ -xy) {x^ - yz^} 
 -\-[^^'-xy) {{z2-xyr--(x'--:.z) (y^-zx)} 
 = (x^ —yz)z[X^-^y^-\-z^ — Bxyz) 
 +&c. 
 +&c. 
 
 = iX'^-j-y^+z^ — Bxyz) {x^-^y^-\-z^ - Sxyz). 
 Exercise xvi. 
 Simplify the following : 
 
 1. {a + b + G)^-\-{a-\-b-c)2 + {h+c-a)^-\-{c-\-a~-b)^ 
 
 2. (a-6-c)2-f(i-a-c)3 + (c-a-6)2. 
 
 3. (a-}-6-|-c-d)2 + (6+c+cZ-a)3-|-(c-f£/-{-«_i)2_|_ 
 (d+a+b-c)^. 
 
 4. {a-\-b-\-c)^-a(b-^c-a)-b{a+c-b)-c{a+b-c). 
 
 ■ 5. (x-\-y+z+ny^+{x-y-z+7i)^+{x-y-{-z-ny-\- 
 (x+y-z-n)^. 
 
 6. (a+6+c)3+(a+6-c)3-|-(6+c-a)34-(c+a-Z>)3. 
 
38 SYMMETRY. 
 
 7. (x-2>/-Bz)^-{-{y-^z-dx)2-{-{z-9x-Sy)^. 
 
 8. (ma-\-u/j-\-rc)^ — (ma-\-nb—rc)^ — (nh-j-rr — tim)^ — 
 ' (rc-\-ma — nb)^. 
 
 9. aib-\-r)ib^-i-c.2-a2)-{-'j(c+a)(c2-{-a^ -b^) + 
 c(a+&)(a2+62_c2). 
 
 10. (ai + 6c-+m)3 - 2rt6c(a+6+c). 
 Prove t)ie following : 
 
 11. (ax-\-b;/+rz)''-{-{hx-\-cy-{-azy +{cx+a?j+hz)^ + 
 {nx-\-cy-]-bz)''^-\-{cx-\-l>i/-^az)^-\-{hx-{-ay-\-cz)^ 
 
 = '2{ci'+b^-{-c--){x^--{-y2+z-^)-^i{ab-{-bc+ca){xy^yz-{-zx). 
 
 12. {a-^b+cy-^{b+c-a)^+{c+a-b)^-\-yu-i-b-o)^ 
 
 13. {a-\-b-\-cy = 2a^ + iZa^b+Q-LaH^ + 12.Za^bc. 
 
 14. (£rt)4 = 2rt* + 42rt3/, + 62a3i2 + 122a3ic + 24SrtZ-(,-t^. 
 
 15. (^a^ -\.b^ +c2)^ + 2{'ib+bc + ca)^ -3{a^ -i-b^ +c^)X 
 f^ab + he + ca) 3 = (« 3 + ^ _|_ c 3 _ ^abc) 3 . 
 
 16. («-/>)2(6-c)3 + (^>-c)2(c-a)2 + (c-a)3(rt-/;)2 = ' 
 ^^2 +63 +c2 -ab- ac - bc)^. 
 
 17. (2a-ft-c)-'(26-c-a)2 + (26— c-a)2(2c-a-6)2-f. 
 (2c-a-&)2(2(*-6-c)2=9(«s + />2+6'2-a6_6c-ci/)2 
 
 18. (rtr2+26?-s-|-cs-)((ifx2 + 26a;y + c^2)_ 
 { arx -{-b{ry + sx) + csy }2 = {ac — b^) (ry — sx) 2 . 
 
 19. {a^+ab+b^){c^ + cd+d^) = {aG + ad + bdy + 
 {ac + ad+bd) (be — ad) + {be — ad)-. 
 
 20. Sbew that there are two ways in which the given product 
 in the last example can be expressed in the form p- +iyq+q^, and 
 two ways in which it can be expressed in the form jJ^ —pg+q^- 
 21. 6(ti'2 +a;2 +?/2 +z2)2 = (w + x)^ + (rt'-a;)^ + (»,-+?/) * + 
 
 {iv-'y)^ + {w + z)^ + {w-z)^ + {x+i/)^-\-{x-y)^ + {x-\-z)'^ + 
 
 {x-z)^ + {y+z)^My-^y' 
 
 22. |{(a+i+c)-' + (a-fe-c)6 + (6_c-a)«+(c-a-i)5}=s . 
 i{(a+i + c)3-fi>-^.-e)3 + (i_c-a)3 + (c_a-i)3|x 
 ^{{a+b + c)^ + {a-b-c)^ + {b-c~a)^-\-{c-a-b)2}. 
 
THEORY OF BfVISORS. 39 
 
 Section II. — Theory op Divisors. 
 
 Any expression which can be reduced to the form ax"-]-hx''~'^ + 
 cx^~^+ . . . . -\- . . . . +Jix + k, in which n is a positive 
 integer and a, h, c, . . . . h, k are independent of x, is called 
 a PoLYNOME in X of degree n. 
 
 The ■expressions ./'(ic)", F{xy, <p(x)"', are used as general symbol.-; 
 
 for polynomes ; the index n. m, indicates the degree of the polj- 
 nome. 
 
 Theorem I. If the polynome/(a;)" be divided by a?— a, the 
 remainder will be /(«)". 
 
 Cor. 1. /(x)" —/(a)" is always exactly divisible by a; — a. 
 (Particular case: ic" — a"is alv/ays exactly divisible hy x—a). 
 Cor. 2. If /(a)" = 0, /(a;)" is exactly divisible by x—a, i.e., /{xf 
 is an algebraic multiple of x — a. 
 
 Cor. 3. If the polynome /(as)" on division by the polynome 
 ^(a;)'" leave a remainder independent of x, such remainder will be 
 the value of /(a;)" when <f){x)"' = 0. 
 
 Examples. — Theorem 1. 
 
 1. Find the remainder when x^ —7x^-\-lSx^ — IGx^ + dx— 12 is 
 divided by x — 5. 
 
 The remainder will be the value of the given polynome when 5 
 is substituted for x. (See Art. III.). 
 
 1 -7 +13 IG +9 -12 
 5-10 15-5 20 
 
 1-2 3 -1 4; 8 
 
 Hence the remainder is 8. 
 
 2. Find the remainder when (x—a)^ -{■ (x — b)^ -\-(a^h)^ is 
 divided by x-\-a. 
 
 For a; substitute -a, then ( - 2a)3 + ( - « - h)^ + (a.+b)^ = _ 8^3. ' 
 
 8. Find the remainder vfhen x^^a^-\-h^-\-{x+a){x+b)[a-\-L) 
 is divided by a; + a+6. 
 
40 THEORY OF DIVISORSi 
 
 For X substitute — (« + 6) and we get 
 ~{a+l,)^-\.a^-\.l^J^ah{ci^h) = -1ah{a-^h). See Formnla [6]. 
 
 4. Find'the remainder when {x^^'-lax — '±a^Y{x^-'lax-%i-) 
 + 32(a;-a)4(j;4-a)4 is divided by x'^ - 2rt.3. 
 
 x^ — 2«2 may be struck out wherever it appears. 
 
 This reduces the dividend to 
 
 {'hnxf{-1ax)->r^1{x-aY[xA-aY= - 16a4.^4 ^32(a;3 -a*)*. 
 
 In this substitute 2a3 for x'^ and it becomes 
 -64a«^-32a8=-32a^ 
 which is the required remainder. 
 
 Exercise xvii. 
 
 1. Find the remainder when 3a;4+60a;3 + 54a;3 — 60.'c4-58 ig 
 divided by a; +19. 
 
 2. Fmd the remainder when ^^x^ — Sr/x-'-f-Sj-rc — s is divided by 
 
 3. Wliat number added to ^x^ A- 34a;4 + 58.6-3+21x3 - 123x- 41 
 will give a sum exactly divisible by 2x+13 ? 
 
 4. What number taken from 10a;' ° ~ 20.'c8 -lOaj^ - -SOa;* — 
 8*9a;3+.20will leave a remainder exactly divisible by 10a;'- — 11 ? 
 
 Find the remainders from the following divisions : 
 
 5. {x-\-\Y-x'> ~x + \, and (a;+a+3)3 - (a; + ,/ + l)3 -^a: + 2. 
 ■ 6. a;"+y" -^ x-y; a;2"+?/2» -r- a;+// ; a;-"+i+y-:'i+i -^ ^j-f^. 
 
 7. (a; + l)3+a;3 + (a;-l)3-=-a;-2. 
 
 8. (a;-a)3(x+^f)3 + (a;3-2i2)3 -i-a;3+Z)2. 
 
 -^a;2+-2^Y2. 
 
 10. (9rt24-6a^+i/;2)(9«3_6a6+462)(81a4— 36a2i3 +16i4)-r 
 (3a-2i)2. 
 
 11. a2(a;-a)3+fe2(a;-i)3-i-a;-a-&. 
 
 12. (rta; + %)3-j-«3y3_|_j3jg3_3(^i;jj^^(-(2;^jy^ -^ (a+&)(a; + y)' 
 
 13. a;3 + a3 +. 63 _ 3Q^jj._i_^_^ _j,^. also -i- a; + a— 6 also-*- 
 
THf ORY OF blVISORS. 
 
 41 
 
 H. Anr polynoine divided by a; -1 gives for lemaindev tlie 
 aam of tlie coefficients of the terms. 
 
 Examples. — Cor. 1. 
 
 1. ar-^-f-!/' is exactly divisible by x-\-y. 
 
 In "a;^ - n^ is exactly divisible by ic— «," siibstitnte -y for «., 
 
 2. m.r'^ - px'^' +qx-{-v} -\-p +q is exactly divisible by x-^1. 
 This may be written 
 
 {v.x-- -jjx^ -{-ox} - {iii{-l)^ - p{~l)- +q{-l)} is exactly divi 
 sible by jc- (-1). 
 
 3. (a;3+6a,vy-I-%3)5_|_(a;2_t-2a;?/+4?/2)5 is exactly divisible by 
 
 {r+2y)^. ¥oi- (x^-hQxy-\-iy^y-{-x"-2xy-iy^)'' is exactly 
 divisible by (x^-\-()xy+'iy^)- ( -x-2 _ 2a;// -4?/3), which is 
 2(a:3-4-4a;.v+4?/3 ) = 2(a;4-27/)2. 
 
 Exercise xviii. 
 Prove that the following are cases of exact division : 
 
 I. a;^"+i+y2«+i ^x + y; a;^" -t/^" -f- a,-+y. 
 
 also -^ X- +?/-. 
 
 3. (rt.a;+/^^)' + {l>'^+('yy ^ (« + '')(a; + ?/)• 
 
 5. (22/-a;)"-(2a;-y)"-^3(2/-a;). 
 
 6. {2y-xf''+^-{-{'2x-y)^'"+^^y+x. 
 
 7. {my — nxY — {vix — ny)^ -=- (jh + w) (y — x). 
 
 8. (.r-{-7/)« + (a;-j/)«-^2(a;2+7/2). 
 
 9. (x2+a;//+7/2)3 + (a;2-a;2/ + 2/2)3--2(a;34-//2). 
 
 10. (7 + ?')^-(«-^)^ -^2i(Sr/3 + Z;2). 
 
 II. (-.«-' + 5ix-+/>2)'+(a;2-Z>u; + ^2)T^2(a;+i)». 
 
 12. (a+6)'i»+--^+(«-6)4"+2-- 2(^(3 _|_ia). 
 
 13. {a;3 + 3a^?/(a;-?/)-y3i3 + {a;3-9a://(a;-v/)-2/3}3^2(x-v/)3. 
 
 14. 3j;a-5x2+4a;-2-r-a;-l. 
 
12 THEORY OP Dn'ISORS. 
 
 15. Any polynome in x is divisible by a;— 1 when the sum oi 
 the coefficients of the terms is zero. 
 
 16. Any polynome in x is divisible by ic+1, when the sum 
 of the coefficients of the even powers of x is equal to the sum of 
 the coefficients of the odd powers. (Tiie constant term is in- 
 cluded among the coefficients of the even powers). 
 
 Examples. — Cor. 2. 
 
 1. Show that a(a+2Z>)3 —h{^a-\-hY is exactly divisible bv (i-f-fc. 
 By Cor. 2, the substitution of —6 for a must cause the polynome 
 to vanish. 
 
 Substituting; o(a-2a)3+a(2.'?-fl)3= -a^^a^ = 0. 
 
 2. Show that {ah — xy^^^ — {ci-\-h — x — y)\ah{x-\-y) — X]j{a-^}))\ it. 
 xa,ctly divisible by {x — a)(T/ — a), also by [x — h){y-b). 
 
 For X substitute a and the expression becomes 
 
 {ah-ayY-{b-y){ah{a-\-y)-ay{a-!rb)}^ 
 aHb-y)^ -{b-y){a^h-y)] =0. 
 
 The expression is, therefore, exactly divisible by a;— a. But it 
 is symmetrical with respect to x and y, hence it is divisible by 
 y—a, and as aj — a and j^ — « are independent factors, the expree- 
 sion is exactly divisible by {x — a){y — a). Again, the given 
 expression is symmetrical with respect to a and b, hence, making 
 the interchange of a and b, the expression is seen to be divisible 
 hj{x-b){y-h). 
 
 3. Show that %{a^+b^+c^)-b{a^+b^-^c^){a» ^-b^ + c*) is 
 exactly divisible by a+fe+c. 
 
 For a substitute — (6+c) and the result which would .be the 
 remainder were the division actually performed, must vanish. 
 
 (3|_(64-c)^+65+c5}_5{-(&+c)3+63+c3||(J^c)?^J3+c9} 
 
 = 6|_(ft+c)s + 6^+c^}+306c(6+c)(62+6c + c2). See [1] and [6] . 
 
 The expansion being of the 6tli degree, and symmetrical in h 
 and c, it will be sufficient to show that the coefficients oih^, h^c^ 
 b^c* vanish, the coefficients of b^c^, be*, c* being the coefficients 
 
THK(mY Of DIVISORS. 43 
 
 of tbe foi-mer terras in reverse order. Calculating the coefficients 
 of tiiese type-terms we get 
 
 6{_5/;4c-1063c2.-...}+30(64f + 263c3-l-...), 
 
 which evidently vanishes. Hence the truth of the proposition. 
 
 In the last example it lias been proved that the dilference of the 
 (quantities here declared to be equal, is a multiple of a + 6+c, i.e., 
 in this case, a multiple of zero. Hence under the given condition 
 they are equal. 
 
 Exercise xix. 
 
 Prove that the following are cases of exact division : 
 
 1. (ax-b7jy + {bx-ay)^—{a^+b^){x^-y^)^a,b,x, y, a-\-\ 
 x-y. 
 
 2. ax^ — (a2 -f 6)a;3 +52 ^ ox—b. (Substitute ax for b.) 
 
 o \ (ax+by)^ - {a- b){x+z){ax+b7j) + {a-b)^xz -^ x+y. 
 ^' \ {ax-by)^ -[a + b){x-{-z)(ax-by) + {a + b)^xz^ x + y. 
 
 4. da^x^—iax^ — l'^^axy-~oa'^xy + 2x^y-{'Oy^ -i-2ax-y, 
 
 5. l-2a*x-16-22a^x^ +^-8a^x^ -{--dax^ - x^ -^ •Gax^2x^. 
 
 6. x^+x^y^+x^y+y^ ■^x'^^y. 
 
 7. {o-d)a^ + 6{bc-bd)a+9{b^c-b2d) -^ a + 3b. 
 
 8. itix-^^yy+yi^^r-^-yy -'^x-y. 
 
 9. a(a+26)3-6(6 + 2a)3 -^a-h, ;dso -=- a+b. 
 
 10. a5-2rt4^)+a363+a3a;3_2rtaa;3+63a;3 -^ (a-h)(x-ha). 
 
 11. a(6-cj3+6(c-a)3-f c(a-6)3 -- (a-b), {b-c), {c-a). 
 
 12. a^{b-c) + b^{c-a) + c^{a-b) -r (a-b), {b-c), (c-a). 
 
 13. a*(6-c) + 6*(c-a} + c4(a-6) -^ (a-b), (b-c), (c-a). 
 
 14. (a-6)2(c-d)3 + (6-c)3(cZ-a)2-(rf-6)3(a-.)2 ~ (a-b), 
 {b-c), {c-d), {d-a). 
 
 15. {{a-b)^+{b-cy' + {«-a)-'}{{a-b)^rJ + (b-c)^a^ + 
 (c-a)363}-{(a-6)3c-i-(6-c)2a + (c-a)2i}3 ^ (a_i), (i-c), 
 
 (e—a). 
 
 16. (:s+t/)(t/+s)(2+a;)+a;2/z-ra:+y+a. 
 
44 
 
 THEOKY OF DIVISOKS. 
 
 18. (cib - bc-ca)l-a^b^-h2c2-c^u^-.a-\-b-c. 
 
 19. (« + 26)3 + (26-.3c)3-(3c-a)3+a34.s^3„27cS + 
 +2^»-3r. 
 
 20. a^b^-hb^c3^cSa3 -Sa^li^c^~<ib^bc-Jrca. 
 
 Examples. — Cors. 3 and 2. 
 
 1. Find the value of Ax'" -^-^x^ - ox^ + 23a;+6 when 1x^ = 3ar - 4. 
 
 Since 2a;2 — 3a; -{-4 = 0, we have simply to find the remainder on 
 division by 'Ix^-'Sx+A, and if it is iudependent of «;, it as th( 
 value sought. Cor, 3. 
 
 
 4 
 
 
 
 9 
 
 — 5 
 
 23 
 
 6 
 
 3 
 
 
 6 
 
 9 
 
 15 
 
 -3 
 
 
 -4 
 
 
 
 -8 
 
 -12 
 
 -20 
 
 4 
 
 2 3 5 - 1; 10 
 
 Hence the required value is 10. 
 
 2. What value of c will make x^ — ox^ +7a; — c exactly divisible 
 by a;— 2. 
 
 If 2 be substituted for x, the remainder must vanish. Cor ^. 
 
 1-0 7-6- 
 
 2-6 2 
 
 1; 2-c 
 
 1 -3 
 
 Hence 2 — c=:0, or c = 2. 
 
 3. What value of c will make 6x^ — 5x*+cx^—20x''^19x-5 
 vanish when 2a;2=3a;_i ? 
 
 By Cor. 8, the remainder must vanish when the given poly- 
 nome is divide by 2x^— dx+1. We may divide at once and find, 
 if possible, a value of c that will make both terms of the remainder 
 vanish, or we may fii'st express cx^ in lower terms in z, and 
 then divide and find the required value of c from the remainder. 
 
 1st. Method, (see page 28), 
 
 6-10 Ac -160 304 -160 
 
 3 18 24 12C+36 ,S6c-420 
 
 -2 
 
 -16 -8c -24 
 
 8 4c+i2 12c -140; 28c- 140 
 
 24c -{-280 
 •24C+12U 
 
THEORY OF DIVISOKS. 45 
 
 Hence 28c = 140 and 24c = 120. Both of these are satisfied by 
 
 2nd Method, x^ = ix{Sx -1)= ^x^ - ix = f (3.t- 1) -^x = 
 2^a;-f — ia;=l|a;-i ; .-. cx^ = IJca;— ^c. 
 
 Substituting for c.v^ in the given polynome it becomes 
 
 6x^-5x*-20x^ + [lic + l9)x-ic-5. 
 Divide and apply Cor, 3. 
 
 6 -10 -160 28c + 304 -24c- 160 
 
 3 18 24 36 -420 
 
 _2 -12 -16 - 24 280 
 
 6 8 12 =ri"40; 28c -140 "^^24^+120 
 
 We thus obtain the same remainder as by the former method, 
 and consequently the same result. A comparison of the two 
 methods shews that they are but slightly different in form, but 
 the second method shows rather more cleaiiy tliat c need not be 
 introduced into the dividend at all, but the proper multiples of it 
 found by the preliminary reduction can be added to or taken 
 from the numerical remainder, and the "true remainder" be 
 thus found, and c determined from it. 
 
 Exercise xx. 
 
 Find the value of 
 
 1. x* -3a;3 + 4a;2 — 3a;+4, given x2 =a;_i. 
 
 2. x^-2x^-4x^ + 13x^-nx-10, given (a;-l)3 = 2. 
 
 3. 2x'' -lx'^-{-l2x^-llx" + '2x~5 given (x-l)2+2==0. 
 
 4. 3a;«+lla:5+10.'c3+7a;2+2a; + 8 given a;? + Sa:^ -2a;T5 = 0. 
 
 ' 5. 6a;'+9a;*5 -16a;* -0x3- 12a;3_ 6a; + 60 given 3a;4+a;— 4 = 0. 
 What values of c will make the following polynomes vanish 
 under the given conditions. 
 
 6. a;* + 13a;3 + 26a;2+52a; + 8c, given a;+ll=0. 
 
 7. a;4-2a;3— 9a;2+2ra;-14, given 3a;+7 = 0. 
 
 8. a;* - 4a;3 - a;3 + 16a; + 6c, given a;3= a; +6. 
 
 9. 2a;*-10x2+4ea;+6, given a;2 + 3 = 3x. 
 
 10. 2a;4+a;3-7cx2 + lla;+10, given 2a; = o. 
 
46 THBORy OF DIVISORS. 
 
 11. 4.1-4 + ra;2-f- 110a;- 105, given 2.t3 - 5a;-}- 15 = 0. 
 
 12. Sx^-lijx^-+cx--i-5x^-lUx+200, giveu x^ = 3x-4. 
 
 13. What values of /_> aud q \f ill make x'^ + ix^ -lOx^ -j'x+q 
 vanish, given x^ = 3(a; — 1) ? 
 
 14. What vahies of p and ^y will make a^^ -5a^" + lOrt » - 15rt '• 
 +29a4 -/^rt2 +,^ vanish, given (a^ -2)2 .=a2 _3 '? 
 
 Theorem II. If the i^olynome /(a;)" vanish on substituting 
 for a; each of the 71 (different) values a-^, a^, a^ . . . . a„ 
 
 f{xY = A{x — a^){x-a^){x~a^) .... (a; — flj 
 in which A is independent of x aud consequently '6 the coefScient 
 of a;" in /(a;)". 
 
 Cor. If /(a;)" aud 9(3;)"* both vanish for the same m different 
 values of x, f{xY is algebraically divisible by <p{x)'^. 
 
 Examples. 
 
 1. x^+ax^+bx+c will vanish if 2, or 3, or —4 be substituted 
 for X, determine a, h, c. 
 
 The coefficient of the highest power of a; is 1 ; 
 
 .-. a;-'^+(/a;^+6a;+c=(a;-2)(a;-3)(a; + 4)=.a;3 -x-3 - 14a;+24. 
 
 .-. a= ~ i: b= -14: <; = 24. 
 
 2. x^+bx^+cx-\-d will vanish if —3 or 2, or 5 be substituted 
 for X, determine its value if 3 be substituted for z. 
 
 The given polynome =(x+3)(a; — 2)(a; — 5) ; 
 
 .-. the required value is (3 + 3)(3-2)(3- 5) = -12. 
 
 3. (ta;3+3/;a;2 + 3ca;4-rf will vanish if for a; be substituted —3, 
 or I;, or 1^, but it becomes 45 if for x there be substituted 3 ; 
 determine the values of a, b, c, d. 
 
 The coefficient of the highest power of a; is « ; 
 
 .-. aa;3 + 36.r2 + 3ra; + ^Z = rt(a;+3)(a;-i)(a;-l^) 
 .-. a(3 + 3)(3-i)(3-U) = 45; .-. a = 2. 
 .'. 2a;3 + 3/'x2 + 3ca; + <.' = 2(a; + 3)(a;-i)(a;- li) 
 ... b = ^, c= -3i, d= 4i 
 
THEORY OF DIVISORS. 47 
 
 4. Ir' x^ -^j'X- -\-(]x-{-r vanish for x^a or b, or c, determine p, q, 
 and r in terms of a, b, c. 
 
 x" -i-px- +qx-\-r={x — a){x - b){x - c) 
 
 = x^ — (a+b + c)x'^ -\-{ab + br + <-Ci)x — abc 
 :. p= — (rt-f /) + c) or —2". 
 q= ab + bc-\-ca or ^nh 
 r = — abc or — 2 "^c. 
 
 5. U x^ +px^ -{-qx+r vanish for x=n, or b, or c, determine the 
 jiolyaome that will vanish for x = b + c, or c + a, or a + b. 
 
 Since x^-{-2)x^ +qx+r vanishes for x = n or I) or c, 
 
 a;3 —px'^-{-qx — r will vanish for x= —a or —6 or — c, 
 
 and — /' -— a-\-b+c; 
 But the required poljmome will vanish for 
 
 x= —p—a, or —p — h, or — p— c; 
 that is, for x-\-p= -a, or —ft. or — c. 
 Hence it is (x+p)^ ~p{x+p)^+q{x+p)—r = 
 x^ +'lpx^ -\-{j}^ + q)x-{-pq —r. 
 
 The following is the aalculation in the last reduction. (See 
 page 31). 
 
 
 
 -p 
 
 9 
 
 — r 
 
 p 
 
 
 
 
 ? ; 
 
 2^q-r 
 
 p 
 
 
 p; 
 
 p^ + q 
 
 
 p 
 
 1 • 
 
 2p 
 
 
 
 6. In any triangle, the square of the area expressed in terms of 
 the lengths of the sides, is a polyuome of four dimensions ; and 
 the area of the triangle, the lengths of whose sides are 3, 4, and 
 5, respectively, is 6. Find the polynome expressing the square 
 of the area. 
 
 Let a, b, and c be the lengths of the sides, and A the required 
 polynome. 
 
 1st. The area vanishes if any two of the sides become together 
 equal to the third side, hence ii a + b = c, A = 0, and consequently 
 A IS divisible hy a -\-b — c. Similarly it is divisible by b-\-c-a 
 and bv c+a — b. 
 
48 THEOKY OF DIVISORS. 
 
 2ncl. Tha area vanishes if the three sides vanish together, 
 hence if a-{-b-}-i- = 0, .-1 = 0, and consequently A is divisible by 
 a-i-h + c. 
 
 We have .thus found four linear factors, but A is of only four 
 dimensions. 
 
 .-. A = vi(a + b + c){b4-c-a){c+a-b){a+b-c), 
 in which w is a numerical constant. 
 But 63 or 3G = (»(3+4+'5)(4-f 5-3)(5 + 3-4)(3+4-5) 
 = 576;;/ ; .•. m = ^^. 
 
 (The above includes all the ways in which the area of a triangle 
 can vanish, for the vanishing of only one side involves the equal- 
 ity of the other two, or if a = 0, b = c, and .*. a-\-h — c, which is 
 included in 1st. ; if two sides vanish simultaneously, the three 
 must vanish). 
 
 Examples on the Corollaky. 
 
 7. Prove that (x+l)'^ —j?' ^ —2«- 1 is divisible by 
 
 2x3 + dx-+x. 
 Factoring the latter expression we find it vanishes for x — 0, or 
 — 1 or — ^. Substituting these values in the former polynome, 
 it also vanishes. But these are different values of x, hence the 
 truth of the proposition. 
 
 8. (x-f ?/+ 2)"' -x^ — y^—z^ is divisible by 
 
 {x-\-}j+z)^ — x^ — y^ —z^. 
 
 The latter expression vanishes if a;= —y, so also does the former. 
 
 By symmetry they both vanish if ?/= — 2 and ii z=—x. Hence 
 they are both divisible by (x-\-y){y+z){z+x). But this expres- 
 sion is of three dimensions, as also is the latter of the given poly- 
 nomes, hence it is a divisor of the former. 
 
 9. Prove that { 0/4-6)^ -f(c + ri) 5} (rt- 6) (c-./) + 
 
 {{b+cy' + {a + dy}{b~c)[a-d) + {{b + dy+{c-{-ay}{b-d)(c-a) 
 is algebraically divisible by {a--b)(c — d)[b — c)[a ~d){b — d){c — a) 
 X (« + 6 + f + '0> ^^^^ ^^^ ^^1^ quotient. 
 
 Let a = b and tlie former polynome reduces to 
 [{a+cy+{'i-^dy'}{a-c){a-d) + {(a+d)-''-[.{c-\-n)^\{a-d){c-a) 
 
THEORY OF DIVISORS. 
 
 49 
 
 which vanishes, the second complex term differing from the first 
 only in the sign of one factor, having (c — a) instead of {a - c). 
 
 Hence the former polynome is divisible by a — b, and by sym- 
 metry it is also divisible by a — c,hya — d,hjb — c,hyb — d, by c — d. 
 
 Again, {a + b)^ +{c-{-d)^ is divisible by (a + h) + {r -r d) ; for, on 
 puttinga+6= -(c-t-f/), it becomes {—{<■+<! )]■'■ +{c + d)^ which 
 = 0. 
 
 Similarly the other terras of the former of the given polnomes 
 are each divisible by a-^b-\-G+d, and consequently the whole is 
 so divisible. 
 
 Now all these factors are different from each other, hence the 
 former of the given polyuomes is divisible by the product of these 
 factors, i.e., by the latter of the given polynomes. 
 
 Both of these polynomes are of seven dimensions, hence their 
 quotient must be a number, the same for all values of a, b, c, d. 
 
 Put.< = 2, 6=1, c = 0, d=-l, and divide. The quotient will 
 be found to be —5. 
 
 ... i^(a + hy'-{.{c + d)^}{a-b){<,--d)-\-{{h + r)^ + {a + d)^} X 
 {h_.c){a-d) + {{h-^d)-''+{c + a)^}{l>-d){c-o)= -o{a-b]{c-d) 
 X{b-c){a-d)il'-d){c~a}{a -{-!>-{-,,• -h'l). 
 
 N.B. — It is not always necessary to find the factors 
 of the divisor, as the following examples show. 
 
 10. Prove that x^ +x+l is a factor of 3;^ 4+j-'' + 1. 
 
 .f 2 4-x+l will be a factor o? z^'^-\-x'' +1 provided 
 
 a;i 4 _}.a;7 _f. 1 = if x2 +X + 1 = 0. 
 
 Ifa;2+a;+l =0 
 
 .-. x^+x-^x =0 
 
 .-. x^+x--{-x+l = l 
 
 .'. «3 =1 
 
 .. .* = landa;i2 = l 
 
 x^ =x and x'^'^ =x'^ 
 .. ^i4+a;'+l =.x^+x + l = ^ 
 ,■ , x'^ 4-a;-f 1 is a factor of x^ * +j;' -{-1 . 
 
50 THEORY Od" divisors. 
 
 Art. XII. Two other methods of proving this proposition 
 are worthy of notice, 
 
 1st. x^+x + 1 will be a factor of x^^-\-x'' +1 provided it is a 
 factor of {{x'^'^+x'' +1) ± a multiple of {x^+x + l)\. 
 
 x''- "^ -\-x' -{-I differs by a multiple of x'^ +x-4-l from 
 x^^+x^'{x^-+x + l)+x^{x''+x-hl)-\-x'+x'^{x'-^+x + l) + 
 x[x'^+x + l) + l 
 
 = x^-'[x^ +x-rl)+x^{x^-\-x-\- i)-\-x'-{x-^-\-x-{-l)^ x^[x-' +x-rl) -{- 
 
 (x^+x+l) 
 = {x^^+x^+x''+x^-\-l){x-^-tx-^l). 
 
 Hence x^+x + 1 is a factor of «*'*+.<;• -fl. 
 2nd. — = . — 
 
 (x^i-l){[ xi^-l)- x{ x^^-'l )} 
 
 (a;^^-l)(a ;'^'-l) _ x{x^^ -l){x ^i -I) 
 {x^-l}{x^^ {x^-l}{x'^'^^; 
 
 Bitt we see at once that on reduction both of these fractions 
 give an integral quotient, hence {x^'*^-\-x'' -\-l) ~x^ -{-x + l gives 
 an integral quotient. 
 
 11. x^-\-x+l is a factorof (x + 1)'' -x^ -1. 
 
 If a;-+.« + l = 0, {x + iy -x'' — 1 will vanish also, for in such 
 case aj+l = —x^. 
 
 .-. {x-\-iy -X' -1 = {-X^)' -X' -1= -x-14_^7 _l^ 
 
 which by the last example vanishes if x''-^-\-x+ 1=1); 
 .-. a;-+.i + l is a factor of [x-[-iy —x'' 1. 
 
 For X substitute — and multiply by i/^ and ?/^ respective! v 
 
 y 
 
 a.nd this example becomes 
 
 ^^-\-X!J+i/^ is a factor of {x+y)'' —x'' —i/'. 
 
THEORY OF DIVISORS. 61 
 
 Exercise xxi. 
 Determine tlie values of n, b, c, d, e, iu the lollowing cases : — 
 
 1. a;3 + 36.u2 4-3(M—|-// vanishes for x = 1, or 3, or 4. 
 
 2. x*^ -it-cx^ +dx + e " " a;=l^or —3 or 4^. 
 ^. x^+hx-+cx + 'i^ " " r/; = 2or-3. 
 
 4. .^6-">-f/;a;2+rx+90. " " a; = 3or-5or2. 
 
 5. ax^+>-jyi -?>Qx + e. " " x= 1|- or -4, or 2^. 
 Q. Qlx'^ + iScx'+Ux + e " " a;=l| or -81- or 1^. 
 
 I. iix'^+1>x^-\-cx'^-'dl " " .^• = for|or3. 
 
 8. ax'^+cx^ + ilx+e ♦* " x = 2 or 1^ or -1 and be- 
 comes 14 for x=l. 
 
 9. ax^-\-cx-\-d vanishes for x=l\., or 2|, and becomes 49 for 
 aj = 3, determine its value for x= —Q. 
 
 Given that x^ - px^ +qx — r vanishes for x = a, or b, or c, deter- 
 termine the polj'nome that vanishes for 
 
 10. x = tt-[-l, ovb + 1, or t' + l. 
 
 II. .f = a — 1, or i— 1, or c— 1. 
 
 Ill 
 
 12. x=l - — , or 1 — -T", or 1 — — . 
 
 13. x = ((b, or be, or ca. 
 
 lA. x = a^, or b^, or c^. 
 
 ( r ) 
 
 15. x = a{b + c), or b(e + a), or c(a+b). Ui{b-\-c) = q 1. 
 
 ( " ) 
 
 a4-b b4-c c-\-a («+^ 
 
 16. a;- — or — - or -f— . ' ^ 
 
 c a b 
 
 Prove that the following are cases of exact division : 
 
 17. (a;-l)i3_a;G4-(a;2-x-i-l)2 -x^ -2:k^ +2x- 1. 
 
 18. (x-l)i«--a;« + (:^2 -a;+l)«^a;3-2a;2 + 2.c-l. 
 
 19. (a;-2)i"(2a;-5)'0-a;i" + 2iO(x2-4a; + 5)5^ 
 
 a;3_6«2 + 13x-10. 
 
 20. (x2 + 4.« + 8)i«-ic'^-3;--5x-3-a;3 + Ga;2+8.f- + 3. 
 
 21. (9x-4)2 i (a;- 1)2 '-•a;2 i - {^x"- - 14a;+4)2 ^ ^(;c_ i) x 
 (9x-4)(9a;'^-Ma; + 4). 
 
 22. {a(.'K-l)f '3_(.2a;3 + 3a;-4)'3 + (2uj2_3;c + 2)i3^ 
 (2a;2 +3x - 4)(2a;2 - 3a; + 2)(a;- 1). 
 
52 THEORY OF DIVISORS, 
 
 23. {2(x+l)(x-2)}'^''-\-{x^-Sx+dy^-(3x^-ox-iy-^ 
 {X'^lrx-2){x^ -Sx+3){dx^ -5x^1). 
 
 24. {6(a;-l)]i6-(2a;3+3x-4)i«-(2x2— 3.c+2)i« + 
 2{2x^+Sx-A)^2x^ -'3x+2)^-^{x-l)(2x^+dx-A)[2x'^ -Sx+2) 
 
 25. {2(a;+l)(a;- 2)[ 20 _ ,,.3 _Sx+B)^'' -('Sx^ - 5x- 1^ + 
 2(x^' -3x + d)\3x^-5x-iy^'^{x+l){x-2){x^ -3x-^'3) x 
 (3a;2-5a;-l). 
 
 26. l+x'^+x^ -^ l-f.r4-a;8. 
 
 27. a;io +0:5^5 + //! -^ a;2+a;?/ + ?/2. 
 
 28. l + a;3+a;«+a;9+x'2 -^ l + x+.t:2+«3*+a:4. 
 
 29. l4-x4+ic«+.fi24-a;ie -^ i+x+x^+x^+x"^. 
 80. a;i^+s;iO//-5+a;5iyio^7is ^ ^3_|.;^2_,^_j.a.^2_}_y3; 
 
 31. x^' +x'>'+x^-\-x+l ^ x^+x^+x^+x + 1. 
 
 32. l+x+x'+x^+x^'+x^'+x^^ ^ 
 
 l+x+x^-\-x'^+.c'^+x'-+x^. 
 
 Find tbe quotient of the following divisions in which D denotes 
 the product 
 
 {b-c){c-a)ia-b)(a-ri){b-d)ic-d) ; 
 
 33. (62c2 + «V2)(6-c)(a-(/) + (c3a2 4.62(Z3)(c_a)(6-rf)4- 
 {a^b^ +c^d'-'){a-b){c-d) -f- D. 
 
 34. (fcc+ad)(62 _c2)(«2_^3)+(ca + 6cZ)(c2-a2){62— d2) + 
 (ab+cd){a^-b^){c^-d^) H- D. 
 
 35. (i + c)(a+'Z)(i2_c2)(fl2_t?2)^the two similar terms -=- D. 
 86. (/>2+c2)(a2+r/2)(i-c)(a_</)+ ". '* -i- D. 
 
 37. {bc{b + c)^+ad{a + d)^}{b-c){a-d)+ " -^ D. 
 
 38. {bc{b + c)+ad{a + d)}{b^-c^){a^-d^]+ " -7- i*. 
 
 39. {bc(b^+c^)+ad{a^-{-d^)}{b-c){a-d)-{- " -^7). 
 
 40. (Z;+c-a-rf)^(&-c)(«-(/)+ " -^ D. 
 
 41. The sum of the fractions |, |, i, \, increased by the 
 
 sum of their products two by two, increased by the sum of their 
 
 products three by three, increased by their product is 
 
 equal to n. 
 
^3 
 
 THEORT OF DIVISORS. O 
 
 42. Ill any trapezium the square of the area expressed in terras 
 of the lengths of the parallel sides and the diagonals, is a poly- 
 nome of foiu* dimensions, determine that polynome. 
 
 43. In any quadrilateral inscribed in a circle, the square of the 
 area expressed in terms of the lengths of the sides, is a polynome 
 of four dimensions, find that polynome. 
 
 Theorem III. If the polynome /(a:)" vanish for more than 
 n different values of x, it vanishes identically, the coefficient of 
 every term being zero. 
 
 Cor. If a rational integral expression of n dimensions be divi- 
 sible by more than n linear factors, the expression is identically 
 zero. 
 
 Examples. 
 
 {x-a ){x-h ) (x-b){x-c) {z-c){:v-u) 
 
 1- ^c_a)(c-/>) + {a-b){a-c) "^ (b-c){b-a) "r^-"' ^^ «' 
 b, and c are unequal ; for this is a j^olynome of two dimensions in 
 X, but it vanishes for x = a, aud, therefore, by symmetry for x=b, 
 and for a; = c, that is, for three different values of x, hence it 
 vanishes identically. 
 
 2. \{a + h)^-h{'-\-<l)^](^-b){c-d) + {(c-hh)2 + {h + d)^ 
 {b-c){a-d) + {{c-{-'()-+('' + dy}{c-a){b-d) = 0. 
 
 Substitute b for a p.nd the expression becomes 
 
 {{b+cy^Mb+dr^}{b-~c){b-d)-h{{c+by' + {b+dy\[c-b){b-d) 
 
 which vanishes, hence the given expression is divisible by a— 6, 
 and consequently by symmetry it is divisible by («—?;), (b-c), 
 (^c — d), {a-c), (b — d), and (a — d), But the given expression isof 
 only four ditaensions, while it appears to have six linear factors, 
 hence it vanishes identically. 
 
 Exercise xxii. 
 
 Verify the following : 
 
54 
 
 THEORY OF nwiSOBS, 
 
 
 1 
 
 {x+a)[x-\-h){x^-c)' 
 
 Q o+x a-\-y a- ]-z ^ a^ 
 
 x{x-y)(x—z) "^ y{v-x) {y - z) z{z - x) {z - y) ~ xyz 
 
 a'{h-c)+b-[c-a]+c-{a-h) 
 8. {iiclf+h.-f-\-he(l-acc)^-\-{hce-\-aed-^acf—hilj)- =3 
 
 {a-b){i,-c){c~<i) ~ 
 
 10. (-a:+?/+z)(x — 7/+2)(a:+y— ■• -^xix — y+z^ix-Vy-z)-^ 
 il{x+y-z){-x + y+z)+z{-x + y\-::)\x-y-\-z) = 'ixyz. 
 
 (a3-/;S)3-|-(/;3 -c2)3_ |_^ C^_ ^2)3 
 
 • (a-+T)(6+c)(c+a) "" ; - 
 
 (a-i)3 + (/y-c)3 + (c-a)3,. 
 
 12. x-{y + zY-{-y-[z^xy'-^z^{x+yY+2xyz{x+y-\-z) = 
 2{xy+yz-\-zx)-. 
 
 Theorem IV, If the polynomes /(a;)", f (a;)'" (n not less than 
 m) are equal for move than n different vAliics of x, they are equal 
 for a// values, and the coefficients of equal powers of x in each 
 are equal to one another. 
 
THEORY OF DIVISORS. 55 
 
 (This is called the Principle of Indeterminate Coefficients, The 
 full use of it cannot be exhibited till the student is able to work 
 simultaneous equations.) 
 
 Examples. 
 
 + 71— :^7T— a71— ,T> + 
 
 {a-h){a-c){a-d) ^ {b-a){b- c){b- d) 
 
 {G-a){G-b){c-d) "*" (7i"(iy(7Z-6)(rf^j ^ ^' 
 Assume 
 
 {x-a){x — b)(x-c){x—d) 
 
 A B CD , > 
 j^ . I 1. (a) 
 
 x — a x-b x—c x—d 
 
 in •which A, i>, C, D are independent of x. 
 
 Mnltiplv by {x-a){;x-b){x-c){x-d). 
 
 :. x^ = [A 4- B-{-C-\- L')a;2 +terms m lower powers of x. 
 
 Now this equality holds for more than three values of x, hold- 
 ing in fact for all finite values of x. 
 
 Again multiply both sifes of (or) by a;—;6 
 
 x^ ,, I B C D \ , . 
 
 (x — b){x — c){x — d) \x — b x — c x — d 
 
 Put x = « 
 
 (a — b){a — c){a -d) 
 
 By symmetry — ^ = b, &c. 
 
 {u—,a)[b — c)[b — d) 
 
 Adding 
 
 «2 62 c2 
 
 + 7Y. :Y7, x-71 T^ + 
 
 {a-b)\^a-c){a-d) ^ {b-a){b- c)[b -d) ^ {c — a){c-b){c-d) 
 
 + id^:aji^b)ld-c) = A^B+C+D = Ohyi&). 
 
 2 a^(a + b)ia+c) b^{h+c)ib+a) c^{c+a){c + b) 
 
 {a-b){a-c) "^ (b-c){b-a) "^ (c-aXc-6) 
 ;=(a-f 6 + c)-, 
 
58 
 
 THEOHY OF DIVI90ES. 
 
 17 a( ^ + h){a+c){a + d) ■ -, . 
 
 *■ ' • 7 7w w T\ + three similar terms* 
 
 a -(a+b)(a+c){a^d) ^^ ^^ 
 
 •'°' {a-b){a-c)(a-d) "*" 
 
 aS(a+b)(a+c){a + d) ^^ 
 
 ^^- {a-b)[a-c){a-d) "^ 
 
 bc{b-\-c) 
 
 20. , Y^, - , +two similar terms. 
 
 [For numerator use x^-\-2px--\-ip^-^q^x-\-('jiq - r).] 
 
 (2a+6:(2a + c) . ' 
 
 21. -7 fw r- + two similar terms. 
 
 (a— f>)(a— c; 
 
 [For numerator use x^ —'lpx^^^^(J)^■\^q)x — (yq — r^^ 
 
 + two similar terms. 
 
 " {(.t — b){a—c) 
 
 [For numerator use a;(a; + ;j).] 
 h-\-c.-\-d ■ 
 
 23. 7 Tw \7 j\ + three similar terms. 
 
 {a — o){a — c){a —a) 
 
 a^(hc-\-cd + db) 
 
 "^- (a-6)(a-c)(a-rf) ^ 
 
 hc-i^cd-\-db 
 
 "^^^ (fl--6)(a-c)(a-J) ^ 
 
 Extract the square-root of (to 4 terms) : » 
 
 26. l+x. .1 27. 1-a;. I 28. l + 2.^+3a;*+4z3 -t &c. 
 
 29. l-4a;+10x2-20a;3 + 35a;4-56ic5+84a;«. 
 
 30. Extract the cube-root oil+x. (To 4 terms). 
 
 Art. XI. 1. Find the condition that px^' + ^qx+r andi p'x^ 
 + 2q'x+r' shall have a common factor. 
 
 Multiply the polynomes hj p' and p respectively, and take the 
 difference of the products, also by r' and r respectively, and 
 divide the difference of the products by x. 
 
 p 'px^ -{- 2p 'qx 4-/J 'r 
 pp'x^ 4-2pq'x+pr' 
 
 2( pq'—p'q)x+{pr' —p'r) 
 
 p7-'x^ +2qr'x-\-7r' 
 p'rx^ + 2q'rx-{-r'r 
 
 (J)r'-p'r)x + 2{qr' -r'q). 
 
 Multiply the former of these remainders by (pr'—p'r) and th$ 
 latter by ^{pq'—p'q), ^'^'^ the difference of the products is 
 ^^r'-p'r)^-^pq'-p'q){(p--r'q). 
 
THEORY OF DIVISORS. C9 
 
 But if the given polynomes have a linear factor this remainder 
 must vanish, or 
 
 ipr' —p'r)^ =4:(pq' —p'g){qr' — r'q). 
 If the given polynomes have a quadratic factor, the linear re- 
 mainders must vanish identically, or (Th. III.) 
 
 pq'—p'q — O, pr' —p'r = Q, and qr< — r'q = 0, 
 
 par 
 ox, ±- = J- = — 
 
 p' q' r' 
 
 2. Find the condition that px^ -\-3qx^ +Srx-^s shall have a 
 square factor. 
 
 Assume the square factor to be (x — a)^. On division, the 
 remainder must be zero for every finite value of x,- and conse- 
 quently (Th. III.) the co-efficient of each term of the remainder 
 must be zero. Divide by (x — a)^, neglecting the first remamder. 
 
 P Sq 3r $ 
 
 a * pa pa^ -\-dqa 
 
 p pa + 3q pa^-{-Sqa-\-3r ; R 
 
 a pa. ^pa"^ +dqa 
 
 \~~p ^a-\-3q ; 3{pa^ -\-2qa+r) 
 
 .•. pa^-\-2qa-\-r = 0; 
 
 :. joaj^+^aj+z- is divisible by a; — a (Th. I. Cor. 2), 
 
 or, px^ +3qx'^ -\-3rx-\-s and jDa;2+25a;+r have a common divi- 
 sor. Multiply the latter polynome by x and subtract the product 
 from the former, and the proposition reduces to 
 
 lipx'^-\-3qx^ -\-orx-{-s have a square factor, ^;a;--f-2g'a;-|-r and 
 qx--\-^lrx-\-s will have the square-root of that factor for a com- 
 mon divisor. 
 
 3. If joa;3 + 3ga;^4-3ra;-|-s vanish for a; = a, or h, or c, find in 
 terms of x, p, q, r the value of 
 
 X — a x — b X — c 
 Eeduce to a common denominator and add the numerators 
 
 • 3x^ -1{a + h + r)x + {ab -irhc+ca) 
 
 [x — a}{x — h){x — c) 
 
60 THEORY OF DIVISOES. 
 
 Multiply both numerator and denomiuator by p and reduce by 
 Tii. II., and Ex. 4 of Th. 11. 
 
 px^ -\-''dqz'^ -\-'6rx-\-s 
 
 a.m+1 y,m+\ rj,m+l ^ 'd{px'"-+^ + 2qx''^+'^-\-rx:^+'^) 
 
 ' ' X — a x — b X — (■ px^ -\-'iqx^ -\-or.c-\-s 
 
 4. li 2}x^ -VQqx^ -\-^rx-\-s vanish foi- x = a., or h, or c, express in 
 terms of/>, q, r, s, the following, a + 6-}- e, a~-\-h~ +c^, a^ + h^ -\-c^ 
 , a"* + 6'" +<;'«. 
 
 Divide a;"'+^ bv x — a. 
 
 1 
 
 a 
 
 a ft' a^ ft™ a"^+i 
 
 1 a a2 ^s ^m . a"^+^ 
 
 Similarly divide x''"'+'^ by x—b and also by x—e. 
 add together the quotients 
 
 /^m+l r^m+1 ajW+l • 
 
 1- ■ 7 + =3x"' + (a + b + r)x'"-' + [a- + b^ + e^hf-^ 
 
 X— a .V— h X — G ^ ' ^ ' 
 
 +(rt.34-63+c3)a;'"--rf- ,1'c. 
 
 Hence, by tlie last example, the required expressions are the 
 coefficients taicen in order, beginning with the second, of the 
 terms in the quotient of 3(/>a;'"'+3 +29'a;""''+^+rx"'+^) -=- {iix^ -^-^qx^ 
 + 3/-a;4-«). These may now be found by Horner's Division. 
 
 5. Writing s^ for a+/'+c, s^ for ii~-\-h"-^c-, &c., express 
 (a— 6)4+(6— c)4 + (c — fl')4 in terms of s^, .Sg, Sg, s^. 
 By actual expansion 
 
 {x-a:)^-\-{x - 6)4 + (a;-r)4 = 
 
 3x4-4(a+/. + c)a;S+G(«2+i2_j.c3)a;2_4(„s_|.^3^c3)a-4- 
 <j4 ^ ^4 _i_ (:4 = ga.4 _ 4s^a;3 -f Gsgic^ - 4.v3rc + s^. 
 
 Puta; = «, = i, =c in succession. 
 
 (a_6)4 _|_(c-fl)4 =3^?4 _4s^a^+6s2fl2_.4.<,^,,^,,_^ 
 
 (ft_c)4 _[_((._rt)4 =3r4 -4SiC3+6s2c2 -4s3r+S^ 
 
 ... 2{f«-&)4 + {i-r')4 +(f:_a)4}=3.S4-4.s,.^3 + 6.s:--4s3S,+3.v^ . 
 in which s^ is written for 3 or 1 + 1 + 1, i.e., a^ + Z^o+'j". 
 
THEORY OF DIVISOBS. 
 
 61 
 
 Exercise xxiii. (a). 
 
 1. Determine the condition necessary in order that x^ frV^-\-9. 
 and x^-{-p'x+q may have a common divisor. 
 
 2. The expression x^-\-Za^x^+Zbx*-{-cx^-\-Mx--\-^e^x-\-P 
 will be a complete cube if 
 
 e d c — a^ 
 
 •^ ~ a ~ b ~ 6a2 ~ ~ 
 
 3. Prove that ax^+bx + c and a+6a;*-f ca;* will have a common 
 quadratic faekir if 
 
 Z)2c2 = (c2 - a2 +&2)(c2 - a^+ah). 
 
 4. Prove that ax^+bx^+c and a+bx^+cx^ will have a com- 
 mon quadratic factor if 
 
 «362 = (rt2 _ cS)(«2 _ c3 +5c). 
 
 6. Prove that ax^+bx^-^cx+d and a+bx+cx^-^dx^ wiilhwe 
 a common quadratic factor if 
 
 {a-\-d) 3 = {b-c){bd-ac). 
 
 6. x^ +px' +qx+r will be divisible by x'^+a^' + b if 
 
 a^ -2pa^-\-{p^+q)a-}-r—pq = 0, and b^ —qb^ -t rpb — r^ = 0. 
 
 7. a;*4-pa;-4-5' will be divisible by x^ 4-ax-\-b if 
 
 rt6 - 4?a3 =^3 and {b^- -¥ q){b^ - qY =JJ-bK ^ 
 
 8. Determine the condition necessary in order that x* +4pa;3 
 -{•Qqx^+Arx+t may have a square factor. 
 
 JI x^+Apx^ + Qqx^ -jr^Li-x+t vanish for a; = a, or b, or c, ovd, 
 find in terms of x, p, q, r, t, the value of 
 
 a;" x" a;" a;" 
 
 x — a x — o x—c x — d 
 
 10. 2 a, 2a^ ^a^, S«*, 2 «^ 2 a". 
 
 11. 2(a-&)S S(a-fc)*. 
 
 12. Determine the values of the expressions in Ex. 9, 10, 11, for 
 the poiynome a;* — 14a;^ +a; — 38. 
 
62 rAOTORING, 
 
 CHAPTER in. 
 
 Section I. — Dieect Application of thk Fundamental Formulas 
 
 Formulas [1] and [21. {x±yy =x^±1xy-\-y^, &c. 
 Art. XII. From this it appears that a trinomial of which the 
 extremes are squares, is itself a square if four times the product 
 of the extremes is equal to the square of the mean, and that to 
 factor such a trinomial, we have simply to connect the square 
 root of each of the squares by the sign of the other term, and 
 write the result twice as a factor. • 
 
 Examples. 
 
 1. 4a;4-80a;3//2 + 400?/4 = (2a;2-20y2)(2a;3_!20?/3) 
 
 2. l-12cc2?/24-36a;42/4^(l_6a;22/3)(i_6ic2y2). 
 
 3. (a_6)2_|.(5_^)24.2(a_6)(&-c). This equals (a- i + ^-c) 
 X (a — 6+fe — c) = («— c)(a— c). 
 
 4. x^ ^xj^ -li-z^ -\-'ixy -1xz--%jz. 
 
 Here the three squares and the three double products suggest 
 that the expression is the square of a linear tnnowial in x, y, z. 
 
 An inspection of the signs of the double products enables us 
 to determine the signs which are to connect x, y, z: we see that 
 
 ist. The signs of x and y must be alike, 
 
 2nd. The signs of x and z must be different. 
 
 3rd. The signs of y and z must be different. Hence we have 
 ^^y—z, or —x—y-\-z= - i^x-k-y-z), and the factors are 
 {x+y-z){x + y-ri). 
 
 Exercise xxiv. 
 
 1. 9m 2 + 12m + 4; c2"'-2c"' + l. 
 
 2. ?/6_2?/323+z«; 16a;2v/2 + -iGa:i/3 + 47/4. 
 
 i). 9^262 4.i2«^c-i-4c^ ; 'di5x'y^-'lixy^'\-iii'^ 
 
FACTORING. 63 
 
 0. (a4 6)2+c3_2e(a4-6) ; S)^'^ - fx4y2 4-^j^y4. 
 
 / a \ -'" / 6 \ ^^ 
 G. ^2^.^^_,^)3_2^(x-y); jy) + (-] -2- 
 
 8. (rc^— x'?/)2— 2(cc2 —xi/)(x)/ — y^) + {x}j — y')^. 
 
 0. (a + ft+c)2-2c(rt+/)-fc)-trc2 ; ff./?'^-2/>3(^2^.y.^4. 
 
 10. (8.c-4/y)3 + (2a;-3//)- -2(3a;-4)/)(2ic-37/). 
 
 1.1. (a;3 -.,;y-i-y2)3 + (a;3 +^^ + i/2)2 +2(s;* +.-^2^3+7/4). 
 
 12. (o.r2 + 2a;?/+:.73)34.(4x-24-6//2)8 _2(4a;'^+6?/3)x 
 ■5x^J^2xy-h7y--^). 
 
 13- (t) +(t) -Mt) • 
 
 14. a? + 63_j.c2_2fl6-26c+2rtc. 
 
 15. a4 + i4+c4-2a-53_2«3,;3+262c2. 
 
 16. {a- i)2 + (t - <')^ + (f; ~ '0^ -f- ^(« -- &)(6 -c)) - 2(a-5)(c-a) 
 + 2(?>-c)(«-c). 
 
 17. 4^,4 _ 12^26+ :)/;3 +.16rt3c + 16o3 -24&C. 
 
 ForjiutjA [4]. a;3 — ?/3 = (cc-}-2/)(a; — ?/), 
 
 Art. XIII. In this case we liave merely to take the square - 
 foot of each ox the squares, and couueet the results with the sign 
 4- for one of the factors, and with the sign — for the other. 
 
 Examples. 
 
 1. {a-\-b)''-{c->rd)^. 
 
 This={(a+6) + (^+(0}{(«+6)-(c+fO} 
 = (,',+6 + 'j + d)(a + 6-c— d). 
 
 2. Factor {x- +^rij-\-i/^)^ -{x^ —xy-^y^)^. 
 Here we hav^e 
 
 {{x^- + oxy+y^) + {x^-x!i+y^-)]{{x"-+5xi,-^y^)-{''^-xy+y^-)\ 
 
 This = a2 - (i> - cf^ = (rt + 6 -c)(«— fc+c). 
 
64 
 
 FACTORING. 
 
 4. EeSolvG (a2 4-53)2_(<j8_52)2_(„2+ftS_c3)2. 
 
 This = 4rt2/,2_(a2 + /,2_,.2')2 
 
 The former of these factoi-s = (a + i)"-^ — c^ = (^a + h-\-<:)(a-}'h — t') , 
 and the latter = c2 — (a — b)^) = (c+a — b){c — c(,-{-h). 
 .•. the giveu expression 
 
 = {a+b+c){a+h — c){c-\-a—b){c--a+b). 
 
 Exercise xxv. 
 
 1. 
 
 49^8 _452^ 
 
 9. 
 
 81a4-l. 
 
 2. 
 
 9a2_ife3. 
 
 10. 
 
 ai-16b*. 
 
 3. 
 
 81a4_i664. 
 
 11. 
 
 a»e-6i6. 
 
 4. 
 
 100a;2-36y2. 
 
 12. 
 
 a2_52^26c-c*. 
 
 5. 
 
 5rt.2&-20&a;22/a. 
 
 13. 
 
 (rt+26)2-(3a;-4^)-=. 
 
 6. 
 
 9x<5- 162/4. » 
 
 14. 
 
 (0,2+^3)2 _ 43,2^2.' 
 
 7. 
 
 9 ^2 1 
 
 y^C — i. 
 
 15. 
 
 {x + yf-iz^. 
 
 6. 
 
 4]/4-|a;2z3. 
 
 16. 
 
 {Sx+5y-{6;e-i'd)^ - 
 
 17. 4a;22/2-(a;2+y2-z2)2. 
 
 18. (^x-2+xy-y-'y -{x^- -ary-7r-)\ 
 
 19. (»2_,y2+22)2_4.,.223. 
 
 20. {a+b-\-c+d)^ -{a-b+c-dy. 
 
 21. (2+ya;+4a;3)3_(2-3a:+4:c-')». 
 
 22. («2+62+4a6)2-(a2+i3)2. 
 
 23. {a^-b^+c^~d2)^-{2ac—2hd)^. 
 
 24. (a;3_y2_ 23)3 _ 4-^223. 
 
 25. (a6-a3t3^iC)3_(^C_5fi3/;3_j_i8)2. 
 
 •20. ai2_/>i2 + Ga9/>3_6i9a3 + SZ?»fi3-8r/S6». 
 27. (a;24-?/2+23 -xy — ijz.—zx)'^ — {xy-^yz+zx)*. 
 !i8. (ic^ +7/2 +22 - 2x,y + 2x2 - 2?/^)-(;?/+s)2. 
 29. 2a2i2 4.2&2^.2+26-2a3_^4_i4_,;4. 
 
 SO. .S4 +^4 _i.24 _ 2::2^2 _ 2^223 - 2»3a;2. 
 
PACTOEING. 66 , 
 
 FoxiMULA A. (x->rr)(x+s)=X' + {r + .i)x+r$, 
 Examples 
 
 2. {x^ij)^+x-y-110 = {x-7j-^h){x-y-10). 
 
 ^(aS _ ab + b2)-^{2a + Sb)} {(fl3 _rt?;_{-62) _ (2«- 3&)}. 
 
 4. (a;2 - 5a;) 2 - 6(x3 - 6x) - 40 = (a;3 - 5a;+4)(a;2 -ox- 10). 
 
 5 . («a; + i?/ + c) 2 — (77?, — w) («.r ■+by-\-c)—mn 
 = [ax-i- by -\-c —ni){ax+by + c-{-n). 
 
 Art. XIV. It will be seen that the first (or common) term ot 
 khe required factors, is obtained by extracting the square root of 
 the first term of the given expression, and tlfat tlie other terms 
 are determined by observing two conditions : 
 
 (1) Their product must equal the third term of tlie given ex- 
 pression. 
 
 (2) Their sum (algebraic) mnltiplied into the common term 
 already found, must equal the middle term of the given expres- 
 sion. Hence, to make a systematic search for integral factors of 
 an expression of the iormx^±bx + c, we may proceed as follows : 
 
 Ist. Write down every pair of factors whose product is c. 
 
 2nd. If the sign before <; is +, select the pair of factors whose 
 sitm is b, and write both factors x+,ii the sign before 6 is + ; a; — , 
 if the sign before 6 is — . 
 
 3rd. But if the sigh before c is — , select the pair of factors 
 whose difference is b, and write before the larger factor x+ or a; — , 
 and before the other factor a;- or a;+, according as the sign be- 
 fore 6 is + or — . 
 
 Examples. 
 1. a;3 + g.x-l-20. The factors of 20 in pairs are 1 and 20, 2 and 
 10, 4 and 0. 'The sign before 20 is +, hence select the lactors 
 whose sum is 9.. These are 4 and 6. The sign before 9 is +, 
 hence the required factors are (a;+4)(a;-f5). 
 
66 FACTORING. 
 
 2. a!2_8a;^.i2. Pairs of factors of 12 ore 1 and 12, 2 and 6, 
 
 3 and 4. Sign before 12 is +, therefore take pair whose sum is- 
 8. These are 2 and 6. Sign before 8 is — , heiice the factore 
 are (x — 2)(x— 6). 
 
 3. a;2-21a;-100. Pairs of factors of 100 are 1 and 100, 
 •2 and 50, 4 and 25, 6 and 20, 10 and 10. Sign before 100 is - . 
 therefore take the pair whose difference is 21. These are 4 and 
 25. The sign before 21 is — , therefore x— goes before 25, the 
 larger factor, and the factors are (a;+4)(a; — 25.) 
 
 4. a;2+12a;-108. Pairs of factors of 108 are 1 and 108, 
 2 and 64, 3 and 36, 4 and 27, 6 and 18, 9 and 12. Sign before 
 108 is - , therefore take the pair whose difference is 12. These 
 are 6 and 18. Sign before 12 is +, therefore x+ goes before 18, 
 the larger factor, and x — before 6, the other factor ; hence the 
 factors are {x — Q)){x-\-lQ). 
 
 Note. — It will be found convenient to write the factors in two 
 columns, separated by a short space. Taking Ex. 2 above, pro 
 ceed thus : Since the sign of the third term is + , write the sign 
 of the second term (in this case — ) above both columns. 
 
 1 12 
 
 (a!-2) (x-Q) 
 
 Ex. 3 above. Since the sign of the third term is — , write tbe 
 sign of the 2nd term (in this case — ) above the column of larger 
 factors, and the other sign of the pair +, above the other column. 
 
 + 
 
 1 100 
 
 2 50 
 (a;+4) (a; -25) 
 
 6. a;'»-84a;4-64. 
 
 Here we have the factors 
 
 1, 64 
 
 x-% a;-32 
 4, 16 
 
 and since the last term has the sign -+•, and the middle term has 
 
 the sign — , we write — over both columns. 
 
FACTORINOc 57 
 
 6. a;3+12a;-64. 
 
 + 
 
 1, 64 
 
 2, 32 
 
 X-4:, x + lQ. 
 
 Here, since the last term has the sign - , we write the sign 
 ( + ) of the middle term, over the column ol larger factors, and 
 the sign — over the other column. 
 
 7. jc*- 10x3 -144. 
 
 Here we have the pairs of factors : 
 + - 
 
 1, 144 
 
 2, 72 
 4, 36 
 
 x + S, u;-18. 
 And since the sign of the third term is — , we write the sign ol 
 the second term (in this ease -) above the column of larger 
 tactors, and the other sign (of the pair +) atove the otiier 
 column. 
 
 Exercise xxvi. 
 
 1. x-^-^5x-U; x^-Qx+U; a;2+7;B+12. 
 
 2. a;2 -8x4-15 ; x^ - 19x4-84 ; x^ -7x-60. 
 
 3. 4x3-2x-20; 9x3-150x4-600. 
 
 4. ix3 4-4*x- 36 : 25x'^ 4-10x4-15; 9x«-27x34-20. 
 y5. ^^x^ + Hx+12; 16x4-4x3-20. 
 
 ^-^6. ^4_u,34-63)x3+.y2/,2; 4(x4-2/)2-4(x4-^)-09. 
 
 7. (a;2+.v-^)3-(a2_fe3)(a;2+y2)_a2i2. 
 
 '^7'(:;rHp4ij[#+2/3)(x+2^ 
 
 10. {a+bf-iab{a + b)-{a^-b^f. 
 
 11. (x3 4-x//4-!/2)2+x3-?y3_5xi/-2^3-2x». 
 
 12. a-^-<ia{b-c)-^{b-c)\ 
 
68 FACTOKINa. 
 
 y^ 3:3. (x^+y^)^-h^a^{x^-\-y^) + a^~b^. 
 
 14. (a;2-10aj)3-4(a;3-10a;)— 96. 
 
 15. (a;2_l4a; + 40)2-25(a;2-14a;+40)-150. 
 
 16. {x^ -x>j + :i^)^+2xy{x-^ -xij-\-y^) ~3x^y^. 
 
 17. z4-3z2 + 2; x^ -2x^-3: 9x^+9x^y^ -lOy^. 
 
 18. c^'" + c™ - 2 ; a;« - a;3 — 2 ; a;^'" - 2a;"'2/» - 8^/'". 
 
 19. a;^"— (a — b)x"Y - « V- 
 
 Art. XV. Trinomials of the form ax^ + bx+c (a not a squai e) 
 may sometimes be easily factored from the following couaiJera- 
 tions : — 
 
 The product of two binomials consists of 
 
 1st. The product of the Jirst terms. 
 
 2nd. " " second '« 
 
 3rd. The sum (algebraic) of the products of the terms taken dia- 
 '^oxially. 
 
 Tnese three conditions guide us in the converse process ol 
 resolving a trinomial into its binomial factors. 
 
 EXAIIPLES. 
 
 1. Resolve 6ic2-13ar7/+ 6^/2. 
 
 Here the factors of the first term are x and 6x, or 1x and 3a; ; 
 those of the third term are y and Qy, or 2y and oy. These 
 pairs of factors may be arranged 
 
 (i) (2) (3) (4) 
 
 » 2» y 2.i/ 
 
 6a; 3x 6?/ dy 
 
 Now, we may take (1) with (3) or (4), or (2) with (3) or (4) ; 
 but none of these combinations will satisfy the third condition. 
 If, however, in (4) we interchange the coefficients 2 and 3, then 
 (2) and (4) give 
 
 2a; 'dy, and 
 
 3x 'ly, where we can combine the " diagonal" 
 products to make 13, and the factors are 
 
FACTORING. 
 
 69 
 
 2a; — Sy, and 
 Sx - 2y. 
 The coefficients of (2), instead of those of (4), might have been 
 iuterchauged, giving the same result. 
 
 2. ea;2-15a;?/+6y2. 
 
 Here, comparing (2) and (3), Ex. 1, we see that their diagonal 
 products may be corabined to give 15, and the factors are 
 Ix—y, and dx—Qy. 
 
 3. Qx--'iOxy+%y^. 
 
 Here, again referring to Ex. 1, we see at once that it is useless 
 to try both (2) and (4), since the diagonal products cannot be 
 combined in any vv'ay to give a higher result than IQxy. But com- 
 paring (1) and (4), we obtaiB by interchanging the coefficients 
 in (4) x—oy, and 
 
 6a;— 2^, which satisfy the third condition. 
 Or, v/e might interchange the coefficients of (3), and take the 
 resulting terms with (2), getting 2x—Gy, and 
 
 8x- y. 
 
 4. iJx^ -i-Sbxy — 6y^ . 
 
 Here the large coefficient of the middle term snows ai once 
 that we must take (1) and (3) together. Interclmnging the co- 
 efficients of (1) we have 
 
 6x— y, and 
 ai + 6?/. The same result will be obtained by inter- 
 changing the coefficients of (3). 
 
 Exercise xxvii. 
 
 1. 6a;2_37a;?/ + 6?/2. 
 
 2. 6x2-f9x(/-6;/-. 
 
 3. 56a;2-76a;?/+20?/2. 
 •i. 56a;2-36x2/-20y2. 
 
 5. 56x2-1121a:*/+207/2, 
 
 6. 56a;2-68£c?/ -1-201/2. 
 
 7. 56a;2-558.-c(/-20?/2. 
 
 8. 5Qx^ + 'dQxy-2Qy-. 
 
 9. 56a;2 -67x^+207/2. 
 10. 5e:i;2+3.c^-20i/2. 
 
 11. Qx^ -IQanj — Qy'^. 
 
 12. 6a;3+5a;//-67/2. 
 
 13. 56a;-^-H562a;//+ 207/3. 
 
 14. 56a;--122ic/y + 20?/^ 
 15: 56a;-'-102a;i/-20?/2. 
 
 16. 56x2 -229a;?/ +20^/2.' 
 
 17. 5 Sa;3 -94x^+207/3. 
 
 18. 56x2- 276a;)/ -20//3. 
 
 19. 3Gx- — 33x^ — 15?/2. 
 
 20. 72x3- 19x1/ -4U(/2. 
 
70 FACTOBINO. 
 
 Art. XVI. More fjeneralbj, trinomials of the form ax^-\-hx-rO 
 [a not a square) iaay be resolved by Formula A, thus 
 
 Multiplying by a we get a^x^ +bax+ac. Writing z for ax this 
 becomes 2=^+ '!/z+'i';. Factor thid trinoaiial, restore the value of 
 z and divide the result by a. 
 
 Examples. 
 
 1. 6.t3 -\-5x-4:. Multiplying by 6, we get (6a-)2 +5(Ga;) - 24 or 
 2^ + 02 -2i. Factoring, we get (-2;-3){c + 3), hence tlio required 
 factors are ^(6x-'d)(Qx+'S) = {2x—l){\ix+-±}. 
 
 2. 6a;2 - 13x1/ + 6»/3 . Factoring z^ - Idzy + 36?/3 we get {z - 4t/) 
 {z — 9y), hence the required factors are i«(Da; — -i^j(6a; — %) = 
 {3x-2y){2x-3y). 
 
 3. 33-14a;-40a;3. Factoring 1320- Ue-g^ we get 
 
 (^30 — 5;)(444-z), hence the required lacfcors are ^^^(30- 40a;) x 
 (44+40^) = (3- ^^(ll + 10.'-c). 
 
 NoxB. — The factors may conveniently be arranged in two col- 
 umns, each with its appropriate sign above it. 
 
 + 
 Ex. 1, above 1 24 
 
 2 12 
 
 ^(6a;'-3)(6a:+8) = (2a:-l)(Sx-f4). 
 
 Ex." 2, above 1 3G 
 
 2 18 
 
 3 12 
 
 ^ (6a; - 4) (6x - 9) = (3j; - '2 ) (2» - 3). 
 
 [Another method of factoring trinomials of the form ax'^~{-lx-\'e 
 is as follows : 
 
 Multiply by 4a, thus obtaining 4:a^x^ +4ff&^-+4ac. Add 6* - 6*, 
 vhich will not change the value, Aa^x^ + 4:abx+b' —b^ + iac ; by 
 [1] this may be written {2ax+by~-—{h^ — 'l:ac). Factor this by 
 [4] and divide the result by 4a. 
 
FACTORIXO. 
 
 71 
 
 Ex. Factor 56x» + 137a; -27885. Multiply by 4x56 or 
 2x112, 1122ic3 + 2.137.1120;— 6216240. Add 1873-1372, then 
 il2-'.«3 + 2.137.112.^+1373- (1373 +P.246240) = (112^- + 137)3 - 
 6265009 = {{112a; + 137)+2503}{(112.r+ 137) -2503} = 
 (112a;+2640)(112.c-23GG). 
 
 Y/e multiplied by 4 x 56, we must, therefore, now divide by that 
 number. Doing so, wc obtain as factors (7a; + 165)(8.£— lCij).j 
 
 Exercise xxviii. 
 
 1. 10a;3+a5-21 
 
 2. 10.c3 - 29.e - 21. 
 
 3. 10.«2 + 29a;-21. 
 ;. 6a;3-37a;+55. 
 
 5. 12«3_5«_2. 
 
 6. 12a;2-37a;+21. 
 12*2 + 37a; + 21. 
 ISaS + lSaS^S-OOS* 
 
 7. 
 
 9. 12.r^-a;-l 
 
 10. 'dxhj^ -'dxy^ -e,y<^, 
 
 11. 4x3+8.v^+3//-. 
 
 12. 662x2 -7ia;3-3.r*. 
 
 13. &x^-x^y^ 
 
 14. 2a;4+a;3_45. 
 
 •35//*. 
 
 ■%*. 
 
 15. Ax^~'61x-y'^- 
 
 10. 4(2; + 2)* -87a;2(:/j+ 2)2-1- 9a;*. 
 
 17. 6(2a; + 37/)3 + 5(Ga;3 + 5.r//-6//-)-6(3a;-2?/)3. 
 
 18. 6(2a;+3?/)* + 5(6.j;3 + 5is,//-6//-)2 _G(3a;-2,v)*, 
 
 19. 6(a;3+a;//+y2)3+i3(a.4+.^2^^2+y4)_385(a;^^-a;'/+?/2)3. 
 
 20. 21(a;2 + 2a:^+2j/2)3-6(ic-^-2x^+2?/2)3-5(a;4+-i^-i'). 
 
 Section il. — Extended Application of the Formulas. 
 
 Art. XVII. The methods of factoring just explained may be 
 a-pplied to tind the rational factors, where such exist, of quadratic 
 multinomials. 
 
 Examples. 
 
 1. Eesolve 12u;2_a;//-20?/2+8a;+41y-20. 
 In the first place we find the factors of the first three terms, 
 !7hich are 
 
 Ax+5y, and 
 Bx — Ay. 
 
 Now, to find the re^naining terms of the required factors, we 
 must observe the following conditions : 
 
72 fACTORINd. 
 
 1st. Their product must = — 20. 
 
 2ad. The sum (ah/ebraic) of the products obtained "by xuulti- 
 plyiug them diagoually iuto the y's, must ==41?/. 
 
 3rd. The sum of the products obtained by multiplyiug them 
 di.agonally into the x's, must =8a;. 
 
 We see at once that —4 with the first pair ah'eady found, and 
 + 5 with the second pair, satisfy the required conditions, and .". 
 tlie factors are 
 
 4:X+5y -4, and 
 
 8a; — 4?/ 4- 5. 
 
 Here the factors oi p^+ V'] —^q^, are 
 
 p+2^, and 
 p — q. 
 Now find two factors which will give - 3r-, and which multi- 
 plied diagonally iuto the p% and qs, respectively, will give 2pr. 
 and Iqr ; these are found to be — r taken with the first pair, and 
 + 3r taken with the second pair. Hence the required factors are 
 
 j9 + 2g— r, and 
 
 ■ ■ . / 
 
 Art. XVIII. But the following examples illustrate a surer 
 
 method. 
 
 3. ic2+a;?/-2?/3+2a:3 + 7?/*-32». 
 
 Reject 1st the terms involvings, 
 2nd. " " 2/, 
 
 3rd. " " jc, • 
 
 and factor the expression that remains in each case. 
 
 1st. x^-{-xu-2y^ = {x~y){x+2y). 
 2nd. x^+2xz-Sz^ = {x+Sz)[x-z). 
 8rd. -2//'^+7//2-3z3 = (-y+Sz){2y-z). 
 
 Arrange these three pair of factors in two sets of three factors 
 each, by so selecting one factor froin each pair that two of each 
 act of three may have the same coefficient of x, two may have the 
 
?ACTORIN(J. 73 
 
 game coefficieut of y, and two the same coefficiant of z {mefficimi 
 including su/n). In this example there are 
 
 X— y, x + 3z, — y-\-?iZ, 
 and x-\-1y, x— z, ^y—z. 
 From the first set select the common terms (inchuling signs^ 
 and form therewith a trinomial, x—y + Bz. 
 
 Eepeat with the second set, and we get x-\-2y—z. 
 
 :.x^+xy-2y^ -\-2xz-\-7yz~Sz^ = {x-y-{-3:){x-\-2y-z). 
 4. 3a;2-acy-8//2-i-30a;+27. 
 
 1st. Sx^-8xy-3y^ ={Bx + i/)(x--By). 
 
 2nd. 8a;2-L30x + 27 = (3a;+3)(a;+9). 
 
 8rd. -%2 ^_27 =(j,^-3)(_3y + 9). 
 
 .'. the factors are {3x+y+'d){x — 3y-\-2). 
 
 6. 6a2-7a6 + 2rtc-2062 + 646c-48c3. 
 
 1st. 6«3_ 7«6-20Z)3 ={2a~5b)(3a-i-'ib). 
 
 2nd. 6rt3+ 2rfc-48c3 =(2r«+6c)(3a-8c). 
 8rd. -2062+ 646c -48c3 = (_.56 + Ge){46-Sc>, 
 
 /. the factors are (2a - u6+Gc)(3a -j- 46 — 6c). 
 
 Exercise xxix. 
 
 1. 7x^~xy-6y9-6x-20y-16. 
 
 2. 20x2-15a;?/-5t/S_68a;-427/-88. 
 
 3. 3«4+a;27/3_4y4 + i0a;2_i7^2_i3, 
 
 4. 20a;2- 20^3 _^9.i-?/ + 28^4-35?/. 
 
 5. 72a;2-8?/3 + 55a;?/-}-12!/— 169a; + 20. 
 
 6. x^ —xy— 12y^ — 5x-~15y. 
 
 ' 7. 8a;3 + 18a;(/+9^3_^2a;2— s*. 
 
 8. 6x^+6y^-13xy-8z^-2yz-{-Sxz. 
 
 9. 6a;*- 10?/^ + lla;2;/2 -2532 -[-107/2 +25!/2z2-15:K2^10a;2z3 
 
 10. 16.c4 -16^4 - 22a;3j/2 j. 1524 _[. 14^222 ^. 50a;222, 
 
 11. 4rt3_i552_4aj_21c2 — 366o— 8ac. 
 
 12. «* + &4+c*-2rt»62-263c2-2c2a». 
 
 OFTHc 
 lift.ii<<___ . 
 
74 FACTORING. 
 
 Art. XIX, Trinomials of the form ax^ + bx^ + c can always be 
 broken up into real factors. 
 
 If a and c have different signs, the expression may be factored 
 hj Art. XVI. 
 
 If a and c are of the same sign, three cases have to be consid- 
 ered : i. 6 = 2v/(ac), ii. 6>2v/(ac), iii. 6<2v/(rtc) 
 
 Case I, b = 2y^{ac). This case falls under Art XII., formula 
 [1] . where examples will be found. 
 
 Case II. i>2y(ac). This case falls under Art XVI., where 
 examples will be found. The following additional examples are 
 resolved by the second method of that article. 
 
 Examples. 
 
 Here we see that (^y^)^ will make, with the first two terms, 
 a perfect square, and we therefore add to the given expression 
 (fy^)'~(f!/^)'- T^^® expression then becomes 
 
 = (2x3+12/2 + 12/ =^)(2a:»+|-y»-f?/2) 
 = (2a;= + 22/3)(2a;' + ij/^') = {x^ + r/^){4x^ +y"). 
 2. 3a;* + 6x2 +2. 
 
 Here multiplying by 4x3, and completing the square as in 
 Ex. 1, we have 
 
 86x* +72^2 + 62 + 24 -62 = (6x2 + 6)2 -12 . 
 
 = (6x3 4-6-i/12)(Gx=+6 + i/12), which divided by 4x3 give 
 the required factors. 
 
 8. ax^+hx^+c. 
 
 Proceeding as in Ex. 2 we have, by multiplying by 4a, 
 
 ax^^bx- +c = {ia^x^-^iabx" +63 - b^ +4ac} -• 4a 
 =:{2ax^+b + y/{b^-Uc)}{2ax^+b--i/{bi-4.ac)}^U. 
 
FACTORINO. 75 
 
 Exercise, xxx. 
 
 1. ic^+TxS+l; 4a;4 + 14a;3+l. 
 
 8. 40:4+10x3+3; S{xA-y)^ + oz^{x+yy +2'^. 
 
 4. a;4 + 7cc32/2+3J?/4; x^ +7x2^/3+8^7/4. 
 
 5. 4x4+9xV+tI^*; 4(rt+6)4+10c2(a+i)2 +-3c*. 
 
 6. 3x^+8x2i/= + 4-T\jr*; 36x^+96x3+55. 
 
 7. 5x*+20x3+2; 4a* + 12rt3 + l. 
 
 8. 4(x+?/)* + 12(x+?/)^z3+24; 5.r4+20x3j/2+2T/*. 
 
 9. 9x4 + 14x2+4; 2x* + 12x3(f/+2)2 + 15(t/+z)4. 
 10. 2x4+12x3 + 15; 7x4+40x3+45. 
 
 n. 8x4 + 36x3?/2+29?/4: 7x4+20x-^2 -20?/*. 
 
 12. 7(a-i)4 + 16(a-5)36-2+5c4; ^a^ + SaH^+b*. 
 
 13. 8x4 + 6x^vM-2//4; S(a+by+Q(a^-b^y +2(a-b)K 
 U. 49a4_84a2i'^+2264 ; 25TO4-{-60w2n3 4.27«4. 
 
 15. 49(m+7i)4-84(»i2 -»43)3+22(m-M)4. 
 
 Case III. &<2;/(ac). This case may be brought under 
 Art. XIII. The following examples illustrate the process oi re 
 duction and resolution. 
 
 ExAJfPLSS. 
 
 1. x4 -7x3+1. 
 
 "We have to throw this into the form a* — 6' : 
 a;4_7a;3 + l = (^2 + 1)3 -9x3 = (x3+H-3x)(x- +1 -Sa;). 
 
 2. 9x4 + 3x3?/2 +42/4 = (3x3 + 22/3)2 -9x3y3 
 = (3x= +2^/2 -8x>/)(3x3 + 2^/2 +3xy/). 
 
 3. x4 +2/4 = (a;S +2/3)2 _2x3|/S 
 
 = (X3 +J/2+XV ,/2)(x3+2/3_x2/-/2). 
 
 4. x4 -ixs^/^* +1/4 = (x^ +2/3)3 -la; V • 
 = (x-^+y3 -^%xy){x^ +.v2-i^?/)- 
 
 5. «x4+ix3+c = (,/«. a;3 + |/c)3-{2v/(ac)-6}a;» 
 = {ya. x3+|/6--|/(2;/^-i)x}x 
 
 il/a. x2+i/c+v'(2v^^— ^)a;}. 
 
76 FACTORING. 
 
 Art. XX. It is seen from these examples that we have merely 
 to add to the given expression what will make with the first and 
 last terms (arranged as in Ex. 5) a perfect square, and to subtract 
 the same quantity. In Ex. 2, e. g., the square root of 9z'^ = 3x^, 
 the square root of 4t/* = 2v/ 3, /. 3a3^+2?/2 is the binomial whose 
 square is requii'ed ; we need .". 12x^1/' ; but the expression con- 
 tains Sx^y'-^ ; .". we have to add and subtract 12x^y'^ - ox^'y^ = 
 
 Hence we derive a practical rule for factoring such expressions. 
 
 (1). Take the square roots of the two extreme terms and con- 
 nect them by the proper sign ; xhis gives the first two terms of 
 the required factors. 
 
 (2) Subtract the middle term of the given expression from 
 twice the product of these two roots, and the square roots of the 
 difference will be the third terms of the required factors. 
 
 6. x*+^\x-y^-\-y'^. Here ^/a;* =a;^, l/y* = i/2, and the first 
 two terms of the required factors are x^-\-y^ ; twice the product 
 of these is -\-2x^y^, from which subtracting the middle term, 
 Y^x^i/^, we get xe^^i/^ j *^^ square roots of this are +izy. 
 Hence the factors are x--\-y^±:-lxy. 
 
 Note that since s/y^= -^u'^, ov -y^, it may sometimes hap- 
 pen that while the former sign will give irrational factors, the 
 latter will give rational factors, and conversely. 
 
 7. x'^ — lix^y^+y*. Here, taking -{-y^, we have 
 
 x^+tj^+xy /13, and a-» +y-^ -xy v/13. 
 But taking —y^, we have 
 
 a;3 ^yZj^Zxy, and x"^ -y^ — 3xy, 
 Sometimes both signs wiU give rational factors. 
 
 8. lQx'*' — l'lx^y^-\-y^. Here we have 
 
 (^ix^Jf-y^J^'dxy){4.x'^+y^ -'dxy, and also 
 {^x"^ ~y^-Yoxy){Ax" -y^ -oxy). 
 
FACTORING. 77 
 
 Exercise xxxi. 
 4. a;4— 7a:-+l, x-4-f9, Ja;4+^4 _ ^3a-.'^2. 
 
 6. 4a;4+y4_8i^.3^,/3^ ^4 +^4_^7^^3_,/-2^ 4;t;44 1. 
 
 7. a;^"' + 64i/*'», a;*"' + %*'", ia;*+y\v/4-5fx37/2. 
 
 9. m^x^ + n^i/^~{2tiin-\-ij)x''^ij2, .t^"' + 2^"'-y". 
 
 10. 16a;'i-2o.62 + 9, 4:;;4 _ le^s 4.4^ ISx'^y^ -dx^ - ii/-'. 
 
 11. 4a;'t-12^|x-2//3-l-()?/4, x4+6.x2+25. 
 
 12. a^~\-h^^(a + b)^, l+«4+(l_|_a)4. 
 Vl3. (;«4-Z/)'^-722(^-+t/)2+24. 
 
 ■\ 14. (rt-f i)4+7(.2(«_|_ft)3+c4, 
 
 15. 16</4-j_4(i_c-)4_9«2(i_,.)3. 
 
 16. 4(a + Z))4 + 9(a-i)4-21(./3_/,3)3, 
 
 17. (a;2+y2_^^)4_7(^34.^3)24.(^.+^)4. 
 
 18. (rt2_|_rti + /;3j4^7(^,3_63)3_|_(a-i)*. 
 
 19. 16rt4 + 4rt2 4-l, ;f;4-41a;2+16. 
 
 20. x4+8l7/8-63a;2i/4, 14-24+2528^ 
 
 21. (a24-l)4 + 4(rt24.i)2^2_|.iea4^ (.«+l)* + 2(a;2 _ 1)24. 
 {»(a;-l)4. 
 
 Art. XXI. We can apply [4] , Art. XIII., to factor expres- 
 sions of the form ax'*'-\-bx^+rbx—r^a. This may be written 
 a{x*-r'^)+bx{x''-\-r)={a{x^-r)-\-bx}{x^+r). 
 
 Examples. 
 
 1. 6x-4 + 4a;3 + i2x-54. This 
 
 = 6(a;4 - 9) + 4a;(x3-f-3) = {x^-+S){6{x^ - 3) + 4a} 
 = (a;2 4-3)(6a;2+4a;-18). 
 
rs 
 
 FACTORING. 
 
 2. 11^4 + 10a;' -40a; -176. This 
 
 = n{x^-lQ) + 10x(x^-4:) = {x''-i){ll{x^+.i)+10x} 
 = (x^-A){nx^+10x+U). 
 
 3. 40a;4 + 30x3+ 60a; -160. This 
 
 = 10(4a;4 - 16) + 1.5a;(2a;2+4) = (2a;2 +4){10(2.t;3 -4)4-15a:} 
 = (2a;2 +4)(20a;^ + lya;- 40). 
 
 Note. — To determine r, take the ratio of the coefficient of x^ 
 to the coefficient of x. 
 
 Exercise xxxii. 
 Resolve into factors 
 
 1. a;4+2a;3 + 6a;'-9. 
 
 2. 2.r* + 2a;3-f6a;-18. 
 .3. a;4 + 3a;-^+12a;-16. 
 
 4. 3a;^+a;3_4a;-48. 
 
 5. 5a;4 + 4a;3-12a;-45. 
 >lf>. 10a;4 + 5a;3+30a;-360. 
 
 3 _ S ^_ 4 
 
 7. ix^-\-20x^+4x-j^^. 
 
 8. 2;"^a;4-40a;3+8a;-l. 
 
 9. 37ia;'i-30.^;3+48x-96. 
 >il0. 63a;4- 39x3 + 52a; -112. 
 
 11. 810x4 + Va;3+|a;-2i. 
 
 12. 242a;4-33a;2-3a;-2. 
 
 13. ix^ + ^y^x^-^2^x 
 
 14. 80a;'' - 32a;37/+64a;?/- 320;/*. 
 
 15. 24a;4 - 12.x-3y+30a;/y3 - ISO*/*. 
 
 16. 2x^ + ^z^i/-8xy^-512ij*. 
 
 17. lla;* + 10a;3-12a;-15fi 
 
 18. 40a;* + 30a;3 + 60a;-160. 
 
 19. 13a;4-12a;3y+72a;?/3-468?,'4. 
 
 20. 3a;* + 3a;3^+12a;//3-48?/4. 
 
 21. oa;4+4a;3//-12a;</3_45(/4. 
 
 22. 4x4 - 14x3/7+28x^3 -16i/*. 
 
 23. x4+80x3?/+16x//3-^i3.y*- 
 
 24. 2x4 -x3?/+6x^3_ 72^/4. 
 
 Art. XXII. Formulas [1] and [4] may sometimes be ap- 
 plied to factor expressions of the form 
 
 ax^-\-bx^-\-cx^-\-7-bx-{-r^a. 
 
 This may be put under the form 
 
 afx* +r2) + 6x(x2 +/-)+cx8 = a{x^ +r)3 + bx{x^+r) + 
 (e — 2ar)a;3, which can sometimes be factored. 
 
 Examples. 
 
 1. x*+6x3+27x2+162x+729. 
 
 We "have x4+729 + 6x(a;2+27)+27x2. 
 
 = (x2+27)3 4-6x(x2+27) + 9x3-36x2 
 
 = {x3+27+3x} 3 - 3Gx3, which gives the factors 
 
 X 
 
 2-3X+27, and x2+9x+27. 
 
FACTORING. 79 
 
 2. x^ + ix^+ix' +20x^-25. This 
 
 = (a;2 +5)2 + 4a.-(a;3 +5) - Ga;3 
 = (a;3 + 5)3 +4a;(a;2 +5) + 4a;3 - 10a;3 
 = {a;2 + 5 + 2x-xVlO}{a;3+5+2u;+aVlO>. 
 
 Exercise xxxiiu 
 Eesolve into factors : 
 
 1. a;4_6cc3 + 27a;3- 162a; +729. 
 
 2. a;H-2a;3+3a;2+8x+16. 
 
 3. a;4+^3 + a;3+a;-j_i. 
 
 4. a:4-4x3+a;3-4a; + l. 
 
 6. 4a;4_i2a;3_6a;2-12a;+4. 
 
 6. a;4 + 14a;3-25a;2-70a;+2?. 
 
 7. 16.T*-24a;3-16x3 + 12a;+4. 
 V8. a;* + 5:«3-16x2 + 2Ux+16. 
 
 9. a;4 + 6a;3 _ 11x3 -12x+4. 
 
 10. a;*+4a;3?/+ic22/2+12a;y3_|.9^^4. 
 
 11. x^+6x^-9z^-6x-\-l. 
 
 12. a;4+-4^•32/-19a;2J/2_}_4,^.y3^,^4. 
 
 13. 4a;4 +4a;3|/- 65x2^2 _io.ri!/34_25,2/* 
 
 14. x*^ +6x^y-9x^y^ -Gxij^-^y*. 
 
 15. a;*-+6.c3// + 10a;22/2+12ic^3^4^4. 
 
 16. ^x^-\-18x^y - 52x^1/- -12xy^ + i7j*. 
 
 17. lla;4+10x37/+39/^V-22/3+2Ua;?/3 + 442,'«./ 
 
 Section III. — Factoking by Parts. 
 
 Art. XXIII. To factor an expression which can be reduced 
 to tlie form a.F{x)+b.f{x). 
 
 When the expression is thus arranged, any factor common to 
 a and b, or to F{x) and f{x), will be a factor of the whole ex- 
 pression. The method about to be illustrated will be found use- 
 ful in cases where only 07ie power of some letter is found. 
 
80 rACTORiNa. 
 
 ' Examples. 
 
 1- Factor acx" —nbz — hc^x + b^c. 
 
 Here we see that only oae power of a occurs, and we therefore 
 group together the terms involving tliis letter, and those not in 
 volving it, getting 
 
 a{cx^-hx)-bc^x+h^c 
 = ax{cx — b) — bc(cx — b) -- (ax — bc)(ex — b). 
 
 2. Factor m^x^ -mna^x — tnnx + n^a^. 
 
 Here we observe that a occurs in only one power («^). 
 Therefore we have 
 
 — a^ [^)imx~ 71^ ) + m^ x^ — 7nnx 
 
 = —na^[mx—n)-\-nix{mx — n) 
 
 — {mx — n)(^wx — na^). 
 
 3. 2x^+4:ax + Sbx+Gab. 
 
 Here we observe that the expression contains only one power 
 of both a and b. W" may, therefore, collect the coeflficieuts in 
 either of the following ways : 
 
 a(4x-{-(jb)+.{2x^+3bx), 
 or, b{3x + Ga)-\-{2x^+Aax). 
 
 Now the expressions in the brackets ought to have a common 
 factor, and we see that this is the case. Hence, 
 
 a{ix + 6b)-\-{2x^+Sbx) 
 = 2a{'2.x + Sb)+x{2x + Sb) = (2a; +36) (a; + 2a). 
 
 4. abxi/ + b^ij^-\-acx — c^ 
 = a(bx!/-[-cx)-rb^y^ —c^ 
 
 = «x(%+5) + (%+c)(%-c) =(by + c){ax+bi/-c). 
 
 5. yS - {2u + b)ij^ +{2ab-\-a^)y - a^b 
 
 = _f,(y2 ^2ay-}-a^) + y^ -2mj^+a^y 
 u: - b{y^-2ay+t(2) + y(y2 _ 2((// + a3) 
 
 = {y-b)iy-a)^' 
 
 6. 2x^y-\-2bx^-bx^y + 4:abx2y -x^y^+iaxy- - 2abxy^-2ay^. 
 = b{2x^-x^y+4:ax^y-2axy^) + 2xhj-x^y^ + 4:axy^-2ay^ 
 =zbx{2x^-x'-^y + iaxy-2ay'^) + y{2x^-x-y+iaxy-2ay-^) 
 
 = (y + bx){2x^ -z^y+iaxy - 2ay^). 
 
FACTORINQ, 
 
 81 
 
 And 2a?« — a;2 }/ 4- 4aa;y — 2a?/» 
 = a{4x!j - 2?/2 ) +2x3 — x^?/ 
 = 2ay{2x - ly) +x2(2x - y) = {2aij+x^){2x - y). 
 
 7, x^ + {2a-b)x^ -{2ab-a^)x-a^b 
 ^h{-x^~2ax-o^)+x^ + 2ax-+a^x 
 
 = -b{x + a)^+x{x + a)^ ={x-b){x+a)^. 
 
 8. px^-(;:p — q)x^-\-{p-q)x+q 
 = q{x^ —x-^l)+px^ —px"^ -\-}>x 
 
 = q{x^-X+l)+px{x^-X + l) ={pX + q){x^-X-il). 
 
 Exercise xxxiv. 
 
 6. x^ —b^x- ~a^x + a^b^. 
 
 7. x' 
 
 -a^x- 
 
 •62a;3 + a362 
 
 8. 8a;3 + 12aa;+ lOhx 4- 15^(6. 
 
 9. a^^{ac-h^)x^ ■\-bcx^. 
 10. a^+{ac-b'^)x^-hcx^. 
 
 1. a;2y— iK^z— "y^ -\-yz. 
 
 2. abxi/-\-b'^y^ -{-acx — c^. 
 
 8. ic322^.flja;2_^322_a3. , 
 
 4. 2a;2— «a; — 46a;+2a6. 
 6. a;2+26a; + 8a^+6rt&. 
 
 12. ;?a;3_(p+y);j.3_j_(^^^)aj_j, 
 
 13. a2-\-ab + 2ac-2b^+7bc-Bc^. 
 \14:. x^+{a + l)x^ + {a + l)x+a. 
 
 15. ynpx^ + {mq — n]))x^ — {))i;r+7iq)x-\-nr. 
 
 16. x^ — {a-\-b-\-c)x'^ -\-{cLb-\-bc + ac)x-abc. 
 
 17. x^ + (a — b — c)x^ — [ab — bG-\-ca)x+abc. 
 
 18. x^ + {a-\-b — c)x^ —(be — ca —ab)x — abc. 
 
 "VlD. a^x^ — a^x^y — a^Xy + a^y^ —ax^yz-{-x^z — xyz-\-ciy^z, 
 
 20. a^bx^-{-nb^xy + acdxy -f hcdy 2 — at'/icz — befyz. 
 ^1. a2a;3-a(6-e)a;2-5-c(a-6U- + c2. 
 
 22. 7Ha;3 —nx^y-\-rx^z — mxy^-\-ny^—ry^z. 
 
 23. amx^ -{-{mby — nay-\-incz)x — nby^ —ncyz. 
 
 24. (aw. — bcm)x- -{-{mn — bcn)x-^an-\-nax. 
 
 25. a262c3-6=c2a;</-a2c2?/2 + 6-2a;^32_a2i22a; + ft2^.3^2^rt222^ 
 
 26. a;5 -m^x^ -{n-n')x^ + {m''-n-m'^n^)x^ -a{x--^n^ -n). 
 
 27. l-(a-l)j;-(a-/; + l).«2_^(a+6-c)a;^-(6 + c)a;4+ric5. 
 
 28. a3a;3 -a^{b- c+d)x^y - {abc-abd+acd)xy^ +bcdy^. 
 
 29. m^npx^ — {n-p — m"}i^ - m^j.q)x^ — (rr' + n] q — m^nq)x — n^q. 
 
82 
 
 FACTOEING. 
 
 — [n^q- -\-n'^q^z)y'^. 
 
 Art. XX IV. Sometimes an expression which does not come 
 directly under the preceding form, may be resolved by first find- 
 ing the factors of its parts. 
 
 Examples. 
 
 1. ahx^ -{-ahy^ —a-xy — h^xy. ' 
 
 Here, taking ax out of the first and third terms, and by out of 
 the second and fourth terms, we hav£ 
 
 ax{bx — ay) — b>j(bx — ay), and hence 
 {ax — by){bx — ay) . 
 
 2. x^-{a+b)x^ + {a^b+ab2)x-a^b^. 
 
 Here, taking the first and last terms together, and the two 
 middle terms together, we have 
 
 {x^+ab){x^-ab)-{a+b)x^+ab{'i-{-b)x 
 = (x^ i-ab)[X' — ab)- (a + b)x{x^ ab\ 
 = [z- - ab) [x- +ab - {a-\-b)K} = {x* —ab){x-a)(x - h)» 
 
 3. a:3»*-4a;"»+3. This eq lals 
 
 3^m _ a;m _, s(x^i -i) - x"'{x""' - 1) - o(a;"' — 1) 
 _ x^(x"'' + 1 ) (a;'"- - 1 ) - 3 (a''" - 1 ) 
 = (a;"^-l){a;"^(a;^'^+l) — 3}-. 
 
 Exercise xxxv. 
 
 1. a^ — ab-^ax — bx. 
 
 2. abx^ + b^zy --a-xy - aby^ . 
 
 3. x^ +ax^ —a^z- «"*. 
 
 L a^x + 'la^x^^^ax^+x"-. 
 5. acx^ + {ad — bc)x — Ixl. 
 0. 'I'ux'*' -5x^+x -1. 
 
 7. a~ — A- +ax — ac — bx + br. 
 
 8. a^ + {l+a)ab + b^. 
 
 9. z^ + 2xy{x^-y-) -y^. 
 '.0. x^-y^+x-'+xy+y^. 
 A. 2b+{h^-4:)x-2bx^. 
 12. a;3-4-3x3-4. 
 
 13. 
 
 P^ 
 
 .p2q 
 
 2pq^ + 2q^. 
 
 U. a3 4-a2_2. 
 
 15. '3an^-2ab^ -1. 
 
 16. //3-3f/+2. 
 
 17. 2a^-a''-b -aV--\-2b^. 
 
 18. &:^'» + i2,7i_2. 
 
 1 7/3n _ 2_,y2»2v _ 2i/"2':^" H-2"-\ 
 
 20. a3_4rt52_j_3i3. 
 
 21. a'^"i-3ft'"f"+2c^". 
 
 22. ax^-{a^+b)z^+b^. 
 
 23. 35.-«^»-6a2a;»_9a4. 
 
 24. ^^3^2_|.2rt/,r3-rt3<---^)-'.a 
 
 25. aia^ —iib^ -\-b^m — )ii^. 
 2ti. i - tta' -4-27::'' 
 
FACTORING. 83 
 
 27. (x-y)^ + {l-x+y){z-y)z-t^. 
 
 28. 2Am9 -28in^n+Gmn^ -7n^» 
 
 29 . a;"*+" + a;"?/" + a;"'|/"' + ?/'" +" . 
 
 80. x^ + 2x^y-a''x^+xhj^ -2axy^ -y*. 
 
 Section IV. — Application of the Theory of DiVvSORS. 
 
 Art. XXV. By Theorem I. we.prove that 
 
 a;" — «" is divisible by x— a ahcays 
 
 jcR — a" *' " " X + a -whenn is even 
 
 af + a''" " " x-\-a yfhen n is odd. 
 
 By actual division we find, in the above cases ; — 
 
 = a;"-^+a:"-^«+ . . . Jca^-'+a"-' (1). 
 
 x —a 
 
 af — a" 
 X +a 
 
 = a;"-'-a;"--<^+ . . +«rt"-^-a"-^ (2). 
 
 '^^'^ = a;'-^-a^'-V(+ . •. -xa^-^+a^' (3). 
 
 Examples. 
 
 1. Eesolve into factors x^ — y^ ; here a;—?/ is one factor and by 
 (1) the other is a;- +a;?/ 4-2/^ • 
 
 2. Eesolve n'^-\-{h — c)^ ; here a + (b—c) is one factor; and by 
 (3) the other is a2_^a(6-c) + (i-r) 2. 
 
 3. Eesolvo a; '5 -f-1024?/^o. This =(x''y + {{2y)^}^, one factoi 
 of which is a:^ + (2_!/)2, and by (3) the otlier factor is 
 
 (a;?)4-(x3i^(4?/2) + (a;3)2(4.v2)3_a;3(4?/2)3 + (47/3)4. 
 = a:i2_4a;9y2^1Ga;'^2/4-64a;3v'5 + 256i/8. 
 
 4. Eesolve (x—2y)^-\-{2x — y)^ into factors. 
 Here by (3) we lia-.e 
 
 :. the factors are 
 
 'dix-y)(7x'^-ldxy+7y^). 
 
S4 FACTORING. 
 
 0. Piesoh-e x^+x^y+x^ij^ +x^ 11^ +x}/^ + If ^ : 
 
 x^ — y^ (x" -{-i/^){x^ —y^) 
 
 -By (!) we see that this = ^ — = 
 
 ■^ ^ ^ x —y x—y 
 
 =^{x->ry){x^ -xy + y'^)(x^ ■\-xy+y^). 
 6. Eesolve x^'^ -x^^'a+x^a^ — x^a^+x'' a*' - x^a^+x^a* 
 
 -x^fl'' +x^a^ -x^a^-\-xa''' -a^K This = r— 
 
 _ {x^+a^){x^-a^ ) __ {x^ +a^){x^ - a^){x^ -\-d^) 
 ~ x-{-a ~ x-j-a 
 
 Exercise xxxvi. 
 
 Factor the following : — 
 
 1. x«-.vS x^-1, x^+S, 8a^-27x'^, S-{-a^x^. 
 
 2. j;5-(i>o, 27a3_G4, ai2_i8^ 3.io_32_y6, 
 
 3. find a factor which, multiplied into 
 
 a*+a3/,_|_rt362^„/,3_f./,4^ will give a^-b^. 
 
 4. By wiiat factor must x^ — ix'^ y-{-16xy^ — My^ be multiplied 
 to give a;* - 250//* ? 
 
 5. Fa.Gtor x^ +x'''y+x-'''y^ +x*yS-\-x^y^-{-x-y^ i-xy~^ fy'. 
 Find the factors of the follow! iv 
 
 6. (32/2 -2a;2)3_(3a;3 -22/2)3, a«~16i*. 
 
 7. x^ —y^ —x{x^—y^)+y{x — y). 
 
 10. a;6-t/6+2x2/(a;4+x22/^+2/''). 
 
 11. (a2_ftp)3 + 8&3c3, a;^/»_a-t'». 
 
 12. x^-dax-^+Ba^x-a^ + bs. 
 
 13. a;3 4-8y3+4a;?/(a;2-2xv + 4?/2j. 
 
 14. 8x^-Gxy{2x+Sy) + 21y3. 
 
 15. l-2a;4-4x2-8.c3. 
 
 IG. t/5+u4it-|-f/3i2c2_Ua253c3^«i4c*+6''r». 
 
FACTORING. 85 
 
 Art. XXVI. The principles iilnstrated *in Section II., chap. 
 II., may be applied to factor various algebraic expressions, ap in 
 the following cases : ^ 
 
 Examples. 
 
 1, Find the factors of 
 
 {a-{-b-\-c){ab+bc-i-ca)~~{a + h){b+e){c+a). 
 
 1st. Observe that the expression is symmetrical with respect 
 to a, b, c. 
 
 2nd. If there be any monomial factor a must be one. Put- 
 ting a = 0, the expression vanishes. .". a is a factor, 
 and, by symmetry, b and e are also factors. .". abc 
 is a factor. 
 
 •8rd. There can be no other literal factor, because tlie given 
 expression is of only three dimensions, and ahc is of 
 three dimensions. 
 
 4th. But there may be a numerical factor, vi suppose, so that 
 we have 
 {a-\-b+c){ab-{-bc-^ca)— {a + b){b-\-c){c + a) =:mabc. 
 
 To find m, put a = h = c = l in this equation, and w = l, 
 .". the expression = aic. 
 
 2. Eesolve a^{h-c)+b^{c-a)+c^{a-b). 
 
 1st. For rt = this does not vanish. .*. a is not a factor, 
 and by symmetry neither is b nor c. 
 
 2ad. Try a binomial factor ; this will likely be of the foi ni 
 b — c ; put — = 0, i.e., h = c in the given expression, 
 and there results 
 a.2[c-c)+c^{c—a)+c^{a-c), which = 0, 
 
 .'. b~c is a, factor, and by symmetry c — a and a — b are fac- 
 tors. Since the given expression is only of three 
 dimensions, there can be no other literal factor ; but 
 there may be a numerical factor, m (say), sq that 
 
 a^ib-c)-^r'{c-a,)-{-c^{a-b) = m{a-b)(b-c){c-a). 
 To find the value of m, give a, b, c, in this equation, any values 
 which will not reduce either side to zero; leta=l, b = 2, c = 0, 
 
8G FACTORING. 
 
 and we iiave 2 = m( — 2), or to= — 1 : so that the given expres- 
 sioii= — (a — b)(J) — c){c — a), or {a-b)[b — c){a — c). 
 
 3. Eesolve aS{h-[-c^)+b\{c + a^)+c^{a+b^)+abc{abc+l). 
 Here we see at once that there is no vwnomial factor : 
 
 put &+6'2 = 0, i.e., h— —c^, and the expression becomes 
 «3(-r3+c2)-c6(c + rt2)+c3(a+c4)-f3«(_c3a+l) which = 0; 
 .*. i + c2 is a factor, and by symmetry c + a^ and a+b^ ars also 
 factors ; and proceeding as in former 'examples we find m = l ; /. 
 the expression = (6 + c2)(c-f-«^)(« + i^). 
 
 4. Eesolve into factors the expression 
 
 (a-i)3+(6-c)3 + (c-a)3. 
 
 As before, we find that there are no monomial factors. 
 
 Let a — b = 0, or a = b, and substituting b for a the expression 
 
 becomes zero ; hence 
 
 a — b is a factor. 
 
 By symmetry, b— c " 
 
 and c—a " 
 
 Hence the factors arc 
 
 vi(a — b)(b — c){c~a). 
 
 To find TO let a— 0, 6 = 1, r, = 2, and we hav« 
 
 6 = 2m, or «i=3. 
 The factors are, therefore, 
 
 S{a-b){b-c){c-a). 
 
 5. Eesolve into factors 
 
 a^{b - c) -f-63(c--a)+c3(a - b). 
 A-s before, we find that there are no monomial factors. 
 Let a~b = 0, or a = b ', substituting b for a, the expression be^ 
 comes zero ; 
 
 therefore a—b is a factor. 
 
 By symmetry b — e " 
 and c — a " 
 
 Now the product of these three factors is of tJiree dimensions, 
 while the expression itself is of four dimensions. There must, 
 therefore, be another factor of one dimension. It cannot be a 
 
FACTORING. 87 
 
 monomial factor, for the expression has no such factors. It can- 
 not be a binomial factor, such as a+b, for then, by symmetry, 
 6+c and c+a wonlii also be factors, which would give an 
 expression of six dimensions. It cannot be a trinomial factor, 
 unless a, b, and c are similarly involved. For instance, if a — b+c 
 were a factor, then, by symmetry, 6 — c-f a and c — a-{-h would also 
 be factors, and the dimensions would be six instead oifour. The 
 other factor must, therefore, be a-\-h-{-c. Hence, 
 
 a^{h-c)+b^{c-a)-^c^{a — b)=m{a—b){h — c){c-a){a-\-o + r). 
 
 To find m, put a = 0, 6 = 1, and c = 2, and we have 
 — 6 = 6v//, ; 
 .*. m = — 1. 
 
 Hence the factors are 
 
 — (a — b){b — c){c — a)(a -\-h+c), 
 or, {a — b){a — c)(b — c){(i-^b + c). 
 
 6. Prove that «• 
 
 a3+b3 + c^-]-S(a-hh){b + c){c + a) 
 is exactly divisible by a-j-6 + c, and find all the factors. 
 
 Let a4-ft4-c = 0, or a= —{b-{-c); substituting this value for n, 
 
 we have 
 
 - (b+c)^ + b^ -\-c^-^Sbc{b-^c), or 
 -(6 + c)3 + (i + (;)3 which = 0, and 
 
 therefore a-\-b-^c is a factor. 
 
 As before, we find that there are no monomial factors. Since 
 a-{-b-{-c, the factor already obtained, is of one dimension, the 
 other factor must be of two dimensions, and cannot, therefore, be 
 a binomial ; for if a-\-b were a factor, by symmetry i + c, and c + a 
 must also be factors. The factors in that case vv'ould give a 
 qi;antity of four dimensions, while the expression itself is only 
 oi three dimensions. Nor can a^-\-b^-\-c^ be a factor. For 
 if so, the other factor must involve a numerical multiple of the 
 first power of a, and, therefore, on taking the first power of a out 
 of terms involving first and third powers, we should have left 
 some numerical multiple of a^-\-b^+c^, instead of wiiich we get 
 
58 FACTOBINa. 
 
 a2_|-3(6-]_e)». Nor can af-i-{h+cy be a factor, for symmetry 
 would require two other factors, viz. : b- -{-{c-i-a)-, awl c- +(« +h)^, 
 tiius giving a quantity of seven dimensions. 
 
 The only factor admissible is, therefore, {a-^h+c)^. 
 Hence 
 
 a^-i-h'-^ -i 9^--hB{a-*-b){b+c){c + a) = ?n{a-\-h + i:\a-Jrb + :)' 
 
 = m{a+b-![-c)'^. 
 To find m, let a = l, 6 = 0, and c = 0, and we nave 1 = »«. 
 Hence the factors are 
 
 (^a + h + '^)(a + h + c){a+b+c). 
 
 7. Simplify 
 
 a{b+c)^ +b{a + cy -^c{a + b)^ — {a-'rb)(a-c){b-c) 
 -{a-b){a-c){b + c) + {a — b){b — c){a + c). 
 
 Let a = 0, and the expression becomes 
 bc"-^cb^-{-bc{b—c) — bc(b-^c) -bc{b- c), vriiich equals zf;ro ; there- 
 fore a is a factor ; by symmetry b and c are also factors. 
 
 The expression is of three dimensions, and abc is of three 
 dimensions, there cannot therefore be any other hteral ff,ctoi\ 
 
 Hence the expression =viabc. 
 
 To find m, let a = b = c=l, and v/e have 
 
 m = 12. 
 .-. the expression =12abc. 
 In the preceding examples the factors have been linear, but the 
 principle applies equally well to those of higher dimensions. (See 
 Th. ii. Cor.) 
 
 8. Examine whether «" + ! is a factor of a;3" + 2a;-"+3.f" + 2. 
 Let a;" + 1 = 0, or x-"=— 1, and substituting, the eiipvessiou 
 
 vanishes, therefore, x"-\-l is a factor. 
 
 9. Examine whether a^ + b^ is a factor of 
 
 2a4+a36+2a2t/2+a^3, 
 
 Let a^ + b- =0, or a^ — —b", substituting, we have 
 2^-i_a63_2i4+a63 which = 0, and 
 therefore <^^-\- 6^ is a factor. 
 
FACTORING. 89 
 
 10. Prove that a^+h^ is a factor of 
 
 a^-^a^^b+a^h^ + a-b-'^ab^ + b'^. 
 Lei a3_}./,3_o, or a^ = -b^; substituting, we have 
 -a?b^ -ab^-b^+a^b^-^ab^-{-b^, wliich = 0, and 
 therefore a^-{-b^ is a factor. 
 
 Exercise xxxvii. 
 Eesolve into factors 
 
 1. {x-i-i/+z)^ — {x^+y^+z'^)' 
 
 2. bc(b — c)—ca{a~(;) — db{b~ti). 
 
 8. (a3-63)3 + (62-f=^)3+(c2-aa)3. 
 
 4. x{y+z)^+7j{z+xy+z{x + ijy-4.xijz. 
 
 5. (^a+b)^ ^{b-\-c)^ + {c- a)^. 
 
 6. a(6-c)3+6(c-«)3 + c-(«-6)^ 
 
 7. (a-i-b-\-c){ab-{-bc-{-ca)—abc. 
 
 8. a^{c-b^) + b^{a-c'^)+c^{b~-a^)+abc{abc-l). 
 
 9. «3(6 + (;)4.i2(f + a) + c2(a + /^)4-2(/ic. 
 
 10. (a -b){c-h){c-k)-{-{b-c){a-h){a-k) + {c- a){b- 'i){b-Ic). 
 
 11. a;4^3+,c2^4+a;4s3_^a;2z4+^423^y3244.2a;2^223. 
 
 12. (a-^)5 + (fe-6-)5+(c-a)s 
 
 13. afc(a+6)+M^+c)+«a(f+rt)+(a3 + 6»+c-3). 
 
 14. a4(c-63)+64(rt_c3)+c'i(6-a3)4-a66-(a262c3-l). 
 
 15. a;4(y3_23)+^4(23_^2)4.24(a;2__,/3). 
 IG. a;* + ^4 ^24 _2aj22/3_ 2^223 _ 222^3. 
 
 17. (6-c)(a;-fc)(x-c)+(c-a)(a;-c)(a;-a)4-(«-^)(^-«)(^-6). 
 
 18. {a+by^ + {h + c)^ + {c + a)s + 
 
 B(rt+26 + 6-)(6+2c+rt)(6-+2a+6). 
 
 19. Shew that a^ +(t^b^-ab^ - b^ has a^ -b for a factor. 
 
 20. Shew that {x + yY -x' -y' =lxy{x+y){.v^ +.'-V/ + ^/-)^ 
 
 21. Examine whether x^ — 5x-\-iJ is a factor of 
 
 ^3_9^2_^26a;-24, 
 
90 fACTOEINO, 
 
 22. Shew that a — b-\-c is a factor of 
 
 23. Shew that a" +Sh is a factor of 
 
 and find the other factor . 
 24. Find the factors of W^ib - c)-i-b^{c ~ a)-{-c^(a—b'). 
 
 Section V.. — Factoring a Poltnome by Teial Divisors. 
 
 Art. X,XVII. To find, if possible, a rational linear factor of 
 the poiynome. 
 
 a;»^6a;"-i+ca;"-'+ ^^nx+k. 
 
 Substitute successively for x every measure (both positive and 
 negative) of the term k, till one is found, say m, that maT\es the 
 polynome vanish, then x - m will be a factor of the polynome. 
 
 Examples. 
 
 1. Factor a;3+9a;2+16a;+4. 
 
 The measures of 4 are +1, +2 and +4. Since every coefS- 
 cient of the given polynome is positive, the positive measures of 
 4 need not be tried. Using the others <• it will be found that — 2 
 makos the polynome vanish ; thus 
 
 1 9 16 4 
 
 -2 -14 -4- 
 
 -2 
 
 17 2; 
 
 Hence the factors are (a3 + 2)(x3-|-7a;-f2). 
 
 The labour of .substitution may often be lessened by arrang- 
 ing the polynome in ascending powers of x, and using 1 -h 
 (measure of k) instead of the measures of k. (This is really 
 substituting 1 -r measure of k, for l^x). Should a fraction 
 occur during the course of the work, further trial of that measure 
 of k will be needless. 
 
FACTORING. 
 
 91 
 
 Examples. 
 
 2. Factor x^ - lOz^ - 63a;+60. 
 
 The measures of 60 are +1, +2, ±3, ±4, +5, &c. Neither 
 4-1 nor - 1 will make the polynome vanish. Try 2 ; thus 
 
 60 -63 -10 1 
 
 1^ 
 2 
 
 63 
 30 
 
 30 -16^ 
 
 A fraction occurring we need go no further. — 5 will also give 
 a fraction, as may easily be seen. Next try 3 ; thus 
 
 60 -63 -10 1 
 
 •63 
 
 20 
 
 20 
 
 -14i 
 
 A fraction again occuring, we may stop. 
 fwaotion. Next try 4 ; thus 
 
 — 3 will also give a 
 
 1 
 
 T 
 
 60 
 
 -63 
 15 
 
 ^10 
 
 -12 
 
 Next try —4. 
 
 -1^ 
 
 T 
 
 15 
 
 60 
 
 -12 
 
 -63 
 
 -15 
 
 5* 
 
 -10 
 
 15 -19i 
 Next ti:yiu^ 5 we find it fails, then try — 5, thus 
 60 
 
 -1 
 
 -63 
 -12 
 
 -10 
 
 15 
 
 1 
 -1 
 
 12 -15 1; 
 
 The remainder vanishes as required ; the factors are, therefore, 
 (a;+5)(a;3-15a;+12). 
 
 Art, XXVIII. When k has a large number of factors, the 
 number that need actually be tried can often be considerably 
 lessened by the following means. 
 
 Add together all the coefficients o£ x (including the constant 
 term k) ; let the sum be called k^. 
 
92 FACTORING. 
 
 From the sum of the coefficients of the even powere of x 
 (including k) take the sum of the coefficients of the odd powers of 
 x; let the remainder be called Jc^. (In the coefficients are in- 
 cluded the signs of the terms). 
 
 1st. If k^ vanish, x — 1 will be a factor of the polynome. 
 
 2nd. If /Cg vanish, x+1 will be a factor of the polynome. 
 
 3rd. If both k^ and Ag vanish, x^ —1 will be a factor of the 
 polynome. 
 
 4th. If neither ky nor k^ vanish, (writing p for " a positive 
 measure of ^ greater than 1 ") ; 
 
 (a) We need not try the substitution of p for x unless ^ — 1 be 
 a measure of ky, and ^-fl a measure of k^. 
 
 {0) Nor need we try the substitution of —p for x unless p-^-l 
 be a measure oi k^, and p — 1 a measure of kc^. 
 
 (In trying for measures, the signs of k^ ky, and k^ may be 
 neglected. 
 
 Examples. 
 
 1. Find the factors of x^ - lOx^ - 63a;+G0. (^ee Ex. 2 above). 
 
 Here ^ = 60 ; ^1= 1 -10-63 + 60= -12, 
 A;2 = -1-10+63+60 = 112. 
 
 Tabulating the trial-measures we get 
 
 12 
 
 1,' 
 
 2, 
 
 3, 
 
 4, 
 
 
 60 
 
 2, 
 
 3, 
 
 4, 
 
 5, 6, 
 
 10, 12, 
 
 112 
 
 
 4, 
 
 
 7, 
 
 
 12 
 
 3, 
 
 4, 
 
 
 6, 
 
 
 60 
 
 2, 
 
 3, 
 
 
 5, 6, 
 
 10, 
 
 112 
 
 1, 
 
 2, 
 
 
 4, 
 
 
 (It is evident that 12 is the highest measure of 60 we need try 
 in the upper table, for the next measure, 15, would give 14 as a 
 trial-measure of 12, and higher measures of 60 would give higher 
 trial-measures. Similarly, 10 is the highest measure that need 
 be tried in the lower table.) 
 
FACTORING. 
 
 93 
 
 In the upper table, 8 is the only measure of 60 that gives a 
 full column ; hence of the positive measm-es of 60 we need try 
 only the substitution of 3 for x. 
 
 In the lower table, 2, 3, and 5 give full columns, hence we 
 must try the substitutions —2, —3, —5 for x. 
 
 On trying the four substitutions to which we are thus restricted 
 we find — 5 is the only one for which the polynome vanishes. 
 (See Ex. 2 above). 
 
 - 2. Find the factors of x^+12x^ -4:0x^ +Q7x-120. 
 &=-120; /.-i=H-12--40+67-120= -80; 
 A, = 1 - 12 -40 - 67 - 120 = - 238. 
 
 80 
 
 120 
 238 
 
 1, 
 
 2, 
 
 2, 4, 5, 
 
 3, 4, 5, 6, 
 
 7, 
 
 8, 
 
 10. 
 
 12, 
 
 15, &c. 
 
 
 
 80 
 120 
 238 
 
 2, 
 1. 
 
 4, 5, 
 
 3, 4, 6, 6, 
 2, 
 
 8. 
 7, 
 
 10, 
 
 i 
 
 16, 
 
 15, 20, 
 14, 21, 
 
 24, 
 
 &c 
 
 The upper table gives us 6 as a trial-measure, and the lower 
 gives us —3 and —15. 
 
 Trying these 
 
 ) we get 
 
 
 
 
 
 
 -120 
 
 67 
 
 -40 
 
 12 
 
 1 
 
 1 
 
 
 -20 
 
 
 
 
 6 
 
 - 20 
 
 U 
 
 
 
 
 
 -120 
 
 67 
 
 -40 
 
 12 
 
 1 
 
 -1 
 
 
 40 - 
 
 
 
 
 3 
 
 -40 
 
 36f 
 
 
 
 
 
 -120 
 
 67 
 
 -40 
 
 12 
 
 1 
 
 -1 
 
 
 8 
 
 — 5 
 
 3 
 
 -1 
 
 16 
 
 - 8 
 
 6 
 
 - 3 
 
 1: 
 
 
 
94 
 
 FACTOEINO. 
 
 Hence a;+ 15 and x^ — Sx^ + 5x - 8 aie the factors. The latter 
 cannot be resolved, for our tables above tell us we need try only 
 x—6, x+3, and a;+15. The first two have been found not to be 
 factors, and 15 will not measure 8. 
 4. Factor a;" _ 27a;2 + Ux+ 120. 
 
 k=120] ^'i =1-27+14+120 = 108 
 /.•2 = 1-27-14 + 120= 80. 
 
 108 
 
 1, 
 
 2, 
 
 3, 
 
 4, 
 
 
 
 9 
 
 
 
 120 
 
 2, 
 
 3, 
 
 4, 
 
 5, 
 
 6, 
 
 8, 
 
 10, 
 
 12, 
 
 15, &c 
 
 80 
 
 
 4, 
 
 5, 
 
 
 
 
 
 
 16, 
 
 108 
 
 3, 
 
 4, 
 
 
 6, 
 
 
 9 
 
 
 
 
 120 
 
 2, 
 
 3, 
 
 4, 
 
 5, 
 
 6, 
 
 8, 
 
 10, 
 
 12, 
 
 15, &c 
 
 80 
 
 1, 
 
 2, 
 
 
 4, 
 
 5. 
 
 
 
 
 
 The upper table gives us 3 and 4, the lower table gives us - 2, 
 —3, and —5. Using these in order we get 
 
 Hence x — 3 is a factor. 
 
 Hence a; — 4 is a factor. 
 
 Hence ar+2 is a factor, 
 and there remains x-\-o, a factor. 
 
 Hence the factors are {x-B){x—4:){x-\-2,){x + 5). 
 5. ¥&ctor x*-px^-\-{q-l)x^+px q. 
 
 k=-q; k^ = l-p + {q-l)+p-q = 0; 
 k„ = l +p+{q-l) -p-q = 0. 
 Since both ki and /c, vanish, the polynome is divisible by both 
 x—1 and x-\-l. 
 
 1 
 
 120 
 
 14 
 
 40 
 
 -27 1 
 18 -3 -1 
 
 3 
 1 
 
 40 
 
 18 
 10 
 
 -3-1; 0. 
 
 7 1 
 
 4 
 -1 
 
 10 
 
 7 
 -5 
 
 1; 
 -1 
 
 2 
 
 5 
 
 1; 
 
 
 
 1 
 
 1 
 
 -V 
 
 1 
 
 q-1 
 -p + 1 
 
 P 
 
 q-p 
 
 1 
 
 1 
 
 1 
 
 -;,+l 
 -1 
 
 q-p 
 +P 
 
 9; 
 -9 
 
 
 
 
 1 
 
 -P 
 
 9; 
 
 
 
 
FACTORINa. 
 
 95 
 
 Hence the othei- factor is x^ —px-^q. 
 
 6. Factor x^+2ax^+{a^ + a)x-'-{-2a^x+a^. 
 
 k^ = 1 -2a+(a2 +a) - 2a^ +a^ ^a^-a^ -a-1. 
 
 The positive measures of k are 1, a, a^, a^. Of these 1 may 
 be rejected at once, since neither k^ nor k^ vanish, and a^ and a^ 
 may also he rejected since k^ or (a + 1)3 is not divisible by eithei 
 a^ + l or a' + l. But k^ is divisible by a-\-l, and k^ is divisible 
 bya— 1; thus we need only try the substitution of -aforx. 
 (See 4 j5, page 92) 
 
 
 1 
 
 2a 
 
 a2+a 
 
 2a3 
 
 flS 
 
 — a 
 
 
 — a 
 
 -a2 
 
 -a3 
 
 -a? 
 
 
 1 
 
 a 
 
 a 
 
 a3; 
 
 
 
 —a 
 
 
 — a 
 
 
 
 -a2 
 
 
 
 1 
 
 
 
 a; 
 
 
 
 
 Hence the factors are {x-{-ay^{x^-\-a). 
 
 7. Factor a;3_(rt+c)a;2 +(&+ac)a; -6c. 
 ^: = — 6c ; 
 
 4j = l — {a+c)-^{b+ac) — bc = l—a+b—c + ac — bc 
 fe, = - 1 - (a+c) — (6+ac) - 5c = — (1 +a+i+c + ac+6c). 
 
 The factors oi k^, other than 1, are b and c. k^is not divisible 
 by either 6+1 nor by c+-l. However, Zr^ is divisible byc-1, 
 and k^ is at the same time divisible by c+1, /. we need only try 
 the substitution of c for x. (See 4 «, page 86). 
 
 (a+c) (6 + ac) —6c 
 
 c — ac ic 
 
 1 —a 6; 
 
 Hence the factors are {x— c){x^ — ax+b). 
 
96 
 
 FACTORING. 
 
 Exercise xxxviii. 
 
 1. a3_9„2_^l(5a_4. 
 
 2. a;3-9a;2+26a;-24. 
 
 3. a;3-7a;2+lGx-12. 
 
 4. x^-Ux+16. 
 
 5. x^ + Sx^ + 5x-\-B. 
 
 7. x^-dx+2. 
 
 8. a;4 + 2a;2+9. 22 
 
 9. m^-Sm^-n+^mn^-27i^. 23 
 
 10. x^-\-2x^-\-2. 24 
 
 11. jftS _ 5m2n-\-8t7in^ — 4n^. 
 
 12. 63+62c+76c2 + 39c3. 
 
 13. m^ — 4mn^ + 3n'^. 
 
 14. a4_7a36 + 28rti3_i664. 
 
 29. x^-18x^ + UBx^-288x+252. 
 
 30. a;4-9a;3i/4-29x22/2_3ga;i^3T-i8y4, 
 
 15. a;3_llx2+39a;-45. 
 
 16. x^-\-5x2+lx-\-2. 
 
 17. «3_3o,2_i93rt_|.i95. 
 
 18. p^-Sp^-6p-8. 
 
 19. a4+3,t3_3rt2_7fl_|_a; 
 
 20. a'''-6a*"+lla"'-6. 
 
 21. aA-Ua^b~ + lGb^. 
 
 rt4_rt3/,2_2a63H-264. 
 
 ^,3_4^;2^67;-'4. 
 
 a;'"+4a;"" - 5. 
 25. y4-5?/M'8i/2-8. 
 26 a4-2a3 + 3a3-2«+l. 
 
 27. a3+a253^a62_363. 
 
 28. 2a''' -a'" -a" 4-2, 
 
 Art. XXIX. To find, if possible, a rational linear factor of 
 the polynome 
 
 aaf+bx''-^-\-car-^+ +hpB + k. 
 
 First Method. Multiply the polynome by a"~^ 
 
 (axy+b{ax)'^^+ac{azy-^' + -^a^-'hiax) + a'^-'k ; 
 
 or writing y for aXy 
 
 ynj^lyn-i ^ acy"-^ + +«"-'*% + a^-^k. 
 
 Factor this polynome by the method of the last article, replace 
 y by ax, and divide the result by a"~\ 
 
 Example. 
 
 Factor 3j;4-|-5a;3 -33a;2+43a;-20. 
 Multiply by 33 and express in terms of Sx. 
 
 (3a;)4 + 5(3a;)3-99(8a;)2+387(3a5)-540 ; 
 or, 2/4+52/3-99^/3 + 387// -540. 
 
FACTORING. 
 
 97 
 
 Here /t=: -540; i5:i = 1 + 5-99+387 -540== -246: 
 
 5 - 99 - 387 -^540 = - 1030. 
 
 k, = l 
 
 246 
 
 640 
 
 1030 
 
 246 
 
 640 
 
 1030 
 
 1, 2, 3, 6, 41, 82, 123, 246. 
 
 2. 3, 4, 
 
 5, 
 
 3, 
 2, 
 1, 
 
 6, 41, &c. (Trying by factors of 246 
 
 5, instead of by factors of 640, 
 
 for convenience). 
 
 The only factors of 540 in full columns are 4 in the upper 
 table and 2 in the lower one ; hence we need try only the substi- 
 tutions 4 and —2. 
 
 1 
 
 -540 
 
 387 
 -135 
 
 -99 
 63 
 
 5 
 
 -9 
 
 1 
 
 -1 
 
 4 
 
 -135 
 
 63 
 
 - 9 
 
 -1; 
 
 
 
 Hence ?/- 4 is a factor. The substitution —2 need not now 
 be tried, since we see that 135 is not a multiple of 2. The other 
 factor is therefore y^-\-2y^ —Q^y-^1^6. 
 Replacing y by 3a; and dividing by 27 ; 
 
 ^V(3a; - 4)(27a;3+-81aj8-189a;-}-135) 
 = (3a; - 4)(a;3 +3a;2 -7a;+5), 
 wnich are the factors. 
 
 Art. XXX. Second Method. Writing m for "a measure of 
 a," and p for a " measure of k, positive or negative ;" 
 
 For aj substitute every value of p-wtill one, say;?'-r-m' bb 
 found which makes the polynome vanish ; then m'x—p' will be 
 a factor. Should a fraction be met with in the course of substi- 
 tution, further trial of that value oi p-rvi will be useless. 
 
 Should k have more factors than a, it will generally be better 
 to arrange the polynome in ascending powers of x and use values 
 of m^p instead of p-i-m, making 2^ positive and ?« positive or 
 negative. 
 
98 FACTORn<[G. 
 
 To reduce the number of trial-measures, calculate k^ and k^, as 
 directed on page 92, tken 1, 2, 3 hold as on that page, but in 4 
 read^ — m for^ — 1 andp-j-m for^j+l. , 
 
 1^^ Examples. 
 
 1. Factor 36a;3 + 171x2 -22X+480. 
 
 ;t = 480, yti= 36+171-22-1-480 = 665 
 ;;;2= -364-171+22+480 = 637. 
 m may have any of the values +1, ±2, +3, +4, ±6, +9, 
 
 + 12, +18, +36. 
 
 In forming the table write out the measures of ^j ; take each 
 measure in succession and add to it each value of m separately, 
 should the sum measure 480, i.e., k, add to it the same value of 
 m, and should the new sum measure 637, i.e., k^, keep the mea- 
 sure of 480, writing above it the value of m used. Should the 
 sum in either case iiot be a measure, another value of ni must be 
 tried ; "^hen all the values of m have been tried, another measure 
 of 665, i.e., k^ must be tried till all have been tested. (Measures 
 of yfc, or 665 have been used in this instance because they are 
 much fewer than those of 480 ; measures of k^ or 637 would have 
 
 done equally well). 
 
 • 
 
 m=+3, +1, +3 -2 -8 -9 -3 
 
 665 1, 5, 7 5 7 19 19 
 
 480 4, 6, 10 3 4 10 16 
 
 637 7, 7, 13 1 1 1 18 
 
 Hence the only substitutions that need be tried are 
 
 8J_8 z^ zl Zl Z^ iox L 
 T' 6' To' 3' 4' 10\ 16' x' 
 
 Arrangement in ascending powers of x. 
 
 By actual trial, as below, we find ^1 is the only one of these 
 giving a zero remainder. 
 
FACTORING. 
 
 99 
 
 4 
 1 
 
 3 
 
 10 
 
 -2 
 
 8 
 -8 
 
 4 
 -9 
 
 480 
 
 - 22 
 360 
 
 171 
 
 86 
 
 10 
 -3 
 
 120 
 
 84i 
 80 - 
 
 
 
 80 
 
 144 
 
 
 
 48 
 
 12-2 
 -320 
 
 228 
 
 -266 
 
 160 
 
 -114 
 360 
 
 133; 
 
 -230 
 
 120 
 
 - 95i 
 -432 
 
 
 
 48 
 
 -45-4 
 - 90 . 
 
 21 
 
 -36 
 
 16 30 - 7 12; 
 
 (The coefficients are written only once, and understood for the 
 other hnes of substitution.) 
 
 Hence the factors are 3a; + 16 and 12a;2 _ 735+30, 
 ■ The latter factor cannot be resolved, for 16 will not measure 
 30, and all the other factors left iox trial by the tables above, 
 have been tried and have failed. 
 
 2. Factor 10a;* -a;3(15|/+4z) -a;3(40y» -6y«) + 
 
 Here m= ±1, +2, ±5, or ±10. k= -2.4t/»z. 
 
 k^ = l0-{15y+iz)-(4:0y^-eyz) + (G0y^-\-16y^z)-2iyH 
 = 10- 151/ - 402/2-1-602/3 -2;s(2-3y- 82/2 -M2?/S) 
 = (5-2z)(2-3y-82/24-12.y5). 
 A-- = (5 + 2z)(2+3?/—8y2 — 127/3), as may easily be found 
 by making the calculation. 
 
 We get at a glance 2z a factor of k, 2z — 5 a factor of Ic^, and 
 2z+5 a factor of k,^ ; hence taking w = 5, we are directed to trj' 
 
 2z 
 the substitution -_ for x. 
 5 
 
 10 -{15y+iz) -{iOy^-6yz) {my^ + 16y^z) -24^/3- 
 
 4z —Qyz . -16y'-z 24?/ "^2; 
 
 ^%2 12^^3'; 
 
 6 
 
 2 -3y 
 
100 
 
 FACTORING. 
 
 Hence 5x~2z is a factor, tlie other being 
 
 2x^-3x^y-8xy2-i-12y3. 
 
 • The latter factor being homogeneous, the method of this article 
 may be appHed to it. 
 
 '^i=+lor±2, k = 12, ki=8, k^ = 15. 
 m = l, 2, 1, -1 
 
 1, 3, 3 The other columns 
 
 3, 4, 2 are not full. 
 
 5, 5, .1 
 
 Hence the trial- substitutions (arrangement m ascending powers 
 of x) are i, |, ^, :±. 
 
 3 
 
 1, 
 
 12 
 
 2, 
 
 15 
 
 3, 
 
 
 12 
 
 -8 
 
 -3 
 
 2 
 
 1 
 
 
 6 
 
 -1 
 
 -2 
 
 2 
 
 6 
 
 -1 
 
 -2; 
 
 
 
 2 
 
 
 4 
 
 2 
 
 
 3 
 
 2 
 
 1; 
 
 
 
 
 Pinal factor is 2>j+x» 
 
 Hence the factors are {z—2i/){2x-3i/){x+2y), and these, with 
 the factor 5x — 20 already found, give the complete resolution of 
 the polynome proposed. 
 
 (The factor 5x — 2z, might easily have been got by the method of 
 Art. XXIII., page 79, but the present solution shows we are inde- 
 pendent of that article. It may also be obtained by rearranging 
 the polynome in terms oiy). 
 
 Exercise xxxix. 
 
 Factor 
 
 l.aa;3_20fl;»+38a;-20; x^ -Ix^y + lQxy^ -I2y^. 
 
 2. 12x^ + 5x^y-\-xy^+dy^ ; 8a;3_i4a;+6. 
 
 3. Bx^-15ax+a^x-5a^; 2x^-\-9x^y-\-7xy^ -Sy^. 
 
 4. 2^*4_7J3c_462c2 + 6c3-4c4 ; 15a3+47a36 + 13a6»- 1263. 
 
 6. UOx'^ -725a;3y+98ia;3y2 ^Q20xy^ - 1152^4, 
 
 7. d6x^-6{9 -7y)x^ - 7{9 + Uy)x^y+3{4:9-40y)xy^ + 180y3. 
 
 8. lOx* -x^l5y + iz) + x^{4:0y^-\-Gijz)+x{60y^ -Hj^-^z)-24.y^z 
 
DIVISION. 101 
 
 CHAPTEE IV. 
 
 Section I. — Division. Measures and Multiples. 
 
 Art. XXXI. When one quantity is to be divided by another 
 the quotient can often be readily obtained by resolving the divisor 
 or dividend, or both, into factors. 
 
 Examples. 
 
 1. Divide a'i-2,ab-{-b^ -c^-{-2cd-d^ hy a— \b+c—d. Here 
 we see at once that the dividend ={a—b)^-{c — d)^, and .•. quo- 
 tient = a-b — {c-d) = a — b — c-^d. 
 
 2. Divide the product of a^-\-axi-z^ and a^-\-x^ by a*'-{-a^x'^ 
 •{■z^. Here a^-[-x^ = {a-\-x){a'^ — ax-\-x^), and the divisor = 
 (a^-^ax+x^){a^ — ax+x^) :. the quotient is a -fx. 
 
 3.*Divide a^+a^b+a^c—abc-b^c-bc^ by a^-bc. The divi- 
 dend is a{a^ —bc)-\-b{a^ -6c)4-c(a* —be) /.the quotient =a+b-Jr-c, 
 
 4. (a3+b^-c^-^3abc)^{a-{.b-c). 
 
 Dividend =a^-\-b^+Sab{a+b)-G^ -3ab{a+b)-\-SabG={a'^b)^ 
 — c^ — Bab{a -{-b — c) which is exactly divisible by a +6— c; quotient 
 =a^-^b^ -\-c^ —ab-\-bc-\-ca. 
 
 5. Divide x^ —x^y-{-x^y^ — x^y^ +xy^—y^ bjx^ — y^. 
 
 The dividend is (Art. XXV.) evidently (x^ —y^) -^ (a'+*')j and 
 this divided by x^ —y^ = (^x^-\-y^) -h (x+y) = x^ — xy-\-y^ . 
 
 6. Divide b{x^ +a.3)-{-ax{x^ - a^)+a^{x-{-a) by {a-^b){x^-a). 
 Striking the factor x + a out of dividend and divisor we have 
 b{x^ — ax-\-a^)-\-ax{x — a)-^a^ = b(x^ — ax + a^y-\-a{x^ — ax-\'0-) 
 = {a-\-b){x'^ — ax+a^) .'. quotient =x^ -ax-\-a^. 
 
 7. Divide apx^ +x^(aq+bp)-\-x^{ar-\-bq-\-pc)-\-x{qc-\-br) +cr by 
 ax'-^bx-i-'' 
 
102 DIVISION. 
 
 Factoring the dividend (Art. XXIII.) we have 
 
 (ax^ + bx+c){px^ -^qx+r). 
 :. the quotient = the latter factoi". 
 
 S. Divide Gx^ - ISax^ + ISa^x^ - 13a^x~ 5«4 by 2a;^ - Sax- a''. 
 
 This can be done by Art. XVII. The divisor is 2x^ ~a^- Sax, 
 and we see at once that 3x'+5a'^ must be two terms of the quo- 
 tient. 
 
 Multiplying diagonally into the first two terms of the divisor, 
 and adding the products, we get -{-7a^x^ ; but -^ISa^x'^ is re- 
 quired. .'. -\-Qa-x'^ is still required, and as this must come from 
 the third term multiplied into — 3ax, that third term must be 
 — lax; :. the quotient is 3x^ -^-ba^ —2ax. 
 
 jJoTE. — By multiplying the terms - lax, — 3ax, diagonally into 
 the ic^'s and a^'s respectively, we get the remaining terms of the 
 dividend ; it is, of course, necessary to test whether the division 
 is exact. 
 
 9. Divide 2«* - a^J - 120252 _ s^js _{.4J4 by «« - ja - 2a6. 
 
 Here, as before, one factor is a^ — h'—1ah; :. ft<7o terms of 
 the other factor are 2a2-462. Multiplying, as in the last 
 example, we get -&a^h^; but -lla^^b^ is required. .•. — Sa^fts 
 is still needed, and -\-3ah is the third term of the requii-ed quo- 
 tient, which is therefore 2a2_463_|.3a/,. 
 
 Prove that 
 
 10. (l-fa;-|-a;2+ - - - -^x-'-^){l-x+x^ - .... -fa;"-^) 
 
 = l-fa;2+a;'i+ .... +x' 
 1 - X" 1 -1- a;" 
 
 ^i»— 2 
 
 Product = 
 
 1—x l-\-x 
 l — a?^ 
 
 z ^ = 1+X^+X*'i .... +^-»-». 
 
 1 —X^ 
 
 11. Divide (a^ -bc-y -{-Sb^c^ by a^-\-hc. 
 
 = (a^ -bc)^ +{2bc)^ hy {a^ -bc) + 2bc 
 « (a3 _ic)3 -(«2 _ be) X 2k-f-(26c)» 
 = a4^.4a3ic-f763ca. 
 
DIVISON. 103 
 
 12. Divide l+23579-i7691a;9 by l-llx+121a;3 
 Dividend =l + (lla;)^ ^ 
 
 = [1 -(llx)3+(ll.'c)«}{H-(lla:)3} 
 
 Divisor= {H-(lla;)3} ^(1+lla;). 
 .-. quotient =a-(ll£c)3+(llx)«}(l + lla:). 
 
 Exercise xl. 
 
 Find the quotients in tlie follo\ving cases : 
 
 1. l-x-\-x^-z^-^l-x. 
 
 2. l-2a;4+a;»H-a;4+2a;3 + l. 
 
 4. x4+4x3j/3-32?/4^a;_2?/. 
 
 5. l-4:X^+12x^-9x'^-^l + 2x-Sx'^. 
 
 6. (rt2 -2ax+x^-)(a^+3a^x+3ax^-{-x^) ^a^ -x». 
 
 7. x^—y^+z^+dxyz^x — y-rZ. 
 
 8. 6a4-a36 + 2,'i262 4-13«63_f.4j4 ^ 2rt» -3a6-f 4fc*. 
 
 9. 4«4-a;32/2+6a;;/3-9^4 ^ 2a;2+3»/3-arj/. 
 
 10. rt*+/)4_c4_2«252 -^rt3_&2_c2. 
 
 11. 21a4-16rt3J4-16a363-5a63 + 264 -r- 3«2 -a&^-J*. 
 12.-2rt3_7a3_4Grt_21 ^ 2a2+7a+3. 
 
 13. {a^{b~c)+b^{c-a)-\-c^{a-b)} -r a-{-i+c. 
 
 14. x^-Sax-+Sa^x-a^+b^ -^x-a+b, 
 
 15. a;* - ^4+4 + 2a;2z2- 2^3-1 -7- a;3 -1/8+22-1. 
 
 16. a:* — (a+c)a;3 + (6 + ac)x3— ica; H- a;-c. 
 
 17. x^-^x^y +xy^-{-y^ -i- a:+?/. 
 
 18. x'' —x^y+x^y^ — x'^y^-\-x^y*-x^y^+xy^—y'' -^z^+y*, 
 
 19. (j44.;,4_c4_2a2i2_2c3_l -H a2-62-c2_l. 
 
 20. a4-rti3_rtc3_2a3&+264 + 26c3 + 3a3c-363c-3c4 
 -5-a + 8c-26. 
 
 21. a^b-bx~+a^x-x^ H- (a: + 6)(a-a;). 
 
 22. a(6-c)3 + 6(c--a)3+c(a-6)3 ^a^-ai-ac+Sc, 
 
104 MEASUKES AND MULTIPLES. 
 
 23 aH^ + 'lnhc--a^c^ -ir-c^ ^ab+ac-bc. 
 2A. x^+y^ + -d.v.i/-l -y x + y-1. 
 
 25. x^-x^-2 ~ x^-x + 1. 
 
 26. a^-2da2-50a-21 -^ a^~5a-l. 
 
 27. (2x-'y)^a^-{x+y)^a^x^+2{x+'i/)ax'^-x^ -^ 
 {2x — y)a--t-(x + y)ax — x^. 
 
 28. {x^-l)a^-{x^+ao^-2)a^+{4:X^+dx+2)a-d{x+l) * 
 -=^ {x-l)a^~{x-l)a-{-d. 
 
 Art. XXXII. The Highest Common t actor of two algebraic 
 quantities majj in general, be readily found by factoring. The 
 H. C. F. is often discovered by taking the sum or difference (or 
 sum and difference) of the given expressions, or of some multiples 
 of them. 
 
 Examples. 
 
 1. Find the H. C. F. of {h-c)x--\-{2ah-2ac)x + a"-h-a'^c, and 
 iah — ac-\-b'''^ — bc)x-\-a^ c -\- ah^ —a^b- abc. 
 
 Taking out the common factor b — c we get (b — c){x^ +2ax-\-ub} 
 and {b — c)\(a — b)x-a^-i-ab} ; 
 .'. b — c is the H. C. F. of the i^iven expressions. 
 
 2. Find the H. G. F. of 
 
 1 — x+y + z — xy + >iz —zx- xyz, and 
 \ — z—y—z^xy-\-yz-\-zx—xijz. 
 
 Their difference is %j-\-2z — 2xy-'lzx = 2(1 — x){}j-{-z). 
 Their sum is 2 - 2x-Y2yz — 2xyz = 2(1 — a;)(l +^2). 
 .'. theH. C. F. is (1-a;). 
 
 3. Find the H. C. F. of a;"' + Sa;^ -8^2 -9a;-3, and 
 
 jc-^ -2a;4-6a;3+4a;2 + 13a;+6. 
 
 The annexed method of finding tlie H. C. F. depends on the 
 principle, that if a quantity measures two other quantities, it will 
 measure any multijjle of thek sum or difference. 
 
MEASURES AND MULTIPLES. 
 
 1 
 1 
 
 + 3 
 - 2 
 
 0-8-9-3 (a) 
 -6 + 4 4-13 + 6 {h) 
 
 
 
 
 + 6 -12 -22-9 (c) 
 
 2 
 
 1 
 
 + 6 
 - 2 
 
 -16 -18 - 6 
 _ 6 + 4 +13 + 6 
 
 3 
 
 + 4 
 
 -6-12-5 (f/) 
 
 
 15 
 16 
 
 + 18 -36 -66 -27 
 +20 -30 -60 -25 
 
 
 
 -2-6-6-2 
 
 
 1 + 3 + 3 + 1 (f) 
 
 
 25 +30 -60 -110 -45 
 27 +36 -54 -108 -45 
 
 
 -2 - 6 - 6 - '2 
 
 (a)x2 
 (6) 
 
 (c)x3 
 (£i)x6 
 
 (r)x5 
 (ti)x9 
 
 1 + 3 + 3 +~1 (gr) 
 E. C. F. = (a;+l)3. 
 The coefficients are written in two lines, (a) and (6). They 
 are then subtracted so as to cancel the first terms, (a) is next 
 multiplied by 2, and added to cancel the last terms. K (c) and 
 (d) had been the same their terms would hav.e been the coefficients 
 of the H. C. F. Since they are not, we proceed with them as 
 with (a) and (6) till they become the same. When {a) and (b) 
 do not contain the same number of terms it is more convenient 
 to find only (r), aud then use this with the quantity containing the 
 same number of terms. The general rule is to operate on hues 
 containing the same, or nearly the same number of terms. 
 
 4. Find the H. 0. F. of Sx^ + 2x^ -Ux + 8, and 
 
 6a;3-lla;3 + 13a;-12. 
 
 8 + 2-14 + 8 (a) 
 
 6 -11 +13 - 12 (6) 
 6~+ 4 -28 +16 (a) x 2 
 
 15 -41 +28 (c) (6)-(«), 
 
 (5_7)(3-4) 
 
 H. C. F. = 3a; -4. (d) 
 
 If (a) and (b) have a common factor its first term must measure 
 8 and 6, and its last term must measure 8 and 12. (c) is not 
 
106 ME4SURES AND MULTIPLES. 
 
 therefore, the H. C. F. Eesolve (c) into factors. 5a;-7isnota 
 factor of (a) and (b). If, therefore, (a) and {b) have a common 
 factor it is 3a; -4. On trial 3a; -4 is found to be a factor of (a) 
 and .-. it is the H. C. F. of (a) and (6). 
 
 6. 1{ X- -{-px+q, and a;2 + >'a;+s have a common factor, prove 
 that this factor is 
 
 x"+ . If x—a be the common factor then the remainders 
 
 p — r 
 
 on dividing; the given expressions by x—a, must be zero, i. e., 
 
 a^+pa+q = 0, and a--\-ra+s = 0, or 
 
 s—q 
 i (^ - r)a = *-?,.-. a = ^3^, and 
 
 s —q q— s 
 
 x-a = x— =x+ r~~Z' 
 
 p —r P — ^ 
 
 6. What value of a will make a^x^+{n-{-2)x+l, and 
 a^a;2-fa2— 5, have a common measure. 
 
 They cannot have a monomial factor. Neither can they have 
 one of two dimensions unless {a + 2} vanishes, i.e., unless a= — 2, 
 in which case the expressions become Ax^ + l, and 4a;'^ —1, which 
 have no C. F. Hence if the given quantities have a C. F., it 
 must be of the form a;+»i; dividing a3a;2+a2 — 5 by x-\-m, we 
 have for remainder, 
 
 5 — a' 1 
 
 fi2„i2 + fl,2_5 = o, orm3= — ~^- ; .-. w= — ^/'(S-a^), in which 
 
 l/{5 — a^) must be possible and integral, .-. a2=4, («2 = 1 gives 
 values to m which on trial fail) and a=± 2, of which the positive 
 ^ alue must be taken, and .-. 2a;+ 1 is the C. F. 
 
 7, If the H. C. F. of a and b be c, the L, C. M. of 
 
 (a+6)(a3-63),and(a-6)(a3+fe3)is ^^— • 
 
 Lei a = mc, b = 'nc, and .-. a^=Tn^c^. b^=n^c^. Thus 
 (a +b ) = c {tn +n ); (« -b ) = c {m —n ), and 
 
 ,«. (a + 6)(a3 _ 53) _ c4^TO+n)(wi3 -n^),. and 
 (a-6Va3+63) = c4(w-n)(m3+?i3). 
 
MEASURES AND MULTIPIiBS. 
 
 107 
 
 The H. C. F. of the last expressions is c'^(m*—n^), .-. the 
 h. C. M. = c4(m6-»6\^ _v — -^_ — 
 
 8. If (x—a)^ measures x^ + qx+r, find the relation between q 
 and r. 
 
 Let a; + m be the other factor, then 
 x^+qx + r=^-^c)"{x+m)=x'^ + im — 2a)x%+(a^ - 2,ou>n)x-\-ma^ 
 equating coefficients, 
 
 m — 'ia = Q,a^—2am — q,ma^=r 
 
 .*.»» = 2a, and .'. a^ — 4a2 =q, 2a3 = ,-, and 
 
 q q'^ r r' 
 
 a* = - Y' or «« = - 2^ ; and a* = -^ or a^ = -^ 
 
 rS ^3 ,.8 qZ 
 
 ''' T=~ 27' °^* ^"^ 27 =^ 
 Or thus : — 
 Dividing x^ + qx+r by (a;— a)' we find the remainder 
 \ {q + 3a^)x+r-2a^ 
 
 and as this vfiil be the same for all values of x, we have, by equat 
 ing eoefOciente, 
 
 2+3a»=0, 
 and r— 2a3 = o, 
 or 3-3=— 27a« 
 
 and r3 = 4a® ; 
 
 therefore ^ + 27 = Of as before. 
 
 Exercise xli. 
 
 Find the H. C. F. of the followmg : 
 1. 2x^+dx^ + ox^+9x-d, dx^-2x^ + 10x^ -6x^3. 
 
 a. a;^ + (a + l)x^+{a+l)x-fa, x^ + {a-l)x^ --{a-l)x+a. 
 
 3. px^-{p + q)x^+{p-q)x + q, px^ -{p+q)x^ +(p+q)x-^q. 
 
 4. ax^-{a-b)x^-{b~c)x-G, 2ax^t+{a+2b)x^-\-{b + 2c)x+o. 
 
 5. l-3|a;-3ia;3+ia;3_a.4, i^i^i^a;_3a;3-[.l^a;3+a;4. 
 
103 MEASt)KE5 AND MUL,TIPLES. 
 
 7. a^x^ i-a'' -2abx^ + b'-'x^ +a5b'* -2a^b, and 
 
 8. {ax-\-hyY — {a— b)[x-\-z){ax->rby)-^{a - b)^xz, and 
 (ax — hy)^ — {a-\- h) {x-]rz) {ax — by) + (a +, b) '^xz. 
 
 9. «(i2-c3) + fe(c2-rt3)+c*(a2-62) and 
 a{b^-G'') + h{c^-a^) + c\a^-b^). 
 
 10. a^"'+a-"'-{-a^-Jfl, and a^m „«'■*'» ^-a^-l. 
 
 11. If x^ +ax^ +bx+c, and x^+a'x+b', have a common factoi 
 of one dimension in x, it must be one the factors of 
 
 {a — a')x^ + {b-b')x+c. 
 
 12 Determine the H. C. F. of (a-fe)^+(6— c)6+(c-a)», and 
 
 18. Find the H. C. F. of 
 
 2(7/3 _2?/2-y + 2)a;3+3(y2_i)a;a_(2?/3_j,2_2y + i^, and. 
 3(2/*-4?/3+5y-2)a;3+7(2/2-2y+l):B-(3i/3-5v=5+2/-fl). 
 
 14. If a;2 4-i^a;4-9, and x2+OTa;+" have a common hnear factor, 
 shew that 
 
 {n—q)^-\- n{m—p)^=m(m—p){n — q). 
 
 15. Find the L. C. M. of x3-3a;2 + 3a;_l.^ a:3-a;8-a!+l, 
 a;4_2a;S + 2a;-l, and x^ -2a;3 + 2a;3 -2a;+l. 
 
 16. Find the L. C. M. of 
 
 a;3 + 6a;2 + lla;+6, a;3 + 7a;3 + 14a:+8. 
 
 a;3+8a;3 + 19a;+12, and x^+dx^ + 2Gx-^24^. 
 
 17. Find the value of y which will make 
 2(y^-{-i/)x^ + illy-2)x+4: and 
 2(^/3^-2/•^)x3+(lV-2^/)a:2 + (2/3^-5y)a; + 5^y-l, have a 
 
 common measure. 
 
 18. The product of the H, C. F. and L. C. M. of two quantities 
 is equal to half the sum of their squares, one of them is 
 2a;3-lla;3+17a;-6 ; find the other. 
 
 1.9. If a;+a and x — a are both measures of x^+pz^ +qx+>\ 
 shew that pq = r. 
 
FRACTIONS. 
 
 109 
 
 20. IS x^+qx+r and x^ -j-mx+n kaye a common measure 
 [x—a)^, show chat q^n^=m^r^. 
 
 21. II' the H. C. F. of x^ +px-^q and x^ +mx+n, he x-\-a, their 
 L. C. M. is 
 
 x^ + {m — a)x^-\-px^+{a^ + mp)x-\-a{m — a){a^+p). 
 
 22. If x--{-qx + l, and i3^jr,^3_|_^aj^l^ liave a common factor 
 of the form x+a, shew that (jb-1)3 — ^(jo — 1) + 1 = 0. 
 
 23. 'iix^-\-px''-^q, and x^ +mx+n, have x+a for theii- H. C. 
 F., shew that their L. C. M. is 
 
 x'^-\-{ni — a-\-p)x^-{-p(in — a)x'^-\-a^{a—p)x + a^{a-'p){m — a). 
 
 24. If x^-\-p)x + l, 2i,udiX^+px"-\-qx-^l, have x — a iox a com- 
 mon factor, shew that a = 
 
 1-q 
 
 25. Find the H. C. F. of {a^ ~b^Y + {b^- -c^yJr{o^ -a^Y, 
 ai,ndia^{b — c)-\-b^{c — a)-\-c^{a-b). 
 
 26. If a. be the H. C. F. of b and c, & the H. C. F. of c and a, 
 
 y the H. C. F. of a and b, and X the H. G. F. of a,.b. and «, then 
 
 a6c5 
 the L. C. M. of a, b, c, is — ^t"* 
 
 27. If a;+c be the H. C. F. of x^ +ax-i-b, and x^+a'x+b', their 
 L. C. M. will be x^ + {a+a'-c)x^ + {aa' -G^)x+{a-cJ{a' -c)c. 
 
 28. Shew that the L. C. M. of the quantities in Ex. 2 (solved 
 above) will be a complete square if x = y^-^z^ -y^z^. 
 
 29. Kind the H. C. F. of x^+^x^+dx"^ -2a;3 + l, and 
 
 ex^+ x'+nx'--7x^-% 
 
 Section II. — Feactions. 
 
 Art. XXXIII, When required to reduce a fraction to its 
 lowest terms, we can often apply some of the preceding methods 
 of factoring to discover the H. C. F. of the numerator and de- 
 nominator. 
 
110 feaotions. 
 
 Examples. 
 
 ^ ac+by+a7j-{.be _ c {a+b)+y{a+b) _ c-\-y 
 aJ-{-2bx+2ax+bf " /(a+5)+2a;(a+6) " /+2a;* 
 
 a(«+i)(a-6)2 a 
 
 g z^+x^y-\-x^y^+z^y^+xy*+y^ 
 x^—x^y+x^y^—x^y^-\-xy^-'y^ 
 
 Here the numerator is eyidently (x^ —y^) -^ (x-y), and the 
 
 denominator is ^. The result is .'. --I^-^. 
 
 a+y x-y 
 
 {x+y)^—x^—y^ 5x^y+Wx^y^ -{-lOx^y^ + 5xy* 
 
 {x+yy+x^-^y^ = ■(cc+^)4-a;V + (^^+2/l)f-aJ,V 
 
 _ 5 a;?/-{ a;^+y^ + 2a;j/(a;+y )} 
 
 ~ (a;S+2/2+a;'/){(a;+?/)^+a;«/+a;2+?/2-iC2/} 
 
 5a;y (a;+y)(a;^ +x y+y^) _ 5.ry(a;+y) 
 2{x^+xy-\-y^y ^X'+xy+y^) 
 
 V *2-12a;+35 
 
 a;3_i0a;3+31a;-30 
 Here we see at onee that the numerator=(a;-5)(a; — 7) ; and 
 it is plain that x— 7 is not a factor of the denominator; we .•. try 
 x — 5 (Horner's division), and find the quotient to be x^ —5a; +6. 
 x-7 
 
 . the result = 
 6. 
 
 x^-5x+Q 
 
 a.4+2a;2+9 
 
 a;* -4x3 + 8a; -21 
 
 The factors of the numerator are at once seen to be a;'+2a;+3, 
 and x^ — 2x+B, of which the latter is one factor of the denomin- 
 ator, the other being (Horner's division) ?:" - 2x— 7 : .'. the result 
 
 x^+2x-\-B 
 
 z^-2x-7 
 
FRACTIONS. Ill 
 
 « 
 
 Exercise xlii 
 
 Reduce the following to their lowest terms : 
 a.2_7a; + 6 Sm/'' -ldxy-\-Ux 
 
 2. 
 8. 
 
 5. 
 
 x^-2x"--8x-96 72/3-172/2^-62/ 
 
 x^+ax^ -a^x — a^' x^ —5x^ -{-Ix-S' 
 
 a;3-3a;+2 x^ +2x ^ +9 ^ 
 
 ^ + 4a;2_5' ^ _ 4a;3 -I- 4:X^ - 9* 
 
 2+J.r xl + 2x^+^ 
 
 26+(62_4)a; - 26a;2' a;5+4a: 
 
 5a^+10a4.r+5ff.3a;^ 20a;4+a;--l 
 
 a3jc+2^^3 + 2rta;3 +a;4 ' 25a;4 + 5a;3 - a; - l' 
 
 a;' — iv^jf +x^y^ —x'^y^ +x^y^ —x^;/^ +^y^ — y' 
 ^' x'' +x^'y+x^y^ +a;*2/H- jc^z/^ ^a;2,/5 _|_ ^yG~^y7' 
 
 I a b 
 
 Sa'^x^ - 2ax^ - 1 
 
 ^'+ r + ^h»z/+2/* 
 
 8. 
 
 
 abc{a — h)(b — c){c - a) 
 
 ^- aS(6-c) + &3(c-a)+c3(a-6) 
 10. From Ex. 4 (solved above) show that 
 
 . (a-6)5 + (6-c)5+(c-a)« ~ 5(a-6)(6-c)(c-a) 
 
 (a;+y)^-a;''-2/^ 
 (a;+2/)^-a;^-2/'* 
 
 12. Shew that 
 
 \a-hy-^[h-cy. + {c-a)'' 7 ,, ^^, ^, ^ 
 
112 FRACTIONS. 
 
 Art. XXXIV. In reducing complex fractions it is often 
 convenient to multiply both terms of tiie eompiex fraction by the 
 L. C. M.. of all the denominators involved. 
 
 EXAMPIiES, 
 
 1. Simplify K^-Mi)-t(l-t:«).. 
 
 Here the L. C. M. of all the denominators involved is 12 ; 
 .-. multiplying both terms of the complex fraction by 12, and 
 removing brackets, we have 
 
 6a;-f8-8+6a; _ 12a; _ ^x 
 21~4a;-17 ~ 4-4* '" 1-x 
 
 a — b 
 
 II. a- 
 
 l+ah 
 
 ^^aia-b)^ Here multiplying both terms by 1 ^-ah, wa get 
 l-\-ab 
 
 a{l+ah)-a+o &(a2 4-l) . 
 
 3. 
 
 Here multiplying both terms of the frac- 
 
 tion which follows x-1 by 4-a;, the given fraction becomes at 
 
 gjjQg , and now multiplying both terms by 4, we 
 
 4 — X 
 «-l+ 
 
 have 
 
 4 
 4 4 
 
 4x— 4+4 — a; 3a; . 
 
 It may be observed that when the fraction is reduced to the 
 
 form -^ -^ — , we may strike out any factor common to the two 
 
 b d 
 
 denominators, and also any factor common to the two numerators ; 
 it is sometimes more convenient to do this than to multiply 
 du-ectly by the L. C. M. of all the denominators. 
 
FE ACTIONS. 113 
 
 la+b a-b\ la^ + b^ a* -h^ \ 
 
 4. Simplify [^^ + ^ ^ [^^^-ZTbi - ^s+li' ' 
 
 Here the numerator of the first fraction is {a + by^ + {a — b)^ 
 and the denominator is a^ -b^ ; tlie numerator of second fraction 
 is (a2+Z,3)a_(«2_j2)3^ and the denominator is a-^-b*", the 
 former denominator cancels this to a^-i-b'^, which, of course, be- 
 comes a multiplier of the first numerator : 
 
 .-. Wehave — ^2_j_^3^2_^^3_^2X2 - 2.(3/yJ 
 
 Occagionally, we ai once discover a common com^^iex factor, 
 strike this out, and sin^plify the result. 
 
 a "^ /' " ^ c M 1 \ 8 1 
 
 5. -:; : ^T ^ : here the den. = V ~t\ —~^ 
 
 ^ "^ iis" - "^ + ^ 
 
 /I 1 IWl 1 1\ 
 
 ss 1- ■-; — I — + -7- , and cancellmg the eom- 
 
 \a b c I \a b c j ° 
 
 mon factor we have 
 
 1 
 
 1,1 1 , and muItiDlying by tibc, this = . 
 
 ■ r ~r — — " bc-^ca — ub 
 
 a o c ' 
 
 Exercise xiiii. 
 Simplify the following : 
 
 l-^{l-^(l-aj^} a-b '^ a+6 
 
 a-\-b a — h 
 
 a~b g+ ft 
 
 ^ - ^{l-i(l -^)T <»+6 a^) 
 
 a — b ~ a + b 
 X jc_ _3_ 1 
 
 x-i-y x—y 1—a \-\-a 
 
 2.f 
 
 ^ - + '- 
 
 8. 1 + 
 
 aj*- y* 1 -a 1 -\-a 
 
 1 
 a x—y 
 
 1+a x-\-y 
 
114 
 
 4. 
 
 FHACTIOWS. 
 
 a'+h^ 2^>2 1 1 
 
 a+b a~h h^ 
 
 c-\-d ^ c-d 
 a+b a~h' 
 
 + :r—i «+&+- 
 
 a 
 
 6. 
 
 a;— 1 ?/.— 1 T*:*?* 
 
 7. 
 
 yz~ zx — xu 1 1 1 
 
 — + —--{- — 
 
 — 4. — A g'^ + fe^-f c* 
 
 ^2 + ^2 + , .3 + ~7ii3,.3 
 
 a /) r; 
 -r— -U -I- 
 
 DC ac ' ab 
 
 /«+6 a^+feg y /„^5 a"--b^\ 
 
 \a-b "*■ rt2-62 I ^ \^+^ ~ a^^^j 
 
 1 1 
 
 — 4- 
 
 '■1 11 ^^ 26c [ 
 
 a b 
 
 ■c 
 
 \x-\-i' I ' .c- a i 
 
 tr^^T" m ranMiTrf-iTi — n — fwn 
 
 (1— .)+! WJ+^+t-J 
 
 2(1 -a;) (l-a;)2 
 
 — >^ 4. > 4.1 
 
 14-x ^ (1+a;)- ^^ 
 l-a; 
 
 
 a; 
 
 + 1 + 
 
 J/_ 
 
 
 1 
 
 — 
 
 y3 
 
 
 
 7; 
 
 
 a; 
 
 
 
 
 x-^ 
 
 ^« 
 
 a; 
 
 
 7' 
 
 • 
 
 
 
 V ' 
 
 
 — 
 
 -1 + 
 
 
 
 
 1 
 
 -p 
 
 
 
 .7 
 
 
 X 
 
 / 
 
 
 
 X • 
 
FKACTIONS. 
 
 ]15 
 
 /a~bY (a--bY {ci~h\ 
 
 ^[^nl -[-v-J - 3(— -7/4-1 
 
 13. 
 
 x^ +yAi/-'rx^y^+x^'^y^-{-xy^+y^ ' \x+y/ 
 
 ' [l^^ + l~x+xV "^ [l+x-hx^ ~ 1+xy' 
 
 16. Find the value of 
 
 Zi^^^v + ^b^^2^x ^l^enc. = i(a+6). 
 
 17. Find the Talue of y'{l -i/{l -x)} 
 
 \l + b} \l+bl 
 
 18. Find value of 
 
 l/ia+hx) + i/{a-h x) ^^^^ ^ ^ 2ae ^ 
 l/{a'{-bx) — i/{a-bx} 6(ln-c2) 
 
 Art. XXXV. When the sum of several fractions is to be 
 found, it is generally best, instead of reducing at once aU the 
 fractions to a common denominator, to take two (or more) of 
 them together, and combine the results. 
 
 Examples. 
 
 1, Find the sum of 
 
 x+y y-x ' x^-y* 
 
 2x-2y 2x+-2y x^+y^ 
 Here taking the firgt two together we have 
 
 i^ +y)' + i^-y)' _ t±yi ; now add this to _ -^IH^'. 
 2(a;3-?/3) x"-y' x^+y" 
 
 and we get 
 
 a;*— 2/* a;*— 2/* 
 
116 
 
 FRACTIONS. 
 
 2. Find the sum of 
 
 l-\-x 4x 8x 1—x 
 
 r^ "^ 1+x^ + T+^ ~ l+i' 
 Here, taking the first and the last together, we have 
 (l+^)^-(l-a;)3 _ Ax 
 l-a;2 - i_a;3' 
 
 taking this result with ihe second fraction, we have 
 
 8a; 
 
 W-^- 4- -^—] - - 
 \l+x^ ^ 1-xV - 1 
 
 .+x^ ' l-aV 1-x^ 
 now take tliis result with the remaining fraction and we get 
 
 ^\i-x^ + iT^j = r::^' 
 
 ar'" a;-" 11 
 
 8. ^;rri - ^r:^ - ^^rzi + ^jqH' Taking in pair. 
 
 those whose denominators are alike, we have 
 
 ic"^ — l a;^" — 1 
 • ^iTZl - ^T^ =x"""+x" + l-(a;"-l)=ar"'+2: 
 
 The work is often m zde easier by completing the divisions repre- 
 sented by the fractions. 
 
 2a;+l 4ic+5 
 
 4. Pmd the sum of 1+ oT^ITTv — o^lTo' By dividing num- 
 erators into denominators, this 
 
 3 1 3 1 
 
 = 1 + 1+ sr^ -2- 
 
 2a;-2 ~" 2a;+2 " 2a;-2 ~ 2a;+2 
 3a;+3-a; + l _ a;+2 
 - 2x3-2 ^ a;2-l' 
 x x—9 x-\-l x—S 
 
 2 2 2' 2 
 
 1+ ^32 +1- ^=7 -^- ^:ri -1+ ^THS'^^ 
 
 2_ ^_ ^_ 2 2(2a;-8) 2(2a;-8) 
 
 fl;-2 ^ a;-6 a;-7 a;-l ~(a;-2)(x-6) (a:-l)(a:-7) 
 
 f _1 1 ) 
 
 = (4a;-16) |a.2_8a.^l2 " a;3 - 8a;+7/ 
 
 = (80 -20a;)-=-(a;4-16a;3 + 83a;3 - 152.r+84). 
 
 [denommator=(a;2 -8a;)2+19(x3 8x)+84]. 
 
FRACTIONS. 117 
 
 6, Find the value of 
 
 TT' + Kl when X = — r \ 
 
 x-2a ' x—2b a + b 
 
 Bydmsion,l+^32;^-h 1+^32^ 
 
 = 2 + 4|-~|j- + ^,\; but the quantity in the brackets 
 
 (a-\-b)x~4:ah 
 = ~{^^2a)i^~2h = «^^^^ {a + b)x = Aab 
 
 :. the value of the given expression is 2. 
 
 Exercise xliv. 
 
 Simplify the following : 
 
 , ct — a x^+ax+a^ x^ — a^ 
 
 5 x+a x^—a^ 
 
 ^ a^+bs a^-Sa^b + 3ab^ -b^ a(a - h) - b(a - h) 
 
 / 1 _1_ 2a X 
 
 \a -\- X a — x a^+x^J 
 
 I 1 J. 2a; \ 
 
 \a + a; a — x a^ +x^l' 
 
 a — 
 
 a; 
 
 1 
 
 
 a — 
 
 « 
 
 h 
 
 
 a — 
 
 b 
 
 2 - 
 
 Bx 
 
 , a h ab ah 
 
 4. _ J_ — -f -. 
 
 a •\- h a — b ab—h^ a'-+ab 
 
 . 3+2a; 2 - 3a; IQx—x^ 
 
 ^' 2-a; ~ 2 +a; "*" a;2-4 
 
 1 1,1 
 
 4a3(«+«) 4a3(a-a;) ^ 2a2(a2+a;3) 
 
 . 1 /3^+2y)\ _ 1 / 3a;-2y l 
 '• 2 \3x-2i/)/ 2 l3a;+22/i 
 
 a;+l a;-l l-3a; , x 
 
 '^' '- .1 ~ 2.C+1 a;(l-2a;) "*" a;(4a;2-l) "^ a;(l'6a;*-l) • 
 
 4 9 x-\ 
 
 -I- 
 
 .2x+2 a;+2 ^ 2(a;+3) (a,-+2)(a;+3) 
 
12. 
 
 H8 FRACTIONS. 
 
 10. ^(^+y) _ %-^) _ 4(a;'^-y") _^ Hx*+1/^) 
 x — y x+y x^-{-y^ «^ — 2/* 
 
 ( 1 1 ) 
 
 \x -\- a X + b f 
 I a+x 4:ax 8a^x a-x \ 
 
 \ a^x "*" a^x^ "*" 'oM^ ~ «+« J "*" 
 
 I ffl2_^2 + a4 + a;* ~ a^+x^\' 
 
 ^o 5a;-4 12a;+2 lO.r+17 
 
 13. -I- — — • 
 
 9 ^ llic-8 18 
 
 ,^ 12a: + 10a , 117a+28x _ 
 
 Jo. + fc H — — lo. 
 
 3a;+a 9a + 2a! 
 
 4a; -17 8a; -30 10a; -3 5a;-4 
 
 ^^- ~^1T " 2a; -7 + ^^T ~ ^T 
 
 17. Find the value of ^:^:^32Z- + ^+b^rd 
 
 nhena-\-h= --_r^. 
 
 ar'" ^"a;"* a;"" «'" 
 
 a;"-?/" af+r/" a;"-)/" ^ a;"H-J^"' 
 
 19. 
 
 {a-bf- (a - by- _ 1 1 
 
 1 1 _ 1 _ 
 
 1+a; 1 -^ 2 _ 2a;3_ 
 
 ^-^- l-a;3 + i+a;3 ~ l-a;3 a;« + l* 
 
FRACTIONS. 119 
 
 Art. XXXVI. The following are additional examples in 
 which a knowledge of factoring and o. the principle of symmetry 
 is of advantaofe. 
 
 Examples. 
 
 Cancelling the common factor z — y-{-z in the two terms of the 
 first fraction, there results ■^ - , hence by symmetry, the 
 
 denominators of the other two fractious will be x-\-y+z, and the 
 numerators will be y-\-z-x, z+x-y; .'. sum of the three 
 numerators = 33 +?/+0, and the result =1. 
 
 ^ „. ,., ab be ca 
 
 2. Simplify 7 ., .. + ,, .,. — s + 
 
 {c-a){c~h) ^ {a-b){a-c) ^ {b-c){b-a) 
 
 The L. C. M. of denominators is evidently {a—b){b — c){c — a). 
 This gives for numerator of first fraction —ab{a — b), and by sym- 
 metry the other numerators are —bc{b-c), —ca(c — a), 
 
 ab(a-b)-i-bc(b — c)-^ca(c-a) 
 
 . : we have — -. txtt — w ^^ ' 
 
 (a — b)[b — c){c — a) 
 
 _ {a — b){b — c){a — c) 
 
 {a — b)[b — c){c — a) ~ 
 
 2. Eeduce the following to a single fraction : 
 
 a h c 
 
 + -nr—xn — z^7:7-^^ + 
 
 {a-b){a-c){x-a) ^ {b-a){b-c){x-b) "^ {c- a){c-b){x-c)' 
 
 Here the L. C. M. is {a-b){b — c){c-a){x—a){x-b){x—c) ; the 
 numerator of the first fraction is 
 
 ~a{b — c){x—b){x — c), and .*. by symmetry that of 
 second is —b{c—a){x — c){x — a), and that of third is 
 
 — c{a — h){x — a){x — b); and their sum is 
 
 — {a[b — c){x-b){x — c)-^b{G-a){x — c){x — a) + 
 
 c{a — b){x — a){x~h)). 
 This vanishes if a = 6, hence « — & is a factor, and .*. by sym- 
 metry b — c and c—a are also factors. Now* the product of these 
 
i'^i) FRACTIONS. 
 
 is of the third degree, while the whole expression rises only to 
 the fourth, hence x^ cannot be involved. The other factor must 
 therefore be of the form nx+n, in which mis a number. 
 
 To determine n put a; = 0, and the expression becomes 
 abc{a — h-¥b—c-\-G — a]=Q; .-. n = 0, or the other factor is mx. 
 
 To determine m put a = 0, h = l, c= -1, and m will be found tc 
 be 1. The numerator is .-. x{a - lj){b — c){c — a), and the result is 
 
 X 
 
 lx — a){x—b){x—c) 
 
 o cj- Tf a + h b+c c + a 
 
 8. Simplify ^ _j_ + 
 
 {b — c){c-a) ^ {c-a){a~b) {a — b){b-c) 
 
 L. C. M. of denominators is {a — b)[b — c){c—a) ; 
 
 .'. first numerator is a^ —b^, and By symmetry 
 
 second " b^—c^,s.ndL 
 
 thkd " c2-a2; 
 
 the sum of these = 0, which is the required result. 
 4. Eeduce 
 
 2 2 2 (x-2/)2+(7/-z)2 + (2-a;)« 
 
 + :-- + - .. + 
 
 x-y y-z ^ z-x ^ {x -y){y - z)(z-x) 
 
 Here the numerator becomes 
 
 ^y-z){z-x)^-^x-y){z-x) + '>.{x-y)(:y-z) + 
 {x—yY-¥{y—z)--\-{z-x)-, which is evidently 
 
 {(•■-2/)+(^y-^)+C--^)p=o. 
 
 ffl3 + 263]3 (2a3+£3,3 
 
 Observe that the denominators become the same by changing 
 the sign between the fractions, and that the expression is sym- 
 metrical with respect to a and b. The numerator of the first 
 fraction is a^* + 6a^i3_^.l2a^6^^-8a369, and by symmetry that 
 
 of the other is -b^^ -Qb^a^ -l'2.b^a^ -Qb^a^ . Their sum is .". 
 ai 3 _ &i 2 _j_6a3/;3(rt6 _ ^fi) _ 8a3i3((^e _ ^6)_ 
 
 — ^^^li^l)S'^(^a'^-.^J^^i^ and since the denoniinatur of the given 
 expression is (a'^—b^)'^ :. the result is a^-\-b'^. 
 
FRACTIONS. 
 
 Exercise xlv. 
 Simplify tht! following : 
 
 121 
 
 1- ^U + //j -^^ \x+y 
 
 \ a — b I \0 — a I 
 
 \ 
 
 a-{-b O-^-r __*' + * 
 
 ^- (ir-7)(c-a) + (7-</)(.i-6) "^ (^-b)(b-c)' 
 
 _ 1 1 1 
 
 *• (a-/;)(a-c) + {b-a)(b-c) + (6--aj(f - i)' 
 
 ((-6 /'~V- c-a (a — b){b — c){c-u) 
 
 (^a:^'}j)^a + c){x + ay(a-{-b){b - c)(a; + b) {a+c){b - c){x-\-c) 
 
 '^' {x-y){x-z} + {>J-x){y- z) "^ {z~x){z-y) 
 
 ,,3 /,3 C3 
 
 11. ^. ' ^, + , 
 
 (b^.~S2^)(^c+a-2b) ^' (c + a-26)(a-i-6-2c) 
 
 1 
 
 (a + b-2c){b-}-c-2a) 
 
 b2-c^ c^a^ a^-l]^ 
 
 1Q " I I 
 
 ■ {a-b){a-c){x-a) "^ {b - a){b - c){x- b) ^ 
 c2 
 
 {c — a]{c - b){z — c) 
 
122 
 
 RATIOS. 
 
 14. ^iy+^) , y(z+x) , z{x+y) 
 
 + iT.'^'z^nrh. + 
 
 (x-y){z-x) ^ {)/-z}{x—y) {z-x){y-z) 
 
 15. [(^ + ir-+{(^-o)'+{a+cy^ _ -^ _ ._!_ a. ---^. 
 
 (a4-6)(6 — c)(a + c) a + c b — c ' a-\-b 
 
 1ft 1 1 1 
 
 x[x — a){x — b) a(b — a){x — a) b{b-a){x—b) 
 
 Section III. — Eatios. 
 
 Art. XXXVII. If -^ = -j ••• ad = hc. Now, 
 
 
 b d 
 dividing ad -he by ca we have - .... 
 
 (1) 
 
 a b 
 
 " ad = bchvcd " — = —.... 
 
 i'2) 
 
 , , 1 ^' c • 
 
 •^ da 
 
 (3). 
 
 ma + nc - 
 Also , , , — each of the given fractions . . . 
 
 (4) 
 
 , mb[ , )+«(/( , ) Imb+nd) , 
 _ wa-\- nc \ b / ' \d / ^ ' b o r 
 
 mb-\-nd mo +nd imi + nd b d 
 
 
 A very important case of this is m = 1, n= ±1, hence 
 
 
 a c n -{-(■, a — c 
 
 b ~ d ~ b+d ~ b — d 
 
 (5). 
 
 a — h e — d. 
 Also . — ; 
 
 (6). 
 
 ■^^s" (,^b cH-ci 
 
 For by (2) and (5) 
 
 
 (f b a—b a+h a — b c — d 
 
 
 c ~ d ~ c — d ~ c -\-d " a-j-b ~ c-\-d 
 
 
 ^ X, a-b _ b _ d c-d 
 
 
 a+b « , 1 '^ , c+d 
 d d 
 
 
RATIOS. 
 
 123 
 
 Generally, to prove that, if — = — , anv fraction whose nu- 
 
 merator and denominator are homogeneous functions of a and b, 
 and are of the same degree, will be equal to a similar fraction 
 formed with c instead of a and d instead of h : — Express the first 
 
 fraction in terms of — , and for — substitute its equivalent — 
 
 b b W 
 
 and reduce the result. 
 
 By (2) the fractions may be formed of a and c, and b and d. 
 
 If — ■ - — = — , . LxL _ — or — or — (7) 
 
 b d J mb-{-nil-\-pj b d f 
 
 I I w 
 
 ma-+-'nc-\-pe 
 
 'ij) + ""(t) + "At) 
 
 mb-{-nd-f-pf mb+nd-l-jif 
 
 a 
 
 {mh-{-nd-\-2)f)Y a 
 
 ~ mb+nd-^pf b 
 
 U± = ^ and ^ = A 
 b d n q 
 
 ma + pr pan^mc ma pa 
 
 nb + gd qb + nd ~ nb qb 
 
 ■n ma DC ma-\-pc ■, ,_. 
 
 For — = :^ = -r^=-^, by (5) 
 nb qd nb + qd 
 
 pa mc pa + mc 
 
 or &c . . . . C8) 
 
 qb ')ld qb±nd 
 
 But !!^ ^ -^, hence the equality stated in (8).' 
 nb qb 
 
 If ^ = _1 = _!. and !!L = Z. = ^, 
 b d f n q s 
 
 rr c-{-pc + re pa -^ re + me „ ma 
 
 -irr-ij—e = 7.7 . , ,- = &c., = — = &c. . (9). 
 nb±_qd+sf qb+sd + nf no ^ ' 
 
 If an upper sign be taken in a numerator, the corresponding 
 upper sign must be taken in the denominator ; if a lower sign, 
 the corresponding lower sign, otherwise all the signs are inde- 
 pendent of each other. 
 
124 
 
 a e 
 
 1- I^T = T 
 
 The given fraction = 
 
 EA'iiOS. 
 
 
 EXAMPI^ES, 
 
 
 i.1. i ^^ ~ 46 Be — Ad 
 
 
 a ^ c ^ 
 
 Fjc - Ad 
 
 a c 
 
 7-r- + 5 7-7 + 5 
 h d 
 
 lc+5d 
 
 2- I^T = -J '^'^^ ^^''^' S^h^Ah^ = Bc^d-Ad^' 
 
 Dividing the given fraction by b^ we have 
 
 , and this becomes, on substituting for -^ its equal --7- > 
 
 2p- + 37J ^ 
 
 2c3 + 3r2^ 
 
 ^3 i.;,R i^s \ 
 
 3. If 3a = 2&, findthe vahie of ^^^^^^^. This = (rj + l) r^ 
 
 — _ — -) [by dividing both numerator and denominator by 
 
 b^ b I 
 
 a 2 
 o^j. But from the given relation — = -3- we have, by substi- 
 
 a 
 
 tuting for -t-» 
 
 4- I^ T = T- ^^'^^^ *^^* 03 + ^3 X -J = (^;:p;/j • 
 
 a b a+h 
 
 We liave ^ = T = c^TT/- ^^'^^ 
 
 — ^- — = -TTTT + 1M--^+ M= Ti-, and this muUiphed 
 by y gives ^=1^-/;. 
 
RATIOS. *■-'*' 
 
 „ r^ x^+rix^ -bx + c x^ + ax—h , . , ^ 
 
 o. if _- -„-— 7 — -— = -;; --, shew that x = — ■-. 
 
 ic* -fl,r- + to+t' x- — ax-\-b' a 
 
 Multiplying both terms of second fraction by x, it becomes 
 
 x^-\-ax^ — bx 
 
 Zz — n^2~L.h^'^ ^<^^ 68,ch of the given fractions = 
 
 difference of numerators 
 
 difference of denominators ; 
 
 = 
 
 c 
 
 ^. =1/. x^+ax-b -x^ -ax+h 
 
 or lax = 
 
 6 
 2b .-. «= — 
 a - 
 
 6. If 
 
 ace ac + ce-\-ea a^ + c' +«• 
 
 For 
 
 ac c<; ea ac-\-ce-\-ea 
 
 bd-df^fb-hd+df+fb' ^y (7)makingm=r.=i, = l. 
 
 Also 
 
 a2 c2 ^2 a,2^.c2_j.^2 
 
 fcs -</2 -yg- ^2 4.(73+^2- By (7). 
 
 
 afl 0.2 
 
 But 73" = T^ hence the required equality. 
 
 The problem is a particular case of (9), with all the signs + 
 and a for m, b for n, c for p, &c, 
 
 (If the fractions given equal to one another have not monomial 
 terms, instead of seeking to express the proposed quantity in 
 terms of one fraction and then substituting an equivalent frac- 
 tion, it is often better to assume a single letter to represent the 
 common value of the fractions given equal, and to work in terms 
 of this assumed letter.) 
 
 „ J. a+b h-\-c c4-a 
 
 prove that 32a+85i+27c = 0. 
 
 Assume each of the given fractions = «, so that a-{-b = 3(a — b)x, 
 b-\'C=i{b~c)x. c-\-a = 5{c-a)x, 
 
126 
 
 RATIOS. 
 
 01' -^- + --J- + -^^ = x{a-b-\-h-c-^c-a) = 0. 
 
 /. adding these fractions we have 32rt+356+27c = 0. 
 
 This example might also be worked as a particular case of (7), 
 thus ■ 
 
 a+b 
 
 h + c 
 
 B{a-b) Mb-c] 
 
 5(c — a). 
 
 20(a+Z)) + 15(6+c) + 12(c+a) _ 32a + 35b + 27c 
 
 60(«-//) + 6U(6-c) + 60(c-a) 
 
 a-\-b 
 
 
 
 32a + B5b+27c = x 
 
 3{a-b) 
 
 = 0. 
 
 8.1f^ + ^ - '^i 
 a + c + 
 
 a 
 
 d I 17 
 
 c 
 If 
 
 e 
 
 + 7 
 
 prove that 
 
 
 «2 4.^24., 2 
 b + d-hjj "^ W-^Ij2~J^}^' 
 
 Transposing terms, &c., we have 
 
 a2 2ac . c2 t'S 2ce 
 
 fe2 
 
 C2 t'2 
 
 i(Z "*" ;^ "^ 7^ 
 
 ^/■^"^ £f2 "^ ^' 
 
 or 
 
 la P \3 / /? c \2 
 
 that is, the sum of two essentially positive quantises = j 
 .'. each of them must=:0 ; heuce we have 
 
 a c e c 
 
 -J- — -y = 0, and -— — — = 0; 
 
 b d ' J (jf 
 
 
 c 
 ~d 
 
 e 
 
 7 
 
 Also 
 
 «2 ^ a2_|.c2+g2 
 
 /a+c+e\-. 
 
 
RATIOS. 127 
 
 Exercise xlvi. 
 
 , Tf « c a"'^ —ab-\-b^ c^ — cd + cl^ 
 
 1. 11 — = — , prove ^ — = — ; — - . 
 
 b d ab-W^ cd-U^ 
 
 2. It — = — , prove = , — , _ I— ^|. 
 
 b d^ b-'-d^ \b-dj - \b+dl 
 
 3. Given the same, shew that each of these fractions 
 
 = yXb-^ + d^l' 
 i. U 2x = Sy, write down the value of 
 
 x^:/ + xj/-^ + 2ii^' [x^-ifi)^ 
 
 5. 11 — = — = — , shew that — = —. 
 
 b d J b mb — nd - pf 
 
 ?}. From the same relations prove that — = ( '■ j • 
 
 ^3 Kb-md—n/J 
 
 7. If J±^ = AfJ±^+^\ then 2.3 = (6_«)-(6+«). 
 1 — x a\l—x-t-x'-'/ 
 
 o- ^ /,' , \ ■// \ = «, prove that a; = ^i — o' 
 
 f. Tr rnx-\-a-\-h mx—c—d b — c 
 
 9. 11 ^ = , prove x = . 
 
 nx-{-a-T-c nx — b — d n—ni 
 
 in 1" '^■^^ b — c c—a a-\-b-{-c 
 
 ay-j-bx bz-\-cx cy + az ax + by-\-cz 
 
 then each of these fractions = , «-f-/*+c not being zero. 
 
 x+y+x 
 
 11. K«t^ = Jp^^ = -^±^,then8a + 9b+5c = 0. 
 
 a-b 2[b-c) S[c-a} 
 
 VC-\- \/{a- x) 1 a — x /I— a\* 
 
 12. If \ , r = — > shew that = hn- 
 
 ya — Y{a — x) a ■ a \l+a/ 
 
 x^ — yz 7/2 —xz 
 
 13. If .. _ "'T = ."{iTZ — V ^^" ^' ^' ^ "® unequal, shew 
 
 x[i yz) y^i. • xz) 
 
 that each of these fractions is equal to x+y-rz. 
 
128 
 
 RATIOS. 
 
 x^-\-2x + l //-+2// + 1 , . 1 
 
 14. If ^ „ , o = .. b . o' shew that each of theso 
 
 x-—2x + o y^—2,i/-{-^ 
 
 fractions = {xy — 1 ) -f {xy — 8 ) . 
 
 25x2-16 8(a;2J,4) , , a;-4 8 
 
 15. If —,(. — ^~Q = o r~' shew that - ~p ■ 
 
 10a; +8 2a; — 4 x + O; 5 
 
 ibc V + ^b y+2c 
 IG. If 7/ = T~-r~ shew that ' ^ + ~o =2. 
 
 25rt2+2762 4-22t;2=o. 
 
 18. If -^~ = -T, = -^ — , shewthat((2^_|./,2^+c32, 
 
 x^ -yz y- -zx z^-xy 
 
 19. n -r^ — = ,-— — = ■ t . ' then wiU (.( - h)x + 
 [b — c)?/ + (c — a)z = 0. 
 
 ^°- -^ T = T "= 7 *^®^ (//^ +c/3 + /•2/ - M -4- ,/4 4-/4 ■ 
 
 hx + ay cy-\-hz az-+cx , ,, , 
 
 21. If --~ = 4-^ = ' shew that 
 
 a—b b—c c — a 
 
 {a-'(-b-{-c){x-\-y-\-z) = ax-\-by-{-cz. 
 
 22. If -- , 2 , 2-r-r = ^r:.' shew that each of these 
 
 expressions =1, 
 
 23. If I (^^3 = j(,'~-^) - M^)^ and a, /., o he 
 different, shew that ]6« + 116 + 15 =0. 
 
 24..If(^-^'f = ^-C pit)vethatx2+^2^22+2a:i/. = l. 
 \y-\-zx) l-x- ^ 
 
 as. If ^* = ^^ = _^, show that a. + /^ + c = 0. 
 x — y ?/-2 z — « 
 
 2G. If _ = — , prove that — ^-- :^ ' .— f- /Vrfx- 
 
RATIOS. 129 
 
 to 
 
 27. ii — - - — , then each is equivalent 
 
 h - d f 
 
 ./'x'^rL^.*, hence shew that 
 
 a h <? 1 
 . _ _ when 
 
 2z + 2a;-?/ 2.c+2?/-2 2?/ + 2z-a; 
 
 X y z 
 
 2a + 21)'^ ~ 26 + 26'- a ~ 2c + 2a^* 
 
 c.r. Tx- <^ c ,1 , /a-6V //a^" + 6''"\ 
 
 28. li _ = — , prove that — - = a(- ^ , . . 
 
 29. If ^^^ = 77-^ = ^-^' PJ^ove that 
 
 ^{y-z)-{-JL{z-x)-^I^{x-y)^0. 
 a c 
 
 30. If - " - '= ,/ , = - * , , . then wiU 
 
 /x(?i^ — ///2) w?/(/z — nx) nz[mx — iy) 
 
 31. If e ^ y ^ -^ •', aua y = *^-^ ', shew that 
 
 y ^ 
 
 X 
 
 z 
 
 »2 
 
 32. If ^--^ = ^q^ = '^ = 1, shew that 
 a3 0^ c^ 
 
 38. If !^ = -!^ = ^, and ^ = ^ . ^1 . 1, 
 prove that __+_ + - =3^:.^y3+^- 
 
 34. If -^ = ^ = 7 = &«•' ^^'^«° 
 
 J 
 
 
130 COMPLETE SQUARES. 
 
 a-, a a^ a„ 
 
 as. If T^ = 7^ = 7^ - . . . = 1^. then 
 
 *] ^o t'3 ^„ 
 
 CTlffg - g^ rtg + . . . ( - ly-^an-ia n ^ «i v/fl,a3 -^-a-iV'a^a^ + &C 
 &i ^2 - ^2^3 + • • • ( - 1)"~^^»-A ^l/i'X + ^2T/^3^" + &<^- 
 
 A+B+C A ^ B ^ C 
 
 36. If 1 = h y- + — , 
 
 abc a b c 
 
 and (J + 5+C)(a4-& + ^) = ^4a+£&+Cc, 
 1+^2 + i^^2 + i+c 
 
 then -will -, , ,„ - + i— rr^ + T~m ~ 0- 
 
 and also — - — + + = 0. 
 
 1 11 
 
 'a b ' c 
 
 xh yk zl x^ ?/- ?2 
 
 37. If -;■ = 77 = -v' and -^ = y^ = -r =1) 
 
 a2 6^ c^ • rH 6^ c' 
 
 I X If 2 \2 a2 1,2 c2 
 
 thenwill^ly + y + vj =,)^ + 7^ + 7^ 
 7 
 
 Section IV. — Complete Squakss, &c. 
 
 1. What quantity must be added to x^ +r>x to make it a com- 
 plete square ? 
 
 Let r be the quantity. 
 
 Then fl;2+;>a; 4- r = complete square = (.c + y7~)^ 
 
 Equating coefficients we have 
 
 2|/r= r 
 
 7'" / /. \ 3 
 
 " = T - \2 
 
 Or thus: Since {a+x)" =a- -^ 'lax + x^ ; we observe, (See 
 Art. XII), thsit four times the product of the extremes is equal to the 
 square of the mean, 
 
 .'. Ax^r=p^x^ ; 
 
 r z= l — \ » as before. 
 
COMPLETE SQUARES. > 131 
 
 Or we may extract the square root and equate the remainder 
 to zero . thus 
 
 P 
 
 X" 
 
 /> 
 
 4 
 
 p2 
 
 ISTow, if the expression be a complete square, this remainder 
 must vanish ; hence we have 
 
 jr)3 I P ^' ^ 
 
 4 \ 2 
 
 2. Find the relation connecting a, b, c, if ax^-\-hx+c is a com- 
 plete square. 
 
 Assume ax'-^ +bx+c = (^ a.x+ -i/c )^ =ax^ +2-i/(ac).x+c. 
 
 Now, since this holds for all values of x, we have 2 ^/ac — h, or 
 6- = iac, the relation required, 
 
 3. Determine the relation amongst a, b, c, in order that 
 
 a'^x^+bx+bo+b- may be a perfect square. 
 As in Ex. 1, we have 4:a'^x-{bc + b-) = h^x-^ ; 
 
 • i _ -1 - 1 
 
 4«3 h ~ 
 
 Or thus : 
 
 Assume a^x^+bx+bc+b^ = (ax+l/'oc + b■^)'' 
 = a^x2+2ayb^b^ + bc+b^ . 
 Equating coefficients, we have b = 1a^bc-\-b'^ ; 
 
 .♦. — — , — = 1, as before. 
 . . 4a" b 
 
 The same result may also be obtained by extracting the square 
 root and equating the remainder to zero. 
 
132 COMPLETE SQUARES. 
 
 4. Show that i{ x*-\-ax^+hx^+cx-\-d be a complete square, 
 the coefficients satisfy the equation c^ —a^d = 0. 
 
 Is it necessary that the coefficients satisfy any other equation ? 
 Extracting the square root of x'^+ax^+hx^ Tcx+d in the 
 usual manner, we have for the final remainder 
 
 Now, if the expression be a complete square, this remainder 
 muist vanish ; and, that it ma^ vanish for general values of x, we 
 must have 
 
 a I a^\ 
 
 1 / a*\ 
 
 (2); 
 
 ^3 
 
 Eliminating * - x' we have <;- -«3f/ = . . . (3). 
 
 The coefficients must satisfy the equations (1) and (2), an J 
 therefore either of these equations, together with the equation (3), 
 which results from them. 
 The same result may be obtained by assuming 
 x^+ax'^ -^bx"-\-cx + d = {x--if^ax-\- y/d)^ 
 
 = x^+ax^ + 2x^y/d 
 
 + ^a^x^ -taxs/d + d. 
 Equating coefficients, we have 2 v/d+i<*2= 6 ... (1) 
 
 and a^/d=c . . . (2). 
 From (2) we have c^ — a-d = 0, as before. 
 
 5. What must be the value of in and n if 
 4x* — 12a;3 +25a;2 — 4???a;+8w is a perfect square ? 
 
 Assume the expression = {(2aj2 - 3x4- \/{Sii)}^ 
 
 = 4:x^-12x3+4x" y{8n)-\-9x-' -6xy/ {8n)+Sn. 
 Equating coefficients, we have Gs/{Sn)=Am .... (1), 
 
 and4N/(8w) + 9 = 25 . . .' . (2); 
 .-. »=2, 
 »i = 6. 
 
COMPLETE SQUARES. 
 
 133 
 
 Or thus : Extracting the square root in the ordinary way, the 
 remainder is found to be (— 4m + 24)a; + 87i-16 ; /. -we must 
 have 4?/; + 24 = 0, or m = 6, 
 
 and 8m- 16 = 0, or ?i= 2. 
 
 6. If ax^-\-hx^+cx+d be a complete cube, shew that ac^ = db^f 
 and b^ = 'dac. 
 
 Assume ax^ +bx^ +cx+d = (x+d\). ' 
 
 Equating coefficients, 
 
 b = 3a^d^ , (1) 
 
 c = 3Jd' (2); 
 
 b ^3 
 
 dividing (l)by(2),_ = — ; 
 
 c d' ■< 
 
 Also, h^=^aU^ (3); 
 
 dividing (3) by (2),^ = 3a.; 
 
 c 
 
 .-. b^ = 3ac. 
 
 7. Find the relations subsisting between a, b, c, d, e, when 
 
 QX* + bx^ + cx- +dx+e is a complete fouich power. 
 Assume ax* + bx^+cx^+dx-\-e = (a^x-\-^)'*' 
 
 = ax'^+4:a^e^x^ + Ga^e^x^ + 4^a*e'x-[-e. 
 
 Equating coefficients, we have 
 b = 4aT cT, 
 
 d = Aai ei ; 
 
 whence fc(/ = 16«t' (1). 
 
 bc = 2iaie^ = QaAa^ei = Qad (2). 
 
 €d = 1ia^ei = QeAa^e^ = Qhe (3). 
 
 8. Shew that x^^-fx^ + qx^ +rx+s can be so resolved into two 
 rational quadratic factors if ^ be a perfect square, negative, a,nd 
 
 ,.2 
 
 equal to —- —. 
 
134 COMPLETE SQUARES. 
 
 Since — s is a perfect square, let it be n^. 
 
 Assume x^ +px^-\-qx^-'rrx-n^ 
 
 = (x^+'mx-{-n)(x" -^vi'x -7i) 
 
 = X* + (in.-\-m ')x^ + mm 'x" — n(m — m ')x -n^. 
 
 Equaticg coefficients, we have 
 
 mm ' = o 
 
 ■J- 
 
 
 r 
 
 m—m' — 
 
 n 
 
 
 ni^+2mm'+m'^=p^ 
 
 
 4:mm'*=4.^ ; 
 
 
 ;. {m-mY=P'^ -"^q = 
 
 r3 
 
 r2 
 
 
 9 . =n-' = —s. 
 
 p^ — ip9 
 
 
 Exercise xlvii. 
 
 
 1. What is the condition that {a — x){b — x) — c^, may be a per- 
 fect square. 
 
 2. Find the value of n which will make 2x"-\-8x-{-n, a perfect 
 square.. 
 
 3. Find a value of a; which will make x^+dx^ + Ux' + dx + dl, 
 a perfect square. 
 
 4. Extract the square root of 
 
 (a-b)^ - 2(a3 +63)(a _ b)^+2{a^+b*) 
 
 5. Find the values of m and n which will make 
 4a;* — 4a;3_{-5a;3 ^ffix+n, a perfect square. 
 
 6. What must be added to x^~V{ix^ -lGx^ + 16)-^x^ in 
 order to make it a complete square ? 
 
 7. The expression x'^-^x^ —IQx^ — 4:x + 4:8, is resolvable into 
 two factors of the form x^-{-mx-^6, and x^-^nx-^8; determine 
 the factors. 
 
 ex 
 
 8. Find the value of c which ^Yill make -ix^ — cx^ + 5x^ -f .y + 1, 
 
 a complete square. 
 
COMPLETE SQUARES, 
 
 135 
 
 9. Oouiiii ihe square root of 
 
 10. If {a-b)x^ + {a + b)-x+{a^ - b^){a+b), is a complete 
 square, then a =36, or b=.'da. 
 
 11. Find the simplest quantity which, subtracted from 
 
 a^x^ +4tnbx-\-4:acx+5bc -{- b^c^ , will give for remainder an es.act 
 square. 
 
 12. a*— 4a;3—a;3 4- 16a; -12 is resolvable into quadratic factors 
 of the form x" +mx-rp, and x- +nx-^q : find them. 
 
 13. Find the values of m which will make x^+niax + a^ 
 a factor oi x^ — ax^-\-a^x^ —a^x-{-a*. 
 
 14. Shew that if x^+ax^+hx'' +cx+d be a perfect square, .the 
 coefficients satisfy the relations 
 
 8c =a{4:b — a^), and 
 64r/= (46-ftS)2. 
 
 15. Investigate the relations between the coefficients in order 
 that ax^-{-bii^+cz'^+dx)j-]-eyz-\-fxz may be a complete square. 
 
 16. If a; 3 + ^2 -f to +c is exactly divisible by {x + dy, shew that 
 
 1(5 -d-)=-^=d{a-2d) 
 
 17. Determine the relations among a, b, c, d, when 
 ax^ — bx^-\- cx—d, is a complete cube. 
 
 18. The polynome ax^ + Shx^ + Sex + d is exactly divisible 
 by (a-x)^; shew that (arf- 6c)3 =4(ac-i3)(fcd_ cS). 
 
 19. Find the relation between p and q, when x^-'rpx^+q, is 
 exactly divisible by {x — a)^, 
 
 20. If x^+nax+a^ is a factor of a;4+ axS-J-aSaj^+a'aj+a"^, 
 shew that n^— 7i— 1 = 0. 
 
 21. Tix^-^-ax^ + bx^ +cx-\-d,he the product of two complete 
 squares, shew that 
 
 (46-a2)2 = 64(?, (46-fl2)rt = 8c, as/{3u^ ~m = Sb. 
 
130 ' RELATION IN INVOLUTION* 
 
 22. Prove that a;* + px^ +qx^ -\-rx+s is a perfect square, if 
 
 ,2 
 
 9 P'' 
 
 p^s — r, and q = — -{. 2^s, 
 
 ■ ■.23. If ax^ 4-36.C- 4-3cx + f/ contain ax'^-\-2bx-\-c as a factor, the 
 former will be a complete cube, and the latter a complete square. 
 
 24. If m"x^ +2^x+pq + q^ be a perfect square, fiud^ iu terms 
 of m, g, and x. 
 
 25. Find the relation between p and g' in order that 
 
 x^+px^+qx-\-r may contain (a;4-2)2 as a factor. 
 
 26. If x^ +px^ -{-qx+1- be algebraically divisible by 
 
 Sx--\-^px-{-q, shew that the quotient is a; + ^. 
 
 o 
 
 Relation in Involution. 
 
 Art. XXXVIII. If «a' = W/ = cc', prove that 
 
 1. (a + b'){h+c'){c + a') = (a' + b)ib' + c){c'+a) 
 a+b')xa' = aa' + b'a' = bb' + b'a'={b + a')xb' 
 b+c')xb' = bh'+c'h' = cc'-^c'b' = {c4-l')xc' 
 c-{-a')xc' = cc' + n'c' = aa' + a'c' = {a-{-c') xa' 
 a-\-b'){b-irc'){c-\-a')xa'h<c' = 
 a'-{-b){b'-\-c)lc' + a)xb'c'a' 
 a-\-b'){b + c')lc + a') = {a' + b){b'+c){c'-\-n). 
 a + b){a-\-bi){a'-c){a'-c') = {a'+b)[a' + b'){a-c){n- c'). 
 a + b)xa' = aai + a'b = bb' + a'b = {b'-\-ai)xb 
 a+b')xa' = aa' + a'b' = bb'+a'b' = {h + a')xb' 
 a' — c) X a = aa' — ac = cc' —ac = (c' — a)xc 
 a' — c')xa = aa'-ac' = cc' — ac' = {G — a)xc' 
 a+b){a+h'){a'-c){a'-c')x{aaY = 
 b'-{-a'){b-i-a'){c'-a){c-a)xbb'.oc_' 
 
 But bb'.cc'={aa')^, 
 
 and (c' — a)(c — a) = (a — c){a — c') 
 :. {a+b){a + b'){af - c){a' -c') = {al + b)(a'+b'){a-c){a-cr). 
 
RELATION IN INVOLUTION. 137 
 
 Exercise xlviii. 
 
 If rta' = 66' = cc' prove that 
 
 1. {a-b'){b-c){c'-a') = {b-a'){a-e){c'-b'). 
 
 2. (b-c'){c-a){a'-b') = (c-b'){b-a){„'-c-). 
 
 3. (^c-a'){a-b){b'-c') = {a-c'){r.-b){h'-ai). 
 
 4. (^a-b'){b -c'){c-a') = {a-c%b- a!){c ^b'). 
 
 (a-b){a-b^ ^ (a-c){a-c') 
 
 (6_c)(6-c') _ (^>-a) (/;-«') 
 
 ^- (;/_,,),,,_«')' - (c'-6)>'-fc')' 
 
 8. Shew that the seven preceding relations may be derived 
 
 trom the single relation 
 
 (a-\-a')(bb!-cc') + ih-^b'){cc'-aa'} + {c+c'){aa'-bb')=^.(i. 
 
138 
 
 SIMPLE EQUATIONS. 
 
 CHAPTEE Y. 
 
 Simple Equations of one Unknown Quantity. 
 
 Art. XXXIX. Preliminary Equations. AHhnnofh the 
 following exercise belongs in theory to tliis chaptei% in practice 
 the numerical examples should immediately follow Exercise I., 
 and the literal examples Exercise III. Likd" those exercises, this 
 one is merely a specimen of what the teacher should give till his 
 pupils have thoroughly mastered this preliminary work. But 
 few numerical examples are given, it being left to the teacher to 
 supply these. 
 
 Exercise xlix. 
 
 What values must x have that the following equations may be 
 true ? 
 
 1. x-b = 0. a;-3i = 0. x-a = 0. x+3 = 0. 
 
 2. a;4-4i = 0. x-]-a = 0. a;+3 = 5. a;-4 = 6. 
 
 3. x — a = b. x + a = c. x-b=—c. 6— a;=3.. 
 
 4. 8~a;=10. 5 + a;=ll. 9+.t = 4. 7-a;=-5. 
 
 5. 8+a;=-6. a-x = 3b. 2a = x + 3h. 8a = 5b~z. 
 
 6. 2a;- 6 = 8. 3a;+8 = 20. ax = a^. mx = bm. 
 
 7. Sx = c. ax = 5. ax = 0. (a + b)x = b-^a. 
 
 8. {a-b)x = b-a. (a + bx) = (a + b)^. {a—h)x = a'^-b9. 
 
 9. {a + b)x = b^-a^. (^a^ -ab + b^)x = a"^ -\-b^. 
 
 10. {n^-b^)x^a-b. (a^-b^)x = a + b. {a-'+b^)x=l. 
 
 11. (a+x — b) = {a-{-b). x — a-{-b = b—x + a. 
 
 12. 2a — x = x — 2b. ax-\-bx = c. ax — b = cx. 
 
 13. ax~b = bx — c. ax — ab = ac. 
 
 14. ax — a^ = bx — b^. ax — a^ = bx-b^. 
 
SIMPLE EQUATIONS. 139 
 
 15. ax — a^ = h^ — bx\ ax+b-\-c = a+hx-\-cx. 
 
 16. a — bx — c = b — ax-\-cx; a + hx-{-cx''^ =ax — h-\-cx^, 
 
 17. hx - ex- -\-e = ex-b- rx~ ; 3a; = | ; 4a; = f . 
 
 18. 10;c=i — 1; rta;= — ; ax= — . 
 
 c b 
 
 in I (i ''' i «c^ ab^ 
 
 19. aoa; = — + • — ^3 oca;= — _i, — .. 
 
 b a be 
 
 20. |a; = 5; |:e=:8; -5.^ = 2; -30;= -06. 
 
 21. 020; = 20: -80;= -2; •4a;=-6. 
 
 a; ax 
 
 22. •lBa;=l-8; — =. 6 ; t = c. 
 
 ax h X ax .. 
 
 a + ft a a — h a-t-b 
 
 *>4 a; = X = 
 
 " ' a- b b ' a-hb b~a' 
 
 a a b —a a — b 
 
 b—a a — rt + o b-\-a 
 
 a-^b a — c 112 3 
 
 a-f-c rt+o a; 2 a; o 
 
 1 11 an hi 1.1 
 
 27. — = -7-5 — = -T' — = — ; — = -^ + -^ 
 
 X ab ^^x b X c X 3 4 
 
 «^ -^ , 4 33 1 « , ft n 
 
 28. H7T + — = ^ IT' — + — =0. 
 
 yu oa; oa; o a; c 
 
 29. A = 6-JL; ^^ =7+-^-. 
 a;-7 a;-7- 3a;-4 4-3a; 
 
 30. (a;-4)-(a;4-5)+a; = 3; 2x- (a;- 5)- (4 -3a;) = 5. 
 
 31. 2(3-a;) + 3(a;-3) = 0; 2(3a;- 4) -3(3 -4a;) +9(2 -a;) = 10. 
 
 32. a(l-2a;)-(2a;-a) = l ; a;- 5(a-a;) = te- 5a. 
 38. wa;(3a-4)+3w<a;-3a+l=0. 
 
 34. a{bx- c)+h{cx — a)-'rc{ax-h) = 0. 
 85. a{ax — b)-\-h{cx — c)-\-c{cx—a)=;iQ, 
 
140 SIMPLE EQUATIONS. 
 
 86. a{bx-a)-\-b{,:x-h) + c{ax-c)=-0. 
 
 37. a(x-2b)-\-h{x-2c)+c{x-2a) = a^-\-h''i+c*. 
 
 38. 3(3{3(3a;-2)-2}-2)-2 = l. 
 
 39. 9(7{5(8a;-2)-4}-6)-8 = l. 
 
 40. i{iaU(^+2) + 2}+2)+2} = l. 
 
 41. i{iaU(x + 2) + 4}+6) + 8}=l. 
 
 42. UUUh^-^)-i}-i)-l=^- 
 
 43. m{m^~H)-n}-n)-n=o. 
 
 44. f|{-A(f{f(!a;+4) + 8} + 12) + 20}+32 = 58. 
 
 45. |{|(f{M-^-+7)-3}+6)-l}=4. 
 
 46. r{5'(jLi{w(?/ia; — a) — 6} —c)—d\—e = 0. 
 
 47. (l + 6a;)2 + (2 + 8a;)3 = (l + 10x)2. 
 
 48. 9(2a;- 7)2 +(4a;- 27)2 = 13(4ic+15)(.'c+G). 
 
 49. (3-4a;)2+(4-4a;)2 = 2(5+4a;)3. 
 
 50. (9-4a;)(9-5a;)-f4(5-a;)(5-4a;) = 36(2-a;)'. ' 
 
 Art. XL. In order that the product of two or more factors 
 may vanish, it is necessary, and it is sufficient, that one of the 
 factors should vanish. ThuB, in order that (x — a)(x — b) may =0, 
 either a;— a must = 0, or a; — 6 must =0, and it is sufficient that 
 one of them should do so. 
 
 Hence the single equation {x~a){x — b)=0 is really equivalent 
 to the two disjunctive equations, either x — a = or x-b = 0, for 
 either of these will fulfil the condition of the given equation, and 
 that is all that is required. 
 
 Similarly, were it required to find what values of x would make 
 the product {x—a){x — b)(x — c) vanish, they would be given by 
 
 a; — a = 0, or a; — i = 0, or « — c = .'. a; = a or 6 or c. 
 
 Hence the single equation 
 
 (x — a){x—b]{x—c) = 
 is equivalent to the three disjunctive equations 
 
 X — u = 0, ox x — b = 0, or x — c — 0. 
 
simple equations. . 141 
 
 Examples. 
 
 1. Solve j;2-x- 20 = 0, 
 
 The expression = {x — 5){x-\-4), which will vanish if either of its 
 factors does, that is, if ic - 5 = 0, or ic+4 = 0, 
 
 .". x= 5, ov x= —4- 
 
 2. Solve a;4-j;3-a;2 4-^ = 0. 
 
 This gives a;3 (a; -l)-a;(.«-l) = <x-l)(a;2_l) 
 
 = x{x—l)(x + l){x—l), which vanishes fo» 
 x = 0, x= 1, x= —1. 
 
 3. Solve aj^+a-it;^ —ax- aS = 0. 
 
 This = x{x^ - a)-\-a "- {x"" - a) 
 
 = (x + a*)(x^ — a), which vanishes foj 
 x+a^ = 0, and a;* — rt = 0, or 
 x= —a", anda;2=a. 
 
 4. Solve x^{a~h) + a^-(h-x) + b^{x-a)=0. 
 
 The factors of the expression are (Ex. 2, page 79) 
 
 » —a, x—b, a — h; hence the expression vanishes if 
 x—a = Q, or a: — 6 = 0. 
 
 5. Solve 221a;2-5a:-6 = 0. 
 
 Here we have the factors 17a; -3 and 13a; +2 ; 
 .•. the equation is satisfied by^ 17a; — 3 = 0, or x = ^, 
 
 and 13a;+2 = 0, or a; = - /j. 
 
 6. Solve 2a;4+2a;3 + 6a; -18 = 0. 
 
 In this case we have 2(x4-9) + 2a;(a;3+3) 
 
 = 2{x^-\-3){x^ —S-{-x}, which vanishes for 
 a;3+3 = 0, ora;2+.c— 3 = 0. 
 
 7. Solve (.r-a)3+(a-6)3+(6-a;)3=0. 
 
 The expression is equal to 3(^x — a)(a — b){b - x), 
 ftnd therefore vanishes for x — a = 0, or a; = a ; 
 and for a; — i = 0, or x = 6. 
 
142 SIMPLE EQUATIONS. 
 
 Exercise 1. 
 
 1. If an equation in x has the factors 2a; — 4 and 2a; — 6, find 
 the corresponding vakies of x. 
 
 2. If an equation gives the factors 2a;- 1 and 3a? — 1, %vhat are 
 the corresponding values of a; ? 
 
 3. If an equation gives the factors dx^ — 12 and 4a;- 5, find the 
 corresponding values of x. ' 
 
 Find the values of a; for which the following expressions will 
 vanish ; 
 
 4. a;2-2a;+l; 4a;2-12a;+9. 
 
 6. 9a;-^-4; x- — {a+bY; x^-2ax + a». 
 
 6. a;8-0a;+20; 4a;2 - ]8a;+20. 
 
 7. x^+x-6: x2-a;-12; Oa;^ ...9a;-28. 
 
 8. 6a;3-12a;+6; 6a;2_13a;+6; 6a;2-20a;+6. 
 
 9. 6a;2- 5a;- 6; 6a;2-37a;+6; 6a;3 +a;- 12. 
 
 10. A certain equation of the fourth degree gives the factors 
 x^ —x — 2, and 4.<;3 — 2a;— 2, find all the values of x. 
 
 Find values of x in the following cases : 
 
 11. x^-2bx''-Sb^x=i). 
 
 12. x^-ax^+a^x~a^=^0. 
 
 13. a;3-2a: + l = 0; a;3-3a; + 2 = 0. 
 
 14. x^ — 2ax^+2a^x—a'^-0- 
 
 15. x^ + {b + c)x'^-bcx-b^c~bS^=0. 
 
 x-a x-b _ {a -by a;«-a^ 
 
 x — b x-a {x—a){x — b) {x~a)(x — b)* 
 
 17. x^-bx^-a^x-a-b = 0. 
 
 18. 3.c3-f4rt6a;3-6«263a;-4a363=:0. 
 
 19. x^{a-b) + a3{b-x)+b^{x-a) = 0. 
 
 ^^- (^a-b){a-c) "^ (6-cj(6-«) 
 
 /a;-2a\ 3 /2a;- o\ 
 
 21. X -.— + <^\^n-i 
 
 = a;2-as 
 
SIMPLE EQUATIONS. 143 
 
 ah , hx , ax 1 
 
 {b-a){x-a) ^ (x-a){a-b) ^ {a~i){b-x) a~b 
 
 24. Form the polyuome which will vanish for x equal 5, or 
 -6, or 7. 
 
 25. Form the polynome which will vanish for x = a, or 4a, 01 
 3a, or — 4fl. 
 
 26. Form the equation whose roots are 0, 1, -2, and 4. 
 
 27. Form the equation whose roots are l-}-\/2, 1— v/2, 1 - ^.'S, 
 and 1 + /3. 
 
 Art. XLI. In solvino; fractional equations, the principles 
 illustrated in the section on fractions may frequently be applies; 
 with advantage, as in the following cases. 
 
 When an equation involves several fractions, we may take two 
 or more of them together. 
 
 Examples. 
 
 1. Solve ^J^ + 'L^-ll^ = '^^±^. 
 14 ^ 6iB4-2 7 
 
 Here, instead of multiplying through by the L. 0. M, of the 
 denominators, we combine the first fraction with the last, getting 
 at once 
 
 7a;-3 7 1 
 
 g^2 ^ n ^ T ■'• '''x- 3 = 3a; 4-1, and 05 = 1. 
 
 2 2^+8| _ lSa;- 2 ^ _ Z^ a; +16 
 
 ~9 17a; - 32 "^¥"12 "36^' 
 
 In this case, taking together all the fractions having only 
 numerical denominators, we get 
 
 8a;- f34+12a;- 21a;4-a;+16 _ 13a; -2 
 
 36 17a; - 32 ' °' 
 
 25 13a;- 2 
 
 18 17a; -32' 
 
 .'. 425a;— 800 = 234a; -36, hence a; = 4. 
 
144 
 
 SIMPLE EQUATIONS. 
 
 It is often advantageous to complete the divisions represented 
 by the fractions. 
 
 4a;- 17 _ 3| - 22a; 6 / x^ 
 
 ^- 9 —33" = *- ^i^- 54 
 
 Here, completing the divisions, we have 
 
 4x 17 1 2a; & x 
 
 9 ~ir~'9""^y^~"^''"'9"' 
 
 10a; a; 6 6 
 
 -9--2 = a;+- - - .-. -2 = --, ora; = 3. 
 
 ax-\-h ex -\-d 
 
 ain+h cn-\-d . 
 a -\ + r + = n-\-c 
 
 X - VI X — 11 
 
 {a7n + h)(x—')i) + {cn + d)(x — vi)=^0 
 {ani-\-h + c7i + d)x={(i+e)mn-i-bn+din. 
 
 6. Similarly may be solved 
 
 ax+h cx-\-d cx^-^-fx—g 
 
 + + TT \ = a-\-c+e- 
 
 x — m x — n \x — m)[x — n) 
 
 am+b cn-{-d {e{)i}+n)+f}x—emn—g 
 
 x—m x — v {x — m){x — n) 
 
 {am-^h){x — n)^{cn-\-d){x — m)->r{e{m.-^n)-\-f]x — rmn—(i — 0. 
 
 {{a-^c)m-\-b + {c-{-e)n-\-d-\- f]x = {a-\-h+e)mn-\-Jin-\-dt)i-^(j. 
 
 6. il^^+l 8x+5 ^ 
 3a; + l ^ x-l 
 
 43 13 
 
 ••• 44 - 3^1 + 8 + — 1 = 52, or 
 
 ; .-. 39x + 13 = 43a;-43, anda;=14. 
 
 a;-l 3a;+l 
 
 - 25 -Xa; 16a;+4^ ^ ^ 23 
 
 a;+l ^ 3a;+2 a; + l 
 
 Taking the last fraction with the first, and multiplying the re- 
 sulting equation by 15, we have 
 
SIMPLiE EQUATIONS. 145 
 
 210x4-08 r-r . 5a,- -30 
 
 ■ — 11. 75 + ■ ; 
 
 Sx+2 x+1 
 
 :. 80 - ^^ = 75 + 5 - ^^, or 
 
 '^L_ = ^ ; .-. 8a; = 27, and * = Sf. 
 3.f+2 a; + l 
 
 8. J- - 4- — = d. 
 
 ., «^» _ 1 + ^-1 _ 1 + ^^ _ 1 = 0; 
 
 6+c «+c' 6-|-ii 
 
 fe+c "*" rt + C "^ 6+rt 
 
 which is satisfied by a; — (a + /y + c) = ; .-. a; = (^/ + A + c. 
 + 
 
 10 
 
 x-\-(i x—b x — c 
 
 vilx — r) nix — f) 
 x—a x—b 
 
 which may be solved as-in Ex. 1. 
 
 3x+5 4^+9 _ 15a;+7 _ 12a;+17 
 
 a;+T ~ 2x+4 ~ 3.K+1 3a;-|-4 ' 
 
 • 3 + — - - 2 - -1- = 5 + J"- - 4 - -J_ , or 
 " ^ x+1 2a;+4 ^ 3a;+l 3a;+4' 
 
 2 1 _ ^_ _ 1 _ . 
 
 :^ ~ 2a; + 4 " 3a; + 1 ~ 3a;+4' 
 
 3a;+7_ _ _ 3a; + 7 
 
 " 2x3+6a;+4 ~ 9a;2 + 15a;+4' 
 
 This can be divided by 3a; -I- 7, giving 3a;+7 = 0. or x= -|. 
 The r^siilt of the division is 
 
 1 1 
 
 — , or 
 
 2a;3 4-6.i-+4 9x2 + 15a; + 4 
 
 9a;2+15a;+4 = 2a;2 + 6a;-f 4, or Ix^ = —9a;, whicl' we can divide 
 by a;, /. x = ; the result of the division is 7a; = —9, or x= —%■ 
 
1. 
 
 146 SUJIPLE EQUATIOirS. 
 
 Exercise li. 
 
 lQa; + 17 12a; + 2 _ 5x-4: 
 
 18 ~" 13a;- 16 ~ ~ir" 
 
 6x-\-lS ' 9a;+l5 ^ 2a;+ 15 
 
 ^' 15 5a;- 25+^= ~5 
 
 7a;+l 35 a;+4 
 
 3 ■ — = — X — ' — 4- 31 
 
 4a;- 7 2- 14a! 3^ +a; _ 10 - 3fa; 19 
 ^- 2a;-9 "^ 7 + "IT" "^ 2 ~ 21* 
 
 2x + a 3x — a 
 
 ^- 3(^^ "^ 2(^2M^"^^' 
 a;-4 3a;— 13 _ 1 
 ^' fo+5 "^ 18a; -6 ~ T* 
 
 3x+ 1 a;— 11_ a;— 9 a;— 5 
 
 ^- 2a; -15 ~ 2a;-10" ' ^^ "^ aT^"^' 
 
 8 ^~^^ 4. ^~ ^ _9 _1_. Sa;-19 3a;- 11 
 
 a;- 7 "^ .^■-12 -""^^--v' a;- 13 + x+ 7"^- 
 a;-2 a;-l _ _5_ . a;+l J^+l. _ ^ 
 
 ^- 2a;+l "^ 3(a;-3) " 6 ' 4(a;+2) + 5a.-+ 13 ^ 2u' 
 5(2a;^ + 3) 5-7a; , / 3 ^ 1 4 
 
 ^^__^ _ 3^^ 3a;^±^. 17 _15 32 
 
 ^^- 9..:!. ^ 2 -^ 2a; ~4' a;-lG "^ a;-18 = ^-1? 
 
 12. 
 
 15. 
 16. 
 
 _ 3-2ia; _ 28-^ _ 10a; - 11 x 
 
 15 14(a;-l) ~ 3 30 ^ Y' 
 
 _L _ ^ +2|a;g- a^3 _ J^ _ _5^ 
 
 ^'^ a;-2 G-5a; + a;2 ' 2*"~a;- 
 
 1, 30 + Oa; G0+8a; 48 , ,. 
 
 14. _L. — ' _ + 14. 
 
 a;-fl ^ X+-6 x + 1 
 
 5a-2+a;-3 _ 7a;3-3a;-9 
 5a;-4 ^ 7a; -lO"' 
 
 X x—9 x + 1 x — 8 
 
 a-- 2 "^ a; -7 a;-l "^ a;- 6' 
 
17. 
 18. 
 19. 
 20. 
 21. 
 22. 
 23. 
 
 25. 
 
 26. 
 
 28. 
 
 SIMPLE EQUATIONS. 
 
 a;2-3a; -9 a;2-7a;-17 _ x^-6x- U 
 x-B'" "^ a;-9 ~ x—Q 
 
 4a;-f7 4a;+9 _ 4a;-f-6 4a; +10 
 Ax+l "^ 4x + 7 "^ 4x+i "^ 4a; + 8' 
 
 2a;- 3 2a;-4 _ 2.r-7 2a;-8 
 2a; -4 ~ 2a;- 5 ^ 2a; -8 ~ 2a;- 9* 
 
 7a;+6 2a; + 4| x _ 11a; Xj-^ 
 
 28 ~ 23a;-6 "^ T ~ "21 ~ 42 ' 
 
 x'^-5 a;3-ll _ a;2-7 a;2-9 
 
 ^2^^ "^ a;2-l2 ~ a;3-8 "'' a;^- lO' 
 
 a; - Iff 2 -6a: _ 5a;-i(10-3a;) 
 
 2 ' ' ~ 13 ~ ^ ~ 39 
 
 l-2a; \+x 1 
 
 U7 
 
 8(a:3-a; + l) ^ 2{a;3 + l) ^ 6(a;+l) 9(a;2+l) 
 
 o. 2a;3+a;-30 x^+^x-A. x"-ll 2x^+7x-V6 
 
 2a;-7 ^ a;-l ~ x-4 ^ 2a;-3 
 
 x — a x — h {a — by 1{a — x) 
 
 x—b ' x-a {x — a)(x — b) ~ a-{-x 
 
 12a;+10a ^ 28a;+117rt ^ ^g 
 '3a;+« 2x+9a 
 
 07 l^|a;-5 13|a;-ll _ ISjx-l 13ia;-9 
 
 13i':e-6 ' 13|a;-12 ~ 13^a;-8 ' 13^a;-10 
 1 1.x 16a; 
 
 + 
 
 2(a;-l)2 "^ 2(a;-l) 2(.t3+1) " ^x-l){x^ + l) 
 
 29. i(|.^+4) - Zi^^ = ^ (-1 _ 1 
 o 2 \ a; / 
 
 3a; 
 
 81a;2-9 3 2a;2-l 57 -3a; 
 
 30. _ - ^^^ ^ = 3a- - 
 
 2 (3a;-l)(a;+3) ^ 2 a;+3 
 
 31 1 + ^ ^ + 1 "^^"fl^, _ a;2 + ^^ + ^ _ 
 
 2(a;-l) ~ %+l) ~ a;3-2a;+l ~ ' 
 
148 SIMPLE RQUATIONS, 
 
 7a: -30 5x-7 2 -21a; 
 
 32. 
 
 83. 
 
 10^ ^x-B 21 
 
 42a;- 171 2a;-9 1 
 
 63 ^^'^ 63-14^ ~ Y^^ ^ 
 
 18^^2 l + Hx 101 -04, 
 
 13 -2x +^^+ ~8~ =131- 'q — 
 
 4 -9a; 5 - 12x 24x2 - 5 
 
 ^^- l-'6z ~ 7- 4x'^^" 7^ 25iB+T2a;2* 
 
 8a; +25 16a; +93 18x+86 6a;+26 
 
 qe I ' - I . 
 
 *^^- 2a;+ 5 ^ 2a;+ll ~ 2a;+ 9 ^ 2a;+ 7 
 
 1 1-1 1 ^ 
 
 x+a + w a;— a + o x-+a— <> a; — a— o 
 
 Art. XLII. The results deduced in Section III., Chapter 
 IV., may often be applied with advantage. 
 
 Examples. 
 ax -i-b m 
 
 (page 123). 
 
 ex + d n 
 
 {ax-\-h)d — {cx-\-d)h md — nh 
 (ex +d)a— [ax + b)c ~ na— mc 
 
 md — nb 
 
 X = • 
 
 na — mc 
 
 ox^-\-hx-\-c a 
 
 mx^-{-nx+p VL 
 
 (rta;2+5a;+c)-r/.r;3 n 
 
 /- — iT . — o = "■ (page 122). 
 
 {mx^ +nx+p) — mx^ m ^^ ° ' 
 
 hx + c a 
 
 :. — r = — &c. 
 
 nx-\- II III 
 
 3a;+J _ ?>x- 1 3 
 .r4-4 ~ X— 4 
 
SIMPI.K EQUATIONS. 149 
 
 By (5) eacli of these fractions = 
 difference of numerators 20 0«+7 
 
 difference of denominators " 8 a;+i x+i' 
 
 or -^ = ---7, •• x = G, 
 
 vrx-\- a + h mx -\- a -\-e 
 
 4. . = ] r» 
 
 nx — c — (I vx — o— a 
 
 mx-\-(i -}- b nx — c — d 
 
 , r— = 7 — -, ; or By 4, page 122, 
 
 mx-\-(t -\- c nx—b- a j ■> i o » 
 
 or (» — >?/)» 
 
 vtx -\-a -{■ h nx — c — d 
 h—c b—c 
 
 — a-\-h-\-r-\-il, .'. x=kc. 
 
 ^ l/{n+x) + i/(>i-x) ^ 
 ''• A^'Jrx)-^/{a-x} • 
 
 Here by (0), page 122, we have 
 
 r" ^-r^ — —, = :: ; or, Cancelling the 2 in left hand mom* 
 
 2p/{a — x) a—1 '^ 
 
 ber, and squaring, 
 
 n+x (a+iy ■ -u /n^ 
 
 = / ,^rj, whence, again by (6), 
 
 a — X {n — ij 
 
 2x _ (rt + l)--(«-l)3 _ 4a 2a^. 
 
 2a2 
 
 6 
 
 ^/{x-a + b) - y{x \n- h) ^ n-b^ 
 
 y^^x-a + h)-^\7{x-{-a-h) ~ a+b 
 
 l/{x — a+h) _ ^. 
 V%«+a-6) ~ b ' 
 squaring and again applying (6), 
 
 1x ^'=+&i 1 _ ^"Vh^ 
 
150 SIMPLE EQUATIONS. 
 
 Exercise lii. 
 
 1+x 
 
 2. - ~ 
 
 6. 
 
 2x^-lx+B ~ a;2-9x-|-2 
 az+b — c (b — c)^ 
 
 ax~h + c {h + cY 
 
 7. If i/{x+n)+^/{x-y) ^ x_^ ^^^^^ ^^^^ x+y^ 
 V'{x+y)-v'(x-y) y x-y 
 
 = 1. 
 
 8. 
 
 0. 
 
 2a:- 7 _ x+l_, 4a- -5_ _ lO x-32 
 
 5 7a; -43 _ ^x-1 . 23a;+5i _ 36a;-- 7 
 19a; 4- 13 ~ 18.r + 2^5 ' 115.r-29 ~ 180a; + 23' 
 
 210a;-73 21a;+7-3 . v»a; — a - i mx-a-c 
 
 3i0a;-86 3]a; + 8 ' nx—c-d nx-b — d 
 
 Zx+^jix-x'^) _ ^ ./(12.e + l)+y(1 2y) _ 
 ■ Sx-^/{4x~x^} ~ ■^■- {/(12;c+l)-|.^(12.tj " ^^ 
 
 12. 
 13. 
 
 x^-{-ax- —hx+c x~+ax—h 
 x^ —ax^ + hx+c ~ x^—ax+b 
 
 ^/{2a''-x-) + b]/(2a-x) _ /a+b 
 |/(2a2_a;3)_&^/(2a-a;) ~ ^a-b' 
 
 ' V(x-+a^)-V{x^-a^) 
 
 15. 
 
 8a;3+12x3-8.'c+r) _ 4 x^+6x- A 
 8x3 -1 2.1-3 +Ba;;+ 5 ~ 4a;2 - 6a; + 4 ' 
 
SIMPLE EQUATIONS. 151 
 
 16 
 17 
 18 
 
 28-K/a; _ 9 +3 yx 
 28 -ya; - 9+2i/.f' 
 
 a^x^+a^bx^ —acx + d a^x-+abx—e 
 
 a^x^—a^bx^+acx+d ~ ci'x''^ -ahx+e 
 
 5T /(2a;-l) + 2t/(3a;-3) _ 
 
 °- 4i/(2a;-l)_2i/(3aj-S) " '^^• 
 
 /2a;+y^(3-2x) _ 3 
 72aJ-i/(3^;^-) - T" 
 
 2£(3aH-3)+jf (7x + 8) _ 
 " ■ 2^(3a;+3)-i^(7:c + S) " ^• 
 
 22 
 23 
 
 20 
 
 24. 
 
 33{13-2/(a:-5)}-3{13+2|/(a; -5)}. 
 
 (l/«+l){l/(».6- + l) - ^oix] = {^/n-l){i/{nx+-i) + y'nx] 
 
 l/{x-hc) + -y/b _ ^x + i/a 
 l/{x+c) -yb ~ ^x - i/a* 
 
 o" V^J^^ _ lAjf38. ^2x + 17 _ f2x+^l 
 l/ic+ 4 ~ ^x+ G' ■^2a;4- 9 ~ -^2^ + ir/ 
 y»+2a _ -1/5+ 4a. 3a; -1 _ 1+j/dx 
 l/'x+ b ~ -j/aH-^i' |/3a; + 1 ~ 2 
 
 ya— |/(a - a;) ^/x + ^/6 
 
 26 
 
 27 
 
 31. 
 
 ■\/a+ ^/{a—x} ' s/x — \/b b 
 
 ax + l + ,/{a^x-^-l ) _ 
 ^^- «a;-t-l -N/(a3.t-3-l) -"' 
 
 on '«+a; _ H-1 . l-fx+a;2 _ 62 14a; 
 
 v/(2aa;+u;3) " 6-1 ' l^^+a;^ ~ 63 T~£ 
 5a;4 + 10a;2 + l _ a^ -\-lQa-^ -\- 1 
 a;^4-10a;s+5^ ^ Sa^ + lOa^ + l* 
 
 Art. XLIlI. Various other artifices may be employed to 
 simplify the solution of equations. 
 
152 simple equations. 
 
 Examples. 
 
 1. Solve 2-f v/(-J:Cc2-9a3-f 8)-2x = : here there is but one 
 surd, aud it is convenient to make that su^d one side of the equa- 
 tion and transpose all the rational terms to the other ; this gives 
 >/(4cc2 - dx + 8) = 2x— 2 ; squaring both sides, 
 
 ■la;2 — 9,t+8 = 1.^-2 -8x-+l, .-. x = 4. 
 
 2. \/{a + x)-{-V{a~x) = 2Vx. We might square this as it 
 stands, but the work will be simphfied if we first transpose, thus 
 
 \/(a+x) = 2 \'^x— \/{a-x) ; 'low squaring, 
 a-\-x = 4-X + (i- x — -i \/{ax—x^), or 
 ic = 2 v' {((X — x^ ) . Again squaring, 
 
 x^=A(ix — 4:X-, whence x = 0, or — -. 
 
 5 
 
 3. Clear of radicals 
 
 -^x + f/// + ^2 = U. Transposing, 
 
 ^x-^ {/y = - f^z ; cube by formula [0] , 
 
 ic + y + of^xy{^x+^y)= -z; and subsiitwtoig wr 
 
 ^x-V-^'y ''-3 val". • —^2, t^is becomes 
 
 iK + ?/ — 3 -^^xyz — —z, ur 
 
 xi-y-\-z = o^''x!jz; :. out lug again, 
 
 {x+y + z)^ = 21xyz. 
 
 . a +x+Vj ^ax+x^) 
 
 a + x ~ 
 
 Dividing and transposing, we iiave 
 
 a+x a-' +2ax+x' 
 
 division in left-hand member, 
 
 fl = (6-l) .-. -^= v/{l-(/.-l)^},or 
 
 (a-i-x)^ a-\-x 
 
 ^ + " = 1 v/il+(fc-l)n. or 
 
 — -f 1 = "fee. 
 
SIMPJ^E EQUATIONS. 153 
 
 5. Solve v/(4.r3-fl9)+A/(4x3-19)= -/dT+S. 
 We have the identity 
 
 (4;t3 4.i9)_(4a;2-19) = 38 = 47-9. 
 
 Now dividing the membera of this identity by those of the given 
 equatioju, we have 
 
 V(4a;3 + 19) - v'(4x2 - 19) = a/47 - 3. Adding this to the giwB 
 equation, then 
 
 2A/(4a;3 + 19) = 2A/47, /. 4^2+19 = 47, and a. = ± ^7. 
 
 6. if(25+a;)+^(25-a;) = 2. 
 
 Cubing by formula [6] , (See Ex. 3), we have 
 
 25+x + 25-a; + 6if(253-a;2)=.8, or 
 T^(f525-x3)=-7, or (625-x2)=-343; 
 .-. a;3 -525-1-453 = 908, and a; = ±22|/2. 
 
 Exercise iiii. 
 
 2. y(3:e + l)+i/(4.r + 4) = i. 
 S. V(2.f-t-10) + i/(2a; - 2) ^6. 
 4. i/{i)ix)~i/{nx) = m — n. 
 
 G. l/-+/(x+3) = ^|^). 
 
 7. A/(aa;+a;3^^(1.-|-a:). 
 
 8. ^(17a;-26)= A. 
 
 , a 
 
 10. 6-i-a;-i/(63+a;2) = c2. 
 
 11. i/(8 + a-)--/a; = 2i/(l+a:). 
 
 12. ■/(2a;-27rO = 9/a-/(2.r). 
 
154 SIMPLE EQUATIONS 
 
 13. ^(l-^cyi-f{H-x) = ^3. 
 
 14. f'i:>+.c) + f/{'d~x) = ^l. 
 
 15. ^(.c + l)_,.X(.t— l) = ^lh 
 IG. ^[a + x) + f(a-x) = -^b. 
 
 17. r(l + iA) + ir(l-l/^) = 2. 
 
 18. |/.^;-/{a-|/(a:c+a;2)} = i/«. 
 
 19. Clear of radicals ^a+^^/ — ^6-, 
 
 20. Solve aj+,/(a2^x2) = 
 
 7ja* 
 
 l/(a2+a--) 
 
 21 . Clear of radicals yx+ \/tj+\/x — |/m. 
 Solve the following equations : 
 
 22. v/(i-^x) + ,/{l-l-.c^-|/(l-a;)|• = l/'(l-»>. 
 
 23. -/(x-+ i/x) - Vi-'-V-^) = '*Vx +17i* 
 
 24. ^(l+x-+u;2)+/(l-.t;+.7;3)^/y,x. 
 
 25. ■i/(a3-.<;2)+.c/(^<3 _!) = ,( -2(1 „^.'). 
 
 2o. —7", — " — = <^' 
 
 • i/{hx)+c- n 
 
 27. i/C-..;-+5)+\/(2a;3-5)= v^l5+ v'5. 
 
 28. -s/(3u;3 + 10)4-a/(3x3-10)= ^17+ /a. 
 
 29. v^(3a;2+'J)- i/(3x'3-9)= \/31 f 4. 
 
 30. A/(3«-o/>+a;3)-t- ^(2(i- 2/^-^x-) = V a+ v js. 
 
 81.. V(4«2_ai-i-2.c2)+;/(3«2_3/.2_^3)=,,4.^. 
 
 32. Clear of radicals, ]^(2x-) - ^{%i] - ^(2z). 
 
 83. |/(«+.t) + /(" - ^) - 2x -: -,/{ -t + i./(a:^ 4-^-) K 
 
 84. y(.. + 2..) + ^(-^-2«.)= -^^ 
 
 85. J f<-^+^-i-v (-"?-) ==^(^^^^^^^^ 
 
 86. ,/{(2«+x)--^-h/'-} f/U2<t-^)-+^-;- = 2a. 
 
SIMPLE EQUATIONS. i£)& 
 
 Art. XLIV. Sometimes a factor can be discovered, and tbe 
 principle of Art. XL. applied. 
 
 Examples. 
 
 1. t+a^^^ji^ = x^Jr{ci-b)x^ + {a^ -ah)x-an. 
 x — a 
 
 Factoring we have 
 
 x — a 
 
 ox x^ —ax-{-a^ = {x — a){x—h) \ 
 
 fl2 
 
 .'. {a-{-h — a)x = ah — a'^,?in<lx = a— — 
 
 Transpose '— and factor, then 
 a 
 
 •+'4 (o+hy) ^1 a\ (a-Vh)-] \ 
 
 «6 
 
 a~+b - ^• 
 
 x+a a; — 6 a; — r ^'4-c 
 
 ^' (^^^)(7^c) " {a^b){b-c) ~ {b - c)(^-^) " {a-o)[b'c){c-a) 
 
 Add term by term the identity (Th. iii., page 54). 
 x—a x—b x—c 
 
 (ir-b){c-a) ^. {a-b){b-c) ^ {b-c)(c-a) 
 
 2x _ h + c 
 
 *• {^b){c-a) ~ {a-b){b-c){c-a) 
 
 1 h+c 
 
156 SIMPLE EQUATIONS. 
 
 Tlie left hand member vanishes for x = 0, and /. oy symmetry 
 for a = and b=0; :. it is of the form mabx in which m is 
 nume7-icaL 
 
 Put x = a = b, and m is found to be 6, 
 
 .*. the equation reduces to 
 
 6ahx = ahc, :. and x = ^c. 
 
 Ix — a\ 3 x — 2ia+b ■ 
 
 5. r) = oz. I '> let x — b = m, x — a = n, and .*. 
 
 \x — bj x — ^b + a' 
 
 m—n = a — h, then we have 
 
 m^ n—{m — n) 2n—m 
 
 «3 ~ m-\-{m — n) ~ 2m — n 
 
 2m* — 7i?» 3 = 2^4 — w^m, and 
 
 2,{m^—7i^) — nin(m^ — w*) = 0, which is divisible by 7n^ — n^, 
 
 .•. mf —n^ =0, or m-\-n = 0; 
 
 Bnt m+n = 2x-a—b = 0, :. x = l{a-\-h). 
 
 1 a;2-4a; + 2 1 x^-Ax+S _2_ a;g-4 a;+3 5 
 6- y a;2_4a;ri + (5 ^2^ 4^~3 ~ 9 * x^-4x-6 ~ 18* 
 
 Let ?/ = a;-— 4a;, then this equation becomes 
 
 1 ?/+2 1 y+3 2 y+3 5 , ,. . . 
 
 — • q+-7r- ^ — TT" ^e = T3' or by division, 
 
 3 y — 1 6 2/ — 3 9 ^ — 6 18 •' 
 
 1 . J_ 1 1 2 2 _ 6 
 
 ■3" "^ 2/"^ "^ T "^ 1/-3 ~ "9" ~ y-a ~ 18'°^ 
 
 _^ : - = ; this may be written 
 
 ^—1 y-S y — (j 
 
 1111 
 y - I y — b y — o y — o 
 
 ^ + ^ =0, .-. 5y-16+32/-3 = 0, or 
 
 y = 2i .-. a;2-4a;=2^, ora;2-4a; + 4 = 4+2^, 
 
 and a; - 2 = + 1. We might assume (as— 2)^ = y, when the eriven 
 equation would take the form 
 
SIMPLE EQUATIONS. 157 
 
 3' y-o '^ a' y-7 9* 2/ -10 18' 
 
 And reducing as before, we should find 
 
 y = 6l = (x-2y, .'. 3^-2= +f, as before. 
 
 Exercise liv, 
 
 2. ?l±i^* = x^ + 2a{a-b)x+{2a-b)x^ -2a2b . 
 x + b 
 
 o a^ a^x _ 2c 
 
 ' a^^ab+b^ ~ a^-b^ ^ ^^ ~ ^^' 
 
 4. ^- _ i- _ J- _ i- = 2ab\x^b)x'^. 
 
 a-\-b-\rX a b X 
 
 1 1 
 
 {x-b){x-c) "^ {a+c){a+b) 
 
 1 1 
 
 + 
 
 {a-\-c){x-c) "^ {a-\-b){x-b) 
 
 ^ bz Bab an^ b^x 2a -b 
 
 9. — — — -7 + 7 TT^ = 3x — — • 
 
 a a-b "^ (a-6)3 "•*' a (a-b)* 
 
 x^-{-2ax x—a 
 
 ' a;* -lla;2a2+«* ^ X'-Bax-a^' 
 
 2 \a;+a/ x+a 
 
 = (.c — rt ) (x — h) {x — c). 
 
 aa; bx ex 2 \bcx acx abxj 
 
 ,, 1-ax l-bx l-.cx 12 2 2\ 
 
 (a — b)^ a a^-b^ I 
 
 12. y^L-^ _ 1 + _ = __ + 1 + 
 
 abc abc \ 
 
 19,, x^J^{h->rcY-\-Bb{b-^f)x = b^. 
 
 a 
 
 . X, 
 
 b 
 
158 SIMPLE EQUATIONS. 
 
 14. x—a-B-^(ahx) = b. 
 
 15. lla;4 +10x3 _40.r = 176. 
 
 in X ac c ax 
 
 Id. , ^— 4- 
 
 -„ « — & 2c.k2 «.— & \ — cx 
 
 a+b ' 1+cx a-\-b l+cx. 
 
 lo 4x*+4rt4— 33a;2a2 w^ , o « o s o "\ 
 
 19- „. -..,.,00 + 
 
 a;3-lla; + 28 ^ x^-llx + lO a:2-14a; + 4U 
 
 20 8 , 8 ^ ^ 
 
 x'^-Qx+5 "^ a;2-14x+45 .r^-lOx-^O 
 
 {a-b){c-a) ~ {a-b){b-c) "^ (fe_c)(c-^,) 
 a + c 
 
 ~ {a-b){h-c){c-a) 
 
 22. (a;-a)34-(a-&)34-(6-a:)3=a;2-a2. 
 
 ix+ 2a\ 3 /a+ 2x, 3 
 
 23. X + « =2m. 
 
 \.r — aj \a — xj 
 
 24. {x+ay -(a +h)s + {b -x)^ ={x + a){x+b){a-\-b). 
 
 25. a;3-(x-6)3-(a;-a+6)3-rt3 + (.c-a)3+(r/-i)3+&3 
 
 = (a-'))c2 
 
 26. {x+a)^-{x+b)^-{x-by'-(-2ayi-{x-a)^+{a+by + 
 (a-by = {a^-b»)c. 
 
 x-\-a x — a (7* 
 
 28. (a; + a-t-5)4 .-{x + n)i-{x + b)*+x*-{a+by + a^ + b^ 
 = 12ab{x-+{a + by}. 
 
 a — x 
 
 b—x c — x. 3.r 
 
 29- ^m^c "^ /)2— ca "^ c2"_«6 ~ a/;+^c + c^" 
 
 30. x^{b - a^)-\-a^{x-b^) ^b^{a-x-) +abj{nbx-- 1) 
 = («-a;2)(62_«4). 
 
SIMPLE EQUATIONS. 159 
 
 81. (l[x+x^-)" =^~-..{l + x^+x^). 
 
 CttLf J. 
 
 •;>o I aj-l-a a — h 
 
 ^ 1. 
 o. 
 
 x + b ^ \2,.c-\-b + c 
 
 34. x/(x3 -f27.j + l 80) --p/(a;3 +26.^ + 168)= J^±l^\. 
 
 35. {{,■ + a + y' (x^ +2ax + b-^)}s + {x+a- ^/{x^+^ax + b^)}^ 
 
 = 14(./j+a)3. (See page 17, Ex. 1). 
 
 36. {x+u+ y{x^ -2ax-2b^)}- + {x + u-'/{x' — 2ax-2b^}}^ 
 
 = .^-b-^+2a{a-b). 
 
 orr , x+a 1^ :c4-2a + 6 
 ■ — ^ / ;<; — «— 2/* ■ 
 
 38. (5a:-7)'-(2.c-4)3^27(a;3-l). 
 
 39. 
 
 _L '^-<^'-l 1 :t:3-6a;-4 2 a;2-6a;-7 
 
 3 * .c2-6x--4 "^ T ' X--6X-9 ~ T ' a;--6x-16 
 
 14 4 
 
 40. 
 
 15 x'^ — &X — 9 
 
 1_ :<;^-2a;-3 J_ a;^ - 2a; - 15 _2_ x^ -2x- 35 
 2.(;-8 "^ 9 *a;2-2.r-24 ~ 13 * a^-2x-~48 
 
 I 
 I 
 
 '_2 
 ~ 585"' 
 
 41. {r-{-a-b+^ (.2^^43_J3)i3_ 
 
 a;+a - 6- -/(.i^s ^ a2 _ ^2) ;. 3 -_: s(x + a - 6)3. 
 42 ^1 , 1 
 
 |.f+a)3 -h~ "^ (.,+6)2 _«3 - 
 
 1 1 
 
 a;3_(a + 6)2 + x^-{a-by 
 
 „ ,./U 45 7.C + 67 \ ^.^^ 
 *3. 41 — — - 4- ^ -r 130 =- 
 
 \ X + 1 .c-r-i I 
 
 ^^ 8a;-;- 57 9.v-^68i 
 39 ^ H 5- 
 
t.60 SIMPLE EQUATIONS. 
 
 44. 51 1?£±15 _?£t^)+e63 = 
 
 . X—1 X—4: j 
 
 45. {x+a){x + da){x+ia){z+Ga) = x'^ + 6a^{x' +7a«-f 6aS). 
 
 4« 1 2 3 6 ' 
 
 4b. + + 
 
 x+Ca X - da x+2a x + a 
 
 Exercise Iv. 
 
 1. a{b — z)'ih{c — x) = b{a — x)"{-cx. 
 
 2. (a-i-/;.c)(a— i) — (flx — 6) = «6(a;+l). 
 
 3. (a-6)(a;-c) + («-f-6Xa;+c) = 2(Z^a; + arf). 
 
 4. (a-')(a;-c)-(a + &)(a;+c) + 2a(6+c) = 0. 
 
 5. (a-6)(fi-c)(a-}-a;) + (a+6)(a + e)(a-a;) = 0. 
 
 6 ( .-6)(a-c4-x)+(a + 6)(a+c-j;) = 2a2. 
 
 (solve in {a;— c}). 
 
 7 {m+a(a + b — x) + {a — m){b -x) = a{m-rb). 
 
 8. m(a + L — a;) = n(a;— a— ^). 
 
 9. (7n+n)(m — 7i—x) + m{x—n) — n(x — m) = m^ — »*. 
 
 m — a; n — x v—x 
 
 10. + S- ^^ — =3. 
 
 m n p 
 
 a^b — x b^c — x c^a — x 
 
 11. -f- 7 + " =0. 
 
 a c 
 
 a — x b — x c—x ^ 
 
 12. -1 V + — T- = 0. 
 
 be ca ab 
 
 1 —ax 1 — bx 1— ex ^ 
 
 18. - -r 1 1 r =0. 
 
 be ca ab 
 
 (Deduce the solution from that of No. 12; 
 
 a—hx h — ex c—ax 
 
 14. — , — H 1 r- = o. 
 
 be . ca ab 
 
 15. {a + b + e)x-—^ = , . + —J.* 
 ^ ' u—b a-\-o a—o 
 
SIMPLE EQUATIONS. ItJl 
 
 Sahc a^b^ ' {2a-hb)b^x _ {b + 3ac)x 
 
 ^^- a+b "^ (a^^ ■*" «(rt + 6)2' ^ a 
 
 10 4 9 2 /^ , . 1 > 
 
 17. — + — = h -5- Solve m — • 
 
 X y X o \ xj 
 
 ^^- X "^ 3 ~ 3a; + 12 ~ ii* 
 
 I. 1^ _ 2(5a;-12) _ 17 10 
 ^^- 3 "^ oa; ~ 3a; ~ 20 "^ T* 
 
 10-a; 13-fx _ 7^+266 4a;+17 
 20.-3 "^ 7 ~ x+21 ~."2i 
 
 6 3 17 
 
 21. -T^ + 
 
 22. 
 
 23. 
 
 a;+3 ^ 2(a;^3) 2 2(a;+3)* 
 
 6a;+5 l + 8a; 1-a; x-% 
 
 8a;-. 15 ' 
 
 15 
 
 
 3 
 
 1 
 
 
 1 
 
 
 a — 
 
 X 
 
 1 *- 
 
 a 
 
 1 
 
 1 
 
 a; 
 
 
 a 
 
 a2 
 
 
 
 
 X 
 
 2.5. (a;-l)(a;-2)-(a;-3)(.r-4) = 3; 
 (a;-3)(a;-4) = (a;-2)(a;-6). 
 
 26. 2(a;-4)(3a;+4) + l2a;-3)(3a;+2)-6(a;-2)(2a;-3)=0 
 
 27. {a-x){b-x)=x^ ; {a-x)(z-b)=x^ -c^. 
 
 28. {a-x){b-^x) = b^-x^.; {x-a){x-b)=x'' -ai. 
 
 29. {a^x){b+x) = {a-x){b-x); 
 {ax-\-b){bx+a) = {b-ax){a-bx). 
 
 30. (if-a;)(6-a;) + (a-c-a;)(a;--6+c) = 0. 
 
IR2 SIMPLE EQUATIONS. 
 
 31. {a~x)(h — x) — (c — x){d — x) = (c + d)x - cd. 
 
 32. {x~a){x-b)-(x-c)(x-d) = {d-^a){d-h). 
 
 S3. {(a2 -h^)x-ah}{a—(a+b)x\ +2ah^x = 
 
 {{a+b)2x+ab}{b-{a-b)x}. 
 34 (ar-f l){x+2)(x-\-B) = (a;- 3)(a:+4)(a:+5). 
 
 35. (a;-fl)(a;+2)(a;+3) = (:«-l)(a;-2)(a;-3) + 3(a;+l)(4.r-?-l) 
 
 36. (a;+l)(a;+4)(x + 7) = (a;+2)(a;+5)2. 
 
 37. {x + 2)(x+5y'={x+Sy-{x-^-6). 
 
 38 (a;-l)(.r-4)(a:-6)-a;(rK-2)(x-9) = 13G 
 
 39. (a+x)(7; +a;)(o+a:) - {a~x){b-.r){c -x) = 2{x^' +abc). 
 
 40- (3^-^)(^-^)(^-^)-(^-^0(^-^)(^^-g) ^ (a;-rf)=. 
 
 a; — (i 
 
 4 1 .^•(x• - a) 2 - (ic - a + t) {x - a-^-r) (x - b - c) = (a 2 + he) (b + '). 
 
 42. {x-fi + b){x- h-}-c){x-c + d) - x-(x~ a + d) = bc{d - a). 
 
 43. (x - a + b){x — b + c){x- c + cl) - x{x - a + c){x- c+d) 
 = bc\d — a). 
 
 44. {x-2(,Yx-2h){x-2c)-{x-a-b){x~b-c){x-c-a) 
 = {a-k-b + c){a^-\-V~+e-)-%abc. 
 
 45. x^ —{x — a-\-h)(x—b-\-c){x—c+a) 
 46..L_i)(„_l)f._i)+i,."+M:£. 
 
 ■ \ X I \ X I \ X I X^ X 
 
 47. (x + a){x + b)+{x+c){x+a) = {x + b){x + d)+(.T + d){x+c). 
 
 48. (aa;+/0(«-«-O-«{^-ic)(«a;+&)=a2(a;-c)(a;-i)_ 
 a((7a; — c)(c— a;). 
 
 2a; -3 3a; -2 _ 5 x2- 29a;- 4 
 *^' ~x-l "^ "ai^ ~ "a;2-12a;+32' 
 
 60 ^'^^ 3a;+2 _ a;^- 30.x- +2 
 
 8(i-fl) ~ 2(a;-l} ~ 6^21^6 
 
SniPLE EQUATIONS. 163 
 
 3x-7 3(^--fl) _ lla;+3 
 
 7x-5 8^-7 _J-^±1 n 
 
 ^^^- 3x--2 "*" 'dx-1 "^ 9a;3-9a;+2~ • 
 
 2a:+7 3x-Q 5(:k- 1) 3x-2 5a;- 8 2^+2 
 ^* 'Sx-7^-Ix-o ' Qx-'Io ~-Zx-6^'J:c-25^-6x-l 
 
 ^^ „^ 4^3 +2x. x — a a; -6 
 
 „ - , a ex c au) 
 
 66. 
 
 2(a;-l) «+8 _ 3(5^+16) 
 
 ^"^^ ~a;-7 "^ a;-4 ~ 5a;-28 " 
 
 _^ ax ex a c 
 
 58. + = ^ _; 
 
 mx—p nx — q m n 
 
 ax+b cx-\-d a e 
 
 nix—p nx — q m n 
 
 „„ 6 — a; c-^x a(c-2x) 
 
 a+x a—x a- —x-' 
 
 a+b b+c a+c + 2b 
 
 x—a x — b x — c 
 
 ax+b bx ax (ax^ — 2b)b 
 
 ax~b ax + b ~ ax—b a^x^—b'^ 
 
 60. 
 
 a 
 
 rt-j ux — b cx — d (bn + dm)x — {bq-\-dp) 
 
 mx—p nx — q {mx—p){nx — q) ~ m "*" n 
 
 ^c, w. n p m n v 
 
 62. -1. + — ^— = -t. 4. -±L^. 
 
 x—a x — b x — c x — c x — a x—b 
 
164 SIMPLE EQUATIONS. 
 
 fi8 
 
 1 1 
 
 ax— 2a ax — 2b ^ a « _ 
 
 1 
 
 X 
 X 
 
 
 
 
 ax— 2b ~ ax-^2a ' 1 1 
 
 1 a + 
 
 a X 
 
 
 
 2a;2-3a;+5 2 
 
 
 
 
 
 7a;2-4a; + 2 ~ 7' 
 
 
 
 
 64 
 
 ax'^—bx-\-G a. ax^ —hx^ ■\-ax — <L 
 
 
 az- 
 
 -h 
 
 
 mx^—nx+p to' mx^ —nx^ + >}tx — q 
 
 
 mx- 
 
 -n 
 
 65. 
 
 i-a; 1 aj . 1 . 
 i+a; •" 4 ^+a; 4 ' 
 
 %^-i 2 2 fx+l . 
 l-a; 3 3 "^ a;-|' 
 
 
 
 
 66. 
 
 21 71 21 71 
 
 X— 98 a;- 94 ~ a;+44 ~ a;-52' 
 
 
 
 
 67. 
 
 7 3 9 1 
 
 
 - 
 
 
 x-6 + a;-ll ~ aj-7 ^ x-12' 
 
 
 
 9 9 2 2 
 
 
 
 
 x-51 x-16 ~ x-Ql a;+81 
 
 5.4' 8 1 
 
 x- 
 
 68. T^ze + ^ = ^ + ^3io' 
 
 70. 
 
 1 
 
 aj-6 
 
 + 
 
 8 
 x-3 
 
 B 
 
 X 
 
 5 
 
 -2 
 
 + i- 
 
 4 
 -5' 
 
 
 m — n 
 
 - 
 
 a — 
 
 X — 
 
 h 
 
 m 
 
 = 
 
 m - 
 
 -n 
 
 a- 
 
 X- 
 
 -b 
 
 x — a 
 
 a;- 
 
 -b 
 
 ■n 
 
 a-\-h 
 
 
 a + c 
 
 
 
 
 b + d 
 
 
 
 69. 
 
 x — a X — m x — o x — n 
 
 x—b x—c x—{a + b+2c + d) x — {a-\-2b+c-\'fi 
 
 71. (x-a + b)^ -{x-a)^+{x-b)^ -x^ +a^ -(a-b)^ -b^ 
 = {a-b)c^. 
 
 72. (x+a+by - (a+by - (x+by ~{x-\-ay +x'^ -\- a" -hbii 
 = 10ahx{2x-\-a + b){x+a+b). 
 
 (m-n){x-a) {n-p){x-b) {p-m)(x- c) 
 o+c c-f-a a'f6 
 
SIMPLE EQUATIONS. 165 
 
 ax—1 bx — 1 cx~l Sx 
 
 74. 7Tnrrh\ + h2(^si7,\ "t" 
 
 a^{c+b) ^ h^{c+a) ^ c^{a-\-'h) ah + bc + ca 
 
 x—2a x—2b x—2c 
 
 75. ,— -f r H -, - =3. 
 
 b-\-c — a c+a — b a + b — c 
 
 X— 2a X — 2b x — 2c dx 
 
 b+c-a c-jra—b a-{-b—c a-^b-\-c 
 
 a — x b — x c — x 6 
 
 77. r,-^- + T-, + 
 
 a^ — be b^ — ' c c^ — ab a+b + c 
 
 x-\-2ab 2aO-x x-2ab x->r2ab 
 
 no '_ I ^ 1 
 
 ' a +b -\-c b -{- c — a a — b-\-c a-{- b — c 
 
 a b a — c b+ c 
 
 x-{-b—(i x-\-a — c x-\-b x+a 
 
 m^(a-b) n^(h-c) , pHc-d) 
 
 80. — ^ + — ^ + -— + 
 
 x—m x—n z—p 
 
 q {pd+{n—p)c-{-{m — n)b— ma} 
 
 x — q ~ 
 
 81. 
 
 (a;-2)(a;-5)(a;-6)(a;-9) + (a+2)(a-4)(a-5)(a-ll) 
 
 X 
 
 + 
 
 (,+l)(6 + 5)(64-8)(6-fl2) ^ (,_43(,_7)(,_ii)_, 
 
 X 
 
 (a2-l)(a-8)(a-10) + (&+2)(6+3)(6+10)(&4-ll) 
 
 X 
 
 Art, XLV. Employing the language of algebra, the princi- 
 ple illustvated in Art. XL. may be stated as follows : 
 
 Definition. — Any quantity which substituted for x makes the 
 expression f(x) vanish, is said to be a root of the equation f{x) = Q. 
 Thus, if a is a root of the equation /(a;) = 0, then /(a) = 0. 
 
 By Th. I., if x — a is a factor of the pohjTwme f{xY, then 
 /(a)" = 0, and a must be a root of the equation f{xY = ; hence in 
 solving the equation we are merely finding a value, or values, of 
 X whicli will make the corresponding polynome vanish, ^u-^. 
 pose/(a;)" = (a;— a)i?>(a;)"~^ = 0, we are re(j[uired to iind a value, or 
 
166 SIMPLE EQUATIONS. 
 
 values, of x which will make (x — a)(^{xY'~'^ vanish. The poly- 
 noma will certainly vanish if one of its factors vanishes, whether 
 the other does or not, and will not vanish unless at least one of 
 its factors vanishes. Hence {x— a)(p{x)"~^ will vanish if a;- a = 0, 
 quite irrespective of the value of (p(ic)"~^ Also, if (p{x}''~^ = 0, the 
 polynome will vanish, irrespective of the value of x — a. It fol- 
 lows, therefore, that if /{x)" can be resolved into two or more 
 factors, each of these factors equated to zero will give one or more 
 roots of the equation /(x)" = 0. 
 
 When there can be found two or mor?; values of x wliich satisfy 
 the conditions of given equations, they are sometimes distin- 
 guished thus : Xy, iCg, x^, &c., to be read " one value of a;," "a 
 second value of x," " a third value of x,'' &c. Thus, if 
 (x — a){x — b){x — c)=^0, , 
 
 Examples. 
 
 1. Solve 2a;3-13a:2-f27a;-18 = 0. 
 
 Factoring, 
 
 (a; - 2) (a; -3) (2a;- 3) =0, 
 
 2. x"-(a + b)x+{a+G)b = (a-\-c)c, 
 ... x^~{a + b)x+{a + c)(b-c)==0, 
 
 :. x^-{(a+c)-{-{b-c)}x + ia+c){b-c) = 0, 
 ... {x-{a-\-c)}{x-(b-c)}=0, 
 .". Xi =(t+c, x^ = b—c. 
 
 3. x-{a-b)-{-a^{b-x) + b^{x-a) = 0. 
 .-. x^{a-b)-x{a^-b^)-\-ab{a-b) = 0, 
 .-. {x-a){x-b){a-b) = 0. 
 
 li a — b = Q, the given equation holds irrespective of the values 
 oix — a and x — b, and therefore of the values of x ; but ii a~b is 
 not zero, x^=a, x^ = b. 
 
z = 
 
 SIMPLE EQUATIONS. 
 
 a;-l ~ b^-{x^l) ■'■ U-i/ ^- 
 
 aJi + l a ^ a+f) 
 
 -^ — - _ -— = .'. X, = r» 
 
 Kj-l 6 ' «-« 
 
 + -— = .'.2:2 = 
 
 167 
 
 iCg + l , ^ _ n - ^^~^ 
 
 ( a-xy + jb-x)^ 34 
 
 {^^^+{a-x){b-x)+{b-x)^ - 49* 
 
 (a-x)^+2{a-x){b-o^ + {h-x) ^ _ 2(49) -34 ^ 
 {a-x)^-2{a-x){b-x) + {b-x)^ ~ 3(34)-2(49) 
 
 ((a-a;)-(6-a;)j " ' 
 
 (a-a;i) + (6-a;i) 
 
 a — b 
 
 -4 = 0, .-. a^i = ^(5&-3a); 
 
 (a-a,,) + ib-x^ + i = 0, ...X, = ^(5a-86). 
 a — ■* ■" 
 
 (a;-«)(x-6) (y-5)(a;-r ) _ 
 "*• (c_a)>-Z>) ^ (a_6)(«-c) " ^• 
 
 Subtract term by term from the identity (See page 53), 
 
 {x-a){x-b) { x-b){x-c ) {x-c){x-a) ^^ 
 
 [c-a){c-b) "^ {a-b){a-c) "^ {b-c){b-a) 
 :. {x — c){x—a) = Q, :. x^—c,x^=a. 
 7. Find tbe ra/Kma?. roots of x* -12x-3+51a;2-90a:+56 = 0. 
 Factoring the left-band member by the method of Art. xxviii., 
 (a;-2)(a;-4)(x2-6rc + 7)=0 
 .-. 0^1=2, 2:2=4, or a;2-6x-}-7 = 0. 
 
 Since x^ — Qx-\-l cannot be resolved into rational factors we 
 know that it will not give rational roots, .". a;^ = 2, a;, = 4 are the 
 only vftluefl that meet the condition of the problem. 
 
168 SIMPLE EQUATIONS. 
 
 Any literal equation of the second, third, or fourth degree, and 
 many equations of the higher degree can be resolved tnto a series 
 of disjunctive equations. A full analysis for the first four degrees 
 will be given in Part II., meanwhile the following special forms 
 of the Theorem in Art. XLV., will enable the student to solve 
 nearly all the equations commonly proposed. 
 
 (A). In order that two expressions having a common factor 
 may be equal, it is necessary either that the common fiictor 
 should vanish, or else that the product of the remaining factors of 
 one of -the expressions should be equal to the product of the 
 remaining factors of the other expression, and it is sufficient if 
 one of these conditions be fulfilled. In symbols this is 
 
 li {x-a)f(x) = {x-a)(p{x), :. Xi=aoYf{x)=(p(x). 
 [B). If an equation reduces to the form (^nx+n)^ =c* 
 (mx-\-n)^ —c^=0, 
 
 ()"a;j+w)— c = and .'. x^ 
 
 or {mx2-\-n) + c = and .*. x^ ~ 
 (C). If an equation reduces to the form 
 
 ni 
 
 — c — n 
 m 
 
 mx-hn] * a^ 
 
 ■px + q) &3 
 
 qa — nb —qa — nb 
 
 tlienx,= ^-^— . X, = -^:,:^- (See Exs. 4 and 5 above). 
 
 (Z>). If an equation appears under the form 
 
 (a — x){x — h) = c, (1) 
 
 then x-i = l{a + b + r), aJs = K^+^-»')» 
 in which r^ = (a — 6) 2 — 4c. 
 
 From the ideu tity {a — x)-i-{x-b)=a — b 
 
 vfe get (a-x)^ +2{a-x){x-b) + {x-by = (a-b)^ (2) 
 
 (2) -4(1) .-. (a-x)^ -2{a-x)(x-b) + (x-h)^ 
 
 = (a-i)2-4c = r2 say 
 .-. {[a-x)-{x-b)}^-r^=0, 
 :. {{a—xy)-(xi-b)}+r = 0, anxd .: Xi=^{a + b + r); 
 or {{a-x^)-(xr,~h)}~r = 0, &nd .-. Xg = l{a+b-)). 
 
SIMPLE EQUATIONS. 
 fill 1 ^ 
 
 169 
 
 x — a x — a 
 
 • • -I ^^ 
 
 1 aa; 
 
 .•. ic — « = 0, or aa;= 1, 
 _ 1 
 
 Applying (.4), 
 
 rt 
 
 9. {:c-^i-\-mx^h^c) = {x-^a^h){^x-?>a^1h-c)\ 
 
 3a + c 
 
 ^[x,~a-\-h) _ x-a-hb 
 '' ^x'^^+V ~ 3a+c ' 
 .♦. {A) x^=a-h 
 
 Page 122. (5). 
 
 ^'^- ic3-2a;~ h' " m{x+2)^ ^n{x^ -'Ix) mn-\"nh ^> 
 But (C) can be applied if vi and ?i are so determiued that 
 m(2;+2)2+7t(a;2 — 2ie) is a square. 
 
 This requires that ^m{m-Jrn) = (2m - n)^, 
 
 Assume ?m = 1, then ?i=8, and (1) becomes, on substitution and 
 seduction, 
 
 fa + 2)2 _ -J" 2 
 
 (3:c-2)3 - a + 86-'" ' '""y 
 
 2(14->-) 2(r-l) 
 
 3r-l ' '*'2~'l + 3/-' 
 
 (a; + 1)4 _ a_ (a;2+2a;-(-l)g _ _« 
 
 ^^- (a;3+l)(a;-l)2 " T" •*• (a;--^+l)(a;3 -2a;+l) " 6* 
 For X- -j-l write a^z 
 
 "*• .c^(:r»-2a;) ~ 6 *** 2(z - 2) 6 
 
 .". a?! — ._, , X.J— -I 
 
170 SIMPLE EQUATIONS. 
 
 This '»quation was solved in Ex. 10, hence z may be treated as 
 known. 
 
 i5ut -^- =z, :. ^^^^ = ^32' 
 
 [x-\-\\ P 2-1-2 
 
 UITil ~ '• — 9? ^ formed solved in (C). 
 
 12. {a-xY^{h-xY^c. 
 In the identity 
 
 Letw = o— a;, v = x — b, .•. n+r = a. - & andii^+v* =c, 
 .-. (a-i)4 = c + 4(a_^,)2(„_a;)(a:_6)-2(a-.r)2(a;-/;)2 
 
 Write 2 for (a—x)(x - h) 
 
 .'. z2_2(a-6)2z+(a-/;)4 = i{c+(a-6)4}=f2, say, 
 
 .-. {2-(a-6)2}3=t2 
 
 .*. by (Z>) 2, =(rt — 6)2— i; Z2=(a— &)' + «, ',*. gis known; 
 
 But (a—x){x -h)-z 
 
 .-. by(Z>) a;i=i(a + i+r);£C2 = l(«+6-r) (1). 
 
 in whicli r^ = [a — b)^- 4z, 
 
 = (a-Z/)2-4{(a.-i)2-«}=4«-3(a-fc)2 
 or («-i)2-4{(a-ft)2+£} = -4i-8(a-6)2 
 and^2 = i{c + (a-/))4}. (3) 
 
 Hence x is expressed in terms of a, b, and r, 
 
 r is expressed in terms of a, b, and t, 
 
 t is expressed in terms of a, b, and c, 
 4nd the expressions for r and t are cases of {B). 
 
 13. (a-jc)(6+a;)4 + («-a;)4(i+3;)=fl&(rt.3+/,3) 
 
 Let a— a; = ?? -2 and i+a; = w4-2 .". n = ^ (a + b) (1). 
 
 The equation reduces to 
 
 (n2 _22)r(„4g)3_^(TO_2)3} =ab{a^-\-b») ' 
 
 :. {n^ -z^)(2n^-[-Gnz^) = ah(a^ +b^) 
 
 (2) 
 
SIMPLE EQUATIONS. 171 
 
 «2 may now be found by {D), and from the resvilt z may be 
 found by (/>'), and from (1) x=-i^{a-h) + z ; 
 
 3z2 = 3(rt _ ly or i(iO«6 - ^<- - h^) 
 .'. x = 0, or a-b, or -^(«-i)+i\/(30a&-3«2 _ 3^3). 
 
 14. {Via-{-x)+ y(a-x)mV{a-\-x)+ ^{a-x}^^cx. 
 
 Divide the terms of tbe identity 
 
 \/{a+x)'^ - y(«-K)* = 2x 
 by the corresi^onding terms of the equation, 
 
 •■• \ (a-u:/ - c-r •• a-"- a; " \c-l/ ' 
 
 (C4-I)4_(c__l)4 
 
 ' ••• *• - «-(,,4:i)~4 + (c_l)4- 
 
 15. ir(.^-x)2 + ir{(*-^)(^-^)} + l^('^--^)' = ^(«^+"^+^') 
 Divide the terms of the identity 
 
 f/{a~x)^--^{b-xy-=a-b 
 by the corresponding terms of the equation. 
 
 .-. ^(.-.)-r('-)= ^(,,^|-+-,.)- 
 
 Cube, using the form (u—v)^ = u^ -v^—3itv{u-v). 
 
 (»-.)-(6-.)- 8#'U«-K'-)[ • ^(^+,,) 
 
 .-. -^{(a_^)i6-a.-)} = 
 
 .-. (a-x){b—x) = 
 a form solved m (D). 
 
 a6 
 
 a363 
 
 (rt«+flZ;-i-62)3 
 
 Assume -i/(" - x) = 2 y (as - 6) 
 
 a — b 
 ,'. x- b = glTpi 
 
1 72 SIMPLE EQUATIONS. 
 
 The proposed equation now becomes 
 
 ^/(x-b){z+lr' 
 
 2-1 
 
 (^_6)(2-l)4 
 
 l/« 
 
 = c. 
 
 
 a form solved in Ex. 11. 
 
 17. {x-2){x-5){x-Q)(:x-d) + (u+2){y-4:){ij-5){y-n) + 
 
 {z+l){z + 5){z-^S){z+i2)=x{z~4.){x-l){x-ll) + 
 (2/+l)(y-l)(z/-8)O/-lO) + (z+2j(z-|-3)(z+lO)(0 + ll). 
 
 Let x' = x^-llx, y' = y'^ -% and a'^^^+lSz, 
 .-. (a;'4-18)(u;' + 30j+(//'-22)0/'+20)+(z'+12)(2'+40) = 
 
 a;'(a;'+2B) + (y'-10)(/y' + S)+(2' + 22)(^'+30) 
 .-. a;'3+48a;'+510+?/'2-27/'— 4'i0+z'2 4.52<;'+480 = 
 
 ic'2 + 28a;' +2/'2-22/'- 80+2'2+522'+660, 
 
 .«. 20x' = 0, .-. a;2-lla; = 0, /. a;i=0,a;3 = ll. 
 
 Exercise Ivi. 
 What can you deduce from the following statements ? 
 1. A'B = 0. 2. A-B-C==0. 3. (a-i):t; = 0. 4. 12a;]/ = 0. 
 
 5. What is the difference between the equation 
 
 {x-5y){x-iy-\-3) = 
 and the simultaneous equations 
 
 x — 5y = and a; — 4^+3 = 0. 
 
 What values of x will satisfy the following equations ? 
 
 6. x{x-a) = 0. 7. ax(x+h) = 0. 8. (a;-a)(&a;-c) = 0. 
 9. ax^-=Sr,x. 10. a;2 = (a+6)a;. 11. a;(a;2 -a3) = 0. 
 12. a^x^^b^x. 13. x-2+(a-a;)2=a2, 
 
 14. ;i;3 4.(«_.^)2 = (a_2.r)2. 15. (a-x)3 + (^ -6)3 =a2a./,2 
 
 16. (rt— ic)(3:-&) + a& = 0. 
 
 17. {a-xy - (a - x){x - h) + {x - hf = «= + o6 + 6^ 
 
 18. x^-{a-'b)x^nb = Q. 
 
 19. x3-(rt+6+c)a:2 . ah->rhc+ca)x—abc ^0. 
 
SIMPLE EQUATIONS. 178 
 
 If X must be positive, wiiat value or values of z will satisfy the 
 iollowing equations ? 
 
 20. (.'e-5)(;f+4) = 0. 21. xS-H 29a; -30 = 0. 
 
 22. x--17x-84: = 0. 23. 3a;3 +10x+3 = 0. 
 
 i4. :<;4_i3^3_f_36 = 0. 25. x^ -^x^ - 5x+6 = 0. 
 
 ■m 
 
 Solve the following equations : 
 
 26. (a-a;)2+(x-6)2 = («_A)3. 
 
 27. (a-a;)3-(a-a;)(x-6)4-(a;-/))2=(a-5)2, 
 
 28. «2(«_^j3=i2(6_a;)2. 29. a^{b~xy = b^{a^xy. 
 
 30. (a;-a)3 + (a-&)3 + (i-x)3=:0. 31. (a;-l)- =«(.x-3 -1). 
 09 (i — x x — a „£. a-j-6 — a; a—c-{-x 
 x — li c-\-x a — c — x ~ a-\-c—x 
 
 84. (a;-a+fe)(a:-a+'') = (a-6)3_a;2. 
 
 85. (a;-a)2-62 4-(rt4.6_a;)(6+c-a;) = 0. 
 
 36. (a+fc+c)a;3-(2^-l-5 + t>+« = 0. 
 
 Q_ a + 6 — x a-\-h — G 
 01. ^ . 
 
 C X 
 
 38. (a - x) 3 + (a - &)3 = (a+J - 2:.)*. 
 
 39. a;(a+&-!B)+(a+6 + c)c=0. 
 
 40. (n—p)x^+(p—7n)x+m — n = 0. 
 
 .-. ax^ —bx-\-c c .0 ax^ — hx+e __ a — h-^e 
 
 v}x^ —nx+p p mx^ —nx-\-p m—n-\-p 
 
 43. 4a;-^+a2-i3_2(a + ft)a; = (rt-x){6+a;)-(rt+x)(fc-x). 
 
 44. (2a-t_a;)3+9(a-&)3 = (a+i-2a;)3. 
 
 45. (2rt+2c-x)2 = (26+£c)(3r(- J-f3c-2a;). 
 
 46. (3a-5&H-a;)(5a-3&-a;) = (7a-6-3a;)3. 
 
 47. {%a-b+x)(9a-\-h-x) = (5a+3i - 3ic)3. 
 
 48. a{a-b)-h{a-c)x-\-c{b-c)x^ =Q. 
 
174 SIMPLE EQUATIONS. 
 
 -r{a — c}^x. 
 
 50. {■'• + l){x + S){x-i)(x-l) + {x-l)(x-3){x-lr4:){x + l) = 9G. 
 
 51. (j--l)i.': + 3){x-^)[x+'J, + [^^:j{x-3){x+o)'^x-^Q}± IS 
 = 0. 
 
 52. ^-1- — =: 8i. 53. ^ + _ ^ _J]_ _^ 
 
 ■'■ ic « — o a-\-b 
 
 KA '^ ^ ^ .„ C' + x b+x ^, 
 54. X— — = — - — 55. -, + =2},. 
 
 X 
 
 1 — tyc. , , -r- 
 
 o a 0-\-x a + x 
 
 m 
 
 KG ^~^ x—b 13 a-x b + x in. n 
 
 Ob, . + = — -. 57. - — — = — 
 
 x~ a—x () b-^x a — x n 
 
 « a; m x^+nx+a'^ 
 
 5b. — + — = — 59. -, , = c. 
 
 X a n X' —ax +a^ 
 
 {a-xY^+ix-by- 5 a~x ^ x-b m 
 
 64. 
 65. 
 66. 
 67. 
 
 {a-x)(x — b) 2 ■ a; — 6 a— iC jj 
 
 (a; + a)2-(a;-i)2 " 2«6 ' 
 (a-a:)3-(a;-Z*)2 4.ab 
 
 {a~x){x-b) {a^-b'') 
 
 (a-x)^ + {a-x )(x--b) + (x-b )^ _ 49 
 {a~x)^-{a-x){x-b)i-(x-b)2 ~ lO' 
 
 2a2 + r/(a-a;) + (a + a ^)2 r+l 
 
 2a^+a{a+x) + {a-xr' " c^l* (^sofo^^ = 5)- 
 
 68. (5-x)^+{2-xy = 17. 
 
 69. a;44.(«_,j.)4=c. a;4_{.(_j._4)4 ^ 82. 
 
 70. {a-xy + {x-b)^ = (a-b)4'. 71. (rt-a:)5^-(a;-5)5=r, 
 
 72. af^+(a-r)»=rtS; a;' -L(C-a-)'^ = 1056. 
 
 73. (a-»)'(x-i) = + (<f-a;)-(x-6)-'=«-//2(„_/,). 
 
SIJIPLE EQUATIONS, .175 
 
 74. {a-x){h+x)'^-\-(a-x)^{b+x)^ + {a-z)-{b+xy^-^ 
 {a-x)^{h+x) = {a + b)c. 
 
 (a-x)^Jx-b)^ _ 41 
 
 _ (a-xV+(x-hY _ 211 
 
 ' . (a-a;)4-t-(a;-i)4 " 97 ('^~'')' 
 
 77. 
 78. 
 79. 
 80. 
 81. 
 82. 
 
 85. 
 86. 
 
 (a-x)^ + ix-b)^ _ a4 + ft4 
 
 (a-a;)2+(x-/;)3 ~ a^ + b^' 
 
 (a—x)^(x-^)i _ rt4 + J4 
 
 (a— a;)54-(aJ— &)^ a^ — 6 
 
 (a -a;) 3 (Z»-a;)3 
 
 
 <7 — re a;— ^ 
 
 a h 
 
 (a;- 6)--^ + (a-x)^ 
 
 ~ b2 a^ 
 
 (^y-.r)4 + (a:-t)4 rt4+54 
 
 (a+6-2a;)2 ~ (^f+fe)^ 
 
 g.j {a-x)^ + {x-b)^ a^-b* 
 
 (a+T-2a;)2 = (a+l^a' 
 
 84 (^i:^'+(x-i)-5 
 
 (a-a-)-(a;-Z)) («-a;)(x— 6) 
 
 (a-x)^+(x-b)^ 
 
 }^ i — - ■ L = c(a—x)(x—b). 
 
 {a-x)^+{x~b)^ ^ '^ ' 
 
 (a-a:)4 + (a;-6)4 " (a-;c)(a;-6)' 
 88. ^l+a;2)3 = (a;3_3)2. 
 
176 SIMPLE EQUATIONS. 
 
 89 ^* + ^ ^ ± 90 (a ; + l)-(a;=^ + 1) a_ 
 
 ix-l)^x a (a;3+a;+l)2 
 
 a 
 
 f7 
 
 93 ('^'+^)' ^ iL 94 (^+1)' _ ± 
 a;(a;+l)3 6 ' ' x{x'+l) ~ b ' 
 
 96. ^(•'^+^)^ ^ 96 a;- + .r + 1 icH-^-1 
 
 (a;-l)4 -ft* ' ■ j^x+1)-' ■ (x-ip" - T 
 
 97 ^IZj^m _ JL 98 ^J^ii+ll _ « 
 
 99 (^ + 1)(^' + 1 ) ^ 100 (^+l)fa ''^-l) _ _^ 
 (a;-l)(a;3-l) * 6* ' (x-l)(a;'' + l) " 6' 
 
 101. (^+11^ = JL 10^ ( x+iy ^ a_ 
 
 x^ + l &■■ "■ x'^ + 1 h' 
 
 103. 2(«.-.7-)4-9(a-a;)3(3;-/;) + 14(a-a;)2(x-6)3--' 
 9(a-a;)(a;-6)3+2(a;-/>)4 = 0. 
 
 104. 4(a-a;)4-17(a-a;)-(a;-6)2+4(x-/;)4 = 0. 
 Find the rational roots in the following equations : 
 
 105. x4-12a;3+49.r2-78x+40 = 0. [Let 2 = a;2 - Oa;] . 
 
 106. x^-Qx^+lx^+Qx-Q. 
 
 107. 2;4_i0x3+35a;2-50a;+24 = 0. 
 
 108. 32a;4-48a;3_i0a;2+21a;+5 = 0. 
 
 109. a;3 - 6x2 +5a;+ 12 = 0. 
 
 5 4 9 4 
 
 — -j_ — _ 
 
 X x — a x — ^2a x— 3a ' x—4a 
 
 14 5 4 14 5 4 
 
 a;+20 ^ a;+5 ar-4 x-s>5 ^ x-40 'x-2{', 
 
 110. 
 111. 
 
 j-|o 2a; + 5a a;+8a 
 
 iC a; — a 
 
 x — a a;+5« 2* — 5a 
 
 + 
 
 + 
 
 9 — 3a x-4.a x—C 
 
ilMPLE EQUATIONS. 177 
 
 ii„ x+i x+2 ic4-4 'x + 3 z-1 x-3 
 
 lis. — _1_ 1— _|_ _J_ a- ' — ^_ < 
 
 x + 2 X x — 1 x~2 x—'d x — 5 
 
 ... 1 31 20 8 20 31 
 
 114. — _ _j_ 4- -^ _L _ — — J- 
 
 X x-l ^ a;-2 ^ x-3 ^ x-4: x-5 ^ 
 ' =0. 
 
 X— 6 
 
 ^^g ^( a^+2x) + i/{a2-2x) _ 
 ■^/{a^-i-2x)-^{a^-2x) ~ 
 
 m^ V{ m^x-\-2) + y/{m^x-2) 
 a2 • 7/(m~'x'+2)--/(w3a;-2) 
 
 117. a/(.t2 -a2) + ^(ajs _i2) + ^(a;2 -c-S) =.r. 
 
 118. {\/{a-x)-\-\/{h-^)]{y\a-x)-W{lj-Ji)\^fi* 
 119 ■^(« — a;) — 1^(0; — 6) _ a+h-'lx 
 
 f{a-x)-\-f/{:x-b) " ~^u^-b 
 
 120. V(a+a;)4-V(«-a;)=y(2a). 
 
 [Write u luJ.' |^\a -x>, and v iov i^(.c — 6)], 
 
178 
 
 SIMULTANEOUS EQUATIONS. 
 
 CHAPTER VI. 
 
 Simultaneous Equations. 
 
 Art. XLVI, There are three general methocls of resolving 
 simultaneous linear equations, 1° by substitution, 2° by compar- 
 ison, 3° by elimination. The last is often subdivided into the 
 method by cross-multipliers, and the method by arbitrary multd- 
 pliers. 
 
 In api^lying the elimination -method the work should be done 
 with detached coefficients, each equation should be numbered, 
 and a register of the operations performed should be kept. 
 
 Ex. Resolve 
 
 u+i+x+y-\-z = 15. 
 u+2o + 4:X+8y + iez = 57. 
 u-\-Sv+dx+ 27?/+81z= 179. 
 u+4:v + IGx-f G-1//+2563 = 45S . 
 M -f 5 <.■ + 2 5x-+ 1 25.// -f G252 = 97 5 . 
 
 Register 
 
 (2)-(l). 
 
 (3) -(2). 
 
 (4) -(3). 
 
 (5) -(4). 
 
 :7)-(0). 
 
 (B)-(7). 
 
 (9) -(8). 
 
 (11) -(10). 
 
 (12)-.(11). 
 
 (14)-(13). 
 
 (15)-- 24. 
 
 ii(13)-G0(W)}. 
 
 A[(10)-{12(17) + 50(16)}]. 
 
 (6)--{3(18)+7(17) + 15(16)}. 
 
 (l)_|(19) + (18) + (17)-i-(16)! 
 
 u 
 
 V 
 
 X 
 
 y 
 
 z 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 = 
 
 = 15 (1) 
 
 i 
 
 •2 
 
 4 
 
 8 
 
 16 
 
 57 (2) 
 
 1 
 
 3 
 
 9 
 
 27 
 
 81 
 
 179 (3) 
 
 «. 
 
 •i 
 
 16 
 
 64 
 
 256 
 
 453 (4) 
 
 1 
 
 5 
 
 25 
 
 125 
 
 625 
 
 975 (5) 
 
 
 1 
 
 3 
 
 7 
 
 15 
 
 42 (6) 
 
 
 1 
 
 .5 
 
 -19 
 
 65 
 
 122 (7) 
 
 
 1 
 
 7 
 
 37 
 
 175 
 
 274 (8) 
 
 
 1 
 
 •J 
 
 61 
 
 369 
 
 522 (9) 
 
 
 
 2 
 
 12 
 
 50 
 
 80 (10) 
 
 
 
 2 
 
 18 
 
 110 
 
 152 (11) 
 
 
 
 2 
 
 24 
 
 194 
 
 248 (12) 
 
 
 
 
 (5 
 
 60 
 
 72 (13) 
 
 
 
 
 6 
 
 84 
 
 24 
 
 I 
 
 96 (14) 
 
 24 (15) 
 
 1 (16) 
 
 
 
 
 1 
 
 
 2 (17) 
 
 
 
 1 
 
 
 
 8 (18) 
 
 J 
 
 1 
 
 
 
 
 4 (19) 
 6 (20) 
 
SIMULTANEOUS KQUATIOXS. 170 
 
 An eXcamination of the Register will sliow how easy it ■would 
 i;ave been to shorten 'the process, thus (lO)'is (7) — (6) which is 
 (:)) + (l)-2(2); similaily (11) is (4)-i-(2)-2(3) ; .-. (18) is (4)-^- 
 H(2)-3(3)-(l), &c. 
 
 A ojeneral systematic arranfrement of the ehmination-metliod 
 will be given in Part II. For two or three simultaneous equa- 
 tions it may be stated as follows. 
 
 r/2.'K-fA._,//-f (-2 =0. 
 Arrange the coefficients thus — 
 ^1 ^1 ^1 ^i 
 
 «2 '^2 '^2 "s* 
 
 Form their products diagonally from left to riglit downwards, 
 thus — a^feg ^'i^2 '*i'^2' 
 
 Form their products diagonally from right to left downwards, 
 thus — ^if<2 ^J'2 "^i^a- 
 
 Subtract the latter products in order from the former, thus — 
 a^b^ — b^a^, b^c^ — c^b^, c^a^—OyC^. 
 
 Divide the 2° and 3'^ remainders by the 1° remainder, the first 
 quotient will be the value of w, the second quotient will be the 
 value of y. > 
 
 [Writing i?^, B.^, 7/, for the three 'remainders' respectively, 
 
 the general result is [iiix + v7j)li^ = niE^ +nR^] . 
 
 Ex. 1. Solve lla;+57/-68 = 
 
 6a;-7?/+31 = 
 11 5 -08 11 
 
 6 -7 31 6 
 
 -77 
 30 
 
 155 
 47G 
 
 -408 
 341 
 
 -1U7) 
 
 -321 
 
 -749 
 
 
 3 
 
 ! 
 
 X 
 
 7 
 v. 
 
1^0 SIMULTANEOUS EQUATIOKfc 
 
 Ex.2. 12 
 
 X 
 
 25 
 
 y 
 
 = 1. 
 
 22 
 
 X 
 
 ^80 
 
 y 
 
 = 17. 
 
 12 -25 
 
 - 1 12 
 
 22 30 - 
 
 -17 22 
 
 360 
 
 425 
 
 - 22 
 
 -550 
 
 -30 
 
 -204 
 
 910) 
 
 455 
 
 182 
 
 
 1 
 
 1 
 
 
 2 
 
 6 
 
 
 II 
 
 H 
 
 
 1 
 
 i_ 
 
 
 X 
 
 y 
 
 .'. x = 2 
 
 and y = 
 
 -.5. 
 
 2° Let the equations be 
 
 a^x+h^if + CjZ+d^ =<> 
 
 Arrange the coefficients thus 
 
 «1 
 
 6i 
 
 "i 
 
 -<^I 
 
 -«i 
 
 -*I 
 
 «2 
 
 ^2 
 
 ^'2 
 
 -t/.. 
 
 -S 
 
 ~6, 
 
 «3 
 
 h 
 
 H 
 
 -dz 
 
 -^3 
 
 -\ 
 
 «I 
 
 K 
 
 <^i 
 
 -d, 
 
 -«1 
 
 -*, 
 
 «2 
 
 \ 
 
 ^2 
 
 -d. 
 
 -«2 
 
 -^2 
 
 Selecting the first three cohimns form the diagonal product** 
 from left to right downwards, thus : 
 
SIMULTANEOUS EQUATIONS. 181 
 
 «i ^1 Ci giving a^b^c^ 
 
 \ 
 \ 
 
 ao ^2 Cj a2*3''l 
 
 \ \ 
 
 «3 ^3 '■$ ^S^l'^fl 
 
 \ \ 
 
 <*1 ^1 ''l 
 
 ^2 • ^'2 ^2 
 
 Form the diagonal products from right to left downwards, thus: 
 «i ^i c, giying c^h^al 
 
 «3 ^3 ''2 
 
 / / 
 
 Cg&ga, 
 
 a, ^3 c. 
 
 Cjjftja, 
 
 a, b-i Cj 
 
 / 
 
 «2 ''2 '^2 
 
 From the sum of the former products take the sum of the latter 
 products obtaining a remainder, which call R^. 
 
 Similarly form a 2° remainder, Eg from the 2°, 3° and 4° columns 
 a 3° " i?3 " 3°, 4° and 5° 
 
 a 4°» « i?4 " 4°, 5° and 6" " 
 
 Then a;=7?3 4-Z?i, y^R^-i-Ri, z=R^—R^, 
 and generally * 
 
 (wa;-{-wt/+;5z)Ei =mR^-\-nR^-^pR^» 
 
 Ex. 8. 3a:+2y- 42+20 = 
 5x-ly-Qz- 1 = 
 7a;+5y + 5i!-24 = a. 
 
:82 
 
 SIMULTANEOCS EQUATIOIiS. 
 
 a 
 
 2 -4 -20 - 
 
 • > 
 
 — ^ 
 
 5 
 
 -7 -G 1 - 
 
 5 
 
 7 
 
 7 
 
 5 5 24 - 
 
 7 
 
 -5 
 
 3 
 
 2 -4 -2!) - 
 
 3 
 
 -2 ' 
 
 5 • 
 
 -7 -G 1 - 
 
 5 
 
 7 
 
 1 
 
 -105 
 
 -288 28 -500 
 
 
 (3x -7x5= -^105 
 
 -100 
 
 700 432 14 
 
 
 5 x5x -4=-- -100 
 
 - 84 
 
 - 20 500 -504 
 
 
 7x2x -6=- 84, &c) 
 
 -289 
 
 392 9G0 -990 
 
 
 19G 
 
 GOO - 15 240 
 
 (-4x-7x7 = 19G 
 
 -90 
 
 10 480 980 
 
 
 -6x5x3 = -90 
 
 50 
 
 G72 -840 15 
 
 
 5x2x6 = 50, &c.) 
 
 156 
 
 1282 -875 1235 
 
 
 -445) 
 
 -890 1335-2225 
 
 
 X y t 
 
 Exercise Ivii. 
 Solve the followiug systems of equati ns : 
 
 1. 2x+Btj = 4:l 
 Sx+2y^S9 
 
 3. 11a; + 12?/ = 100 
 9a;+87/ = 80. 
 
 5. nx+ly = 7 
 
 5a; + 3?/ = - 36. 
 
 7. 5a:+%+2 = 
 3.^;+2?/ + l=0 
 
 9. 10a;+77/ 4-4 = 
 .Gx + 5// + 2 = 0. 
 
 11. ^x+hj = ^' 
 
 3a;-4y = 4. 
 
 2. 5a; + 7?/=17 
 lx-5y= 9. 
 
 4. 18x-?j5y+ld = 
 15a; +28?/ -275 = 0. 
 
 G. 3a;+lG/i-5 = 
 28// = 5a;+19. 
 
 8. 21a; + 82/+66 = 
 23?/ -28a; + 13 = 0. 
 
 10. 23a;+15?/-4i = 
 32.r+21?/-6 = 0. 
 
 12. lx-ly = l. 
 
SIJIUI^TANEOUS EQUATIONS. 183 
 
 13. ly = lx-l. 14. §x+§y = n. 
 
 15. l-5x-2y = l. 16. lx=10y + -l. 
 
 2-5x-dy = G. Ux=lGy + -l. 
 
 17. 5x-iy-{-l = 0. 18. ■lGx--Oi>j = l. 
 
 l-7x-2-2y+l-d = 0. •ldx--lly = l. 
 
 19. 3-5.c + 2^v/ = 13+Ua;-3-57/. 
 2i*-+ -8^ = 221+ -Tu; - 3|!/. 
 
 20. 1 + JL = A. 
 
 ic y 6 
 
 Jl J_ _ J_ 
 
 a; 2/ 6 
 
 22. i:? = ?:! _ 1. 
 
 « y 
 
 •8 3-6 
 
 — + — = 5. 
 
 X y 
 
 24. |, A.,,. . 
 
 IT + y - '^^• 
 
 20. ix-\{:y+\.) = \. 
 k{x+l)+l{^y-l)^'d. 
 
 1 o 
 
 28. 
 
 3x-+l 5// + 4 
 1 2 
 
 4a;- 3 ~ 7//- 6 
 
 80. l^^^±i == 8. 
 
 45-1/ 
 
 21. 
 
 ^ + '^ = 3. 
 
 X y 
 
 
 15 4 
 
 — — — = 4. 
 X y 
 
 23. 
 
 •3 
 17a; - — = 3. 
 
 y 
 
 
 •4 
 IGa; - — = 2. 
 
 2/ 
 
 25. 
 
 ';= + ■' = 6. 
 
 
 lOa; 9 
 
 - - -j = 31 
 
 
 5 7 
 
 27. 
 
 .^+2^/ ~ 2a;+2/ 
 
 
 7 ■ 5 
 
 
 3a;-2 ~ G-y' 
 
 2'J. 
 
 x+^y _ g 
 
 
 7a;-13 
 3//- 5 
 
 
 31. 
 
 3a;+l 4 
 4-2y - 3* 
 
 
 {B+|/=l. 
 
= 4. 
 
 J 84 SIMULTANEOUS EQUATIONS. 
 
 32. 1--^ = 1. 33 ^^±^1±} _ 2 
 
 5-3?/ 2 2a;-?/ + l ~ 
 
 7-22/ _ _2 8.r-y+l 
 
 5 -3a; ~ 3* «-!/ + 3 ^ ^' 
 
 3^ a;+3y+13 x+1 y+2 2(a;-y) 
 *" • •4X+-52/-2-5 = ^^' ^^- ~3 r = ~5^ 
 
 •8a;+ajH--6 _ J_ ^-3 y-3 
 
 6a;H-%-23 ~ T' 4 "" 3 = ^^ ~ '■ 
 
 36 ^^-y+^ a;-2y+ 3 
 3 4^~ 
 
 3a;- 47/+ 3 4a;-2?/-9 
 
 i — + — y— = ^• 
 
 87. 20(.c+l) = 150/ + l) = 12(a;+7/). 
 
 38. (a;-2) : (y+l) : (a;+.v-3) :: 3 : 4 : 5. 
 
 39. (a;-5) : (y/+9) : (a;+^ + 4) :: 1 : 2 : 3. 
 
 ^^- ^ = ^- ^1- (^-4)(.'/ + 7) = (^-3)(i/+4). 
 
 X+i 2/r" 
 
 ^) = 5W (.+6)(,-2) = (.+2)fa-l). 
 
 42. (a;-l)(5y-3) = 3(3x+l). 43. (a;+l)(2// + l) = 5a; + 92/ + l. 
 (a;-l)(42/ + 3) = 3(7a;-l). (a;+2)(3//+l) = 9a; + 13y + 2. 
 
 44. (3a;-2)(5i/ + l) = (5ar-l)(2/+2). 
 (3a;-l)(s^+5) = (a;+5)(7y-l). 
 
 45. a; + 7/ = 37. 46. 2a;+2z/ = 7. 
 
 ?/+z = 25. 7a; +192 = 29. 
 
 z+a; = 22. i/+8«=17. 
 
 47. l-3a;-l-9</ = l. 48. 5x-+3i/+2z = 217. 
 
 17//-Mz = 2. 5a;-3s^ = 39. 
 
 2-9z-2-lx = 3. 3?/ -22= 20. 
 
 49. \x-\xj = ^. 50, l^a;+l|?/ = 10. 
 
 ^a;-:|z=l. 2§x+2|0 = 2O. 
 
 ^«-jy = 2. 3i-s/+3iz = 30. 
 
SIMULTANEOUS EQUATIONS. i^" 
 
 //+z-a;=l3. x+2y + iz=li. 
 
 z+x-y= 7. a;+3/y+y2 = ^B. 
 
 53. ^.a.^^^^= 3. 54. 7x+6?/ + 7i = 100. 
 
 x-y 
 
 = 0. 
 
 2^-+4?/-f- F3=13. :c-2(/+ z = 
 
 3a;+%4- 272 = ;U. ?ix+ y ~2z = G, 
 
 Sr,. 3j5+2!/ + 82-110, 60. .c + y+z = d. 
 
 5x-\- ?/-42 = 0. a; + 2y + 3c-14. 
 
 2x-3y+ z = 0. a:+3.v + Gz = '20. 
 
 57. aj+2// + 82 = 32. 58. x + y + 2z = %4t. 
 2^ + 3y-(-z = -42. «+2.v+z =33. 
 
 3x+ ?/ + 22=10. 2a; + ^i2^82. 
 
 59. 3.C+3//+ z = 17. 60. c+'ly-z= 4-6.. 
 
 3a;+ y4-Cz=15. y + -2z~x=l(yi. 
 
 x+Sy-\-Bz = lS. z + 1x-y= 5-7. 
 
 61. x4-2i/--73 = 21. 62. x+^^l^z + S. 
 
 3a; + -27/- z = 24. </+z = 2f/y-U. 
 
 •9;c + 7^- 2^ = 27. z +a; = 3|.,--82, 
 
 63. ix+il/ + |3 = 30^. 64. %x + ?>\y^A\z=\m 
 
 ix + iv + 12 = 27. 3^x + 4iy/ + .^^z = ? 75. 
 
 1:^ + ^2/ + ]2 = 18. 2§.c + 3|^+l?2=iuV. 
 
 65. -+i = 2. 66. -^_p^- = ^ 
 
 z+3 _ , gg + ^ ^. 9 
 
 :^ - ^' 2/ + 1 " ' 
 
 67. -_±^ = 10. 68. ^^ = 2. 
 
 a + z 
 
186 
 
 SIMULTANEOUS EQUATIONS. 
 
 69. i - A ^ 3. y 70. — + :l + :l = i 
 
 v y 
 
 2 '6 
 
 X z 
 
 y g 
 
 = 4. _ + — + — = 4. 
 
 X y z 
 
 ^ 12 10 
 
 V. 0. '- + = ^r 
 
 71. ^- . 4- 72. 
 
 J/2 _ 1 
 j/+z ~ "6" 
 
 za; 1 
 
 2+a; ~ 7 ' 
 
 73. (.r+2)(2?/-i-l) = (2a;+7)//. 
 
 (a: - 2)f32 -^ I) = (x+3)(3z - 1). 
 (y + l)(2+c =(2/+3)( z + 1). 
 
 74. (2a;-l)(2/+i;=2(a;+l)(v/-l). 
 (a,+4)(z+l) =(a:+2)(0 + 2). 
 (2/-2){z+3) .-=(y-l)(2 + l). 
 
 75. (rc-ui)(%-3) = (7.t-+l)(2//-3). 
 (4a;-l)(z+l) = (a;+l)(2z-l). 
 (i/+3)(2+2) =(3^-^)(3z-l). 
 
 76. 21a;+31?/+42z=n5. 
 6(2aj+i/) = 3(3a;+2) = 2(^+2) 
 
 77. 15(a;-22/) = 5(2a:-3;?) = 3(?/+2). 
 21a;4-31^ + dlz = lSrj. 
 
 78. 6a;(i/+2) = 4?/(z + x) =^ 32(x ^ vj. 
 1 1 1 
 
 79. 
 
 6 
 
 4 
 
 y 
 
 6 
 
 + — ■ 
 z 
 
 — + 
 
 X 
 
 8 
 
 5 
 
 ft/ 
 
 '- + 
 
 X 
 
 12 
 
 10 
 
 y 
 
 z 
 
 xy 
 
 iy — 3x 
 
 = 
 
 20. 
 
 xz 
 2x - Si 
 
 .-= 
 
 15. 
 
 yz 
 4»/ - oz 
 
 = 
 
 la. 
 
 a; 2/ ^ 
 
 = 9. 
 
 
 ^ 
 
 3aj +7/4-2 = 20. 
 3«+a; + 4?/ = 30. 
 
 3?f. + 6.c-fz = 40. 
 
 6u-f8;/-f3^ = 50 
 
 
 60. 
 
 a-+;'-f S// = 33. 
 
 5» f ?/-f-2 = ii, 
 
 i^z+A-f 2 = 11. 
 
 3?. ^ a'+^ = J.l. 
 
ST:\ItTLTANEOUS EQTTATIOXS. 187 
 
 81. a-{-x + i, + z = lU. 82. ■u + .r-\-jj-^z = ^4:. 
 It +2j;-{-2ij + 2z=-2,67. ti + 2x+ 3i/ - L'z :- 0. 
 
 n + 2x+Bu + 3z—SrA). Su-x-5ij-^z = J. 
 
 7( + 2x-^3i/+iz = 110. 2u+3x-4:y - 5z = 0. 
 
 38. u-{-x + y+z = GO. 84. :i-{-x-\' ,/-\-z = l. 
 
 u-\-2x+3ij+-iz = 100. 2u-\-4x ^ 8// + IQz = G. 
 
 u+Sx+G>j-\-10z=150. 32i + 9x+27.(/ + 8l2 = lo. 
 
 ?t + 4.«;+10// + 202=210. 4?H-16x-f 04// + 25Gz = 35. 
 
 85. ^x+y-U = l. 85. ht-^x-\-iy-:^.^Vi. 
 
 lx-^P-lu = l. ^u-Sx+l-,j-l^^l7. 
 
 f.'/- 12-^(6 = 0. pi^}^x-y + i?=n. 
 
 Art. XLVII. Tlie principle of symmetry is often of use in 
 the solution of symmetrical equations. For from one relation 
 which may he found to exist between two or more of the letters 
 involved, other relations may he derived by symmetry ; also, 
 when the vahie of one of the unknown quantities has been deter- 
 "lained, the values of tlie others can be at once written down, &o, 
 
 1 • (x4- y) {x+z)=a. 
 
 {x+y){y+z) = /^- 
 (x+z){y+z)=c. 
 
 Multiply the equations together and extract the square root. 
 
 •■• {x+y){y+z){z+x) = i/{ahc). 
 Divide this equation by the third. 
 
 .'. x + y = }— i— — i-, and therefore, by symmetry. 
 
 .*. J/+Z = 
 
 .". z-\-x = 
 Hence we get 
 
 X = 
 
 c 
 
 1 
 
 a 
 
 V[ahc) 
 
 nh — bc-\-ca 
 
 2Viabc) 
 wlienee y and z may be derived by symmetry. 
 
188 SIMULTANEOUS EQUATIONS. 
 
 2. x.+v-^-z-O (1). 
 
 vx + by + cz = (2). 
 
 br.r+caij-^ahz-{-{a-b){b-c){c-a) = 0: (3). 
 
 cx(l)-(2) gives {c-a)x+{c-b)y = 0. 
 
 .'. y = ^ L, and similarly, 
 
 b — c 
 
 _ (a-b)x 
 
 — r • 
 
 b — c 
 Substitute in (3) these values of y and z, and reduce, 
 
 ,-. x{a — b){c —a) = (a — b)(b — c){c — a), 
 
 .'. or x = {h — c), .'. y = c--a, z = a — h. 
 8. a{yz—zx—xy) = b{zx — xy — yz) = c{xy — yz — zx) =xyz. 
 Divide tlie first and the last equations by axyz ; 
 
 .-. — — — _ — _ — , and hence, by symmetry, 
 axyz 
 
 J- _ i- _ -1 _ i- 
 
 b ~ y Z X 
 
 1 _ JL _ 1 _ 1, 
 
 c z X y 
 
 ,♦. _)_ — = — —, and by symmetry, 
 
 be X 
 
 L i_ - _ A 
 
 c a y 
 
 !_ J_ _ _ A. 
 
 a b z ■ 
 i. ax -Y by -\- cz = \ (1 ) 
 
 «2.f+A3^+r32 = l (2). 
 
 a^x + b^y-\-c^z = l (3). 
 
 <;x(l)-(2) gives a{r-a)x+b{c -h)y = c-\ (4). 
 
 cx(2)-(3) " a^-{c-a)x + b^{c-b)y = c-l (5). 
 
 4x(4)-(5) " ah{c.-a)x-a^{c-U)x = h{c-l)-{c~l), 
 ox a[a — b){a — c)x=-{G-l){b—l), 
 
 ... X = (1-^Hi-c) . 
 
 a[a — b)[a — c) 
 wlienae y and z may be derived by symmetry. 
 
SIMULTA?^SO'JS EQUATIONS. 189 
 
 6. Eliminate a;, y, z, u (wLich are supposed all diflferent) trom 
 the foUowiug equations : 
 
 x=^hy-\-cz-\-du. 
 y-cz + du+ax. 
 e = du-\-az+by. 
 u = az-\-hy+cz. 
 
 Subtracting the second equation from the first» 
 .•. x — y = hij—ax, or 
 (1 + a)x = (1 + h)y = (by symmetry) (1 + '^)z = (l+dju. 
 
 These relations may be &lso obtained by adding ax to both 
 members of the first equation, by, to both members of the second 
 squation, &c. 
 
 Now divide the first equation by these equals, 
 
 1 b__ c_ _d_ 
 
 " \ + a ~ i + 6 "•" r+c "*■ \-ird 
 
 And since = 1 — we have 
 
 1+a 1+a 
 
 ^ _ a b c a 
 
 Exercise Iviii. 
 
 b-c-bc' 
 
 1 . (jriven ax -\- bi/ = c and that x = ,— t~i* 
 
 '' b a -ba' 
 
 a'x+h 'y = c' derive the value of jr 
 
 a(dm— en) 
 
 2. Given bx = ay and that x = — ] t — » 
 
 "^ be — ad 
 
 dx + ""? = cy-\-n c derive the value of y. 
 
 8. Given ax-{-bi/-^cz -d. and that a; = 
 
 a(d-b)(d-c) 
 a^x-{-h^ij-\-c^zz=d- "-T pr/ \. write down 
 
 <3iSx-\-b^y -\- e^z =d^ the values of y and z 
 
190 
 
 SIMULTANKOUS EQUATIONS. 
 
 4, There i? a set. of equations in x, y, z, u, and w, -witli corres- 
 coucling coefficients {a to x, &c.), a, b, c, d, and e ; one of the 
 equations is 
 
 x = by + rz-\-du-\-ew, write down the others. 
 Solve the following equations : 
 
 K ^ , y V z ^ X 2 
 
 m n n p m p 
 
 6. x-\-a)/ + J>z = 7n, y + az ^hx = n, z + ax-\-hij =p. 
 
 7. x + rtv = /, y + bz = )n, z + cu = ii, u-\-dw=p, w + ex = r. 
 
 8. Eliminate x, y, z, (supposed to be all different) from the 
 following equations : 
 
 z=by-\-cz, y = cz + ax, z = ax-\-liy. 
 
 9. Eliminate x, y, z, from 
 
 X y z 
 
 = a, — . — = i), = c. 
 
 y-]-z ' z-\-x * x+y 
 
 10. Having given 
 
 x — hii-\-rz-{-dv.-\-ruj, 
 y = cz + dn-\-en--\-((x, 
 Z =dit + (ii-\-ax + by. 
 u = ew ■]-ax-]'by + cz, 
 w = ax-^by-^cz-\-dii, 
 
 abode 
 
 Shew that ,-7— + ip- —7 + tT' + i~r~) + rT~ = ^^ 
 1+a 1 + 1+c l+(t 1+^ 
 
 Art. XLVIII. Eesolution of Particular Systems of Linear 
 Equations. 
 
 Ex. 1. • x+y-\-z = a (1) 
 
 y+z-^u = b (2) 
 
 2-fH + . r = c (3) 
 
 ii-{-x+y = d ^ (4) 
 
 (l) + (2) + (3) + (4) B(u + x-\-y-^^)=r> + h+r+d (5'] 
 
 8(1) 3(a;+7/+z) = 3« (C) 
 
 |t{(5')--(C')} tt =i(-2rt4-AV + ''-) 
 

 SIMULTANEOUS EQUATIONS. 191 
 
 The values of x, y and z may now be written down by sym- 
 metry. 
 
 Tiie following is a variation of the above method, applicable to 
 a much more general system. 
 
 A-Ssume the auxiliary equation 
 
 u-\-x-\-y-{-z = s, (5) 
 
 .-. (1) becomes s — a-.a, (6) 
 
 (2) " s-x=^b, (7) 
 
 (3) " s-ij = c, (8) 
 
 (4) '• . s-z = d, (9) 
 (5) + (6) + ^7) + (8) + (9) 4.s = s + a + h^-c+d. 
 
 :. s = \{a ->!■■}> -\-c-^d), 
 
 s is now a known quantity, and may be treated as such, 
 in (6) giving u = s — a 
 
 (7) " x = s-b 
 
 (8) " y = s~c 
 " (9) " z = s-d. 
 
 Ex. 2. yz = a{y-\-z), (1) 
 
 z.c = b{z+:^, (2; 
 
 xy^c{x-\-y), (3) 
 
 111 
 (l)^ayz, ^ -\ = — , 
 
 y . z a 
 
 /o^ / 111 
 
 Z X o 
 
 111 
 
 {Sy-i-cxy, 1 = — . 
 
 X y G 
 
 This may now be solved like Ex. 1, using the reciprocals of a 
 
 b, c, X, y and z instead of these quantities themselves. 
 
 Ex.3. ai«+ii(a;+7/+2) = r, (1) 
 
 a„x-\-b2{y+z + u) = c^ (2) 
 
 '^3.'/ + ^3(2+ i + 'i^) =^^3 (3) 
 
 a4Z + b^{u-^x+y) = c^ (4) 
 
 Assume the auxiliary equation 
 
 u-tx+y+z = s. (5) 
 
V\;iJ SIMULTANKOUS EQUATIONS. 
 
 \i; becomes b^^s—ib^— a ^)it = c^ 
 
 by ' r, 
 
 «i — «! Z>i - a, \ / 
 
 Similarly from (2) , ^ s -x = j — ^ (7) 
 
 2 "2 2 ~ '2 
 
 ^ ' ^B-'^3 ^ h-^3 ■ ^ ^ 
 
 From (10) we can at once get the value of s, which may there- 
 fore be treated as a known, quantity. 
 
 bjS — Cy 
 in (6) giving u=, _ 
 
 "\ "i 
 
 'and the value of a;, y, and z may be obtained from (7), (8) and 
 (9), or they n'va^y be written down by symmetry. 
 
 Ex. 4. ax-\-b(}j-^z) = c ' (1) 
 
 ay+-h{z + n) = d (2) 
 
 az-\-hl(ii-\-x) = e (8) 
 
 au-irb{xJf-y)^f (4) 
 
 Assume tt+.T + i/ + z = s (5) 
 
 (l) + (2) + (3)-f(4) (a + 26)s =c + ^ + r+/ (6) 
 
 Hence s is a known quantity and may be treated as such. 
 
 From (1) and (5) hs—bu-\-{a — h)x = c, 
 
 :. bi( ~(a - b)x=bs—c, (7) 
 
 Similarly from (2) and (5) bx -{a — h)y = bs—d, (8) 
 
 " (8) " " by-{a-b]z^bs-e, (9) 
 
 *• (4) " " lz~{a-b)n^bs-f, (10) 
 b{7) -{-{a- b){8) i" a - (a - b) -y = abs -bz- {a - b)d,[l 1 ) 
 6(9) +(^i - b){\i)) b-^y - (a - h)"- a = ab>i - hi,^ {a-Oy, (12) 
 
 
SIML'LTANEnuS EQUATIONS. 
 
 1'15 
 
 -«(/>2./ + ("-6)3/-}-6{63(c-fZ) + (a-6)2(.-/)} (13) 
 
 The values of x, y, aud z may now be written down by sym- 
 metry. 
 
 Ex. 5. a^ + a^.v-\-ai/+z = 0, 
 
 b^ + b^x + by-\-z = 0, 
 
 The polynomc t^ + .<;;- -{-yt+z vanishes for i = a, t = h, t = e, 
 .'. by Th. II., p. 4G, for all values of t. 
 
 t^ +.rJ2 +yt+z --= {t-a){t- h)(t - r\ 
 = t^-{a + h-\-c)t-+{ah + hc + ca)t-ahc. 
 ,-. Th. III., p. 53, x= -{a-\-h-\-c), 
 
 y = ah-\-bc-^ca, 
 2= —ahc. 
 
 Ex. 6. a;+?/-|-2 + ?/ = ], 
 
 ax + /^^ -\-fZ-\- du = 0, 
 
 Employing the motliod of arbitrary multipliers, 
 
 (4) + /(3) + »»(2)+?i(l) a^x+h^y+ c^ 
 
 + ma + ?.'fi + VIC 
 -\-n I 4- 71 I -{■ n 
 
 To determine x assume 
 
 63-f /Z/2-l-H?7> + n = 0, 
 
 + d^ 
 + ld'^ 
 +md 
 + n 
 
 u = n 
 
 (2) 
 (3) 
 (^) 
 
 (5) 
 
 iC = 
 
 (7) 
 
 {^) 
 
 (9) 
 
 But the system (6), (7), (8) has been solved in Ex. 5, from 
 which it is seen that 
 
 1= —{h-\-c-\-d), m = hc + cd + db, n= —bed, 
 and a^+a~l-]-ain + n = {a -b)(a-c){a- d) i 
 
'i;H SIMULTANEOUS EQUATIONS. 
 
 •. using these values in (9) 
 
 -hcd 
 
 ( a — h){^a— c){<:i — d) 
 
 The values of ?/, z and u may now be written down by sym- 
 metry. 
 
 Ex. 7. ^^ + _L ^ _^ = 1. (1) 
 
 VI — a m — m — c 
 
 - +^J'+^^^l, (2) 
 
 n — o n — h n~c 
 
 -f- — ^V + — = 1- C'^) 
 
 p — a p—b ]j — c 
 
 Assume 1 - ^_- _ . L' _ _^_ _ t^+Bt^~-\-Ct+D 
 t — a t-h i-c ' (t-a){t~b){t-c) 
 
 But in virtue of equations (1), (2) and (3), the first member of 
 (4) vanishes for t = m, t = n, and t=p, and .-, t^ + Bt- + Ct + D 
 ♦janishes for the same vahies of t, and .•. Th. II. p. 46, 
 
 ti-\.Bt^ + Ct + D = {t-m){t-n)(t-p), 
 
 .'. (4) becomes 1 — 
 
 X y z 
 
 t — a t — h t — c 
 
 _ (t-m){t~H){t-p) 
 '" {t-a){t-b){t-c) ' 
 
 To obtain the value of x multii^ly both sides of this equation 
 hj{t-a), 
 
 t-a-x- y^^~^) _ z{t-a) ^ {t-m){t-n)(t-p) 
 t-b t-c ~ (t-b}{t~c) 
 
 Now t may have any value in this equation ; let « = a. 
 
 (a — ni)(a — n){a — p) 
 ~ {a—b){a — c) 
 
 The substitution (xyz\abc) will give the values of y and t, 
 
 Ex. 8. ^^t^ = ?^ = ^±i . (1) 
 
 p q r 
 
 lx + 7Hi/4-nz = s^ (2) 
 
simulxa! 
 
 By Art. XXXVII., 
 
 x+a y + b z-i-c 
 
 ??5 
 
 v 
 
 (2) 
 
 q r 
 
 s^-{-la + mb-\-nc 
 
 Ix-\-))iy-{-fiz-{-ia-{-miy-\-nc 
 lp-\'-viq-^nr 
 
 = II, say 
 
 lj)-\-niq-\-nr 
 ,'. x=pR-a, y = qR — b, z = rR—c. 
 
 Ex.9. 
 
 yz+zi«+xy = (a-fi +Cjxyz 
 
 
 (1) 
 
 
 yz +zx ZX+ xy xy + yz 
 a "^ b ~ G 
 
 
 (2) 
 
 {l)-xy» 
 
 11,1 
 
 — + + - a+b + c. 
 X y z 
 
 
 (3) 
 
 {2)^xyz. 
 
 11111 
 
 — + — — + — — + 
 
 X y ^ y z _ z 
 
 y 
 
 X 
 
 (4) 
 
 
 a b c 
 
 
 
 Page 122 and 
 
 (3) " 4- 2 + 2 
 
 - X y 2 _ 
 
 a-\-b-\-c 
 
 2 
 
 c^) 
 
 (4) and (5) 
 
 11 11 
 
 .-. h — = 2rt, — + — = 
 
 X y y z 
 
 20, 
 
 
 
 1 1 
 
 — + — = 2c. 
 
 Z X 
 
 
 (6) 
 
 (3) -(6) 
 
 111 
 
 — a — b + c, —a-\-b—c, — 
 X y z 
 
 — a 
 
 + h+c. 
 
 Ex. 10. 
 
 aJ + c , y+b 
 a+b a-\-G 
 
 
 W 
 
 
 "-'' + ^'-f - 2. 
 «— c a—b 
 
 
 (2) 
 
 (1) 
 
 x-\-c y + b 
 a-\-b a + c 
 
 
 
 
 x—a—b+c a+c — b—y 
 a+b ~ a+G 
 
 
 (3) 
 
1C6 SIMULTANEOUS EQUATIONS. 
 
 Similarly from (2) 5Z/^1±£ ^ a-h + c-y_ ^^^ 
 
 a—c u—b 
 
 (3) and (4) ... x-a-h-\rc = '^-(a-h-\-c-y) 
 
 a-\-G 
 
 But unless ^- — — ^ ~^., this cannot be the case except for 
 a + c a — b 
 
 a-b+c-y = 0, 
 
 in which case z—a — b-{-c = also, 
 
 giving x = a + b — c a.Tady = a — b+c, (5) 
 
 Ti; a + b a~c 
 
 If ---— =: r- .-. a3-£3 = ,,2_,.2 (6) 
 
 a.+-c a — b ^ ' 
 
 63-c2=0, or (6 + c)(5-6-)=..0, 
 
 & = c, or b— —c. 
 But if &= +c or — e, (1) and (2) are one and the same equation ; 
 hence if (1) and (2) are independent, (6) cannot be true, thus 
 leaving only the alternative (5). 
 
 Ex.11, "lax^ib + c-aYii-^z), (1) 
 
 2by = {c + a-b){z^x), ■ (2) 
 
 (a;+?/ + z)-'+a;2 + ?/-'+23=4(a2+62 + f2) (3) 
 
 (1) and page 122 (5)^_ = Ip- = ^+l±f (4) 
 
 b + c — a 2a o+c+a 
 
 (0) u u y _ x+z _ x + !j-\-z .gv 
 
 c + a-b 2b ~ c + a+b ' ^ ' 
 
 ^4), (5)and " ... ^+?/+^ =. ^ = ^ = __J__ 
 « + /' + (^ 6+c — « c+a-6 rt-|-6-c* 
 
 «3 (a;+?/+z)2+x2+?/2_(_22 
 
 Reduction and (3) = 4(^ 24.^,2^^2) = ^* 
 
SIMULTANEOUS EQUATIONS. 
 
 1&7 
 
 1 1 1 ... 
 
 Ex. 12. . ax =by = cz= ~ + — ^- ^ \^) 
 
 .a h c _ ff+6+ c 
 
 {l)^xyz :. - = ^ = :^ • • - ^H^+---» ^^ 
 
 a 1 /I 1 1\ a;(/ + ?/z + z.c ' 
 
 (2)x(3) • 4\- = ^i^ • 
 
 :. a^x^ = a+b-\-G. 
 
 Ex.13. 
 
 a ~ b ' 
 
 c 
 
 (n 
 
 
 xyz = m^ 
 
 
 (?) 
 
 (1) 
 
 z X y 
 
 a+ b ~ b+c ~ c + a' ~ 
 
 m 
 
 suppose 
 
 (3) 
 
 then 
 
 xyz m^ 
 
 
 
 {a-\-b){b + c){c + a) rs 
 
 
 .-. r^ = (a + b){b + c){c+a) 
 
 Hence the value of r is known and from (8) 
 
 rx — m{b-\-c). 
 
 Ex. U. y+z = 2axijz (1) 
 
 2+u; = 26.<;^z (2) 
 
 a; + ^ = 2rjc2/« ,: (3) 
 
 y-\-z z+x x-\-y x+y + z 
 
 •*• ^y^^ ~2^ = ~2r "^ ~2r "^ ^+y+o 
 
 a; 7/ z 
 
 b + c — a c-{-a~-b a-{-b — G 
 
 xyz 
 :. x^yH^ = (^f)^c-a){c + a-l)){a+b-c) 
 
 1 
 
 .-. x^yH^ = ^^r^ a) (c -I- a- b){a + 6 - c)/ 
 
 (-t> 
 
t9B 8IMULT4NE03S EQUATIONS. 
 
 Hence the value of x^y~z^ is known, call it —^ and substituto 
 
 in (4) • 
 
 1 X 
 
 r ~ b -{- G —a 
 .-. rx = b + c—a, 
 in which r^ = [b-{-c — a)(c-^a — h)[a + b — c). 
 
 Ex.15. y^+z^-x{2j + z) = a (1) 
 
 z^-+x^-y{z + x) = b _ . (2) 
 
 a;3+2/3-2(.x-+^) = t (3) 
 
 (l) + (2)+(3) 2{x-+y^+z^-xy-yz-zx) = atb-\-c (4) 
 
 (1) may be written , x'-^ +y- +2- —x{x+y-]-z) = a (6) 
 
 (2) " " x^ + y"+z-^-y{x + y-\-z) = b (6) 
 
 (3) " " x^+y"+z^-z{x+y+z):^c - (7) 
 
 .•. x + y+z ■= 
 
 a — b b — c c — a 
 
 y — x z — y x — z 
 
 \y -x)- ^{z -yy + {x-z)^ 
 
 a'i^h^j^c^-ah-bc-ca 
 x^+y'^-{-z^ -xy -yz — zx 
 
 2(a3+&3_f.c3_3aJc) 
 Write r2 for 2(a3 + &3+c3 _3^,;,c^^ 
 
 (9) 
 
 (9) ....+,+. = __L__ (10) 
 
 Eeturning to (8) {x-^y^zY = ^^"^ + ^^t+r+t "'^"''^ <^) 
 
 (4) 2(x-^ +r"+2"-^i/-2/2-2-0 = ^^^±^ (11) 
 
BlMUL'rANEOUS EQUATIONS. 1'^ 
 
 i[(8) + (n)} x^'-^^r-^z-^ = ^l±}l±^ (12) 
 
 (5) ami (10) x-+7/-^z"^- — ^_ = a 
 
 a + b-\-G 
 
 (12) ^a^j^h^-j^c"_a{a + h-\-c) 
 
 -h^'+c^ -a{h+c). 
 
 (5), (6), (7) are symmetrical -with respect to (.-ryzjfiroc); (10) sliowg 
 this substitution does not affect r, and consequently the values 
 of 2/ and z may be written down at once from that of «» 
 
 
 
 
 Exercise '. 
 
 lix. 
 
 1. 
 
 az+h]/ = c, 
 mx+ny = d. 
 
 
 
 2. 
 
 (iz~\-hy = c, 
 mx — ny = d. 
 
 3. 
 
 ax + by = c, 
 mx-\-ny = c. 
 
 
 
 4. 
 
 -2- + ■' = 1. 
 
 a b 
 x+y = G. 
 
 5. 
 
 a '^ b 
 
 1, 
 
 
 6. 
 
 a ' b - ^-^ 
 
 
 X V 
 b ^ a 
 
 1, 
 
 
 
 X y 
 b a 
 
 7. 
 
 ax+bc = by-{- 
 x + y =c. 
 
 ac. 
 
 
 8. 
 
 a h 
 
 b a 
 
 — + -=«. 
 X y 
 
 9. 
 
 {n-^c)x-{a- 
 (« + %-(a- 
 
 ■b)x = 
 
 ■- 2ab, 
 = 2ac. 
 
 10. 
 
 x—c a 
 y-c ~ b' 
 
 x — y = a — b. 
 
 ll" 
 
 X a 
 
 - /■' 
 
 y '' 
 
 x+m c 
 
 1 
 
 
 12. 
 
 x+y a+h-{-c, 
 7/-|-l ' a — b+c' 
 y—1 a — b^c 
 
 y + n d x+l a+b — c 
 
200 SIMULTANEOUS EQUATIONS. 
 
 y-a+h c ' a+b ^ a+c ' 
 
 y-^b c + a x — b y-c 
 
 " " = 7 — • 4- — 2 
 
 x-\-c h-\-a a — c a — b ' 
 
 vi-a '^ m-b ~ ' (b-^c)x-\-{a-\--)//~\-{u-{-b)z 
 
 X y ^ =0, 
 
 + - ^ = 1 
 
 « - (7 n-b ' hex-\- acy + ahz = 1 . 
 
 17, x-{-y+z = l, x-a y-b z-c 
 
 15. 
 
 -» 
 
 r 
 
 19. 
 
 ;s-a y-b z-c 
 P q ~ r ' 
 
 
 hc-{-my-\~vz = l. 
 
 21. 
 
 x-\-y-\-z = a-\-l>JrC, 
 
 
 hx\cy + nz = a^-\-b'^+c^. 
 
 
 cx-\-ay-\-hz = a^+b^ -f c2. 
 
 ax + hy + cz = m, ^^- 'p q 
 
 Jl_ s V_ , « , l{£-a)^m{l^-b)^-n{z -c) 
 
 I- a ■•" Z-i + TZTc " ■^- =1. 
 
 20. rf(a;-^)=%-/j) = r(2-c), 
 ax\by-\-cz~ in'^. 
 
 22. a;+?/+2 = 0, 
 
 ox + 6y + c« = rt/)+Sr f r/7, 
 (^; - c).« + (c - «)^ -f (a - b)z 
 = 0. 
 
 23. a;+^-}-2; = /n, 21. ax+by + cz = r, 
 
 X : y : z = a : b : c. inx= <ny, qy=pz 
 
 25. xy + yz^zx = 0, ayz-\-hzx^cxy = 0, 
 
 hcyz+a.cxz-\-ahxy ^{a -b){b — c) {c — a)xyz = 0. 
 
 26. (a + 6)x4-(?'+c)?/ + (c + a)z = a6 + 6c+ca, 
 
 {h-irc)x-\-{a + c)y-\-(a + b)z = a^+b^+e'». 
 
 27. *ox + ny+jiz + qu = r, 
 
 X y z u 
 
 a b c d 
 
SIMULTANEOUS EQUATIONS. JOl 
 
 a b c o ^b-c 
 
 1 1 1 
 
 — + — + — =■- a-\-h-\-c. 
 
 X y e 
 
 29. {<i-b){c-\-c)-a:j-{-hz-={c-a)[ij -{■})) -C2-faaj = 0, 
 
 x-\-ij +z = '\'< +^ + c). 
 
 30. a:'-+h// = l, 31. lf/+mx = n, 
 
 b// + cz = l, ' vx + lz = m, 
 
 cz + ax —1, viz-\- ny = L 
 
 82 x + y = a, 33. ^+2-r = '-^^ 
 
 , /)i 
 
 2/+2 = o, H-.C -y = — f 
 
 x-^-z-c. z-\-y~z = — , 
 
 n 
 
 84. 1 = 2a. 85. ^ — , 
 
 y Z y z X a 
 
 1 ^ o. 1112 
 
 « X z X y b 
 
 a; 2/^' ^ y z ~ c 
 
 86. (a + ?>).r4-(a -5)^ = 2ic, 
 (6 + c)7/+ (& — c)a; = 2ac, 
 (c4-n)z4-(c — «)?/ = 2a6. 
 
 87. a: + f - - = a. ^8. ^ + -^ = 6-.,, 
 
 /j c b-\-c c-\-a 
 
 Z ^ _ -L ^ JL ?- 
 
 T ~ T " ' T+7 "'' ~^b 
 
 y + — - — = ^y -TTT + TTTIT = ^-^* 
 
 XV y z , 
 
 a 
 
 b ' c+a a + 6 
 
'202 SIMTTLTANEOUS EQUATIONS. 
 
 39. a:+Z/ -2 = ", 40. jt-f-v-x = a, 
 
 y -^-z — v — h, v+ X — y ~ b, 
 
 z+v — x=c, x+>/-z = c, 
 
 v-\-x-y=d. y+x-u = d, 
 
 z + u — r = e. 
 
 Exercise Ix, 
 
 Eesolve 
 
 1. {a+b)x+(a-b)y = 2{a''-\rb^) 2. x + y = a, 
 
 {a-b)x + {a + b)y = 2{a^-b2) x- - y^ =b. 
 
 3. 2x — 3y = m, 4. {a — b)x-l-(a-\-b)y = a + h, 
 
 2x- — 3y^ = n^ -{-xy. x y \ 
 
 a + b a — b a-{-b 
 
 K V 7\ . a^-h + 1 {a-irb-c)x-{a-b^c)y 
 
 5. [a ~- b)x+y = — — rr ' "• i n \ 
 
 a — b-^1 X a-\-b—c 
 
 Z-\-ia-\-b)y= T~ ' — = T~7~' 
 
 ' V ' /:/ a — b y a — 0+0 
 
 x-\-ii a x — a a — b 
 
 7. — ^ =- j . 8. = — T- 
 
 x+c y-\-^ ^ a^~b^ 
 
 a + b ~ a-\;-0 y a^ + b^ 
 
 x-y+1 _ .Q ^+y_±l _ "+} 
 
 x — y—1 z—y + 1 a — 1 
 
 x+y+1 ^ ^ ' x+y + 1 _ 1 + h 
 
 x+y-1 ~ ' x — y-1 1 — b' 
 
 x+y—1 a+b a—b ' * 
 
 x+y+1 , ^ , ?/ o 
 
 x—y—i a 6 
 
 13. {a + c)x+{a-c)y = 2ah, 14. rt^+r/.r 4-7/ = 0, 
 
 (a+6)?/ — (a-fe)jc = 2ac. b'^ + bx + y = 0. 
 
* SIMULTANEOUS EQUATIONS. IZHB 
 
 15. y + Z — :f = rt. 10. 7:r-{-ll;/ + z = a, 
 
 j: + i/—z = g. 7z-{-11x + ij — c. 
 
 X 1/ z '^' ' • . 
 
 — < — _ IL ^2hc (c-a){>/-rb)-ci,+ax = 0, 
 
 X 
 
 y 
 
 c a b x+y^z = 2[a + h^c). 
 
 — + — — — = Aca. 
 
 z X tj 
 
 •0 , y - %->n ^ y ^ 
 
 19. T—- + -r—. =«+^ 20. — -- -4. ^ ^^0, 
 
 + c c —a o-\-c c — a a— b * 
 
 y ^__ _ / ^ y ^ 
 
 c-\-a a — b ' h — c c — a a'^b~ * 
 
 . Z X X If ^ 
 
 21. ^ + -J^ + -^ = 1, 22, ^ . „. 
 
 a a — 1 a — 'A x+y 
 
 T + 6-1 ■*■ &-2 - ■"' i/+s - *» 
 
 c ^ c-1 ^ c-2 ~ ^' z+x ~ "" 
 
 23. ± + f + i--+^ + A^ 
 
 rt c oca 
 
 a; ,y 2 1 1 1 
 
 04 ^ •'/ ^ 
 
 "" ' rt b c 
 
 u 
 
 d' 
 
 25. ax = bt/ = cz:r=ifi 
 
 mx + ny-]-pz-\-qu = r 
 
 
 7/3 — 23 _^„..jj. 
 
 26. ?/+z = att, 
 
 r 
 
 27. x+!/ = m, 
 
 x + z =4)U, 
 
 y-^z = n. 
 
 x+y = cn, 
 
 
 z+^i = a, 
 
 1-x a 
 1 — u ~ b 
 
 
 u — x=:b. 
 
•201 
 
 SIMULTAXEOUS EQUATIONS. • 
 
 28. Ux+9i/-\-z-u = a, 21J. x + ay + a^z + a^u-^a* = 0, 
 
 Uy + Vz+u-x = h, x-{-bi/-i-b"z-^L^u+ij'^ = 0, 
 
 11z+9u+x-y = c, a;+«/ + c22+6-3w + 6-4 = 0, 
 
 nu-{-9x+y-z^d. x+dy + d-zi-d^H + u^ = Q. 
 
 30. .x + y = a, 31. x + Iy = a, 
 
 v+z~b, i^^niz = b, 
 
 ii-\-i- = d, u^pi: = ,i^ 
 
 v-\-x = e. v+qx = e. 
 
 82., xf//+3 = a, 33." x- !i-\-z = a, 
 
 y + Z + H=:b, y^z-\-H=b, 
 
 u + V'\-x = d, u—v-{-x=d, 
 
 v-\-x-\-y = e. v—x + y = e. 
 
 34. x+y+z~u = a, 35. , x+y+z-u — v = a, 
 
 y+z + u-r = b, 'f^' y^z + u-v-x = b, 
 
 z + u + v — x = c, z-\-u + v-x-y = e, 
 
 u + v+x — y = d, u + i- + x-y~ z = d 
 
 v+x + y-z = e. v + xj-y-z-u = e. 
 
 86. 2x-y-z+2io v = 3a, 37. v~ 2:c + SH-2y + z = a, 
 
 2y-z- H + 2v -x= ob, x—-2y^Sv -'■22 + 11= I,, 
 
 2z ~ u — c -\-2x- y = 3c, y-2z + 3x-2u+v = c, 
 
 2u-v- x + 2y — z = 3d. z- 2u + oy — 2v+x = d. 
 
 2v-x-y+2z - u «= 3e. u—2c+'dz - 2x+y = e. 
 
 Exercise Ixi. 
 
 Eesolve the following systems of equations : 
 
 J l+x + x^ 
 
 " i+u + y' 
 
 1 + y+x^ _ x-+x-{-\ ,„,.c-l 
 
 i+*-+^^ ' y'-^y+^ \y-l 
 
 (\+x )a+y) _ l±a 
 (l-x)(l-y) " l-a 4. 
 
 x+1 
 
 ix-l\ 
 
 i/+l - 
 
 
 X-+X+1 
 
 
 i/'-i-y+i 
 
 :* b2 
 
 x + y 
 1+xy ~ 
 
 
■SIMULTANEOUS EQUATIONS. 205 
 
 (l+a;)(l- //) _ 1+6 x-y _ b^ - g- 
 
 a;+?/ a ,, x-\- y 2a 
 
 1+a;// ~ 6+6-' 1—^ ~" l — a^ 
 
 x—y b — c -^ ^—y 25 
 
 1— icy ~ a l + a;y ~ 1 - 6' 
 
 1 — a;// ~ a^ — a^' ^ + y l + -<^// "'- 
 
 a; - ?/ _ 2.''/3 1 - a;// x - // 2/^ 
 
 l+'V/ ~ 62_^3" ;j;_^ "f" 1_ ^y ^ ,t 
 
 9. :yU+-^') _ 10 Z/ + z = 2rta;;v2, 
 
 a;(l + ?/2j ~ » a;+z = 2ia;?/^, 
 
 i/(l-a;3) x + y = 2cxyz. 
 = 0, 
 
 11 
 
 41 -r') 
 
 ?/+5 — .1- z-{-x-y x+y-z 12. ax-hy = cz, 
 
 a ~ b ~ c ' I 1 i 
 
 = — 4- — 4- . 
 
 xyz = m^. ' X ' y z ' 
 
 13. y- +z'^ -x{y + z] = a, 
 
 x^+z^-y(x + z)^b, 
 x^-ty^-z{x-\-y) = c. 
 
 14 2ax = {b-\-G — u)(y-^z), 
 ^hy = {c + a — b){x-\-z), 
 {x+y-\-zy +x^+y3-^z^ = i{a2 +b^ +c^). 
 
 x^ +xy-\-y^ x^ -\- 11^ xii 
 
 f 
 15. 
 
 x—1 a — 1 
 1/-1 - 6-1' 
 
 
 a;3-l a3_ 1 
 
 
 j,3_i - 53_i- 
 
 17. 
 
 x^ \-x^y^+y^ = a, 
 
 
 x2+xy+y^'=b. 
 
 19. 
 
 xy + ^ - «(a;2 + j/2) 
 
 16. 
 
 x'^ — xy-\-y'^ a 
 
 IQ 
 
 '-• c^ + fr = 
 
 3 _ 
 
 a 
 
 "> 
 
 a;^y - r//2 = 
 
 x — y 
 
 ' cl 
 
 20. x^=a{x'^-iry^)-b.nj. 
 
206 SIMULTANEOUS EQUATIONS. 
 
 21. 4.c[x-^ + l) = {a + b){x-ii)2, 
 4.c{y^-l)^{a-b){x~y)\ 
 
 22. x^-ti^ = -^ {x'^xy + y^'){x-^y), 
 
 23- r^ = a, 24. ~^^ = a, 
 
 ll-ry xy ' 
 
 ic+2/2 - ^- xy~ = *• 
 
 25. % + 2) = a., 26. (x-+7/)(a;+0) = a, 
 
 «(a;+ //) = (% (z^x){z^y)=c. 
 
 "in. x{x+y+z) = a-yz, 28. a;^ - (3/- «)2 ^«^ 
 
 ^ V{x+y-rz) = b-,-:x, y2_^z-x)-=b, 
 
 z{x-}-y+z) = 6- - x^/. z'^~{x- y) 2 = c. 
 
 29. a;2+^2^,,2 1 1 2a 
 
 30. ^ + — = — . 
 X- y 22 
 
 a- + ?/==/^z, 1 1 26 
 
 ^3 0,2 
 
 -» 
 
 ai^* ?/^ z3 
 x-y = cz. l^ 1^1 
 
 X y ~ c' 
 
 32. xy = ^--i, 
 
 x+y = I'z, . (a; - y)(3 + 1 ) - 2a,. 
 
 x-y = cz, {x^-y-'jiz+iy-i^iht. 
 
EXAMINATION PAPERS. 207 
 
 CHAPTEE VII. 
 
 EXAITINATTON PaPERS : EdUC\TION DEPARTMENT AND UxiVSRSITT 
 
 OF Toronto. 
 
 L 
 
 1. State the rules for the addition and subtraction of Algebraic 
 riiiautities. Express in the simplest form. 
 
 (b+c — a)x+ {c-\-a. — b)y-{-{a-\-b - c)z 
 {c+a-h)x + {a + b-c)y-\-{b+c-a)z 
 (a-}-b — c)z-{-{b +c — a)y + {c+a — b)z 
 
 2. State and prove the Index Laws. Assuming these to be 
 general, interpret a;"™. 
 
 Find the products in the following cases : 
 
 (1) (x^ + 6x^y + 12xy^ +8y^){x= -6x'^y+12xy^ - 8yS). 
 
 (2) {a + b-\-c){b-\-c-a){c+a-b)(a + b-c). 
 
 3. Prove the rule of signs in Division. 
 Divide : [Apply Horner's Method to (1)] 
 
 (1) a;6_22x4+60a;3-55a;3 + 12a;+4 by x^-^Gx+l. 
 
 (2) a;4+9-l-81a;-4 by a;2-3-f9a;-2. (3) a;"' - 1 by a;* - 1, 
 
 4. Find the square roots of 
 
 4 ] 
 
 (1) 4a;4™-— a;^"* + — x""" 
 
 • o J 
 
 b^ c^ a^ c b a 
 
 5. Distinguish between an algebraic equation and an identity. 
 Solve 
 
 (1) 1^/(1 -2.1)4-1^(1 + 2a;) = 3. 
 
808 
 
 EXAMINATION PAPERS. 
 
 (2) '-^2 _^ .r^ ^ ^.+, 
 
 x+1 ' x-2 ~ x-3 
 
 6. A person bought a certain number of oxen for $320. If he 
 had been able to i^urchase four more for the same sum, each 
 would have cost him $4 less. Find the number |of oxen. Ex- 
 plain the negative result. 
 
 7. (1) If ^ = ^ shew that «_!±M±!^' _ ^i^J^ 
 
 b d c2 + 2c(/+3d2 - ^(c-3f/)* 
 
 (2) Find the value of a;6 -200a;^-fl98a;4+200a;3 -197;r2 
 -397a; when x=199. 
 
 S. Three towns, A, B, C, are at the angles of a triangle. T'rom 
 A to C, tlirough B, the distance is 82 miles ; from B to A, through 
 <; is 97 miles ; and from C to B, through A, is 89 miles. Find 
 the direct distances through the towns. 
 
 11. 
 
 1 . Prove x^ -=- a;" = x"'~". 
 
 Simplify {a+h + c)^ -S(a-{-h-\-c)'^c + ^a-\-h-]-c)c^ _fS. 
 
 2. Prove the rule for finding the L. C. M. of two quantities. 
 F«a theL. C. M. of 
 
 a^-\-h^ + c^-dabc, and (a + b)^ +2{a + b)c+c^. 
 
 ^ a c ac 
 
 8. Prove -^ x —r = —;• 
 b d bd 
 
 Pimphly [^^^ + -^— ^,) . (^-^_^-^-, _ ^-^^. 
 
 4. Eoduce to their lowest terms — :r„7-; — ;;; — ir, and 
 
 (I- '-{-a. — 2 
 
 a(a^.2h)-^b(b + 2c) + c(c + 2a) 
 a2_^2_(.2_26^ 
 
 5. (1.) li a^-pa^+ga- r = 0, then x^-pxl+qx~r is exactly 
 divisible by a; — o. 
 
 (2.) Prove that {n + b + c){bc-{-ca + ab)-{b+c){c+a){a^b)is 
 divisible by abc. Is there any other divisor ? 
 
6. Ux = 
 
 EXAMINATION PAPERS. 209 
 
 la + 6\ 2^ , a - -b- /« + b\ "+" 
 
 7. Solve the equations — 
 
 ^ ^^ l-"2x ~ 7^^2i ~ -^ ~ 7^1fe+lx--^* 
 
 (3 ) •^•+^ ^+1 _ 4x-l-9 12a; +17 
 a; + 4 ~ x+2 ~ 2uH^7 ~ Gx'-PTu' 
 
 8. A pei'sou going at tlie rate of ^j miles an hour, and desiring 
 fco reach home by a certain time, finds, when he has still r miles 
 to go, that, if he were continuing to travel at the same rate, he 
 would bo g Inufs too late. How much must he increase his 
 speed to reach home in time ? 
 
 9. Of the three digits comprising a number, the second is 
 double of the third ; the sum of the first and third is 9, and the 
 sum of the three digits is 17. Find the number, 
 
 10. A owes B $a due in months hence, and also .$6 due n 
 mouths hence. Fiud the equation which determines the time at 
 which both sums could be paid at once, reckoning interest at 5 
 per cent, per runum. 
 
 III. 
 
 1. If 3;= 10, ?/=lL 2=12, find the value of 
 
 \ x'^ — (i/-[-z)- X — , — ; and subtract 
 
 ( V^ ' ' j x+)j + z' 
 
 (ij — 2)a2 + (z - x)ab + [x — j'/)b'^ from 
 
 {u-x)u^ — {y—z)ah~{z — x)b^. 
 
 2. Divide a + (a + 6).f + (a + i + c)a;2 ^^a+b+c)x^ + ib^c)z* 
 + rx^by l+x-{-.i^ -\-x^ ; and find the square root of 
 
 9 - 2-l.iH-58.c2 - llC./;3 + 120^1 - UOr^ + lOf 'a;«. 
 
 c. r. ^ ■-, •^•c+5 a; + 5 2x-\-') 
 
 3. Solve (1; ^T + ^7~( = .> - TT" 
 
eio 
 
 EXAMINATION PAPERS. 
 
 i. A boy bought a number of oranges at the rate of 45 cents a 
 dozen ; if he had received 20 oranges more for the same money 
 the whole would have cost him only 40 cents a dozen. How 
 many did he buy ? 
 
 6. A farmer took to market two loads of wheat, amounting to- 
 gether to 75 bushels ; he sold them at difierent prices per bushel, 
 but received on the whole the same amount for each load ; had he 
 Bold the whole quantity at the lower price he would Lave received 
 $78.75 ; bnt had he sold it at the higher price he would have re- 
 ceived $90. Find the number of bushels in each load. 
 
 6. Show how to find the square root of a -h y^b. 
 Find the square root of 1 + |/(1 — w^) 
 
 ^^+^ 4x— 1 7a; -1-1 
 
 7. bolve ^ — "^ + X = ^ ; and find the value of i 
 
 ^x — / X — S X — o 
 
 when ax^—^Qx + Sl =0, has equal roots. 
 
 a+b _ \/{ac)+ \/{bd) 
 *^a*a_i - -/(rtc)- V{bd)' 
 
 9. Show that a3(6 — c)+i3(c — «)+c3(a — i) is exactly divisible 
 by a-{-b + c ; and resolve the expression into its factors. 
 
 IV. 
 
 1. Multiply a^-f 5=- c2 + 2a6 by a^ -b'^ +c^ + 2ac, and divide 
 the product by a^ — 6^ - c- -f 26c. 
 
 2. Simplify 
 
 18a362 _ ^3ab{x-y) li(c-d) S(c^ -d^)\ 
 
 ■u 
 
 ] ~1(^4 " \ 2T^3~ -^ a{x^ - j/^^l r 
 
EXAMINATION I'ATl^RS. 211 
 
 3. Find theL.C.M. of 4x•--9//^ 4x2-10x'y + G»/e. and 6.r'- 
 13a;?/+6?/3, and the G.C.M. of l+x^+x{-x^ and 2x + 2x' + 
 
 Sx'^ + Sx^ 
 
 4. Obtain the square root of i — fj/A^, and find the value of c 
 when 4:X^ — 12x^u+cx^l/^ — 12x7/^+4^* is a perfect square. 
 
 5. Distinguish between an equation and an identity. Give an 
 example of each. What value of m makes (a;— 3)2 —{^x — V){x—'6) 
 = m an identity ? Can any value of m make it an equation ? 
 
 6. Eeduce to its simplest form 
 
 l/(2 + :e)-T/(l +x) ^ l+ y^ll -1 -f- (l+a-) \ 
 1/(1+0,-) -v'x "^ l + i/'{l + l-(l+x-j[ 
 
 7. Solve the equations 
 
 (2) 73^-5rr=:.;-5^)(.T + 3^), 
 
 2 ' S _ 7 
 
 X— 5p x + '^!/ ~ 33 
 
 8. A person performed a journey of 22. V miles, partly by car- 
 riage, at 10 miles an hour, and partly by train, at 36 miles an 
 hour, and the remainder by walking, at 4 miles an hour. He 
 did the whole in 1 hour 50 minutes. Had he walked the first 
 portion, and performed the last by carriage, it would have iaken 
 him 2 hours 30^ minutes. Find the respective distances by car- 
 riage, train and walking. 
 
 U^^, ^^^v 
 
 ° 
 
 
 
 9. Solvo 
 
 
 
 • 
 
 a: + 3 
 
 x+1 
 
 4a--f9 
 
 12a;+17 
 
 x+4 
 
 ic + 2 ' 
 
 " 2x-+7 
 
 6x+16 
 
 10. What value of y will make 2x*-^Sxy +Qy^ eracily divisible 
 by a:- 3? 
 
 If a and ?^ are lae roots of the equation x^-i-x + l = 0, show 
 that fl3_?,3^0. 
 
212 BXAMINATION PAPERS. 
 
 V. 
 
 1. Multiply 
 
 Ax^'-^.x+\\ by 2ar + |. 
 Prove that 
 
 {h'' — '>j)^ — (x — ^y)^ is exactly divisible by x+y. 
 
 2; Express in vrords the meaning of the formula 
 
 {x + a){z + b)=x" + {a-\-h)xi ab. 
 
 Retaining the order of the terms, how will the right-hand 
 member of this expression be affected by changing, in the left- 
 hmd member (1) the sign of b only, (2) the sign of a only, (3) 
 the signs of both a and b ? 
 
 C. S:m\My {a + by-\-{a-h)^-2{a^-b^)» ; and show that 
 {a+b + c){b + c - a]{a-\-c - b){a-{-b - c) = iaH^ 
 when a^+b^ = c». 
 
 a c (id 
 
 4. Prove that -, — =---■ = -r— 
 u a he 
 
 Simplify 
 
 a^-\-h^ \ I ah^ \ ^ 4«(a+6) 
 
 2a6 ^ I \a^ + b^j ' a^-ab+b» 
 
 5. I went fi-om Toronto to Niagara, 35 miles, in the steamer 
 " City of Toronto " and returned in the " Eothsay," making the 
 r:>und trip in 5 hours and 15 minutes; on another occasion I 
 went in the " Eothsay " (whose speed on this occasion was 1 mile 
 an hour less than usual), from Toronto to Lewiston, 42 miles, and 
 returned in the '• City of Toronto," making the round trip in G 
 hours and 30 minutes ; find the usual rates per hour which these 
 steamers make. 
 
 6. Solve 
 
 ^ ' X y a X y a 
 
 (2) a;3 + 5a; = 5^/(^2+5x+28)-4. 
 
 7. Find three consecutive numbers whoso product is 48 times 
 the middle number. 
 
EXAMINATION PAPERS. 
 
 £18 
 
 8. If m and n are the roots of ax^ +bx + c = Q, then 
 
 ax^ -}-hx + c = a{x — vi){x -n). 
 
 Show that if ax^ + bx-i-c = has ecjual roots, one of ihem is 
 given by the equation 
 
 {2a^-2ah)x^ab-b"=0. 
 
 m n , x"^ y^ 
 
 9. If — = — and -r- + -r-^ =1, prove mat 
 
 ??)3 7;-- III- -\- 71^ 
 
 VI. 
 
 1. SimpHfy 
 
 2. Divide a^—h^ — c^—Sahc by a — b — c, and show, without 
 expansion, that 
 
 (1 +^+a;2)3 _ (1 .-x-{-x^)^-Gx(x^ +a;2 + 1) -8a;3 =0, 
 
 3. Eesolve into factors x'^—^x-y^-i-i/^, and 
 
 7^2 - G;/3 - x;/ + 19x + 33// - 36 ; and prove that 
 b^{c-\-a)+c^{a-\-!j) — a^{b-\-c)+abc is exactly divisible by 
 fc + c — a. 
 
 4. Apply Horner's method of division to find the value of 
 5a:»+497a;* + 200a;3 + 19Ga;^ -218a; -2000 when a:=-99, and 
 the va ae oi Gx^ ^Sx"^ -llx^ -Qx^+lOx-2 when 2a;- = -3^; + !. 
 
 6. Find what 
 
 ^\ -r-c)-]- — 1^ -J becomes when x = -. 
 
 V{a+x)- V{a-x) 1+6* 
 
 6. If a and b be any positive numbers, prove that 
 la a b 
 
214 EXAMTNATION PAPERS. 
 
 7. Solve the equations — 
 
 (]) ,.* + 7/- = 5, 
 
 .1 
 
 « +2/ 
 
 K' 
 
 (2) a;+27/+3z=i4, 
 2a; + 3?/ + 2 = ll. 
 3a;+?/+2z = ll. 
 
 (3) (x+l)(x + 3)(.T+4)(«+6)=.iG 
 
 8, There are three consecutive numbers such that the sum of 
 their cubes is equal to 16|- times the product of the two higher 
 numbers : find the numbers, 
 
 9, (1) Form an equation three of whose roots are 0, \/{ — 2), 
 
 and 1 — ^/2. 
 
 (2) If one of the roots of the equation a;-4-i''C-f-g = 0, is a 
 mean proportional between p and q, prove that 
 p^=q{l+py. 
 
 10, Two trains start at the same instant, the one from Dto A, 
 the other from Aio B; they meet in 1^ hours ; and the train for 
 A reaches its destination 52^- minutes before the other ti-ain 
 reaches B : compare the rates of the trains. 
 
 VJI, 
 
 1. Give some application of tlie '• rule of signs " in Algebraic 
 alultiplication and Division. 
 
 2. Find the numerical value of the quantity 
 
 hc(c — a){a — h) — cii[((, — b)(b — c) + ab{b - c)(c — a), 
 when a a 10, b = -01, c = 0; and prove that if 
 
 X = , then will (a-+o) . — ; 
 
 a+b a+b—c+x 
 
EXAMINATION PAPERS. 215 
 
 8, Investigate a method of finding by inspection the remainder 
 after dividing any rational and integral function of x by x+a. 
 
 Show that the quantity 
 
 a^b^-ab^x-{a^+2b^)x^+ax^-{-2x'^ 
 is divisible by each of the quantities x-\-a, x-\-b, a—2x, b-x. 
 
 4. Investigate the rule for finding the H.C.F. ef two algebraio 
 quantities, showing under what limitations factors may be intro- 
 duced or suppressed at any step. 
 
 Find the H.C.F. of 
 
 (1) 6.c4-7x-3_l3«3 + I9a;-G anda;3 + 2x3-l. ' 
 
 (2) {x-\-y){ax^-bir)-xy{a-b){x-\-y), and 
 {x-y)[ax^-by^-)+x>j{a-b){x-y). 
 
 6. Prove, by general reasoning, that the value of a fraction is 
 
 not altered by multiplying or dividing both the numerator and 
 denominator by tlie same quantity. 
 
 13 7 X \ 
 
 Simplify (1) 
 
 12(2a;-3) 12(2.f+3) 4:X^-\-% 
 
 ^^^ \{x-\-a){x~b) + {x-a){x-^b)) ' 
 
 1 1 
 
 + 
 
 Xx+a){x + b) {x — a){x — h) 
 
 6. Solve, with respect to x, the equations 
 
 a;-18 2a;-24 lla;-34 _ 1_ 
 
 -> '^T' ■'" ^Tl^^ "^ 22 " 44" 
 
 5. ^2 +x- 3 7a;2 -3a;-9 _ a;-3 
 
 ^^1 5a;_4 ~ 7.C-10 ~ 35ar3-78a:+40' 
 
 (3) X- = ax-^by, and y^ = bx-\- ay. 
 
 VIII. 
 1. Definfi the terms "power," "root,'* " index," and "coeffi- 
 cient ; explain also the reasoning by Avhich it is shown that 
 
 <t ~ (6 — c) = a — 6 + c. 
 
219 EXAMINATION PAPERS. 
 
 2. Multiply (a;3+x?/ -1-2/2)2 ^y (x—y)'*. 
 
 ■ Find the values of a and b which will make 
 
 x^+ax-\-b divisible 'byx-\-p, and also by xi-q. 
 
 a. Divide x^ +7/^+2x^,0^ by (.c-(-//)^ and 
 
 4. Investigate a rule for the extraction of the square root of any 
 algebraic quautit}', and deduce the rule for the extraction of the 
 square root of a number. 
 
 If to any square number be added the square of half the num- 
 ber immediately preceding it, the sum will be a eomplete square : 
 viz., the square of half the number immediately following it. 
 
 6. Find the square root of 
 
 (1) a2a;6 + 2a6x4 + (62-f2a,)„2-j-c3a;--+2/jc. 
 
 (2) ix^ - ix' -h ix^ + lx^ - lx'^ + ^\J. 
 
 6. If x^+ax-\-b and x^+a'x-b have a common measure, it 
 
 will be a;+ — k — , and the condition that they may have a com- 
 
 mon measure is Ab — a^—a'^. 
 
 Find the H. C. F. of x^ +p^ x^ -\-p* and x"^ +2px^-\-p^x^ -p^. 
 
 Find the L. C. M. of 2^{x^ + x-20), d^{x^ -x-BO), and 
 mx^-lOx + 2,4). 
 
 7. Find values of a and b which will render the fraction 
 
 3x^-{'ka + b)x-\-a-{-2b^ 
 5x- -{Sa-fb)x-a + 4:h3 
 
 the same, for all values of x. 
 
 8. Solve the equation 12 - i/{x+])[x+(}) - i/{x-'i)[x-{-5} = 0. 
 and account for the circumstance, that the values of a;, determined 
 from it, apparently do not satisfy the equation, 
 
BXAMINATION PAPHRS. 217 
 
 IX. 
 
 1. Prove thata(2?i-}-l)(a2+7i-M + l)-n(2</-f l)(7i2+«-a-i-l; 
 = (a — n)^. 
 
 2. If a, b, and s are positive quantities, and if a>fe and c>a — 6< 
 
 prove that 
 
 e — {a — b)=c — a + b. 
 
 Assuming this equation to hold good when a, b and c are unre- 
 stricted, prove that the expression-( -a), occurring in an algeb- 
 raic operation, is equivalent to +a. 
 
 3. li x^-\-ax^+h and x^+px+q have a common measure of 
 the form of x^ + ni,c-\-n, then a^bq-{b — q)^ 
 
 4. Find the H. C. F. of 
 
 a--b--abxy + abx~^y~'^, and a-x^ -b^y-^ +a'bx^y-b^xij''i', 
 
 5. A and J5 are two numbers, each of two digits. The left- 
 hand digit of A exceeds that of B hy x; the excess of A above B 
 is y ; but the sum of the digits of B exceeds the sum of the digits 
 of Ahy z. Prove that y-\-z = 2x; and give an example of two 
 such numbers as A and B. 
 
 a b c 
 
 6. If -r- = — = -r, prove that each of these ratios 
 
 ^a a+b-^-c 
 
 = „ r-, and also = ,— — :-,♦ 
 
 7. Solve the equations 
 x + a x — a b->rx h — x 
 x — a x-\-a ~ h — x b+x 
 
 (2) a{x'-+y^)-h{x^-y^) = ^a 
 
 (fl3-63)(a;3-7/3) =4rti. 
 
 8- A farmer buys a sheep for $P and sells h of them at a gain 
 of 5 per cent. ; at what price ought he to sell the remainder to 
 gain 10 per cent, on the whole ? 
 
 9. The sum of three numbers is 70 ; and if the second is divided 
 by the first, the quotient is 2, and the remainder 1 ; but if the 
 third is divided by the second, the quotient is 3, and the remain- 
 der is 3 ; whg^t are tlie numbers. 
 
 (1) 
 
218 EXAMINATION PAPEB3. 
 
 X. 
 
 1. Divide ax^ + 1cxijz-\-hij^ +ax- {y-\-z) + hij- {z-\-x) + 2cxy{x + y) 
 by x+y-\-z. 
 
 2. Prove that if x*^-{-p.v--^qx+a^ be divisible by x^ —1, it 
 is also divisible by x^ —a^. 
 
 3. Explain the reason for introducing or suppressing factors in 
 the process of finding the H.C.F. of two algebraical quantities. 
 
 Why is the name " Greatest Common Measure " objectionable ? 
 Find the H.C.F. oi x^-x^-x^ -x-2 and Qx^ -lx^+dx-2. 
 
 4. A traveller leaves A for B at the same time that another 
 leaves B for A ; the former walks at the rate of 3 miles an hour 
 till he has performed half the distance ; he then rests for an hour ; 
 after which he resumes his journey, walking now at the rate of 4 
 miles an hour ; the second traveller goes at the rate of 4 miles an 
 iiour till he has got over one-third of the distance between B and 
 A. ; he then rests for 40 minutes ; after which he resumes his 
 ]aurney, walking now at the rate of 3 miles an hour. The tra- 
 vellers reach A and B respectively at the same time. Find the 
 distance between A and B. 
 
 6. Show by examining the square of a-\-b how the square root 
 of an algebraical quantity may be found. 
 
 Find the square roots of 
 
 (1) 25a;4_30rtx3-f49a3a;2-24a3x+16rt4, and 
 
 (2) ^+Kl _ (^ + JL)^2 + i. 
 2/2 a:- \y xj 2 
 
 6. Show that a = \/aJ^, when m and n are integers, and m is 
 divisible by.7i; and state the principle on which you would main- 
 tain the truth of the equation for all values of m and n. « 
 
 7. Solve the equations 
 
 ^ ^ 5a: -4 ~ ' 7a;-- 10 
 
 (2) (y.c- 1)2 -f (-U- - 2)- = (ox - 3)3, 
 
EXAMINATION PAPERS. 
 
 219 
 
 B. Two regular polygons are so related that the number of 
 their sides is as 2 to 3, and the magnitude of their angles as 3 to 
 4 ; find the figures. 
 
 XI. 
 
 1. State in words the several operations to be performed in 
 order to obtain the result expressed by the following algebraical 
 expression : 
 
 V 
 
 'ma^ +nb^ 
 
 Also find its value when a = b = 4:. 
 
 2. Two men, A and B, dig a trench in 3f- days. If A were to 
 do more work by one-third than he does, and B more work by 
 one-half than he does, they would dig the trench in 2|f days. In 
 what time would each dig it alone, at his present rate of work ? 
 
 3. Perform the multiplications in 
 (1) 
 
 / 2a;^-f 3/ j / 2x^-2/ I ( 4x^-\. 6xV+9z/' ) ( ^^^ " ^^V + %* 1 
 
 (2) ax^+\xy^pj^^){ix^-lxi/+iy^). 
 
 4. Divide 
 
 (1) a;4-f9+81a;-* by x^-3 + 9x-^. 
 
 (2) x^-{a-{-h+p)x^-{-(^ap-\-bp — c+q)x^-{aq-{-bq-C]>)x — qc hj 
 x^ —px+q. 
 
 5. Show that x~"'+^-x'^''~^ is always divisible by x±l, m and w 
 being any positive integers. 
 
 6. Define a fraction ; and from yonr definition prove a rule for 
 adding together two fractions with different denominators. 
 
 Add together the fractions, 
 
 a^ —be b^ — ca c^—ab 
 
 (a + b){a+c) (b + c){b+a) {c. + a){c + b) 
 
220 EXAMINATION PAPERS. 
 
 7. Solve the following equations : 
 ,^. 3;2^2x+2 x^ + 8x-{-20 x^+4tx + 6 x^ + Gx+12 
 
 x+1 ^ x+i a;+2 ^ a;-f3 
 
 X 100 , ^ o. ?/ 
 
 (2) (x^+y^)-^ = ^^, (X3--/2) ^ 21 
 
 XII. 
 
 1. When OT and n are whole numbers, and m greater than n, 
 
 a™ 1 
 
 show that — - a'"-" and that -^ is correctly symbolized by a~" . 
 
 2. Multiply (a '-6)(a+^)(«-+^^)(«''+^'') • • • to (n + 1 ) factors. 
 
 3. Divide 1 — x by 1 — 2a;, to 5 terms, and write down the 
 <'7-+l)th term, and the remainder after (r+l) terms. 
 
 4. If the number three be divided into any two parts, show 
 that the difference of the squares is three times the difference of 
 the numbers. 
 
 5. Find the L. C. M. of l-8x+llx^+'^x^-24^xA, and 
 
 ^ 1 -2a;- 18;<;2 +38.1-3 -24x4. 
 
 6. What relation must there be between the coefficients m, n, 
 p and q, in order that 
 
 (x3 4-wx+7i)2 -f pa;2 + gx 
 may be an exact square for all values of a; ? 
 
 7. Solve the following equations : 
 
 (1) (T+^ys + (l-x)2 " "• 
 ox — h^ ^/{ax) — b 
 
 (2) v{ax]-\-h = ^r~ -'• 
 
 C3^ — "- = 1, — ^- = 2, and — = 3. 
 
 8. Given x4-?/+2 = «a; = %, find (x+?/+z) -z. 
 
 9. Find a number expressed in the decimal notation by two 
 dio-its, whose sum is 10 ; and such, that if 1 be taken form its 
 double, the remainder will be expressed by the same digits in a 
 reversed order. 
 
EXAMINATION PAPERS. 221 
 
 XIII. 
 
 1. Find the value, when a = 2 J, b-.3^, c- 4+ of 
 
 2. Show that the vahie of the expression, in the preceding 
 question, is not altered by changing a into a-^x, h into b-\-x, and 
 c inte c-\-x. 
 
 3. Multiply (1 -fa ia;)(l 4- rt 2^) (1+^3^) ••• (l+«„a;) to 3 terms. 
 
 4. A speculator borrows a sum of money at the yearly interest 
 of 7 per cent. ; part of the amount he Invests at 8^ per cent., and 
 the remainder at 9 ; and, at the end of the year, he finds that he 
 has made a profit of $75 ; but, had the former part been invested 
 at 9 per cent., and the latter at Qh, his profit at the end of the 
 year would have been only $65. Find the whole sum borrowed. 
 
 5. Given ax+hy = c, a'x+h'y = c', determine the value of 
 nix-\-ny, and find the conditions under which the value becomes 
 indeterminate. 
 
 (I I da 0„ 1 
 
 6. If-!- = ^= = -^^, 
 
 then will a, +^2 +^3+ . . . + a„-=-^ 
 7. Eliminate x and y from the equations 
 
 S , -7. 5 
 
 X -^r y = a 
 
 « = x-k-^x^'y 
 
 a\-a^a^ 
 
 «! — aj 
 
 1 a 
 5 ■^ 
 
 P = ?/+3xSA 
 
 8. \iax- ■\-'bx-^-c = ^ -A d .<i»^ + 6ia;-fc, =0, then will 
 
 («5j —a^b)(bci —b^c) = (aCj — a^c)^ . 
 
 9. Find that number of two figures to which if the number 
 formed by changing the places of the digits be added, the sum is 
 121 ; and if the same two numbers bo subtracted, the remainder 
 is 9. 
 
222 EXAMINATION PAPERS. 
 
 XIV. 
 
 1. SimpKfy 
 
 a(i + c)3 + 6(r+rt)2 + c(a+&)2-{(a-?;)(a-c)(64-c) + 
 {b-c){b-a){c+a) + {c-a)(c-h){a-j-b)}. 
 
 2. State the law of Indices, and prove it for positive integral 
 indices ; and assuming it to be general, interpret the expressions 
 
 z~^, X , where m and n are positive integers. 
 
 3. Having given the equations, 
 
 prove that a^ [yz -y'z ') + ''^ [z-c- -z'x')+c'^{xy-x 'y ') = 0. 
 
 4. A traveller P sets out to walk from A to B, proceeding at 
 the rate ot 3 miles an hour ; and, 32 minutes afterwards, another 
 traveller Q sets out to walk from B to A, proceeding at a uniform 
 rate. They meet half way betwixt A and B. P then quickens 
 his pace by 1 raile an hour ; and Q slackens his 1 mile an hour. 
 Q reaches A at the same time that P reaches B. Find, the dis- 
 tance between A and B. 
 
 6. How are equations classified ? 
 Solve the equations — 
 
 (1) mnx+a'm.n = n^x-\-aiu^. 
 
 (2) x^-x^+y*-y^ = 84:, 
 x-+x^y^-\-y'^=49. 
 
 6. What two numbers are those whose difference, sum and 
 product are to each other as the three numbers 2, 3, 5 ? 
 
 XV. 
 
 1. What is the meaning of the symbols a, a-, a^ . , ? 
 Show a priori that a°=l ; how do you know that ab = ba ? 
 How is it proved that the multiplication of hke signs gives a 
 positive, and that of unlike signs, a negative result. 
 
EXAMINATION PAPERS. 223 
 
 2. Find the value of 
 
 (^b~c)^+2{c-ay + {a-h)^-S{b-c){c-a){a~b) 
 
 when a = l, b= —^, 
 
 3. Simplify the following expression : 
 {ac-b^){ce-d2)-\.{ae-c2){bd-c^)-{ad-bc){be-ca) 
 
 4. P and Q are travelling along the same road in the same 
 direction. At noon P, who goes at the rate of m miles an hour, 
 is at a point A ', while Q who goes at the rate of n miles in the 
 hour, is at a point B, two miles ha advance of A. When are they 
 together ? 
 
 Has the answer a meaning, when m— n is negative ? Has it a 
 meaning when jn = )i7 If so, state what interpretation it must 
 receive in these cases. 
 
 5. Show how to find the Least Common Multiply of two or 
 more algebraic quantities, 
 
 (1) x2—ax~2a^, x^+ax^ andax^ — x^. 
 
 (2) x^ —x^y—a^x+a^y and x^+ax^ —xy^ —ay^. 
 
 In what algebraic operations is the Lowest Common Multiple 
 of two or more quantities required ? 
 
 6. State and prove the principle upon which the rules of Addi- 
 tion and Subtraction of fractions are founded. 
 
 Simplify the following expressions : 
 
 (a _!_& _ c)2 - ^3 (^b+c -a)^-d^ (c+a-by - d^ 
 
 (1) (^^6)2_(c+d)2 + {b + c)''-W-fd)2 + (^+u)-^b+d)-' 
 J.2 ^y2 -z^ + 2xy a^-\-a^h a{a—b) 2ab 
 
 (2) ^2_\j2-z- + 2yz a^b - i>3 ~ {a+h)b ~ a^-b^' 
 
 7. liax-by-{-c{x-y) = {a-b){a + b-c}, 
 
 by - cz+a{y-z) = {b- c){b + c-u), 
 cz--ax+b{z-x)=^{c-a){c+a — b) 
 then will a^{b-c)+b^{c-a) + c2{a-h) = 0. 
 
 8. P is a number, of two digits, x being the left hand digit, and 
 y the right. By inverting the digits, the number Q is obtained. 
 Prove that 11 {x + y){P-Q) = i) {x-y) {P+Q). 
 
224 EXAJIINATION PAPERS. 
 
 XVI. ' 
 
 1. Show that 
 
 {(ax + by)2 + (ay-bx)^}{{ax-{-hy)^-'{a!/+bxy} = 
 (a4-64)(:<;4_,^4) ; and that 
 
 2{a-b){a-c)+2{b-c){b-a)-i2{c-b)ic-a) 
 is the sum of three squares. ' 
 
 2. It s = a + b-{-c-\-&c. to n terms, then 
 
 s — a s — b s — c,j, " 
 
 + + + &C, r= n — 1. 
 
 s s s* 
 
 3. Show that a — b, b — c, and c~a cannot be all three positive' 
 or all three negative. 
 
 4. Extract the square root of 
 
 Ax^ + dx^ - V2x^ +16x- +9 -2x{Gx<^-8x* +9x^-12). 
 
 5. Given ab - ^{a+b){p + q)-{-2jq = 0, 
 
 cd-i{c+d){p-{-q)+2)q = 0, 
 
 find the value of p—q, and show that if either a or 6 is equal to r 
 or d, then p is equal to q, unless a+b = c + d. 
 
 or * 
 
 6. Find the value of — , having given 
 
 y 
 
 ic^" — ay^" x^ — Mx—yY 
 
 7. Prove that {a — b){b—c){c — a) is a common measure of the 
 quantities 
 
 c4(a-6) + a4(/v-f)+i4(c-a). 
 
 8. Find the conditions that a^x+h\y = Cy, cf^x+h^y = c^, aocl 
 aoX-\-b^y = c^ may be satisfied by the same values of x and y. 
 
 9. Two persons, A and B, start at the same instant from two 
 stations (c) miles apart, and proceed in the same du-ection along 
 the line joining tlie stations with velocities (a) and {b) miles per 
 tiour. Find the distance {x) from the stations where A over 
 bakes B, and interpret the result when a z b. 
 
EXAMINATION PAPEKS. 
 
 225 
 
 XVII. 
 
 1. Ex^Dress in symbols the result of subtracting from unity the 
 quotient obtained by dividing the sum of a and b by their product. 
 
 2. Multiply together x + '^/a + b, x— Vn-\-b, x+^/a-h and 
 X- Va-h\ and divide 24a3-..22«2^, + 2fl2c_5a^,2^27a6c-34ac2 
 4-6^3 _22i2c-fl6&c2 + 8c3 by '6a-2b-\-4c. . 
 
 3. If x-\-a be the H. C. F. of x^+px+.q and x^+p'x+i', 
 their L. C. M. will be {x-\-a){x-\-p- a){x-\-p' - a). 
 
 Show that the difference between 
 
 X x X a b c 
 
 + 1 + z — lana- — - + 
 
 x—a x — b x — c x — a x—b x — c 
 is the same whatever values be given to x. 
 
 4. Prove, if the four fractions "" 
 bx-\-cy-\- dz cx-\-dy + az dx+ay '+ bz ax-{-by ■\-cz 
 h^c + d — a e-\-d-\-a— b' rf-^rt-f-Z> — c' a-\-b+c ~d 
 
 are equal to one another, their common value will be equal to 
 
 x + y+2 
 
 — ^ — as long as a + h-\-e-\-d does not vanish. 
 
 5. What do you mean by solving an equation. Show that 3 is 
 a root of the equation 
 
 T/(a;3-3x+4) = -^^^ 
 
 6. Eliminate x between the equations 
 
 a;3 4--^ + 3 Lc 4 ]=m, ani 
 
 a; * \ X I 
 
 a;3— -r — 3 a; = n. 
 
 x^ \ X I 
 
 1111 
 
 7. If — + -— _ — = — -—r , a, 6, c are not all different. 
 
 a b c a-\- — c 
 
 8. A cask, A, contains m gallons of wine and. n gallons of water ; 
 an another cask, B, contains p gallons of wine and q gallons of 
 water, how many gallons must be drawn from each cask so as to 
 produce by theii- mixture h gallons of wine andc gallons of water ? 
 
226 EXAMINATION PAPERS, 
 
 XVIII. 
 
 1. Multiply together the f.^rtors 
 
 1-x, 1+x, 1+x"-, l+a-*, arc! 1+x^. 
 and show that if n is any uneven number, the s im of the nth 
 powers of any two numbers is always divisible by the sum of the 
 numbers. 
 
 2. Find the numierical value of the expression 
 
 c \'n + \/c 
 b \/a— \/c 
 
 where a, h, c are connected by the equation a'b — c)^— c(h+c) '^ — 0. 
 
 3. A has a younger brother, B. The difference between their 
 ages is ^ of the sum of their ages. By adding twice B's age to 
 5 times A's, we obtain the age of the father; and by subtracting 
 twice B's age from 5 times A's, we obtain the age of the mother. 
 Show that the age of the mother is -^^ that of the father. 
 
 4. Find the II.C.F. of 
 
 x^-(2a+b)x^+a(2a+?>)x-a2(a + b), and 
 X* - {2b+a)x^ +b{2b+a)x-b2(b+a). 
 
 114 
 
 5. If — 4- = — . shew that 
 
 b c a 
 
 6. Show fully how the rule for finding the square root of a 
 «flven number is obtained. If w + 1 figures of the square root of 
 a number have been obtained, prove that the remaining n may be 
 obtained by division. 
 
 Extract the square root of 
 
 7. Find the value of the expression 
 
 t ^ when X = , y = — 
 
 l+xy «— « a 
 
EXAinNATION PAPERS. 
 
 227 
 
 8. Solve the equations : 
 
 (1) ),{.v-'la) -l{x + Ba) + l{x-6a)=0. 
 
 (2) V(2x2+l)+v'(2.;3+3) = 2(l-a:). 
 
 9. Divide 21 into two parts, so tliat ten times one of them may 
 exceed nine times the other bv 1. 
 
 XIX. 
 
 1. Multiply together 
 
 X' + \^ax-a2 - |.«+|«-i. 
 Divide this product by 
 
 lx^+^ax-2a^-^x-{-2a-^; 
 and extract the square root of the quotient. 
 
 2. If x+y+z= 1 1 = 0, shew that 
 
 •^ X y z 
 
 (x'^ +y^ -\-z^)-^ [x^ +y^ -hz^) - xyz. 
 8. Find the H. C. D. of 20a;*+.r- -1 a.nd'ii>x^-i-lox»-3x-3; 
 also of (x+y)' -ic^ -y' and {x^-y^}'. 
 
 4. Given that ab - (a-^b)(x+y)-^4:xy = 0, 
 cd — {c+cl){x—y)->r4:xy = 0, 
 find the value of (j"-i/)^. 
 
 6. Having given 
 
 x^=y^-\-z- -2ayz 
 
 y^=z^-{-x^-2lzx 
 z^=x^+y^-2cxy, 
 x"- y"" _ 2^ 
 
 Show that i-a^ ~ 1-b'^ ~ 1-^c^' 
 
 6- ir^+^{2x-\-x'^) - ^^• 
 
 7. Determine a; in terms of a and b in order that x^-\-2ax^ + 
 3523.3 _ 4^ 3a._j. 454 may be a perfect square. 
 
 8. A company of 90 persons consists of men, women, and 
 children ; the men are 4 in number more than the women, and 
 the children exceed the number of men and women by 10. How 
 many men, women, and children are there in the company. 
 
228 EXAMINATION PAPEB3. 
 
 XX. 
 
 1. Divide (l-{-m)x^ — (in-{->i)xy{x — y) — (n—l)y^ hy 
 
 x^ -xy-\ry^. 
 •I. If x'^+px'^^+qx+r is exactly divisible hy x'^-\-mx-\-n^theti 
 nq — n^ = rm. 
 
 3. Prove that if m be a common measure of j? and q, it will algo 
 measure the difference of any multiples of ju and q. 
 
 •Find the G.G.M, oix^-px^ + {q-l)x--\-px-q and 
 
 x^ — qx^-{-[p — VjX^+qx —p. 
 
 4. Prove the rule for multipKcation of fractions. 
 
 simphfy -IjiS^yz^ X T-(?-^ ^ z^^ziix-zy): , 
 
 (2/+2)'--^2 {^+^Y-y- (a:+^j2-23 
 , a a a^ 2a^~b^ — ab^ 
 
 5,, What is the distinction between au identity and an equation / 
 li x — a=:.y + h, -piOYQ x — u = y-\-a. 
 
 Solve the equation 
 
 16a;- 13 40^-43 32^-30 20a; -24 
 
 4a;-3 ' 8x-9 8x'-7 ' 4a;-5 
 
 6. What are simultaneous equations ? Explain why there must 
 be given as many independent equations as there are unknown 
 quantities involved. If there is a greater number of equations 
 than unknown quantities, what is tlie inference ? 
 
 Eliminate a; and ?/ from the equations ux-tliy = c, a'x-\-b'y=.c^. 
 a"x + b"y = c". 
 
 7. Solve the equations — 
 
 (1) ^{n + x)+^{n-x) = m. 
 
 (2) 3a;4-Z/+z=13, 3//-f z-|--j; = 15. 32 + x+z/=17. ' 
 
 8. A person has two kinds of foreign money ; it takes a pieces 
 of the first kind to make one £., and b pieces of the second kind : 
 he is offered one £ for c pieces, how many pieces of each kind 
 must he take ? 
 
EXAIMINATION PAPEES. 229 
 
 9. A person starte to walk to a railway station four and a-half 
 miles off, intending to arrive at a certain time ; but after jvalking 
 & mile and a half he is detained twenty minutes, in consequence 
 of which he is obliged to walk a mile and a half an hour faster in 
 order to reach the station at, the appointed time. Find at what 
 pace he started. 
 
 10. (a) If — = — then will — ' — = — . 
 
 (b) Find by Horner's method of division the value of 
 a;«+290a;*+279a;3-2892a;2- 586a;- 312 when x= -289. 
 
 (c) Show without actual multiplication that 
 {a+b-\-c)^-(a + b+c){a^-ab+b"-bc-{-C'-ac)-3abc = 
 S{a-\-b)[b-\-c){c-ra). 
 
 Note. — In. Ex. 6, p. 87, after proving that a+h+c is a factor, 
 we may proceed as follows to discover the remaining quadratic 
 factor : 
 
 The quadratic factor must be of the form 
 
 m{a^ + b^+c^) + n{ab+bc+ca), 
 in which m and n are independent, being either zero, or a positive 
 or negative number. To determine them put <:- = 0, then the 
 given expression gives 
 
 {aS + b^-{-3ab{a+b)}^{a + b) = a2-^b^-+2ab, 
 but also = m{a^-\-b^)-\-nab. :. ?n = l andw = 2. 
 ,*. a^ + h^ +e^ +S{a+p)(h + c){c + a)} ^ {a + h+c)» 
 a^+b^-i-c^ + '2{ab + bci-m) = {a-i-b + c)''(. 
 
230 EXAMINATION PAPERS. 
 
 . XXI. 
 
 1. ?ind the value of a:3- I— 1 r] ^' + \~u ~)*+"p 
 
 when a = ^, b=^, x = 2. SimpHfy 
 
 2. Fiud, by symmetry, the sum of {a-{-b-\-G)^ —{a + b — c)^ - 
 (a-b-\-c)^-{b-a+cy, and of (a^-^r/^.t-f Sa^ajS - 2«x3 + 3a;4)2 
 and {a'>= +4:a3x+2a^j;- + 2ax^+Sx^)-. 
 
 3. Exphiin and illustrate the signs > , < 
 
 Prove: x^+y^'>2xi/, (x+y+z)- >3(.c//+//z+zaj), and 
 
 s J i 4 J ^ 
 
 4. Determine the value of a; 4-2/ — ^ + 3a;' //"z , when a; +y — z — 
 
 0, &c. ; of a7+7f/x-^ + 8a;--3a3-(a;* + 7«a;3-8a;3-3a'-'), when 
 x= —1. 
 
 p Tnp 
 
 5. Show that (a"") «' = a7 . 
 
 SimpUfy I (-—J *| ' X (-^) ^ X t/(256), and divide 
 
 a; —Qax -f 5a x+la^x —2a by a; —2a a;+a . 
 0. If u = ^ ix-^—\ and v = jl y + ~) pi'ove that 
 
 7. Gold is 19j times as heavy as water, and silver 10^ times. 
 A mixed mass weighs 4,160 ounses, and displaces 250 ounces of 
 water. What proportion of gold and silver does the mass con- 
 tain ? 
 
 8. Shaw that l+px+qx^-i-rx^^is a perfect cube if p-=dq, 
 and (]' =3pr. 
 
 9. Solve the equations : 
 
 ix-2 lx+2 
 
 m ^.+-2 + ^'.-2 = *- 
 
 (3) — + --p^ =20 - , x+Q = ^y. 
 
* 
 «XA.l\nNATTO^ PAPBR3. 231 
 
 10. A person buys two bales of clotb, eacb contaiaing 80 yards, 
 for $240. By selling the first at a gain of as maeh per cent, as 
 the second cost bim, and tbe second at a loss of as mucli per 
 cent., be makes a profit of $16 on tbe whole. Find tbe cost 
 price per yard of each bale. 
 
 SECOND CLASS TEACHERS, 1880. 
 
 XXII. 
 
 1. Find tbe value of x'^+x^ -IQGx'^ -lCn).c^ +81x + 81 when 
 x= —'•7 ; and the value of x^ — dpx- + {3p'^ +g)x — pq when 
 x=:a+p. (Arrange the latter result according to powers of a). 
 
 2. What is tbe condition that x-\-b shall be a factor of 
 ax^+hx-\-c? 
 
 Find the factors of 
 
 (a). (a^^-ah) + 2{l>^-ah)^3(a^~h2)-{.4:(a-h)^ ; and 
 
 (b). (ax+b){bx + c){cx-\-cb) — {ax ■}- c)(bx -{- a)[cx -j- b) . 
 
 3. What must be the relation among a, b, c, chat a.v-+bx-\-c 
 may be a perfect square ? 
 
 (a). Extract the square root of 
 
 (a-b)i-4:{a^+b^){a~b)2+4:(a4^+b*)-r8aH-\ 
 
 (b). If 5 be subtracted fi.'om tbe sum of tbe squares of any four 
 consecutive numbers, the remainder wiU be a perfect .square. 
 (Prove this.) 
 
 a c c h In 
 
 4. If T~ = ~r = "I' a^f'l IT = — = — ■ 
 
 b a t « in p 
 
 (a+c + e){h + L-^n) ah-\-cl-\-en 
 
 Pi-o^etbat |6 + J,-J:/-)(A;+iM+;;;) = bk+d^^' 
 
 ab{x^-y^)+xy{a^-b^) 
 
 5. (a). Reduce ^^^.^^^2^^_ry(^a2+b^) *° ^*^ ^°^®^* *^^°^^- 
 
 (6). II xi/-\-yz + zx = l prove that 
 
 X y z 4,xyz 
 
 + 1-^ + 
 
 l_a;2 -r i_y2 -r i_^i - ^i^^2>^i^i_yi)^i_z^) 
 
232 EXAMINATION PAPKB8. 
 
 6. Prove that 
 
 2{x-+2+4/(x2-4)} . 
 
 (h) {b+c-a)a'' + {c + a-b)b^ + {a-^h~.c)c a 
 
 a A -1- -^ ■■< a 
 
 {a + h-4-c){a'' + //' + c')-2(a' + 6' +c')., 
 
 7. Solve the equations — 
 
 (a), (b - c){x - a)^ + {c - a)(x-b)3 +{a~b){x ~c)^ = 0. 
 (b). x + ij = 4:Xi/; )j-\-z = 1yz; z+x-'dzx. 
 (r). x+y+z = 0. 
 
 ax-\-bii-{-cz = 0. 
 
 bcx + cay-{-abz-\-(a — b)(b—c){c — a)=^(j. 
 
 x — 1- »; — 3 
 (^) ,1^ + . + 1 + 2 = 0. 
 
 FIRST-CLASS TEACHERS, 1876. 
 
 XXIII. 
 
 1. Investigate Hornei-'s method of division. 
 
 Divide x^ ~dx^ -Six' +'2rjx<' +dx^ -8x^ -\-19x^ -{-Sx-^IO by 
 3.** — 21a;3-f 9a;— 6, showing the " final remainder." 
 
 Find the value of 2x^-^803x^ -3d^x^ + 1605x^- -1204x+i22, 
 when x= —402. 
 
 2. Ii/{x), a rational and integral function of x is divided by 
 
 X' +px + q, the remamder is a'—B ' 
 
 where a, (3 are the roots of x^ +px+q = 0. 
 Examine the case where p- =4:q. 
 
 3. Show without actual expansion that 
 
 a^{b-c) + b^{c-a) + c^{a-b) 
 
 (a3 _ /;2 )3_|_(i2 _ c 2)3 + (c2_a2)3 
 
 (a-i)3+(6-c)3 + (c-a)3 
 
SXAMINATION PAPERS. 233 
 
 4. rind the value of x aud y that will render the fraction 
 
 o 9 , , n 5^;^ ^TT the same ior all values ot z. 
 
 ^« + (2/ ~ '^)'^ + ^^'(i/ "■ ^^) 
 
 5. Show how to find the suro of n terms of a series in Geo- 
 metric progression. 
 
 (1) Show that the sum of n terms of the series 
 
 l + r + (l4-2r)(H-r) + (l+3r)(i+r)2 4- . • ., isn (l+r)". 
 
 11 1 
 
 (2) Sum to infinity the series o.^.p + 4^./-.o + p Q.in + • • • • 
 
 6. Explain the notation of functions : prove that if 
 
 f (,„) = i + „,a;+ ^^^.7 •^' +<^c-» then/ (w) x/ (?i) =/(w.-}-70. 
 
 Show that in the expansion of (1 -[-«)" the sum of the squares 
 
 1-2-3 .... %i 
 
 ot the co-efificients = . _, ^^ .-, ,-^« 
 
 (l-2*3 • • • • ny 
 
 7. Solve the equations — 
 
 ,,N x - « x— i x — c 
 
 ^ ' h~-^c "•" l^c + a-f-A "" ^• 
 
 (2) a;4-10.6-3 + 35x2 -50.^ + 24 = 0. 
 
 1 1 
 
 (^) 21x2 - 13x+2 + 28x8 - 16x4-2 "■^^•^^"'^■''■^^' 
 
 8. Give a brief account of mathematieal induction, and show 
 that a square of a multinomial is equal to the square of each term 
 together with twice the product of each term into the sum of all 
 that follow it. 
 
 Find the sum of the products of the first n natural numbers 
 taken two and two together ? 
 
 9. If — - = ?/ + «, ^= 2 + X, — - = X 4- y. pi'ove 
 
 (1) 
 
 1 1 1+a 1+i 1+c 
 
 a 1) G \ — ah\ — 6c 1 — ca 
 
 ^^ a{l-bc)~b{l-ca)~c{l-ab) 
 
 VT^^ Vl-ca . V\-ab ^1—bc Vl-ca VV^ 
 
234 EXAMINATION PAPERS. 
 
 10. AB is divided in C, so that AB, BC = AC^ ; from CA is 
 cut off a part CD equal to CB ; from DC is cut off a part DE 
 equal to DA ; from £1) is cut off a part equal to UC, and so on 
 cd inf. Sbow that the points of section continually approach a 
 point C such that AC' = BG. 
 
 14. Eliminate x, y, z and n from the equations 
 
 a^x+hiy + c^z+d^u = 0. 
 
 a2X-\-h2y + c^z-\-d.2U—0. 
 
 a ^x+ h ^y -{-c ^z + d ^n = 0. 
 
 a^x-^-b^y+c^z + d^K = 0. 
 
 12. A rail^vay train travels from Toronto to Colling-wood. At 
 Newmarket it stops 7 minutes for water, and two minutes after 
 leaving the latter place it meets a special express that left Colhng- 
 wood when the former was 28 miles on the other side of New- 
 market ; the express travels at double the rate of the other, and 
 runs the distance from Collingwood to Newmarket in ,1^ hour; 
 and if on reaching Toronto it returned at once to Colhngwood, 
 it would arrive there three minutes after the first train ; find the 
 distance between Toronto, Newmarket and CoUingwood. 
 
 FIEST CLASS TEACPTEES, 1877. 
 
 XXIV. 
 
 x-{y--z)+y''{z—x)-j-z^{x-y) 
 
 x^y^ -\-x^y^+x'^z- -j-xH^-\-y'^rJ+y''z^ + 2x^y^z^ 
 
 ax+m-{-l nx-\-n ax + m ax-hn+1 
 
 2. Solve (1.) ^^^^—i + ^;^qr7i32 = ^ + ^ - 2 + ax^n:^! ' 
 
 (2.) iri:^7^+rr:^y^=2. 
 
 3. A, B, and C start from the same place ; B, after a quarter 
 of an hour, doubles his rate, and C, after walking 10 minutes, 
 diminishes his rate one-sixth ; at the end of half an hour, ^ is a 
 quarter of a mile before B, and half a mile before C, and it is 
 
EXAMINATION PAPEKS. 235 
 
 observed that the total distance walked by the three, had tLey 
 continued to walk uniformly from the first, is 6J miles. Find 
 the original rate of each. 
 
 4. {1) investigate the relations that must exist between the 
 constants in order that Ax'^-\-Iiij- +Cz- ■^-aijz + bxz + cxy shall be 
 a perfect square. 
 
 (2) Find the conditions that the values of x and y derived from 
 
 the equations ax+hy= — -4- — = c^ maybe rational. 
 
 X y 
 
 5. 11 x~+px+g and x^-\-mx+n have a common factor, then 
 will {n — q)--\-ii{m—p)- =in{m—p){it-q). 
 
 6. Prove («"*)" = «"*", whether m and n be positive or negative, 
 integral or fractional. 
 
 Show that (x--'"+ar")»» =a;"» ^ " x (af-" + -*;"-'")«» 
 
 7. (l.)Ifx = 4-then V^^5?! = {-^X^ 
 
 1 
 
 of these fractions = — (a"4-i"+c"+t^). 
 
 8. If a; be very small, show that — 
 
 , — =2 - 4..!-, very nearly. 
 
 2+ox-(l+4.«) 
 
 ^2(^2 _ 12) 7l2(,j2_12)(-^2_23) 
 
 9. Prove that l — n^ + "Ts " 02 12 — 2^ — P — + -. = 
 
 10. If a debt $a at compound interest be discharged in n years by 
 
 . a 
 annual payment of $ — , show tliat (l+rj"(l — m/-) = 1, where r 
 
 is the interest on $1 for a year. 
 
236 EXAMINATION PAJfUAa, 
 
 11. Solve— (1.) 8a;2-2a:2/-55. 
 
 x--5xu + 8y^ = 7. 
 
 5 5 
 
 3 a p + ,j 1 
 
 i, 5 
 
 (8) a^b'^x" -4:a b'xm ={a-by-x 
 
 V 
 
 FIEST CLASS TEACHERS, 1878. 
 
 XXV. 
 
 2 ..3 
 
 1. SimpHfj W'^-±f_V-4-)'- (V^-V^r-7-"---, 
 
 ■ a; a + a' \ « x ■ a{a-x) 
 
 x^-ii/-zr- y^-Jz-x]^- zl-J^-][Y 
 
 ^""^ {x-\-z)^-y^- ^ '{x+y)--z' + (y + ^P-^c^ 
 
 X b b^ b h'i 
 
 2. Divide — — 1 — — —-7,+ — +— , bvic— a; 
 
 a a a" X x^ "^ 
 
 shew that (-9a2)* = i{ ^(6rO-t-A/(-6rt)}. 
 
 m n r x^ v^ z- 
 
 8. If — = — = — and-5r+7-^+^7 = l, prove that 
 X y z a^ b^ ' c' ^ 
 
 4. Fiud the relations between the roots and co. efficients of the 
 equation ax^+bx-\-c = 0. 
 
 If m and n are the roots of the equation ox^ -bbx-\-':--0, show 
 that the roots of the equation acx'--i-{2ao— b'-^)x+ac = are 
 
 ni n 
 
 — and — • 
 ■II m 
 
 6. Solve the equations : 
 
 (1) a-24-2l/a2_2a;=2x+8. 
 
 ,^, x^ y^ X y 
 
 (2) — -— = 10|, — - — = 4. 
 
 y X *' y X ^ 
 
 (3) X2 = i/S x+y+z=12, x^-^y"'+Z'=[n. 
 
EXAMINATION PAPERS. 287 
 
 6. Two men start at the same time to meet each other from 
 towns which are 28 miles apart ; one takes five minutes longer 
 than the other to walk a mile, and they meet in four hours. Find 
 each man's rate per hour. » 
 
 7. If P, Q, R be respectively the pth, ^th, rth terms of a G.P., 
 shew that 
 
 12 3 
 Sum to infinity the series — + 773 4-":^+ ^^' 
 
 JC tL- Jb 
 
 8. Find the amount of $f ufc compound interest for n years, r 
 being the interest on $1 for one year. 
 
 Supposing %2^ to be withdrawn at the end of each year, what 
 will be the amount at the end of n years ? 
 
 9. Determine the number of combinations of n things taken r 
 together. 
 
 The number of combinations of n things taken two togethej; 
 exceeds by 6 the number of combinations of n — 1 things taken 
 two together : find n. 
 
 30. (1) Find the limit of (l-f^)"^ when x increases without 
 ''imic. 
 
 (2) Find the (7*+l)th term in the expansion of (3 — 6a;) 
 
 x^ - 3a; - 8 
 
 11. Determine the limits between which lies o .3 lo^ii for all 
 
 possible values of x. 
 
 FIEST CLASS TEACHEES, 1879. 
 
 XXVI. 
 
 7. Prove that ^{{0 -hy ■{■(h-c)-' + {c- ay] =l{a-b){h-c) 
 {c-a){{a-bY->r{b-cy + {c-ay). 
 
 2. Extract the square root of ^<6—2a-j/(r/& — a8), and find the 
 simplest real forms of the expression 
 
 ,/(3 + iA/-l) + v/{3-V 1). 
 
238 EXAMINATION PAPF.K». 
 
 8. Solve the equations : 
 
 (1). 2a;4-f^3_ii^s_l_^4-2^-0. 
 
 (2). u;2+.y2+2-^=«3 
 yz-\-zx -^x^ = h'^ 
 z-f- y— z—c. 
 
 (3). y (»3 + 5x+4.) + -^/(x- -f-Sx- - 4) = x+i. 
 
 4. Prove that the number of positive integral solutions of the 
 
 c 
 equation ax+hi/ = c cannot exceed — r + 1. 
 
 In how many ways may £11 15s. be paid in half-guineas and 
 half-crowns ? 
 
 5. If xy = (il>{a + b), and x'^ —n-y + i/^ =:a^-\-b^, shew that 
 !^ j[\ I ^ V 
 
 \ a o I \ o a j 
 
 6. Given the sum of an arithmetical series, the first-term: and 
 the common difference, shew how to find the number of terms. 
 Explain the negative result. Ex. How many terms of the series 
 6, 10, 14, &c., amount to 96 ? 
 
 7. Find the relation between p and q, when x^+px+x = ('* has 
 two equal roots, and determine the values of ni which will mane 
 u^ +max-{-a^ a factor of »■* —ax^ +a"x^ — a^x-\-a^. 
 
 8. In the scale of relation in which the radix is r. «hew that 
 the sum of the digits divided by r — 1 gives' the same remainder 
 as the number itself divided by ?•— 1. 
 
 9. Assuming the Binomial Theorem for a positive integral 
 
 index, prove it in the case of the index being a positive fraction. 
 
 « 
 
 Shew that the sum of the squares of the co-efficients in the ex- 
 pansion of (l-fx)" is [2n-j-d w )2^ n being a positive integer. 
 
 10. Sum the following series : — 
 
 ' (1.) l + 3x-\-5x^ +lx^ +&C, to n terms. 
 
 (2.) o — o+Q — i'Q+ &c. to n terms, anil to infinity. 
 
11. Shew that 
 
 EXAMINATION PAPESS. 239 
 
 be, — nc, — ah 
 
 b^^c^, a^ + 2ac,-^r^ -2nb is divisible hy 
 c2, 0^, (a+i)3 
 
 FIRST CLASS TEACHERS, 1380-~Grade C. 
 
 XXVII. 
 
 1. Kin ax^+2bxy+ci/^, hi+lv be substituted for a; and W7?(+n^; 
 for y, the result takes the form Au^ + IBnv + Gv'^. Find the value 
 of (£2 — ^Q'^.^(j)2 _ac) in terms of k, I, m, n. 
 
 2. Eesolve a{b—c)^-\-b(c — a)^+c{ri-b)^ into factors. 
 
 Prove that = ^ 
 
 «i;i/; xi/z 
 
 i{ It = x{-By^ — Cz^) , v=-y(Cz^—Ax^), w = z{Ax^ — By^). 
 
 5. Extract the square root of 
 
 (a-i)2(6-c)2 4-{&-c)2(c-rt)2-f (c-«)8(fl_/,)3, 
 
 and tiy, cube root of 
 
 4. Eliminate a;, y, e from 
 
 ' -^ ' X y z 
 
 k{x"-i-y- +z^)+2[Ix-\-my+7iz) + /i = 0. 
 
 ^' Shnphfy ''|^1;|;^'^^ {l/(4 + 3i)+i/(4-3i)}2, 
 
 / -1+.?V3 \^ -14-JV3, ^ 
 and (^ 2 j + 2 +^' 
 
 in which j= \/{ — l). 
 
 6. Given the first term, the common difference and the number 
 of terms of an arithmetical progression, find (i.) the sum of the 
 terms, (ii.) the sum of the squares of the terms. 
 
240 «XA5fD:s,TI0N PAPEES. 
 
 7. Solve tbe equations 
 
 ah 
 (11.) aa;+%=— + — =1. 
 
 —1 -1 —1 
 
 (iii.) oc{y-\-z )=a, y(z + x ) = h, z(.r + y ) = ^. 
 
 8. What value (other than 1) must be given to q that one of 
 the roots oi x^ —2x+q = 0, may be the square of the other. 
 
 If a, b, c are the roots oi x^ — fx^ -\-qx — r, express 
 
 'iah + 26c+ 2ca -a^-h^ -c"^ 
 in terms of/-", q and r. 
 
 9. A vessel makes two runs on a measured mile, one with the 
 tide in m minutes and one a.gainst the tide in n minu^es. Find 
 the sjjeed of the vessel through the water, and the rate the tide 
 was running at, assuming both to be uniform. 
 
 10. Five points. A, B, C, and P lie on a straight line. The 
 distances of ^, B, and C, measured from the point 0, are a, h, 
 and c ; their distances measured from the point P are x, y, z. 
 Prove that whatever be the positions of the points and F, 
 
 «* {b - c) +2/ -' {c-a)-^i^{a-b)-^{b-c)l^c-u) {a -b) = 0. 
 
APPENDIX. 
 
 Section I. — Elementaey Theorems on Polynoaies. 
 (See page 39, et seq.) 
 
 Theorem I- If the polynome/ (ic)" be divided by x — ti, the 
 remainder will be /(«)". 
 
 D'Alembert's Proof. J\xY is the dividend, a; -a is the divisor : 
 let /i (a;) "~^ be the quotient, which is necessarily a polynome o* 
 degree n—1, and let R be the remainder. Then, since the pro- 
 duct of the quotient and the divisor added to the remainder re- 
 produces the dividend, 
 
 But R does not contain x, hence it will remain the same, not 
 merely in form but in actual value, whatever value be given to x.. 
 Take the case x = a, then («— «)/i (a?)"~^ vanishes for its factor x—a 
 does so. heucB R=:f{ay\ Thus the remainder is the value of the 
 dividend when x has the value which makes the divisor vanish. 
 
 It has been objected to the above proof " Division can be per- 
 formed only when there is an actual divisor, therefore in assum- 
 ing R to be the remainder ol f{xY -i-{x — a) it is assumed that a- is 
 not equal to a, and although R will remain unchanged for all 
 values of x that fulfil this assumption, it cannot thence be inferred 
 that it will do so if the contradictory assumption be made. In 
 such case the only legitimate conclusion is that there being no 
 divisor there is neither quotient nor remainder. Therefore, 
 although /(a)" may be the remainder in the case in which x is 
 not equal to a, yet the above argument does not prove it." This 
 objection confuses arithmetical or numerical division- with alge- 
 braic or formal division, division by a definite quantity with divi- 
 sion by an undetermined or variable quantity. The following 
 proof does not involve the assumption x-a, and consequently is 
 not open to the foregoing objection. 
 
242 APPENDIX. 
 
 Lagrange s Proof. Lemma, af* - a" is divis\ble by .* -i, if « 
 be a positive integer. 
 
 By actual division ——zr~'^ -_^---- - , 
 
 .-. «" -a» is divisible by x.—a if a;"-^-«»-i is so divisible, 
 hence u;'-^ - a"^^ " " x-a " ^''-^-^"-^ .... 
 Thus we can reduce the exponent unit by unit until at last we 
 arrive at, x^ — a"^ is divisible by x-a ii x—a is so divisible. But 
 x~a is certainly divisible by Itself, .-. x^-a^ is divisible by ic - <*, 
 .-. x^-a^ is also divisible by a; -a, .". so also is x^—a'*' and thus 
 we may go on to any positive Integral exponent whatsoever. 
 
 Theorem. Writing /(re)" m polynomial form arraugpa in 
 ascending powers of x, 
 
 f{xy=A^+A^x+A^x^+A^x^-h +A,a^, 
 
 .-. /(a;)"-/(«r=^i(a;-«) + /l2r.-/;3-a2)_^.^g(^3_a3)+ .... 
 
 +.l„(a;"-a"). 
 
 But every term of this polynomial is divisible by a; — a, and the 
 highest power of x in the quotient is a;"~^ got from the term 
 A^{j:" —a"), so the quotient may be represented by/i(a;)"~S 
 
 fix)" f(a) 
 
 x — a -' 1 ^ ' 'x — a 
 
 Theorem II. If the polynome /(^)'' vanish on substituting 
 
 for x each of the n different values a-^, a^, a^, . . . . «.„, 
 
 i\\Qnf{xY =A{x-a^){x — a^) {x-a„), 
 
 in which A is independent of x and consequently is the coefficient 
 of.7f in/(.t;)". 
 
 Since /(ai)=0, ■• fixY ={:x-a^)f (a;)"-^ In this substitute 
 a, ior X, :. ^ineef{a^Y =0, it becomtrf = («., -rtj)/i(a2)"~^- Of 
 this product the factor a^ —a^ does not vanish since by hypothe- 
 sis a^ is not equal to a^, therefore the other factor fiia^Y"^ must 
 vanish that the product may vanish, and consequently/j(a;)"~^ is 
 
APPENDIX. 243 
 
 divisible Ly x—a.,. Let the quotient I)e denoted by /2(.'»)''~^, .. 
 f{xY=(x — a^){:x~a,^f^{xY~". Substitute tig for ./• and proceed 
 as before, and it will be proved that x—a^ is a factor of /(jc)" . 
 Continuing to n factors we get a quotient independent of a;, since 
 each division reduces the exponent of a; by unity, .-. finally 
 
 fixY = Ai;x — a^){x-a^^ (./j - a" ). 
 
 Cor. \ij\xy- and <p(x)"' both vanish for the same r> different 
 values <Ax,j\xY is algebraically divisible by (^{xY^. 
 
 » 
 
 •Let «j, «2> ^3' «m 1^6 t^6 "'■ different values of x for 
 
 which the poiynome.s vanish, 
 
 .'.J{xf ={x-a^)[x-a^) {x-a^F{xY-^- 
 
 and ^(./')"' = .i(»; — «i («—<<„ ) (.c — a„) 
 
 .-. 1\xf -=-(Z)(.r)"' = F(a;)"-'"-4-J , 
 w.'iich is an integral function of a siuce A does not contaiu x. 
 
 Theorem III. If the polynome/(a;)" vanish for more than 
 n different values of a; it will vanish identically, the coefficient of 
 every term being zero. 
 
 Let (7 J, Oa,*a^ a„, «„+i be v+1 different values of a; 
 
 for which /(a;)" vanishes, 
 
 .-. f{xY =A(x-a^)(x-a2)(x-a^) (x-a„) 
 
 Substitute (1,1^1 iov X, and since /(a„+i)" =0, 
 
 .-. 0= ^l(«„+i - rt J )(«„+! - «._,)(rt„4.i - u^) .... (a„,+i - n„ , 
 But none of the factors On+i — a^, a„+i — a^, &,e.. vanishes, 
 ,'. A must be zero, or 
 
 f(xY =0{x-''ii)(x — a^)(x-a^) (x — a„) 
 
 and the factor, zero, will be a factor in the coefficients of every 
 term. 
 
 Theorem IV. If the polynomes f{xY , (p(.r)"' (n not less than 
 m) are equal for more than n different values of x, they are equal 
 for all values, and the coefficients of equal powers of x in each 
 are equal to one another. 
 
244 
 
 APPENDIX. 
 
 f(xY'^-A.,-i-A,x+A.,x^ + A^x^+ .... +A„x'' 
 (p{x)^ =B^-^B^z + B2x'' + B^x^ + . . . . +5^ic^ 
 .'.f{xr-<i>(x)"^ = A,-B, + {Ai-B,)x + {A^-B^)x^-{- 
 {A^-B^x^^+ .... +(J„-i?>- 
 + J,«+ia:"^+i-f.^^+2a;"'+2 -]-A„x\ 
 
 aud this is a polynome of degree n at most. i>nt/(a;)" =(p(a;)"' foi 
 more than n different values of x, that is /(a;)" — ^(a;)'" vanishes 
 for these vahies, .-. hy Theorem III. /(a;)" - (p{x)"' vanishes identi 
 
 cally, aud*the coefficients yl„ — 5o, ^ J— 5j, A^—B^, 
 
 A„- B^, A„i^'i, Am+i, ^„ are all equal to zero, 
 
 .*. Aq ~ Bq, a j =B^, A^ = lJ2,.--Aj„= B^, Afn^i =0, Afn4-2 — ^ ■■■ 
 
 Note to Art. XVII. To find, where such exist, the factors of 
 
 ax~ + bx;/ + Cxz+ey^ +fnjz+hz^. 
 Multiply hy 4a 
 
 4:a^x^ -^iaJ>x>j + 4:'^icxz+4:riey^ -^4:ar/yz-}-4:nhz^ . 
 Select the terms containing x and complete the square, thus 
 Aa^x^ +4:aJ)xy-i-4:acxz + b^ij^ +2bcxz + C^Z' 
 ■^(b2-4:ae)tj2-2{bc-2arj)yz-{c^-4.fth)2^ = 
 (2ax+b]/ + Cz)^ - {(62 -4:ae)y^ + 2(bc-2ag)i/z + {c^-4:ah)z^} 
 
 If the part within the double bracket is a square say {my + nz)^ 
 the given expression can be written 
 
 {2ax+bii + Cz)^ - (mf/+nz)' 
 which can be factored by [4J . Factor and divide the result by 
 4a. If the part within the double bracket is not a square, the 
 given expression cannot be factored. If b and c are both even, 
 multiply by a instead of by 4a and the square can be completed 
 without introducing fractions. If e is less thau a it will be easier 
 to multiply by 4e instead of by 4a and select the terms containing 
 y. A similar remark applies to h. 
 
 This method can evidently be extended to quadratic multino- 
 mials of any number of terms. 
 
appendix. 245 
 
 Examples. 
 
 1. Eesolve x- ■{-x)/ + 2xz-2i/-+7ijz-iiz'^ iuto factors. 
 Multiply by 4 
 
 4x^ + 4:X>j+8>jz-S!/^-{-^S>jz-12z^ - 
 Complete the square selectiug i-nns m x, 
 
 4a;2 + 4<:.y+8x-2+.'/2+4?/s+4z3-9^2+24j/z-l(;z3« 
 
 (2x- + 7/ + 2z)"-(32/-42)2 = 
 
 {(2x+v/ + 22) + {3?/-4^)}{(2a;+7/+2z)-(3^-4r)}=- 
 
 (2x-l-4?/-2z)(2x--2^+6z)=.4(^4-2^-2)(a;-^+3.^) 
 . • . the f iictors are (u; + 2// — s) (a; — // + 3z). 
 
 2. 6rt2 -7rt/;+2^tc-2062 + 64Z;c-486-2. 
 Multiplygby 4 x 6 = 24 
 
 14ia2-168a6 + 48^c-480/^2+1586&c-1152c5=s 
 
 (12rt-7i+2'-)^- 529^2 _^1504/>c-1156c2 = 
 
 (12„— 7i+2f)2-(236-34f)2= 
 
 (12rt+16?^-32r)(12a-306 + 36r)=B 
 
 24(3«+4&-8c)(2«.-5/^ + 6c), 
 .'. the factors, are 3fl+4& — 8c and 2a— 5/> + Gc. 
 3= a;2_|.i2.r;/+2x2+26^2_8^2_923 = 
 
 (a;2 + l2x//+2x-2 + 3Gy2 4.12//Z +2-') _ 10^2 _ 20^2-102'* = 
 
 (x+6y + z)--{0/+2)v/10}2:= 
 
 {:r+(6+ v/10)^+(yiO+l)z} < 
 
 ^a: + (6-A/10)y-(i/10-lM 
 
 4. 3a2+10r^6-14«p+12«rf-8/>2_8W+8';2_8(;f/. 
 
 Multiply by 3, not 4x3, since the coefficients of the other terms 
 in a, are all even, 
 
 9rt2 -i- 30ai - 42«c+36a6i - 246^* - 24^*d + 24c 2 - 24efi. 
 
246 
 
 APPENDIX. 
 
 Select tho terras containing a and com^Jotc the square 
 
 706c - MIkI - 25c ' + GOcd - 3Gd^ - 
 (3a + 5h-7c-{-Gd}^ - {Ib-iic + Gd)^ = 
 {3a + 12b-12c + lM){'da~2b~2c) = 
 ii(a-^U-4:C + 4d){3a-2b-2c), 
 
 ,•- the factors are a + ib —4c-{-id and da — 2b — 'Ac. 
 
 Work lixercise XXIX by this method. 
 
 m p m _ 
 
 _ P 
 
 q 
 
 
 m 
 . b~n 
 
 nt m 
 
 {a -f h)Ti = an - 
 
 m 
 -ITn 
 
 m p mp 
 
 (a'n )'q = am 
 
 
 Section II. — Indices and SuPvDS. 
 The general Index-laws are 
 
 in p m i_ p 
 
 an , a g = a n V (1) 
 
 (3) 
 
 (^) 
 
 {-) 
 Tho law connecting the Index and the Sura byiubjls is 
 
 m 
 
 oJ = ^"/(a™) (G) 
 
 [The indices 1-, i, i, &c., are generally used to denote ' either 
 square-root,' ' any of the cube-roots,' ' any one of the fourth- 
 roots,' &c. 
 
 The surd symbols -j/, -^, V, &c., are by some writers re- 
 stricted to indicate the arithmetical or absolute roots, sometimes 
 called the positive roots. Thus 
 
 ^4 = 2, but4*=±:2, .-. 4^=±y4 
 Also, V{{-2f-}= i/4 = 2. 
 
 lf27 = 3, b'lt 27^= 3 or 3/ ^ j /. S ' =(]/')-3/^': 
 
 J. 
 
 i ,X 
 
 4/16 = 2,but IG =+2or ±2j, .-. 16" = {r)t/16 
 
APPENDIX. 24V 
 
 With this restriction the geueral connecting formula -wonltl be 
 
 a7 = (111) ;'(«"•) 
 In the following esercises this restriction need not be observer,] 
 
 EXEKCISE. 
 
 1. What is the arithmetical value of each of the following : 
 
 8G*, 27^", IG*, Q2\ 4^ 8', 27*, G4^, ?.2', 64*, 81*, (3f)^ 
 
 i I h I -5 -2 -75 
 
 (5:Vr) , (IfV) , (-20)-, (-027)% 49 , 32 , 81 
 
 2. Interpret ar'^, «», a^' , («2) ^, aS ", r^, («"')-% a^ a"*. 
 
 3. What is the arithmetical value of 
 
 36~', 27~*, (•1G)~^ (-OOIC)^, (1)~^ {4^)~^', {^%)~\ {5^\)~^ 
 
 1 -L 
 
 4. Prove {iry = (a")" ; [cC^f = (a" )"^ ; a"^ = (a-^)"* ; 
 and express these theorems in words. 
 
 6. Simplify a J, c^ x^ ,ni .vi'^n .n~^^\ (7i)*-(2^)*-(Bl)* 
 a c a e e 5 1 \h 
 
 1' T ^' "T"' "^' (2fr .(G^r-^(i) 
 
 a c a ex 
 
 6. Eemove the brackets from 
 
 (ae)^ (Z;)"^, (cJ)~^ ('^V', («~^A (.r^)~^ 
 
 {an--)\ {al}y, («.^c-i)~*, (a--'^/')~^ (a;'^/"')""^ 
 
 7. Eemove the brackets and simplify 
 
 1 .13- 1 1 2 1 1 i / 1 I \ 2 / 1 1^2 / 1 1 \« 
 
 (.« ) (x* ) [X ) ', X X X ; 
 
 r A' ; a? ;r : 
 
 {x''~' -^x-^~' ] {x''--''-^x-- ^}. 
 
248 APPENDIX. 
 
 8. Sim^\i{j-x{x~^{-x)-^}^, x{{-x)~^{-x)-'-}~^, 
 
 { — x) {x~^x } 
 
 9. Determine the commensurable and the sua.-cl factors of 
 
 12% 24^ 18"^, (-81)*, 12^ 64^ {^^J, {Gif^. 
 
 (The surd factor must be the incommensurable root- of an 
 integer.) 
 
 10. Simplify 8*-fl8*-50^72V(T¥;5/-(To5)'"^i 
 {(6 + 2")(6-2^)}S (2V3V+(^'-3S'; 
 
 (2 +3^)(r+9^-6^); (7 -3^)-(7" + 3-)-. 
 
 [{{a-^x)[x + h)y'- {{a-x){x-h)}^] 3 ; ' 
 
 {a'+(a3 a;3)^[* . {J -{a'-^ -x^)^}^ 
 Express as surds, 
 
 i.1. a , x , jj , c , h 
 
 n + ^ _^ + § .25 -« + ,^ 
 12. a; , ^/ , a , b 
 
 a m—H n~^ 
 
 18. (ax-bf,{x^-4:x+l) 4 , (p -7:^) *• 
 Express with indices, 
 
 14. if a-, Vc^ "/a;"', if/r-", Vf^/.-K), 7/1-* 
 
 15. -^(a^+iB)^ ^(^34.^3)2, {ir(«3+63)p, .-y|^^_;).^>, 
 ;/(a-ia;)"-\ V (a" -Z>" )'"-'" 
 
 16. (rt ) , (6 ) , (c ) , (x- ) , (a^x) , {a ^x ) , 
 
 1 — i 14 
 (a? v/ ') . 
 
 Simplify the following, expressing the results by borb v^^-Vk- 
 tions, 
 
APPEND1.V. 240 
 
 1 X ■ t 2 1 
 
 17. ii-(i' ",a^.a ', a .a , a.a , a '-^(1,0,' ■^"'n^ , a ^a 
 
 a h c . a b c 
 
 •^ "^ *^ a; y oc- c[ab) 
 
 X i 3 3 -^n Sn 
 
 It' + a " a —a * a~ ^ — (t^ a'-+l + a~'^ 
 
 1Q - - , ., I 
 
 4 —h h —h _"~ -" « + l + «~^ 
 
 a —a a — a a '^ -\-a'^ 
 
 1 1 * 3-2- ■ ,■' \ I ¥ 
 
 20. Divide x-y by a;" — ;v^; ic -\-a x" + a by "-+-« .c -f'* ; 
 ic+.'/H-^— 3.C // 2 by* +?/ +;s 
 
 i .\ i 
 2a6+2/;(;+2c;a-a!»-fe2 -q- by <(" 4-^''+f 
 
 ExERCISE. 
 
 1. Express the following quautities i. as quadratic smds, ii 
 
 as cubic surds, iii. as quartic surds. 
 
 ' J. __ " 
 
 a, 3a., 2a2, a-x, a;", y . a~"\ -, mx p, •!, '01, I'l.c'-. 
 
 y 
 
 2. Eeduce to entire sui'ds, 
 
 xVx,a^a, l>^^h\ 3if3, 4if2, V^, ilT^, if%, ^\ 
 
 
 n + 
 
 xJ^y I /j- + ?/\ 
 
 .'//. 
 
 
 (x-y) ''■^{x^+2xy + y^} ^ (x-x ')f{x^ + iy, 
 
250 APPENDIX. 
 
 3^ Eeduce to their simplest form 
 
 1/12, 1/8, ^,^50, -^16, 4if -250, Vh ^h i/A. 5ir(-320), 
 
 ,/{a3(l-^^)[, ^r,,2(,,._i)4|, ^(,,i), »^,,.-.i^ »,^»+»^ 
 y«^«-^^ I'ya'™-, V('f2x + «3), ^(a34.2.,4^^,,5^2)^ 
 l/{(x-l)(^2_i)|, -^{(a2 + 2ffa;+;c^)(«3+;c3)}, 
 
 ,/(8.c3 -16^+8), ^{{x^-^+x-^){x'-'lx^-^l)f, 
 
 N \~^+2+.<;-i / N \27j;2 +18u; -r 3 N ( ^^ j 
 
 4. Compare the lollowing quantities by reduciug tlicm to the 
 same surd index : 
 
 2 : ^/3 ; 2 : -jf 9 ; 1/2 : -^3 ; ,/10 : -,^30 ; 2 ^/2 : ^^22 ; 
 a2 : Va^ : V-^ : iy// ; ^x : ^// ; '"^£' . y.c^; i/a : 'yb : ::/c ; 
 
 5. Reduce to simple surds with lowest integral surd index 
 
 Vif'a), f^{i/b), ^{^c), ^{i/x% t/ifx'), 4/(r-^")» 
 ' f'it/'^''), 1^(1/27), A/(ir81), irif/81), yi.n/a), 
 ■ f{aya), VWx), iTC^'-yf), iflov/S), |/{3-^'3), 
 V(3#/3), e'(^C/-^), V{a^'{h:yc)], xV{x-Wx-'), 
 y^{y-'fr''),z^/{z-''-fz-^-),y{x:y{yiyz)],x-^f{x"^Vx^) 
 
 6. In the following quantities, combine the terms involving the 
 same radical ; 
 
 3|/2+5v/2-7a/2; -/8-'/2; ^nQ+Qf^; 
 
 ^16+ \/ 2; cq/aj-i/ic; a'2/x—bl/x; 
 
 8^/a + 5Vx-7i/a+ V{4.a)-3y{Li-) + iV{dx) ; 
 
 /.c;+3/(2.f) - Zy'{dx)+y^{ix) - v' (S^j + v'(12..) : 
 
APPENDIX, 
 
 251 
 
 4.\/{a^x) + 2^(b^x)-Sj/{{a + h)^x\; 
 
 ^/{{a-b)^x}+^/{{a+by-x\-V{a^:^^ + l/{{l-a)^x}-Vx; 
 
 ^/ (a _ 0) + ■,/ {lQa-Wb) + i/{ax^-bx^)- V { 9 {a - b)} ; 
 
 l/{a3+aV>)—i/{b^+ab-^); 
 
 j/{a^+2a^b + ab^)- ^(^a^ -2a"'b-Jrah^) - i/{AaJr-). 
 
 7. In the following quantities, perform, as far as ]DOSsible, the 
 indicated multiplications and divisions, expressing the results in 
 their simplest forms : 
 
 ^/2.|/o; a/3, a/12; |/14. a/S-j.-j/IO; j/a.y'[3a); 
 
 l/c.A/(12c); V{Gx)y.{8x); Vy^Vu^ ; -^y' -V^u' I 
 
 fa.^a^yb: iA.a/(^); Va.vi}]; l/-3^'- y (£) ' 
 |/r/"+i.-i/«"+^ ; f/b^^\^lr-^^ : v/12-- A^3 ; ^/{Qx) - x/(2a;) ; 
 
 {a^x)~V{a-\-x); (a2-a;2)_=.v(a-aj); (.^^ _i)_^^y(a-^l)3 . 
 
 (3v/8-6s/2+>/18+\/32+x/72-2v'50).i/2; 
 
 (7i/2-5 A/r)-3v/S-}-4 a/ 20) '/18) ; (v'5+i/3)(^5-v 3) ■ 
 
 (i/2+1)(s/6-a/3)- (3-V2)(24-Bv/2); 
 
 (5V3+v/6)(-V2-2) ; {^n- Vb){ya-\-y^b) ; 
 
 («y'/;4-?,yrt)(/; Va-aVb) ; 
 
 { ^/(:/;-hl) + v/(a;- 1)} { a/(^;+ 1) - n/(x- 1)[ ; 
 
 {A/(3rt-/>) + |/(3&-a)}{v/(3ff-&)- A/(36-rr)}; 
 
 V(<' + ^/b).V{a- Vb); V{^/x-\- ^ ij). V^x- Vy) '. 
 
252 APPENDIX. 
 
 {a + y\l -«2)13 . !y^a + b-.r)-y'{a-h + T)}^ ; 
 
 [{V(«+^)(x-Z>)} + v/{^z-a:)(;.+i)}]2;|4i^)-4|^)[' 
 W {{a-^x){x+h)] + ^ {{a-x){:c-h)]Y^ ', 
 
 {V^V10+1)-V(V10-1)}2; 
 
 ■\y{a-\-^{a"- -a:-')}4- V{^/ - V(«2 _a-2)l j^ ; 
 
 (V./;+Vy)* + (V.^- v/.v)4 : [a^+ah^2+h2){a^ -ah s/'l-^-b^); 
 
 B. Find rationaliziug multipliers for the following expressions, 
 and also the products of multiplication by these : 
 
 a-{-Vh, \/a-\-l\/c, aVb — bVa, a + V{a^ —x^), 
 
 V(a-x) - V(ri+x), V(a^ + Vr)+\/(a2 -Vc), 
 
 V{8 + V(24 + Vo)}-V{8 + V(24-V5)}, V-v + V^ + Vc, 
 
 3 + V2+V7, v'6 + V5-V3-V2, ^a + J-b + ^c + ^d, 
 
 y{l+a)~^{l-a) + y(l + b)-^{l-c), f/a+f/c, 
 
 yx + l + ^x-^', y{aI-')-^{a-'b), f-^ + fS-fB, 
 fu + f'b + ^c, a + ^b + fc. 
 
APPKNDIX. 253 
 
 « 
 
 9. Eationalize the divisors and the denominators in the follow- 
 ing, and reduce the results to their simplest form: 
 
 l^(2-^/3), 3--(3+i/6), 5-^.(■^2 + V7), 
 
 (,/3+v/2)-=-(v/3-i/2), (7|/5+5|/7)--(|/5+V7), 
 
 «-=-(s/rt4-a), {x — a)-^{-i/x — i/a), 
 
 aVx + bVii 2V6 1 + 3 |,/2-2a/3 
 
 TVsT-^^' 1/2+1/3- V6' 1/2'+ 1/3+1/6' 
 
 a/6 -1/5 -1/3+^2 2 
 
 • V'6+i/5-a/3-V2' i/(fl + l)- A/(a-l>» 
 
 2c a.+ a;+\^(a2+a;2) 
 
 |//(rt-j_c) + i/(a-c)' a+x-v\a^-\-x^y 
 
 ^/{a-\-x)-^V^a—x) 1 
 
 V(a+a;)-V(a-a;)' ai/(l + h^)+b ^Jl'^aFy 
 
 \/(l-62)+ i/(l-a2)' av'(l-c-^) + cV(l-a2)' 
 l/ {(l+a)(l + Z>)}-A/{ (l-a)(l-6)} 
 . l/{(l + «)(l + ^)}+l/f(l-«)(l-i)}' 
 
 (a-x)i /(&^ + //2)-(6-y)i/(a2+a;2) 
 («+a;)i/(62 + 2/2) + (6 + 7/)i/(a2+^2)» 
 
 A/( l + a)- l/(l-a)+i.l /(l+//)-vX l-^>) 
 l/(l + a)+ v'(l-a) + s/(l+/>)+V(l-^)» 
 
 % /(x + fl)-N/( y -a)- ^/{x+b ) -\-^{x - ft) 
 l7(u; + «)+ V{x-a)-\-y{x+h)+ V{x-~by 
 
 V6"^l/a' ^U-W ^U+W' Na-V-C' ^Vj^Vy' 
 1 1 a/« \/a; 
 
 •4r-i/(a3-l)' 1 1 ' A/a v/a;' 
 \/j; •«/ V \/ic A/a 
 
254 APPENDIX. 
 
 10. Find the values of the following expressions for n=l, 2, 
 3, 4, 5, respectively. 
 
 V5\ \ 2" / - \ 2 / [• 
 
 1 f(2+ i/6)"+^-(2+V6) (2-\/6)"+^-(2-VG) ] 
 2761 1 + 1/6 ~ 1-V6 I 
 
 11. Show that 
 
 2(x — 1) 
 is a square for n=l, 2, or 3 respectively. 
 
 12. Extract the square roots of 
 x+y-<2.A/{x}j), a-{-c+e+2V{ac-\-ce), 
 
 a+2c + e + 2^/{{a+c)(c+e)}, 2a+2V{a^ -c2), 
 
 2{a2-|-63-^/(rt4+a2/>2+64)}, x-^ + x'^, 
 
 ^x-\-2 + Vx-^, x+3x^ +x^+2x^/x-^2x^ y/ X, 
 
 x^-xi/ + iy^-hV{Ax^y-8x^-ir'+xy^), 2x+^ {?,x^ -y^), 
 
 5-2i/6, 10+2\/21, 9+4a/5, 4-v/15, 7+4 v'3, 
 
 12-5V6, 70 + 3;/451, 4-x/15, 
 
 9 + 2;/6+4(i/2+v'3), 15.25- 5 V.6. 
 
 13. Find the value of 
 
 • ^^¥+te^'gi-ent.l=:j^^^^^andy=^,^, 
 y(a;2+i/2), given x=^{a^c) y = ^{a^e)-y 
 
 V(l +x)-V(l-a; ) . _ 2ah 
 
 V, r+;^V(l -a;)' ^^^^ ^ - «3 +/;2 J 
 
 2«V(l + a^) _J |l_ l£l. 
 
APPENDIX. 
 
 255 
 
 14.. JiV{x+a-{-b)-rV{x + c-td) = y/{x-i-a-c)+V{x-b-\-cl), 
 
 *(1.4-V5)a;-2 ^(l-V5)a;-2 
 
 15. Simplify :,y^^i^v5)x+l'^^-h[l-Vo)x-^l' 
 
 Complex Quantities. 
 
 Quantities of the form a+bV — l in which neither a nor i 
 involves \/ — 1, are called complex quantities. The letter J (or i) 
 is frequently used as the symbol of the ditensive unit •/ — 1, so 
 that rt + ij/ — 1 would be written a + hj. So also V —x^JVx, 
 y/-x.V -y ^p_ V {xy) = - Vxij, and p = -j 
 
 EXEKCISE. 
 
 Simplify the followmg, writing j for >/ — 1 iu any result in 
 which the latter occurs : 
 
 1. y_4, |/-3G, 1/-81, V-8, i/-12, 1/-72, ^-8, 
 v/-5.i/-G, V-QV-8. V-8.VI-2, ^/-8.lr-8, 
 l/-5.j/-20. 
 
 2. /-x, i/-a;^ v/-a3, ^-a^"*, V{-a)^ ■/(-a)^, 
 
 ■«/5. \/ — rt. 
 
 Q o2 ;3 i4 iS ,;9 -;i5 ,*1G ,-17 ,'18 ,'4'> ,'ln+l -/te+2 „'4n+3 
 
 4. cj-bj, jVx.Ji/y, 5j, j-^i/5, ji/~a,ji/-a^, J-i/a.-i/ -a. 
 
 6. V-r, V-p, i/-j\ l/-i^ -/-i^", i/-p\ 
 
 ^_6 a/-6 |/G i/a |/ a 1 
 
 6. 
 
 V 3 ' V-3' i/-3' i/-/.' -i/-6' ■/-!' 
 a a^ \/{—ax) —V-1 a^ 
 
 y/ —a v/— a^ y' — x V — a ' f/ — 
 
 a^' 
 
256 APPENDIX. 
 
 J_ J_ J_ -JL 2_ _J_ ^iL ^ 
 
 ^' j ' P ' F' i ' j' ' i*"+'' i^"^'' i^"-'' 
 
 
 2 
 
 8. A/(a-Z,)V(/-'-a), a/(3*-4?/)V(47/-3x), (3 + 5;-)(7 + 4j), 
 tS-9i)(8-7;), (7-jV5)(7+jVlO), (v'3~iV6)(V2-jVU), 
 
 (/a4-il/6)(l/a->^/c) (a+/i/'j («-/;;'), («;' + &)(«y-&), 
 ( v'"+y|/^)(/a-iV'6), {(iVl)+GJy/x){a]/b — (jVx), 
 l/(l+yjV(l-yj, V(3+4i)V(3-4i), 
 v/(12+5;jVil2-5;-), (l+i)3, (ya-jV6)3, (5-2JV6)", 
 {a + hjf+[a-hjy\ ^a + bjy^-ia-bjr-, (a+l>jy + {aj-by, 
 {^/(4 + 3Jj+v/(4-3i)}^ {^(3-4;-)-v/(3 + 4i)}^ 
 
 [i{y(30-6N/5)-l-i/5} + i;{N/15+ ^/3+ ./(10-2|/5)}]» 
 tor all positive integral values of n. 
 
 4 64 21 5 1-20;V5 " 
 
 1+JV3' 1 -iV7' 414- 3;- V6' V2+Jv3' 7-2;V5 ' 
 
 i+yv3' 1-/ i+i' i-i' (i+j)3' i-y' u;-^/ 
 
 a+jy/x JV(i+V-b a — hj (i-j-jv'(l-£2) 
 a—[y/x \' -<t-j\/b aj-{- 1) a-jV\^ 1 - x - )' 
 
 9. 
 
APPENDIX. 257 
 
 V{x-y)-V(>/—x) 1 1 l+i 1—,; 
 
 v{y^x)+v[x~^y 1+7+ r^'' i-j+i-h/ 
 
 1,1 1 1 x-^yj ^' — vj 
 
 (l+^-)2-^(l-j)3' (l+ij_4 (i__^-)4' «_|_j;-r^, _ij 
 
 a; + yj a; — y j Vx +JV1J Vjz+jV^ 
 a -\-hj a —hj Vx—j^/ij ^y—jVx' 
 
 y(l +a)+7V(l-a) _ y(l_--«.) -|-^-y(l+^) 
 
 y(i+«) -Ml - a) y(i -rtO -jV(i +rt)' 
 
 10. ^/(3+4y)+|/(3-4i), ^(3 + 4i)-A/(3-4y), 
 
 v^(4 + 3i) + v/(4-3i), |/(l+2;V6)±v/(l-2;V6) 
 V(5+2;V6)±a/(5-2jV6), 
 v/(2,/15 + 3q;)±v/{2v'15-30j), 
 A/{x/3+iv/105)+ A/(i/3-ii/105), 
 
 11. Prove that both i(-l+j|/3) and i(-l -jV3) satisfy the 
 
 a;^-l 
 equation . _^ =0, 
 
 that {x-\-icy -^-w-zY = x^ ^-y^ -\-z^ + ^{x + wy){y\- wz){z-{-n-x) 
 and that (a^ + ^ +«)(«+ it;^-}- it; ^z) (a; +m;^ ?/+«;«) = 
 x3-j-?y3_}.23 — 'dxyz, vo. which 10 represents either of the pre- 
 ceding complex quantities. 
 Hence, prove that 
 
 (i) {2a-&-c+(5-c];V3}3 = {2&-c-a+(c-«)j/V3)}3=: 
 {2c-a-6+(a-%V3}3; 
 
 (ii) u^-\-v^-\-w^ —Quvii:=-{a^ +h''^-\-c^ — ^ahc)x 
 
 (a;3 4-7/3 ^gS -3a;!/2), if u = ax-{-by+cz, v = a^-f &2'+c3«j 
 u; = az + bx-\-cy, or if w = rtx-f-c?/+/;2, 
 t; = ca;+% + rt2, w = hx-\-ay-\-cz. 
 
258 
 
 APPENDIX. 
 
 12. Prove that i U/^ + t+j v' (10 — 2 -/S)} satisfies the equation 
 
 x^ + 1 
 x+l 
 
 Writing v for tlie preceding complex quantity, prove that 
 (7+'r+?<'2_|_:3„,3)(7_ti.4_„;3„3^(.2) = 71^ 
 
 and {x ■}- >/ -\-z){x+w^y — ir3z){x - iv^jj — icz)(x - icz-{-rv*^z) 
 (x-^w^ij + ic-z) = x'^ +1/''' +z'' - 5x'''yz+5xy^z^. 
 Prove that {4« + (/;-,-)(,/5 - 1) + (6+c)jV(10 + 2 \/5)}5 = 
 
 {[{a + b){-l+jVii/5 + 2)} + ia-b){V5+jy'(V5-2)}] 
 X y5-4c}^ 
 
 Section III. — Pure Quadratics. 
 
 Examples. 
 
 J x + S(a -b) _ a{Zx + % a -lb) 
 x-S{a-b) ~ b{3x^a+\9b)' 
 
 m p m-\-n X''^^ . 
 
 Apply, if — = — , .". = 5 
 
 ^••^ •" n q m — np — q 
 
 X 3.r(ff + ^>) + 9«2_i4,,?; + 9J2 
 
 '*• B[a-h)^ 3a;(a-6)4-9(a2Tr^ 
 
 Dividing the denominators by 3(a — b) 
 
 .-. a;2 = 9a2_ 14^6 + 9^3 
 
 'a;_2ff-f-4M 2 5a;-9r7+3& 
 
 x+4a-26/ bx + 'da-Sib 
 
 m p n — m q—p 
 
 Apply, if — = — , .'. — - — = — — , and factor the numerator 
 
 (a;+4a-2i)3-(a;-2fl + 46)2, 
 
APPENDIX. 25i) 
 
 mx-\-aA-h){a -h) \^a-h)_ 
 (a;+4«-2i)3 -5^+ 3a- 95* 
 
 x+a-\-b .T4-'4a-26 ^a-'h) 
 
 :. 7^ = T- — ,—7^ Ki = "a ^. oy taking differ- 
 
 •• x-{-^a-^h 5a;+3a-9Z; 4ai-a-lV ^ ° 
 
 ence of numerators and difference of denominators. To the first 
 and third of these fractions, apply if 
 
 m p m p ^ 
 
 n ~ q ' " ** — m~~ q—p 
 
 x-\-a-\-h ^a — b) 
 
 ''• S{a - b)^'4x--4:a — A¥ 
 
 .-. x2^x^i(^a + b)"}+d{a-by^}. 
 l/{Sx-'- l)-\-V{B- x'- ) _ a_ 
 
 3a;- -1 a-{-b' 
 
 4 
 
 3_a;2 -a-b' 
 
 • 3a;2-l (a + b)^ 
 " 3-x' ~{a-by' 
 
 S{a + h)^ +{a-b)^ a^ ^ab + b ^ 
 •■• ^^ = '(^j^Ij)2 ^ s^a-by-^a^ -ab + b^' 
 
 4. 7nV{l+x)-nV{l-x)=V{m^+n'i) (1) 
 
 Square both members and reduce 
 
 .-. {m^-n^}x-'imnV{l—x-)=0. (2) 
 
 Transfer the radical term and square both members, 
 
 .-. (m3_^3)3a;3=4^2„2(l_a;2) (3) 
 
 .-. {m^-{-n^)^x"=4:m^'fi^ (4) 
 
 The above follows the usual mode ol solving equations involv- 
 ing radicals, viz., make a radical term the right-hand member 
 gathering all the other terms into the left-hand member, square each 
 
260 APPENDIX. 
 
 member, repeat, if necessary, until all radicals are rationalized. 
 This method is convenient but it does not explain the difficult^ 
 
 that only one of the values of x in (4) satisfies (1) viz. ~jT~Va 
 
 — 2m7i 
 The otber value, „ , ^ satisfies the equation 
 
 w \/(l 4-a;) +?i a/(1 -ic) = l/(m2 + /, 3). 
 
 The exi")lauation is simple. Squaring both members of (1) is 
 reully equivalent to substituting for (1) the conjoint equation 
 
 {wV(l+x) + /.V(l-a;)- V(/?t2+n2)|^0 (5) 
 
 which reduces to (2) above. 
 
 Treating (5) or (2) by transferring and squaring is equivalent 
 to substituting for it, the equation 
 
 {m V {'i- -i- x) - 71 V {1 - X) - V{>ii'^+n")^- X 
 
 {7«|/(1 +a;)+» \/{l -x)- v/(m3 +^2)} x 
 
 {mA/(l + x)+W|/(l-a;)+i/'(m3+w3)} =0 (6) 
 
 ^s'hich reduces to 
 
 Urn^ -n^)x-%nny/{l-x^)] {m'' -n^).c-\-2mnVil-x-} =0 (7) 
 which further reduces to (3) 
 
 Thus the whole process of solving (1) is equivalent to reducing 
 it to an equation of the type ^ = and then multiplying the 
 member A by rationalizing factors. Thus instead of solving (1) 
 we freally solve (6), i.e., a conjoint equation equivalent to four 
 disjunctive equations. (See page 140, Art xl ) Now the values 
 given in (4) will satisfy (6), the positive value making the first 
 factor vanish, the negative value making the' third factor vanish, 
 while no values can be found that will make either the second or 
 the fourth factor vanish. 
 
APPENDIX. 
 
 261 
 
 Hence, if one of such a set of disjunctive equations is proposed 
 for solutiou, the conjoint equation must be solved, and if there be 
 a vakie of x which satisfies the particular equation proposed, 
 that value must be retained and the others rejected. 
 
 (This process is the opposite to that given in Arts. XL. and 
 XLV. : there a conjoint equation is solved by resolving it into its 
 equivalent disjunctive equations. The two processes are related 
 somewhat as involution and evolution aie). 
 
 Further, it should be noticed that just as there are four factors 
 in (6) while there are only two values in (4), it will in general be 
 possible to form more disjunctive equations than there are values 
 of a; that satisfy the conjoint equation, and consequently it will 
 be possible to select disjunctive equations that are not satisfied by 
 any value of x, or, in other words, whose solution is impossible. 
 
 This will perhaps be better understood by considering the fol- 
 lowing problem. 
 
 Find a number such that if it be increased by 4 and also dimin- 
 ished by 4 the difference of the square-roots of the results shall 
 be 4. 
 
 Keduced to an equation this is 
 
 ^(a;+4)--/{a;-4) = 4 (8) 
 
 Eationalizing this'becomes 
 
 {4--/(a;-H4)+v/(a:-4)}{4-/(a;+4)-i/(x-4)}x 
 {4+v/(a;+4) + i/(a;-4)}{4 + i/(a;+4)-V(;c-4)}=0 (9), 
 which reduces to 
 
 {24-8v/(a;+4)}{24+8i/(a;-f4)} =0 
 ie. 9— (a;-+-4) = 0, oric = 6. 
 
 Now »= 5 satisfies (9^ because it makes the factor 
 4-|/(a;+4)-A/(a;-4) 
 vanish and it is the only finite value of x that does satisfy (9), or, 
 in other words, there are no values of x which will make any of. 
 the factors 
 
262 APPENDIX. 
 
 * 4-i/(a;+4) + |/(a;-4), 4+ V{x^^)+ \/{x-4.), 
 or 4 + A/(a; + 4) -V (^ - 4) 
 
 vanish. There is, therefore, no number that will satisfy the con- 
 ditions of the problem. 
 
 [It will be found that as x increases, ■j/(.'c+4) -|/(a;— 4) 
 decreases, hence as 4 is the least value that can be given to x 
 without involving the square-root of a negative, the greatest real 
 value of -i/(x+4) — -/(a; -4) is -j/S which is less than 4. We see 
 by this that our method of solution fails for (8) simply because (8) 
 is impossible] . 
 
 5. V.{{a + x){b+x)}-i/{{a-x){b-x)} = 
 ^{{a-x)(b+x)} - V {{a-hx){b-x)} (1) 
 
 Collecting the terms involving i/{a-{-x) and i/(a-x) respec- 
 tively the equation becomes 
 
 {l/(a + x)-y/{a-x)}{^{h + x)-i-V{b-x)}=0 (2) 
 This is satisfied if either 
 
 l/(a+x)-i/ia-x) = (3) 
 
 or Vib + x)+\/{b-x) = (4) 
 
 The rational form of (3) is (a + x) — {a-x) = Q which is satisfied 
 by a; = and this also satisfies (3). 
 
 The rational form of (4) is {b + x)-(b—x)=0 which requires 
 x=0, but this does not satisfy (4). Hence the second factor of 
 the left-hand member of (2) cannot vanish. 
 
 Therefore the only solution of (2) and /. of (1) is a; = 0, derived 
 from (3). 
 
 6. f{a+x)-]-^ia-x) = f{2a) 
 
 Cube by the formula (u+v)^ = u^ -\-v^ -\-Snv(u-{-v) 
 
 :. {a+x)-h{a-x) + 3f'{2a{az-x^)}='2a. 
 .-. 2a{a^-x-^) = 0, 
 :. x=±a. 
 
 Both these values belong to the proposed equation. 
 
A1>1'ENDIX. i 
 
 The rationaliziiog factors of 
 
 f/^a+a-)-{-^{a-x)-f/{2a) = 
 are -^{a + x)-\-o>]V{" -x)-(^)^-^{2a), 
 and i/(a4-a;) i-o^-f/{a -x) -(^^y{2a). See page 257. 
 
 The remarks on Ex. 4, will apply wwfaiis mutandis io equation •; 
 of this type. 
 
 ^' f/(a+x)^ -f{a2-x^-)^+^{a-xy "' ^^' 
 
 Assume -^{ft+x) = w and ]^(a— a;) = v 
 .'. u^ +v^ = 2a and u^ — v^ = 2x, 
 
 and .'. o , ^ -0 = —- (2) 
 
 Also (1) becomes 
 
 ll^ + W + V^ 
 
 = e (3) 
 
 m2 —UV+V^ 
 
 U — V 
 
 Multiply both members by ^— ^ 
 
 ^3 — y3 u — v . ,^^ X u—v 
 
 Again adding and subtracting denominators and numerators 
 in (3) 
 
 u^ -\-\v^ c+1 
 uv ~ c — 1 
 
 Adding and subtracting 2 (denominators) and numerators in this 
 
 ti^-2uv4-v^ 3-c /M-r\ 2 B-e 
 
 or ' ' "~ 
 
 M2_j_2Mu + t;3 3c— 1' \u-\-vl 3c- 1 
 
 a-2 3-c 
 
 .'. substituting by (4), -^ = c^q^_ i* 
 
 |8-r 
 
264 APPENDIX. 
 
 8. ^V{x + a) + V{x -a)}^ {r{x+a) - \/{x--a)]^1c {\) 
 Assume u=\/(x-\-a) and v='^y{x--a), and (1) becomes 
 
 (?i+?.-)3(M-r) = 26- or (?t+i-)2(?i2_r2) = 2c (2) 
 
 Also ?t4 -r4 = 2a or («3 + y2)('u3 - ^2) = 2a ■ (3) 
 
 and u'^-\-v'^ = 2x. (4) 
 
 From (2) and (3), {u - v) 3 (nS _ t-2 ) = 4^ _ 2c • (5) 
 
 .-. (2)X(5), (l(.2_t-2)3(^2_^3)2or (M3-t;2)4 = 4c(2«-c) [<6) 
 
 Also (3) 3-1- (6), 
 
 |(M2 + ,.3)3_^(jt2_^,3)2|(ji2_^3)2=4(«2_|.2aC-c2) 
 
 or (?(44.^.4)(^i2_^,3j2 = 2(a3 + 2ac-c2) 
 Substituting by (4) and (6) 
 
 ^a;|/(2ac-c2) = «2_|_2ac-c3. 
 
 Exercise. 
 
 1. (a;+a + 6)(x- a + &) + {x + a - h){x -a-b) = 0. 
 
 2. {a + bx){h — ax) + (/; -(-oa;)(c— hx) + {G+ax){a—cx) = 0. 
 
 3. {a-^bx){ax—h)-\-[b-{-cx)[bx — c)-\-[c + ax){cx — a) 
 
 = ^{a^-{-h2+c^). 
 
 4. (a+a;)(i-a•)-4-(l-fa;«)(l-^*a:) = (a + ^))(l+•^•3). 
 
 o. {a-rx){b-{-x){c — x)-{-[a-\-x){b — x){c+x) + {a—x){b+x){c + v) 
 
 + {a-x){b-x){c+x}-\-{a-x){b-^ x) (c -x)-\- 
 
 [a-\-x)(b — x){c — x) = 5abc. 
 J. {n + x){b + x){c + x)-\- {a +x){b + x){G-x) -^{a+x){b - x){c + a;) 
 
 + [a—:x){b + x){c-\-x)-\-{a+x){b — x){c — x) + {a — x)[b + x){c — x) 
 
 -t-[a-x){b — x){C'\-x)-\-{a—x){b — x){e-x) = Qx^ 
 
 7. (a4-5&+a;)(5a+6+a;) = 3(a-|-i+a;)3. 
 
 8. {a + llb+x){lla+b-\-x) = %a+h+xy. 
 
 9. (9a-7i+3a;)(9i-7a-f3a;) = (3a+36+a;)^ 
 
APPENDIX. liuj 
 
 ab cd x — a x-\-n 
 
 a^ — b^x" c-^ — d^z^ x + 1 x—1 
 
 V 
 
 a + x x+b ax+b cx-\-d 
 
 a — x~x — b' ' a-{-bx~ c+dx . 
 
 a—x 1 — bx a — x -k—x 
 
 14. z = -, . • 15. 
 
 22. 
 
 25. 
 
 26. 
 
 1 — ax b — x' ' 1 — ax 1-bx 
 
 x+a + 26 h-%(-\-1x a + 46+a; 'd b-a+ x 
 
 ^^' a; + a-2ft^6 + 2a-2.c' . a-4.b^x^ 2>b + ^'^x' 
 
 x-\-6a4-b x — a-\-b a—lbA-x a+5b-\-x 
 
 ^°- x-3a-{-b~a-x + Sb' '^' Ta-b-x 5a-tb-\-x 
 
 3a~b—x 5b-3a+x Sa-2b + Sx x-a+2b 
 
 20. "771-— = .— oT^.. 21 
 
 32. 
 
 a — '6b-bx oa — 'db+.r' ^ ' a — 'Io + x 'dx — 'da-j-'Ab' 
 
 3a-2b+Bx x-7a + 8h^ 
 a — '2>b-\-x ~ 3a;— 5a 4-46 
 
 5a-6b + x da — 5b-\-3x a + b~x S(a-h+x) 
 
 23 ! — 24 = ^^ ' 
 
 a-^x a+b+x ' ' Sa—b — Sx a — ^b-^-x 
 
 la + b — x 3{a — h+x) 
 
 5a-\-db — 'dx a—llb + x 
 
 5a-b + x 2(2g— Z)+a;) 
 2(a + 26-a;)'^^ll/>-a; * 
 
 7a — b-\-x a{a-{-5h-[-x) x + a — h a(x + a-{-5by 
 
 27- Tb-a-{-x^b{5a + b+xy x-a+b"^ b{x + 5a+'Fj' 
 
 .5a-Sb+x\ 2 7a-9b + Sx 
 29. 
 
 30. 
 
 5b-da + xl 76 -9a 4- 3a; 
 la + 5b+x\ "- a+17i+a; 
 
 \5a + b-\-xj \la+b-\-x 
 
 na-h-\-x- 3 ^ ll a + b-x 
 31- [rjJZ-^x] rfb + a-x 
 
 lla+b-x a^(a+nb + x) 
 
 a+llb—x 62(17a+6-i-aJ) 
 
son Arpr.Nnn:. 
 
 33. 
 
 {5x + 'Sa-llb){:v-a + nb} bx+la — bUb 
 
 g^ (1 + Sx+]5x^){x^ + 3a;+5) _ ^ 
 (l + 2.c+3a;2)(a;2-|-2^ + 3)~ 4" 
 
 35. 
 
 8G. 
 
 37 
 
 38. 
 
 39. 
 
 40. 
 
 y(l+^2)_j_^(l_a; 2) ,, 
 
 V'(l+.f-^)-l/(l-xf)~ ^ 
 lf(l+a;2) + ^(l-a; 3) q. 
 
 4/( l+.,.2)^_4/ (l_a.2) ^ 
 
 t/(i+«2)-V(i-a;-)~ir 
 V(l4-x^)+! y(a;3-l) _ jt. 
 
 'y(i+x^-)-'^{x-'-i)- b" 
 
 y(a;2+l)+y(a;2-l) a 
 
 41. -,/(4rt + 6-4.i;)-2V(.r + Z/--2x)=v''^. 
 
 42. -,/(3n-2/>+2a;) - %/ (3a-26- 2.^) = 2 \/a. 
 
 43. A/(2a-/)4-2.r)- A/(10a-96-6a;) = 4v/(a-6). 
 
 44. i/{Sa-4:b + 5x) + ^/(x--a) = 2V(x+a). 
 
 45. V(3a-46 + 5.ri + |/(a;-fl) = 2v'(2.«-2&). 
 
 46. i/(5x -- Sa -{-ib) + -^/{5x-Sa - Ab) = 2V (x -{-a): 
 
 47. i/(2a+/^ + 2a;) + i/(10^/+9/>-Ga;)=2v/(2a + 6-2a;). 
 
 48. 2'v/(2r/+6+2.r) + ,/(10rt+6-6«)- N/(10ff+96-6.'c). 
 
 49. y'{2a-13b+Ux)+V {S{b -2a-{-2x)} =fiV {2.a-b + 2x) 
 60. i/{3(7a+6+a;)-N/(a+76-x) = 2i/(7a + 6-a:). 
 
 51. V{{a+x){x+b)} + V{{a-x){x-b)}=2i/(ax). 
 
 62. ^{(a + a;)(a;-l-6)}-i/{(a-a:)(a;-6)[ = 2|/'(6a;). 
 
 68. ^(aaj+ic^j- -t/(ax-a;2)= ^{2ax-a^). 
 
APPENDIX. 207 
 
 1 1 _2_ 
 
 x+V{ax) a + y^(«.c) x — a 
 
 56. 7- — .-{- 77 — r = ■• 
 
 a — V [ax) X — -i/{ax) a 
 
 l/{ {a+x){x-^ b) }+ V{{a-x){x-b) } U 
 ''^' V{{a-{-x){x + b)} - V{{a-x)i:x-b)}--<b ' 
 
 t Sa-U + 'i.e {^/a-^^/(2ft-26)}" 
 
 58. ^3,f_2i-2.<;~ 26-^* 
 
 59. ^{a-{-x) + f{a-x) = 2^a. 
 
 62. ir(H-x)--'+iril -x)^ = ^h^{l-x-'). 
 
 63. i^'(3+*-) + lf^3-.c) = ^6. 
 
 6i. -,K(i+^)='+r(i--^-)'=5{i^(;i+^)+f/(i--*^)i'- 
 
 65. f{U+x)^ -f^{196-x-^) + f'{U-xy^ ^7. 
 
 66. {if(9+x-)+i^(9-a;)}if{81-a;3) = 12. 
 
 67. {f{u-^.ir-f^{u-xy-}{^{u+x)-f^{u-x)} = ie. 
 
 68. {irt57+.«)^+#/(57-x)3}{if(57-a;) + iK(57+a;)} = 100. 
 
 69. 5{t/(-il+'-^) + V(41--^-)}'=8{^/(41 + '^-)+v/(41-^)}- 
 
 70. {t/{x+5)+t/i'^-5)}m/(x+5)-t/{x-5)}=± 
 
 71. {V(-*^+i)+M^-i)}{-/(-«+i)+v/(-«-i)} = 
 
 26{V(^+l)-t/(a;-l)}. 
 
 72. r\^^ + r\^^ = a. [y+y-'=a]. 
 
 73. 2{f'(l+a;)3+^(l-x3)} = (c2+l){ir(l^''^) + #'(l-^)}^ 
 
208 APPENDIX. 
 
 74. f^(a+x) + ^{a-x) = fc. 
 
 75. {^(a + x) +f'(a - x)}f'{a"--x^-) =c. 
 
 76. ^ia+x}-=^{rr- -x2)^^(a~x)- = fcK 
 
 77. {f'{a + xy'-f/{a-xy}{f{a+.v)-f\a-x)}=c. 
 
 78. {f/[a+xy-+f{a-xy-}{f{a+x) + fia-x)}=c. 
 
 79. [a-^x)-^{a ~x) - ia-x)f{a+x) = c{f/{a + x)~f/(a-x)}. 
 
 80. (a + x) f/ia+x) - {a-x)f{a-x) - e{^(a + x) - |f (a - u;)}. 
 
 81. {^ia-^xy^-f{a^~x^') + f[a-xr-}^- = 
 c{f'{a+x) + f/{'i-x)}. 
 
 82. {V(«+^) + M«-^)} '=(« + !){ \/(«4-aj)+A/(«-.c)}. 
 
 Sectioi? IV. — Quadratic Equations and Equations that 
 
 CAN BE RESOLVED AS QUADRATICS. 
 
 Examples. 
 1. x^ + (ah^iy = {'i^+b^)ix^+l)+2{a^-b^)x-{-l, 
 .-. x^+aH^' = {a^ + b^yx- + 2{a^-b^)x+[a-by 
 :. X* + 2abx^- +an^- ={a-{-by-x- +2{a-2 -b^)x+(a-h)^ 
 .-. x^.^ab=±{{a + b)x+{a-b)}, 
 or x"+{a-\-b)x + ab-±{a — h), 
 :. x^^{a+byx-{-^ia + by = ^a-by ±{a-b), 
 .-. x^i{a-\-b)=W{['^~byziz^{^a-b)l. 
 
APPENDrX. 
 
 269 
 
 {a-x)^l/(a-x) +(x-b)W (x-b)__^_f^ 
 2k. ' 
 
 {a - x) i/{a ~x)-\-)x-b)^^{x~b) 
 
 Writer — 6 in the form {a—x)-r{x~h) and multiply by the 
 denominator of the left-hand member, 
 
 .-. {a~x)^yia-x) + {x-by \/{x-b = 
 
 {a - x) V(« - ^) -^ {ci-x)(x-b){^{a-x)+ ^{x - b) } -L 
 (x~hyW{x-h), 
 :. (a~x)(x-b){V{a-x)+V{x^b)\ = 0, 
 :. {a~x) = 0, or x-b = 0, 
 or \/{a — x)-\-V(x — b)=0, 
 
 x^ =rt, .fo = ^• 
 The equation V{a-~x) -\-V[x -b)=0 has no solution for the 
 Bum of two positive square-roots, cannot vanish. 
 
 The solution x = ^[<i-\-b)) belongs to the equation 
 V{a - X) — V{x — b) = 0. 
 
 ax+b mx—n 
 
 3. T—~ = 
 
 bx -{- ci nx — m 
 
 Add and subtract Numerators and Denominatorg 
 
 {a+h)(x + l) _ {m-\-n){x-l) 
 {a - b){x — l)~ {m-n){^x+V) 
 
 ^_l/ {a+h){m-n) "" 
 
 •'• ^1 = s"Zi' ^-3 - .s4-l* 
 
 b-\-x a —x 
 Square both members, subtract 4 and extract the square-root. 
 
270 
 
 APPENDIX. 
 
 /fg —X 
 
 
 a — x 
 
 " b^x 
 
 = 63 
 
 1 - ^3 
 
 ••• x=h{{a~b) + {a+b) j-^^3^ 
 
 Or thus, cube both members, 
 
 a—x b+x „ 
 
 .-. - -- + 3c+ = c3 
 
 b-^-x a—x 
 
 {a-xr- + {b+xY 
 
 = C3-3C 
 
 •• ^a-x){b±x) 
 
 ( {b+x)-{a-x) Y g^-3c-2 (£+1)2 (c- 2) 
 
 2a; - (g- 6) c + 1 |c-2 
 a +"6 ~ c - i Nc + 2' 
 
 Wrt-a;)-V(6-a:) ^/{{a-x){b-x)] 
 
 ■ + 
 (Prove that ^~ 3 
 
 V(<i-x)-j-V(6-a;) 
 
 {v/(a-.r)-V(6-a;) }g v{(a-3;)(5-a;)} 
 {a—b) - c ' 
 
 { V(a —x)-V{b-x )] - a-h 
 V(a-a;)|-V(i-:c)) 3_ a-6 
 
 . Eationalize Deuom. 
 
 (-1) 
 
 " iV(a-a;)H-V{Z»-a;)) a-i+4c' 
 
 V{(a-a;)(^-a;)} _ I _a-b^ 
 c ~ ^c7^b + Jc' 
 
 Also from (A), 
 
 a+b — 2x a — b-\-2c 
 
 V{{a-x){b^^'x)} "^ c ' 
 
 (^) 
 
APPENDIX. 271 
 
 Multiply (B) and (C) member by member 
 
 ;.{„+,_(._, .+2.) 4-- 
 
 h 
 
 ■ 6. a;4-4=-5^^^;a;e.-2.7;4- 5a;3-12 = 0. 
 aj- — 2 ' 
 
 Find the rational linear factors of the left-hand member by the 
 method of Art. XXVII., page 90. 
 
 .-. (a:-2)(x-f2)(^* + 2a;3+3)=0, 
 .-. x-2 = 0, ora; + 2 = 0, or x'^-\-2x^+3 = 0. 
 The last of these equations may be solved as a quadratic giving 
 a;2 = -l±2A/— 2, :. x=±l±V-2, 
 ,'. x^=2, x^ = -2, x^ = l + V-2, x^ = l-\^-% 
 a;5=-l + V-2, a;6=-l-V-2. 
 
 I^.B. — In solving numerical equations of the higher orders, the 
 rational linear factors should always be found and separated as dis- 
 junctive equatiom, before other methods of reduction are applied. 
 Such separation may always be effected by the methods of Arts. 
 XXVIl. to XXX., and unless it is done the application of the 
 higher methods may actually fail. Thus, if it be attempted to 
 solve as a cubic the equation, 
 
 a;3-9a;-10 = 
 
 the result is aj= {5+ V- 2} +{6-V-2} , which can be reduced 
 only by trial. The left-hand member can however be easily 
 factored by the method of Art. XXVII.. and the equation reduces 
 
 to 
 
 {x+2){x^-2x-5) = 0, 
 
 which gives a; = 2 or l±y'6. 
 
272 APPENDIX. 
 
 7. {x-2y -x'-\-2' =0. 
 
 Factor, (See No. 20, p. 89), rejecting constant factors, 
 
 .'. x=-0, ov x-2=^0, ov x^--2x + 4: = 0. * 
 
 The last equation gives a; = Izt \/ —.3. 
 
 Exercise. 
 
 ' Solve the following equations : 
 1. {x + a + b)^=x^+a^-hb3. 2. (x + a + b)^ =.r:^ + a'' +b». 
 
 3. {a—b)x^ + {b-x)a3 + {x-a)h^ = 0. 
 
 4. {a-b)x^+{x-b)a^+{x+a)h^ = 2abx. 
 
 5. (x-a)^ + {a-b)^ +{b-x)^ =0, 
 
 6. (x-ay -\-{a-by+{b-x)' =0. 
 
 7. {a^-b)x'>^+{x^-a)b^-\-{b»-xyi* = abx{a^b^X' -1). 
 
 8. {x-a){x-b){a-b) + {x-b){x-c){b-c) + 
 (x — c){x^ a){c — a) — 0. 
 
 x^-1 x^^-1 
 
 ^^' x^-1 x^ — 1 
 
 13. a;'' +5a;3-l6x^4-20,x-16 = 0. (See Art. XXII.) 
 
 14. a;4 - 3.6-3 + 5.C2 + 6.C + 4 = 0. 
 
 15. (x-a)4+a;4 + «4=0. 16. 2.r--' = (.r-G)». 
 
 17. x{x-2y{x+2) = 2. 18. (4;.-3-17)x+12 = 0. 
 
 19. a;* + (ai+l)= = («"-+^')(^' + l) + 2(«'-&^)-i^ + l. 
 
 20. a;2(x-169)2+17x = a;3-3540. 
 
APi-ENDIX. 273 
 
 21. 6x(a;2 + l)3 + (2^'-+5)3 = 150x+l. 
 
 22. 2x{x-l)^+^ = [x + l)^. 23. x^ = 12x+5. 
 
 24. 5x^ = 12x^+1. 25. (x+4)3 = 3(2:«-l)=. 
 
 20. V(.c2+w2)-fv[(7i-2;)2+m2}=V{(a;-4«)^ + (i«'^3-wO}- 
 
 27. 
 
 31. 
 
 33. 
 
 34. 
 
 35. 
 
 86. 
 
 (x + iy . m 28. 1-^+1)^ ''J^L 
 
 (.6-" + lj^.t;-l)~ n x{x^+l) n 
 
 (^3 + 1)0:3 + 1) m . (:,3 + l)(,;."+l) m 
 
 -^- {x + l){x^ + l)~ W "^^^ {x^-l\x-^-l) n 
 
 x2(u;+l) ~w* ''"■ x(x'-^ + l)(x--l;-~ ?t 
 
 x{x-\-l)' n{n—m) 
 
 (a;3 + l)(:c-l)2-2»i(2m-?2) 
 
 (a;3 + l)2 4m2 
 
 .<;- - 1)- 7/i- — ?4'' 
 
 t>-l)(a;2+l)2 2(»?- -n)3^ 
 (j;3-l)(a;+lj'"~ '"^'^ 
 
 a;^ - 1 2h?. 
 
 (.c+l)(a;^- l)'"2m- 
 
 71 
 
 ^ x^-l){x + l )^ _ in+n (x+l){x^ + l ) ^ m + n 
 
 39. x^ =7 •• • 40. x^ = 7 
 
 bx-a hx-a 
 
 ax-b ,^ , ax3 + ?»ar + c 
 
 41. ^^=i^i::^- 42. a.^=^:;-j^qr^- 
 
 43. a;3 = (a;-l)2(x2+l). 44. aS^jS = («_a;)2(«2 _^2). 
 
 45. x^ = {x-ay{x^--l). 
 
 46. aA/(2;2 + l)-a;/(a;2 + r72W^, 
 
274 APPENDIX, 
 
 47. l^(a3-f-.,.-3)4-^(,(3_a;3)^^(a6_a.6)3. 
 
 48. tn{x+7n~7i){x — m + ln)^ =n{x — 7n+n){x-\-lm.—n)^. 
 
 49. vi^{x + m+nn){x-m- 5n)^ =u^{x+nm+7i){x- 5m+n)^. 
 
 50. )7i^{.r-i-m + n7i){x-7n + l7i)^ ^n^{x + nm+7i]{x+lin - ii]^. 
 
 V{x-a) + i/{x-b ) _ Y x-a 
 V{x-a)--i/{x-b)~^x-b 
 
 V{x — a) + V(x — b) \a-x 
 
 52. 
 53. 
 
 V{x-a)-V{x-b) 
 
 ]a — X \b4-x \a — x,\h-x 
 
 \r^~-J =c. 54. , +V = ( 
 
 'vj'^+x' ^u — x ^b — x ^a — x 
 
 oyCi—x o/h+x 3 / la — x\ ^ lb—x\" 
 
 b -\- X a—x ' \b —xj ^ \a — x! 
 
 ^,a-x b + x 4/«-a; 4/i-a; 
 
 o+u; a — a; b — x a — x 
 
 .,a — x . b + x riA—x .,b—x 
 
 59. y.-— +^ — =c. GO. V^ — -e^ ^c. 
 
 b+x a — x b — x a — x 
 
 61. V7n— +v'^-=c. G2. V, -V = c. 
 
 w+;<c ax b — x a—x 
 
 ^^ ^{a-xY-^sf(b-xY __ 
 ^{a-x)-\r^{b-x) 
 
 ^/(a-xy^+ i/{b-x)^ _ 
 ^^' {^^a-x)+y{b-x)}^-'' 
 
 V{a-x)^+ l /{h-x) ^ 
 -\/{a—x) — V{b—x) ~ 
 
 _ W{a-x)+^/(b-x) y 
 '^^' ^/(^a-x)-V{b-x) -"' 
 
 y^{a-xy+V{x-by 
 ^'' V{a-x)+/{x-b) -"' 
 
APPENDIX. 27/) 
 
 68. 
 
 70. 
 
 it^. 
 
 76. 
 
 
 ■x/{ a-x)^ + V{x + h)5 _ ((? + />) 2 _ 
 
 ~~V{a -x)+V{x + b} ~4:V{{a — x) {x + bj}' 
 
 x-+(a-x^)V(a-x~) 
 a;+V(a— a;^) 
 
 72. ^ , 77-0-—- o-x '' ^ ^-^"^i^' --x^). 
 
 x+V{a^ —x^) 
 
 73. if(«-x)-^-ir{(«-.T)(.r-/>)} + |3/(a:+6)2 = ^/(a2-a5+i') 
 74 b^/{^-x ) + aV{x-b) 
 
 75 ^\/(fl-a;) + ^l /(a;-/ ^) _ 
 l/(a-aj) + i/(a;-ft) ~ 
 
 V{x- a) + -/( a;+q) - .7(2^) _ y^'+f 
 v'{x — a) — y(.x-H-rt)+v/(2a) ~ a;-c' 
 
 77 -y(^^ - a:) 4- y ■ _ 4. ^ -a: 
 
 78. ■|y(rt. - a:)2 - fV | (a -x){x + h) } +-^/{x+h)^ = iy{a^ - ab + h^). 
 
 79. {^{a-xY-^^[{a-x){x-h)-\ + ^3/(a;-/.2)2p = 
 
 (^,-i){r(«-.^) + l>n:« -/')[. 
 
 80. {x1/(rT-a;)2+ir(/;+.'^)2l2 = (a + 6){^3/(„_.^)4.^/(/,^^-jj 
 
 81. f/{a-x)^^{x-h)=f/c. 
 
 82. -,3/(a+^)2-T^/(^'-^')'=if(2^a;). 
 
 83. f/{a-x) 2 + 1^/ J (a. _ a,) (/, - a:l } + f{h -x)^= f/c^. 
 
 84. ■,f(a-a;)2--,f{(«-x)(a: + o)}4-^(a;-fi)3 = 
 
276 APPENDIX. 
 
 85. { ,3/(a - x) + f/{x + h) )f{{a-x){x-\- h) } = c. 
 
 86. f/{a-xY +f/{x-hy^ =c{f\a~-x)-{-f{x-h)]'^. 
 
 87. x+f/{a^-x^) = "' . 
 
 89. (a+x)V(fl+^) + («--''')M«--'') = «{t/('-« + -'^) + M«-2j)}. 
 
 90. (a+x}^/{a-x) + {a-x)t/{a-\-x) = a{'^{a+x) + t/{a-x)]. 
 
 91. 4/l^G-»^) + t/(^-10) = 2- 
 
 93. {a- x)t/{(Ji'-x) + {x-y)t/i:^ -^>) = 
 
 {a-b){J:/{a-x)+^{x-h)]. 
 
 95. {^{a-x)-\-if{b-x)]{s/ (a-x)+ ^ [b-x)y~ ^ 
 
 c{t/{a-x)-if{h-^x)}. 
 90. rt\/(l+.r=)-3;i/(a;2+rt3)^g. 
 
 97. {a-x)^{x-h)+{x-h)-^{a-x:)=e{^(a-x) + f/{x~h)]'-, 
 
 98. {Ty(a-^)+ir(&+a:)}^=c{ir(«-a?)2+r(/>+^)2}. 
 
 99. {^{a-x) + f{h+x)]'^=cf^{{a-x){h-^x)]. 
 100. ir(a-a;)^-i«/(?>-rc)2=r|r(fl + /;-2a;). 
 
 101. V('^-^) + M-'^-^) = ^''- 
 
 102. ^{a -x) + '':/{x-h)= ^c. 
 
 {a-x)^{n-x) + {x-lAt/(^--h) _ 
 ^^^' (^.-a;)i/(a;-&)+(^—*)^> -•'«)"''• 
 
 104. ly^a-x)^i/{h-x) ^■''• 
 
 105. 
 
 V(a-a;)4 -V(->--fc) ^ g 
 4/(a-a;)- t^(3;— i) a+^— 2a;' 
 
APPENDIX. til 
 
 « 
 
 107. [a-xyi/{a-x)-{x-hy^{x-b) = c{:i/{r,-x)-^{x-h)}. 
 
 108. {a-x)^{x + b) - {x+h)^{x- a) = c{^y^a -x) - i^[x+b)}. 
 
 109. {^{a-x)^ +i/{x-b)^}-l/{ia-x){x-b)} =c. 
 
 110. {:/(a-a;)-y(^-i)|3{5^(a-:c)^-M^-5)2}=c. 
 
 111. {^y{a-xy- ^{x-by-}^- {y{a-x) + -Ifix-b)} =c. 
 
 112. {^(«-u;)^+:/(x4-Z^)3}2=c{.V(a-a:) + y(a;+Z^)}. 
 
 Section V. — Quadratic Equations involving two or more 
 
 VAUIABLES. 
 
 1. (x + y){x^+y^) = a, I. 
 
 X^-^j/+XlJ^ =c. II. 
 
 I + 2TI. .-. {x + yy = a + 2G 
 
 .•, x-{-y = -^{a+2c). (Any one of the three cube-roots). III. 
 
 ic^ c ' * ' U+2// a+2c' 
 
 Tj TTT ■i/(a-2c) 
 By 111. a; — y = -'— ) — -; . 
 
 Ai . i/(«+2c) 
 
 ^^^° "+•'' = ^/I;h:2^)' 
 
 . ^ ^ y(a+2g) + y(a-2c) 
 26/^a + 2c) 
 
 = A/(a+20 --!/(« -2c ) 
 ^ 2V(" + 2c) 
 
 (Not any one of the six. sixth-roots of a -f 2c may be used indiffer- 
 ently in the denominator, but only any cube-root of whichever 
 equare-root of a-t-2c is used in the numerator. Thus if the radi- 
 
278 APPENDIX. 
 
 cal sign be restricted to denote merely the aritlimetical root, if it- 
 be defined by the equation k^-k+l = 0, and if m and n indicate 
 any integers whatever, equal or unequal, the value of x may be 
 written * 
 
 {l-"'s/(a+2c) + k"'-^ v^(a-2c)}-4-2V(« + 2c). 
 
 2. 8x^-5x>/+d>/^=9{x-\-y) !• 
 
 llx^-8xii+5y^=lB{x+7/) , il' 
 
 1st Method. Eliminate {x+y). 
 
 .-. 104^2 _65.r//+397/2= 99x2 -72x!/ + 45;/». 
 
 .-. 5.r2-f 7a;// -6^/2=0, 
 
 .-. (5x-32/)(:/:+2»/) = 0, 
 
 .-. a: = §?/ or -2?/, 
 Substitute these values for a: in I. 
 
 .-. 72?/2 = 360.y or 45//2 = -9^ 
 
 .-. ?/ = 0, or 5, or — ^, 
 anda; = 0, or 3, or f . 
 2nd Method. Take the sum of the products of I. and 11. by 
 arbitrary multipliers h and /, 
 k{Qx^- -5a;2/+3z/2)+^(llx2-8:r?/4-5?/2) = (9/,- + 130(x+?/). HI. 
 
 Determine h and /. so that the left-hand member of III. may, 
 like the right-hand member, be a multiple of x+y. This may 
 be done by putting x=-y in III. from which 
 
 16/C + 24^ = 0, .-. 2k=-dl 
 .«. if ^ = 3, /=-2. 
 
 Substituting these values in III., it becomes 
 2.^2 +a:7/-?/2 =«-+•?/ 
 .-. {x-¥y){'ix-y)=x+v, or (.r+?/)(2ar-u-l) = 0. 
 .-. either x+y = 0, or 2a;-?/ - 1 = 0. 
 .. t/= -ic, or 2a;-l. 
 
APPENDIX. 279 
 
 Substituting these values for y in I., it becomes 
 16^--' =0, 01- 10a;--' - 7a; + 3 = 27a;- 9, 
 .'. x = i), or 3, or | ; 
 and y = 0, or 5, or — ;^. 
 
 a;^ + ?/3 ~(^3 -j. ^3 ' 1. 
 
 a;* —x'^y + x^y^ — xy^-{-y* 
 
 I.~I1., . 
 
 {a-i + b^-y^-a'ib- ' 
 
 x^y+xy^ a^b-\-ab^ 
 
 • * (x^+ir)- - x^-y^ ~ (a3+/.3j2 _ ,,2^2 
 Write 2 for -;r-,^ — 5 and k for 
 
 z k 1 
 
 I^i- ••■ 1372 = 1313. ••• 2 = /'^or— ^ 
 
 a;;/ ab ct" -^-b'-^ 
 or i — > 
 
 v/(II.-3IV.) &nd^x-y=±{a-b), 
 
 M-+ab + b^ 
 
 ,*. 3;- ±a, +6 or 
 
 III. 
 
 a;;/ «6 a^ -{-b^ 
 
 IL, .-. x^v = a5,or («2 + i2)-__X_ iv 
 
 n/(IL+IV.), .-. a;4-?/-+(a-fi) 
 
 orV(2a3_«5 + 263) y^^' +'^ -r^' . 
 
280 
 
 APPENDIX. 
 
 y=+b; ±« or 
 
 .a"--{-ab + b^ 
 
 Putz = 
 
 xy 
 
 1-z 
 
 l-^. 
 
 lU 
 
 lY 
 
 a 
 
 T 
 
 : %tz^-bz—{a-b)^0 
 : Aaz'^=b±:V{8a^ —S(ib + b-)^b-\-y say. 
 xy _ b+r 
 
 X' + ?/3 
 
 . x + y 
 
 4a 
 
 [ 2a 4-^ + 
 --Nl'ia-/;- 
 
 x-y 
 
 . x_ _ |/(2c/-f/>+r)-}-v/(2«-/'-r) 
 " ~y ~ ]/(2a + /;^+r)— v'(2rt-i — /•) 
 
 2(6 + r) 
 {x^^-y^--^1xy){x''-\-y^Y{{x^+y^)-xyY^a 
 
 Ua+2b + 2r ] I 4a \ ^ (ia-b-r^ 3 
 
 M' 
 
 b+r 
 
 I Aa 
 
 b + r 
 
 — a- 
 
 .1 
 
 X 
 
 1 
 II. 
 
 in. 
 
 IV 
 
 32u2{2a+b+r) { 4:a-b-ry ) ^ ^ 
 . (b+r)^ ' ~ 
 
 X' 
 
 - [yl t32(2a + /'+r)(4«-/>-r)2| 
 
 JT/(2a + 6+r) + V^ (2<< - 6— r )} \' 
 102i(2.t-f A + '•)(•!'.< - i-r)2 
 
APPENDIX. 2S1 
 
 _ V{''^'-i-\-lj + r)+ V{-2,a — b-r) 
 
 in wliich r^±s/{Qa--iDab-\-b-). 
 
 The value of y may be deriverl from that of x by the first form 
 in lY. 
 
 6. X'* =ax — bij, 1, 
 
 ?/4 = ay - bx. II. 
 
 X.I. — 2/.II. x'^ —y^ =a(x^ — y^) 
 
 y.l.-x.Il. xy{x^-y^) = b(x^-y^), 
 
 .". either x — >j = from which x = y = 0, or ^(a — 6) III. 
 
 or x^+x'^y-^x^y^ -\-xu^+y* = a{x-lry) IV. 
 
 and a?^(ie2-[-a;//+y^) = i(.«4-^) V. 
 
 (IV.+V.) (^^ + y)2^^'^+y2'^ = a.{.b VI. 
 
 V. » (a,^^J4_(^-2 4._y2)3^4/,(^^.y) VII. 
 
 l/(VII^ + 4.VI). (^+^)4 + (^'-'+^2).^2|(u; + y) VIII. 
 
 in which t= y{ia-\-bp -j-ib^}. IX. 
 
 i(m.+VIIL), .. ^x+y)^^{2h + t){x+u) 
 
 :.{x-{-y)^ = 2b+t 
 
 . :.(x+y) =f''{2b+t) X. 
 
 VL-.X^ •■-^+^^-rPT7) ' ^^- 
 o VT V / ^•^ 2(a + b) 2a — t 
 
 x-y = 
 
 y{2a - i) 
 
 
282 APPENDIX. 
 
 in which t= V{'-i'^ + '^ah-\-5h^), 
 6. x^—G*'=m{x-\-y)*y 
 
 Letz= -, .'. z-{-l= audz-l = — ^ III. 
 
 x-y' ^ x-y x-y "^* 
 
 I. + 11. x^-\-y'^ = m{x-{-y)^-^n{x-y)'^ 
 
 ,'. (2 + l)4 + (z -1)4 = 16(W724+W) 
 
 { 3 + ]/{9-(8/ra-l)(8TO-l)f , 
 •*• * - >J . 8m- J ~ J^« 
 
 11. & III. (?-l)*(x--7/)4+lGc4-16w(.c-^)* 
 
 2c 
 
 •■•'^~^~i/li6/i-{2-i)4} V. 
 
 2f2 
 
 Kz+1) c(g+l) 
 
 ' C(2-1) 
 
 and 2/ = 4^ ( i6« - (zHtTT' ^^^ *^® ^^^"® °^ * ^^ S^^®^ ''^^ ^^• 
 7. a;2 +2/2 = 1 (2»7.+n2), 
 
 and {x-\-y) 3 — o.(://(^- + ?/) =•. mn. 
 
APPKNDTX. 283 
 
 Let u = x-\-y aud v = xy, and the equations become 
 
 U^ —3llV = 17171. 
 
 Eliminate r, .-. u^ — {2m+n^)u+2>nn = 0, 
 
 .'. w* - {2m+n^)u^ +2mn ti = 0, 
 
 .*. u^—%nu^-\-m^ =n'^u^ — 2innu-\-m'^ . 
 
 .'. u"^ —m = ±{uu—in), 
 
 .-. u = n, (the value it = was introduced by the multipiica- 
 fcion by m), 
 
 or u^ -\-nii — 2m ~0, 
 
 ,'. u=l{ ~n± \/(n^ 4- Fyni)} 
 
 .'. v=\{n^-m) or l{;j3+8/« + 3;?|/(w24-8w;} 
 
 .% tt and V are completely determined. 
 
 Also x-\-y = u, x—y=V(u^—4.v) 
 
 r 
 
 If ?« = 7 and n = 5, the above equations become 
 x-+y^=^13, and a;^ +^3 = 35. 
 
 Solving, as above, gives 
 u = 5, or 2, or —7, 
 2j; = 12, or -9, or 36, 
 
 .'. x+y=:5, or 2, or —7, 
 
 x-y=±l,or±V22, or ±ji/2n. 
 ,:. a; .-=3, 2, |('2±s/22) or K-7+/\/2B); 
 
 _7 = 2,, 3, i(2+v/22) or i(-7q=,;V23). 
 
284 APPENDIX. 
 
 8. x^-\-y = U; 
 
 « +2/** = - 
 
 4" 
 
 •■• H-^/=(f-?/')^ 
 
 Testing tliis for rational linear factors it is easily reduced to 
 
 li/-l)nz/^+2z/+4-) = 0, 
 
 .-. 2/ = l or^(-2+v/2); 
 a; = i or i(-l±4V2). 
 
 ;«. (2.r— 7/+z)0r+?/+z) = 9; , I. 
 
 {x + 2y-z){.r + >/+z) = l', n. 
 
 (a;+7/-2z)(x+^ + z) = 4. III. 
 
 Let s = x+!i+z and the equations may be written 
 
 (s + a:-2//)s = 9' IV. 
 
 (s-\-y-2z)s = l ' V. 
 (s-32)s = 4. VI. 
 
 IV. + 8.V. (4s +X+.V - 6z)s = 1 2, or (5s - 7z)s = 12 VIL 
 
 R VII-7.VL {(15s-21z)-(7s-21z)}s = 8, 
 .-. 8s2=8, .. s=±l. 
 
 Substituting in I, II. and III. they become 
 
 2x-y+z=±9, x+2y-z=±l, x-{-y-2z= ±4., 
 
 .-. x= ±4, y= +2, z= ^1. 
 
 10 x^+y^ = a; 
 u^-{-v =b; 
 xy+uv = c; 
 xu + yv = e. 
 
 Let t = xy — nv. 
 
 :. {x + yY=a + c+t, .'. x = \{^{a-\-e^t)+y/{a^':-t)} 
 {x-y)''=u-c-t, y=lU/{a+c+t)- V(a-i-t)\ 
 
APPENDIX. 285 
 
 (u-\-vy = b + c-t. u = l{^/(b + c-t) + ^/(h-r-{.t)\ 
 (u-v)^ = h-c-\.t, v=l{y'{b + c-t)~V{b-c-{-f)} 
 Also 2{xii +yv) = (a; 4- ?/)("+■'') + (•«— Z/)(« —■*-') = 2e, 
 
 ••• ^^W'+<' + t){h+c-t)}+^/{a-c-t){b-r-\-t)}=2e, 
 .-. {4:e^-\-{a-c-t){b-c + t)-(a + c+f){b+c-t)}^ = 
 
 IGe^a-c-tXh-r + t). 
 ,'. {(a-i)? + 4r3}i2-2(a2-i2)rt-f 
 
 (rt4_/,)3c2-4e2(a6+c3)+4^'4=0, 
 
 (a^-62)c-j-2cv/ [{ab - e^ ){{r, -by ~4:{c^ -e^)\] 
 •*• * = (a-Z))'-^+4^ " " ■ 
 
 11. xy = uv I. 
 
 aj+y + w+t-^a II. 
 
 jc9^y3+u^ + v^=b^ III 
 
 a;S+y5^„6+^;5^c5 IV. 
 
 het x+y = k{a+z). :. u-^v = l{a-z). V. 
 
 Also let r = xy = uv VI. 
 
 .'. a(302+a2) = 4(fc3 + 3rt,.) VII. 
 
 Also (a; + //)-^ =^\+^'^+6u:.v(a;3+,y3)^10^2^2(a.^^) 
 
 .-. a(5z4 + 10a2z2 4a4) = 16{c5 + 5'j3,-4.io.ir2} Vlll. 
 
 Eliminating r between VII. and VIII, 
 
 45„224 _30a(fl3+2J-^)52_|_„6 _20a3/;3 _80/;6 + i44,,cS =0 
 .-. Ir./22_5(a3+2^j^)=±2v/{5(a34-5^3)2_i80,,c5} IX. 
 
 . a ^ 4 2b^±2y [^{(g s + 56^)2 - SQac^ }] 
 
 2= V- 3a 
 
 X. 
 
286 APPENDIX. 
 
 VII. & IX. 12ar = a^ -U^ + Saz^ 
 
 5(a^ - b^) ±V{5{a^ + 5b^)^ -ISOac^} 
 ''' '' ~ ^ 30a ^• 
 
 X. aud XI. give the values of z and r which may now be treated 
 as known iu V aud V. 
 
 ^+l/~^(a+z), and xi/ = r 
 
 .-. x-y = ^v{{a+zy-16r} 
 
 x=^{a+z±V{.{a + z)Z-16r}); 
 
 i/ = i(a+zqiV{(a+z)2-16r}). 
 
 The values of u and v may be obtained from those of x and y 
 respectively by changing z into - z. 
 
 Exercise. 
 
 1. 6{(7 -a;)3 +2/2} = 13(7 -x)?/, x^- +Ay = y^+4:. 
 
 2. 10x3-9?/3 = 2a;^ '8x^ -6y^ = ldx. 
 
 3. xy={S-x)^ = {2-y)K 4. x^ ^y^ = 8x + 9y = U4. 
 
 5. x2_|.^2=a;+y+12, a;?/+8 = 2(a:+y). 
 
 6. x+xy+y=5, x"" +xy + ij^ =7. 
 
 35 28 
 
 7. x^+y^ = 7xy==2S{x-^-y). 8. a;2+cc^ + ^/» =^^-^2 =— • 
 
 10. {x-]-7j){x"-+y'-) = 17xy, {x-y){x^ -y^)=9xy. 
 
 11. 25(a;3 +2/3) = 7(ic_f.^)3 = 175a;?/. 
 
 12. 2x^-y^ = U{x^-2tj^) = U{x-y). 
 
 13. 2a;2-3a;_y = 9(a;._3//), 3(a;2 -St/^) = 2(2a;2 - 3a;!/). 
 
 14. 2x^-xy + 5y^ = 10{x-\-y), x^+ixy+Sy^ = U{x+y). 
 
 15. (2x-32/)(3a;+42/) = 39(a;-2i/), {3x+2y){ix-Sy = {99(x- '2y) 
 
APPENDIX. 287 
 
 16. {x-\-^y){x+^) = ^x-Vy), {%x^y){^x+y) = 1Q{x+y). 
 
 17. x+y = ^, a;4 + i/* = 706. 18. x+y = 5, .t^ + z/^ = 275. 
 
 19. x-\-y = % 13(a;S + ?/5) = 121(a;3+2/3). 
 
 20. a;+^ = 4, 41(a;3 +i/5) = 122(.c4 + ?/4). 
 
 21. x^ — 5xy-\-y^ +5 = 0, xy = x-\-y — 1. 
 
 22. x^+y = 5{x-y), x+y'^ ='l{x-y). 
 
 23. 3(x^+2/) = 3(.:c + //2) = 13.c//. 
 
 24. 10(a;3+//) = 10(.>;+.y-^) = 13^x--^+^3). 
 
 25. a;3+.v=y. ^J + y^^V- 26. 9(a;2 4-2/) = 3(z-+^3) = 7. 
 
 27. x4-^//+// = 5, a;3+.r//+//3 = 17. 
 
 28. .«+i/ = 2, (^-}-l)-'+(i/-2)5 = 211. 
 
 29. 3(.-i)(,+i) = 4(.+i)(,-i), ^;,::^:^i = ^(^._^;i) 
 1 • 
 
 80. x-+//= — , x-y = xy. 
 
 31. x--F// + 1^0, u;6+i/^+2 = 0. 
 
 32. .^+^ = 1, 3(u;«+^«) = 7. 
 
 33. 4.f//a=5(5-.«), 2(a;3+2/3) = 5. 
 
 34. 'ilxy = 17, 9{x^-\-y-^)=-8. 
 
 35. (.^.■•^+//^)^ + 4x2^^=5-12r/, y(aj3 +^2) 43 = 0. 
 
 36. x-\-y=--xy, x^+y" =x^-\-y^. 
 
 37. x^ Cxx-W{y'-x^) -16y^ = 9x^, 
 {x^~+-A)^=4.{2 + x^]/{y"+x"-)-y^-}. 
 
 38. x{y-^+Sy-l) = tiy^-\-2y-\-d, y(x-^ + ?> r -l) = 2x^ +2x+S. 
 
288 ■ APPENDIX. 
 
 39. i^ , ^ ^ 2c3, ^ + 1- . .(-^ _ r\. 
 
 a^ ^ b^ a ^ b \a b j 
 
 « 
 
 40. x^+xi/^=a, y'^ -\-x^y = b. 
 
 41. x+y = a, J. ^ = c. 
 
 b-y X 
 
 42. a;^+«!/2='^, aa;2+^2 =(a3 -1)^. 
 
 a—1 
 
 43. a; + T/2=aa;, x^-^y = by. Ai. x -{■ y- = mj^ , x^+y = bxS. 
 
 45. a;* — y^=a'^(x—y}^, x^ -x^y-\-xy- —y^ = b^[x-\-y). 
 
 46. (a;+2/)(a;3 + 3?/3) = »i, (a;- ;/)(a;2+3//2) = n.' 
 
 47. x^y^ =y(a—x)^ =x{b — y)^. 
 
 48. x3(6-^) = 2/3(a-.c)^(a-a;)2(6-y)3. 
 
 49. a3(^t:+^2) = i2(^.+^)2^ a^-iy^+e2) = c^{x + y)^. 
 
 50. x3-2/3=a(a;2-2/3), a;3+^3 = 6(u;+«/). 
 
 51. x+y = a, x^-\-y^^bxy. 
 
 52 I— - I— = ^~^ , ^(^1+^) ^ ^3 
 
 53. a;+?/ = a;2/ = a;2+2/3. 64. a;-2/= — = a;^ -^^ 
 
 55. x^{l-¥y^)(l-Vy^)^a, x^{\-y^){l-y^)=h, 
 
 _^ a;'^+a;i/ + 2/2 x^+y"^ xy 
 
 Ob. =- = = —. — 
 
 x^-xy+y^ a b 
 
 57. x''y+xy^= -^^, x''y+xy^=b. 
 
 58. x^y-\-xy^=a{x^-+y^), x^y-xy^ = b(x^ -y'^), 
 
 59. (^ + JL\ix+y):=a, fl ,+ HI ^ t, 
 \y ' xj y i'- 
 
AT^!- vilX. 289 
 
 60. x^ +.'/'^ = nx^]!^ = a;//(.r+?/). 
 
 61. abxii = (i {x •' + // 3 ) = A {x + // ) ^ . 
 
 62. xi/(x+y) = a, x^i/^{x^ +1/^) = h. 
 
 63. (1 ^ l)(x-3-yS)=.^, [1 _ lj(.rs+^3^==ft. 
 
 64. a;*+_?/4 =7?7(.r2+7/2), x^ -\-xy+y'^ '—n. 
 
 65. fl6(a; + y) = .r;/(a+i), a;2+v3=a2 + i^ 
 
 66. x3+?/3 = a(a; + t/), .r4+?/* = 6f:r-l-?..)* 
 
 67. x^+v-=o., x^A-ir^=b(x^-^V^\, 
 
 68. xy = a, x^-^y^ =b(x^+'i/^). 
 
 69. (x-?/)(:r=5+?/3) = (a_j)(fl3+53)^ a;2_^2=(^2_^.j_ 
 
 70. x^—y^ = o, x^-^7j^ = h{x-y). 
 
 71. a;+?/ = «, x*+//*=6. 72. a;+^y = a, a;^+i,'5=ft. 
 
 73. a;+2/ = o, x^-^y- =h^-x-y^. 
 
 74. a; + ?/ = a+^ (a-i)2(a;*+?/4) = fa;-7/)2(a4 + 64). 
 
 75. x^ry = a, c{x^->ry'^)=xy{.c^+y''). 
 
 76. (a;4-?/)3=rt(,i;2-f-?/2)^ a;.?/ = c(a;+i/). 
 
 77. x^y+xy^ =a^, c^(x^-\-y^) = x^y^. ^, 
 
 78. x^ = a{x^-\-y^) — cxy, y^ = c{x^-\-y^) — axy. 
 
 / 1 1 \ 
 
 79. a;2-?/2 = fl2. 3.3 _y3 ^^4 |____ 1 . 
 
 80. ic4-y''=«2a;?/, (a;- +?/2)3 = ^2(3.2 _2^2\_ 
 
 81. [x+y)x^y'^ = a, x^-hy'^ = b. 
 
 82. {x+y)xy = a, x^+y^ -hxy. 
 
290 APPENDIX. 
 
 83. x^-i y^ = a{x+y)'^, x^+y^ = b{x-\-y)^. 
 
 < 
 
 84. x*+x-y'^-\-y^=a, x^-xy+y^ = l. 
 
 x'^iX+x^y^) a 1 + xy 
 
 85. {x"-+>i^)xy = x^ ~2/^ yi{l+xy)^ =X ' r^y' 
 
 ■j/(x^i(/^)— a; 
 
 86. x+y = {x-y)V.{^y), V(^5y7)+^ = «- 
 
 87. ^ + ^ = ~ i/{l-x)-V{l-y) = b. 
 
 88. «-+,'/-=«(»; + ?/), a;4 + 2/'^ = i(a3 + 2/3). 
 
 89. x3+y^=a, {x + y){x^ +y'^) = b{x^+i/^). 
 
 90. ('a;2 + (/2)(a;3+,^3) = rta;;/, (:«+?/)(«* +//'^ ) = ^'-^.V. 
 
 91. ix+yy'{x-'+y^) = a, {x^--Jry^)\x^+y '>') = (>, 
 
 92. (x-2/)(x2_^3)(a;4_j/4) = 4aa;y, 
 
 {x+y)(x^+y''^){x^ +y^) = K^ - ?/)• 
 
 93. x'^y +xy4 = a(x3^ -hxy^) = h{x'^ +*/*). 
 
 94. a{x'^ -f ?/5) = ai(a; +?/) = hxy{x^ +?/^)- 
 a;3 - y^ a^ - h^ x^ -y^ a^ - &* 
 
 97. .r^ = 2a.'<; - &?/, y^ = 2a?/ — /)»;. 
 
 98. {x+y){x^+y^}=^'', {x-y){x^ ■-y^) = h. 
 
APPENDIX. 
 
 201 
 
 100. {x + y){x^-*- ij^) = axi/, {x — y){x^-y'^)=bxy. 
 
 101. (^ + 2/)(^2+i/3)-a(a;^+.V^j, ^x-y){x^-y^') = h{x^-^y-). 
 
 102 {^+y )^{x ^+y^ ) ^\., 
 
 [x^+xy + y-){:x'-\-y-) 
 
 '(x-y)^{x^~y^y ^ ^^ 
 
 {x"'-xy+7j^){x-'+y-') 
 
 105. xy{x+y)[x^ + y^)=a, xy{x-y){x^ -y^) = b. 
 
 107. ^{x-x]f)^V{y-xy) = a, y(^x-x^)^V\y-y-') = h, 
 
 108. (.c4-l)(2/-l) = «(a;-l){2/ + l), 
 
 {xo + l){y-lY=b^{x-lY{y-^+l). 
 
 109. a;+t/ = rt, V(iC-^) + V(2/-'^) = <^. 
 
 110. x+y = a{l+xy), {x^yY = b^{l±x^y^), 
 
 111. a;+?/ = «(l+a;?/), a^^+y^ = 63(l+x'i/5). 
 
 112. (a;+l)(w-l)=a(a;-l)(y+l), 
 {x^-l){y-l) = b--{y^-l){x-l). 
 
 11^- (l-x)(l-y}-"' (l-a;3)(l-^3) 
 
 (c+a;)(c+2/)_ { c^^x^)(G^ + y^ ) _^ ' 
 
 { x + m)iy + n) _ ' (a;^+m^)(y4 4. ?i4) ^ ^^ 
 
292 
 116. 
 
 APPENDIX. 
 
 119. 
 
 120. 
 
 121. 
 
 
 (a;+l)(y + l) a_ (a;^ + l)(y^ + l) ■'• 
 
 117 i 'K- • -- / c/yx- rx-; 
 
 •'"'• :.(l + ^^)=«' ..3(^+^4) = *. 
 
 118. ,^±^ = « \^, yil+^±^)_^ 
 
 414-y-)-''- x^(r+yio) = 6. 
 
 122 ^£±J()M±J:)__ «'^+^^ :%2-fl)_a-6 • 
 (^-2/)(^^^^^ 2(<6 ' (/(a;2-lj=.7+d . 
 
 123. ^^-^^^^-^\-aHb--l) ^(i-^"-')-fe 
 
 124 ( ^+y)(l+ 'a-y)^ (a;^+2/3)(l + a:V)_. 
 
 125 (^+2/)li+^/)^ (a;3+y 3)(i^^.3^3 ^ 
 
 126 C-^' + ?^*)(l+^V) {x + y)(l + x>,) 
 
 127 (ii±jO(l+£//)^ (a5^+Z/^)(l+a;V)^/, 
 
 128 (^!±^^l^l+a:^+^^2/=')_ 
 
 {^->jY{i-xyr~ ••-*• 
 
 129. x^ ^x-^u + oa^x-\-y"=0, y--x^~y^-2a'^r = 0. 
 
APPENDIX. 
 
 293 
 
 130. 2x(y^-2x)^=a, yiif- -2x)-^{y^ -4:x) = h. 
 
 (Hence deduce the solution of x^ — 5x^-\-2 = 0). 
 131 2xij(x^+y ')''-= a, {x- -y-){x' -^y^)- =h. 
 132. v^x-+//2) + V{(a-x)2+^3|=V{(>V3-^)- + (ia-a;)^}. 
 
 Q{x- -//^) = «(6x-2^vo + -0- 
 
 ExEltClSE. 
 
 1. (2./-+7/-4z>(.r+7/+c) = 24, 2. x^-iz=l, 
 
 {x + 'hj -2z)(:c+.V+z) = 6, y^-xz = % 
 
 ( — 2a; +3^H- 52)1^0; + 2/4- z) = 30, z^-xij = '6. 
 
 3. (x-+2//-83)(a;+ // + 2)-2(a;y+i/s+2.i-^ = - 12, 
 (2a; - %+2)(a;+.7 +z) + (a;(/ + 2/Z + 2./.) = Gl, 
 {;dx-y + '2z){x-Ty-tz) — o(xij+yz+zx) = 5. 
 
 4. a!2_^2,^^ 6. (a;5^^5^.2o)3_j_(a.^.,/)2^31. 
 a;+2/+z = 7, (a;^+^^4-2^)3 + u-+^./ + 2)3 =729. 
 x3+i/2+22 = 21. (a;4-i/;2-}-(a;+y/4-z)3---=31. 
 
 6. a;2-(,'z=0, 7. x+yz = U, 
 
 • a;+.y-f2 = 21, ^+za; = ll, 
 
 (a;-2/)2+(i/-s)2-i-(2-.tj2 = 126. z+xy = l(i, 
 
 8. a;+|/ = Sz, 9. a;+?/ = 5z, 
 
 a;=*+i/3 = 134.'23, x^+y''=S9z, 
 
 a;2+-y- +2^ = 134. a^ +^3 = 105^2. 
 
 10. x+y = 7z, 11. a;+?/ = 78, 
 
 a;2+^3=25z2, a;3+?/3 = 25z3, 
 
 a;4+7/* = 674z3. a;» + t/« = 20272z. 
 
 12. aj4-2/'.V+2:3+i<'-:^:^.<5» 13. '■x+y:,/-{-z:z-\-x::a:b:c, 
 {n + b+c}xyz = 2,, {a + b + c)xyz = 2{x+y-\-z:) 
 
294 APPENDIX. 
 
 14. ax = b)/ — rz— — _j_ — -f. — 16. zl — _[_ — 
 
 Z X 
 
 — 1. 
 
 15. {x + y — z)x = a, ^Jx 
 
 {x — y+z)fi = b, 
 i-x+y^z,.=:c. ^(f + ~) ^'* 
 
 17. (y + z){1x-h!f-j-z) = a, 18. x{^j+z):y{z + x):z{x+y) = 
 [z+x){x+'Iy-'rz)=b, b-\-c:c-^a:a+b, 
 
 {x+y){x-{-y + 2z)=c, xy+yz+zx={a-^b+c)(x+y-rz). 
 
 19. {a + b)x+[b+c)y + {c + a)z = {a-^b+c){x + y-i-z), 
 a{x+y) = c{y+z), 
 
 {x + yy' + {y + zy- + {z+x)2=4:{a^^b-+c-^), 
 
 20. c{x+y)-^b{x-z}-a{y+z)=0, 
 b{x-z) = {a-c)y, 
 
 21. x-{-y—az = x — by-'rz= —cx + y-\-z = xyz. 
 
 22. {a+b+c)(x-y)+a{x+z)-b{y-\-z) = 0, 
 {a + b-\-c){x-z)+a{x + y)-c{y-\-z) = 0^ 
 
 ax^ by^ cz^ ^ 
 
 (6+c)3 "^ (c+aj3 ^ (a + by^ 
 
 23. xy + — = a, yz + 1- = b, zx + ~ = c 
 
 z X y 
 
 24. y-\-z'.z-\-x:x+y::b+c:cJra:a^h, 
 {x+y-\-z){xyz) = {a+b-\-c){xy+yz+zx). 
 
 25. n^—yz = a, y^-xz = b, z^-xy = c. 
 
 26. a;2+(i/-«)3=a2, y^^^^.^^. =/,2^ z'^ +{x-y)^ =c». 
 
APPENCIX. 295 
 
 28. x^+y^ —z^+Sxyz = a(x+y — z) 
 x^ —y^-{-z^ +Sxyz = b[x — y-\-z), 
 
 — x^ +]/^ +z''-{-3xyz= c{ — x+y -^z), 
 
 29. x+y+2az=--0, 30. x-^y-az = 0, 
 
 «"+?/" +2" = <-•". x^-\-^J^=c^. 
 
 31. a;0/-l)(2-l) = 2a, 32. x{y-l) = a{z-l), 
 
 a;3(y3_l)(23_l) = Cv22 a;3(2/3_i) = c3(z3_i). 
 
 33. x{y-l) = a{z-l), 34. a;(2/-l)=na(z- 1), 
 
 35. a;(2/-l) = fl(2-l), 36. {x-y^- =nz{x+y), 
 
 x^{y^-l) = h^{z"--l), {x^~y^=bz{x+y)^, 
 
 x^y^-l) = c^{z^ - 1). (x-y)^ = cz{x^+y^ 
 
 87. x—y — af 88." a;-|-2/ = <*) 
 
 w — i; = 6, w + ?; = 6, 
 
 xy = uv, x^ + H^ =c^f 
 
 x^ —y^-\-u'^—v^ ^c^a + h). y--\-v^=e^. 
 
 89. xy = icv — a^, 40.. xy = uv = a^, 
 
 x+y+'u.-\-x = h, x-^y+u-\-v = h, 
 
 a;3 ^^3 4.,t3 _J.i;3 = c». a-4 +7/4 ^„4 _,_j,4 =(.4. 
 
 il. xy = uv = a^f 42. a;^ = w'y = a3, 
 
296 
 
 APPENDIX. 
 
 43. xy = uv = a^, x-\-y-\-K-\-v = h, (.c + u)^ ->r{y + vY =c^. 
 
 44. xy = nv, 45. xy = uv, 
 
 x + y-}-ii-}-v = a, x-*-y-\-n + v = a, 
 
 X- +?/"+"- +r2 = h^, X^ 4.,y2 +„2 ^^,2 ^ ^,2^ 
 
 x^ +2/^+"'^ H-*^^ =c^. X* -\-y^-Yu^-\-v^ =0"*. 
 
 46. xy = uv, 47. xy — ^tv, 
 
 x-\-y-\-n-\-v — a, x + y + u-\-r = n, 
 
 iC^4-?/^ +M^ +1-^ =C^. X^+y^ -f-W^ + f* =C*. 
 
 48. a;//-?<?; = 0, 49. a:2+?/2 = fl,3^ 
 xii+yv — a- , ii--^v^ =71(1^^ 
 
 iJC-\-y + v+v = h, iix+vy = c^f 
 
 x^-{-^^-\-7i^+v^ =c^. vx + uy = n^. 
 
 50. x+y+7( + v = a, 51. 7/{l+x^) = 2x, 
 xy + uv = b", ' u{l+y^) = 2y. 
 
 u^+v^=n^. x{l+v^) = 2v, 
 
 52. x + y-i-ii+v=:a, (x+y)" +{u+v)^ = b^, 
 
 53. 
 
 X la — u y 2b— u z ' 2e—u 
 y + z~ a — 2n z + x fc — 2i*' x+y~ c—2u 
 
ANSWERS. 
 
 Exercise i. 
 
 1. 9. -P.O, 1, 0. 1206, -29, 1|. 2. -160,106,41,108 
 8> -^,i-h -25, 125, -.v, -31, -4^V0. -1- 4. 9,8 
 7,-^1^. 5. 176, 82, 254II-, -37-=-7^3. 6. 18 each. 
 7. 146,14,-72,-270,896. 8. Eacii = 0.^ 
 
 Exercise ii. 
 
 1. -1. 2. -166542. 8. 100. 4. -2967511. 
 6. 968. 6. -162. 7. 10. 8. -8. 9. 0. 
 
 10. -20. 11. 706440254900. 12. each. 18. Each 0. 
 
 Exercise iii. 
 
 1. 0, 16a4. 2. a, a^/B. 3. 2a, 0. 4. 26^«, -26«^ 
 6. 0. 6. 4^7*. 7. 6a4. 8. #. 9. c. 10. 0. 
 
 11. a-f-(a+6). 12. a2c{b-h2c) -^b^. 13. a2+/,2_|_c2. 
 14. 0. 15. (12a2i-24rt^2_j.2863)-f-(3/>-a)3. 16. 0. 
 17. 0. 18. -b^c. 19, 20, 21 and 22, each 0, 
 25. 2{b-{-l)h, 4:x^. 82. <Z2=3i3. 33, i= y'^^ojg). 
 
 85. wr^ <;-+r')(r-r')- 
 
 Exercise iv. 
 
 1. 2{bx+cy). 2. 3(aa;-6?/). 
 
 3. a2{x-z)-ab{x-i/)-b^{y-z). 
 
 4. (a;+2/+2)(a + 6 + c). 5. (a + b+c){z^ +y^-i^z^). 
 6. 2(a;+?/+g)x{«-+63+c2-a&-6c-ca). 7. 0. 
 8. 2/'7;r4-i3/ + c?). 9. a^+^s+c^. 
 
 10. 2a;"(a-2fe). . 11. a-[-b-c. 
 
11 ANSWERS. 
 
 Exercise v. 
 
 1. 2(r? + 9//4), 4rt2ft2. 4. 4(«.3-S3)2. 
 
 5. x^+4:x, -3ia;4-4a;3.72_|.3iy4. 6. a2. 
 
 8. a;2 -6.1-3 +9^4+ 2x// .-6x//3 -6a;3y + 18x3//3 +7/3 -6»/3 +9^/4. 
 ■ 9. ixu{x^-y^), 2(1 + 12a;2 +16.1-4). 
 
 10. yV^^- 11- ^'2-2*2, 8«6(fl + &)2. 12. 2(a-c)(i-d), 
 13. ia;2+i//= + i2= + K.^7/ + Z/« + ;^a;). 15. (l+x--)^. 
 
 16. A(x;/+yz + zx)-2.{x^+y"+z^-). 17. .x^. 
 
 18. (a'-^+2Z.2_2c3)2. 19. 16a;2;v3. 20. -4rt&. 
 
 21. 4(« + A-fc)2. 23. 4(1+^2 +a;i+;c^^). 
 
 24. (rt^X-- +63^2)2. 
 
 Exercise vi. 
 
 1. l-4:x+10x^-20x^ + 25x^ -2ix- -f 16a-.^, 
 l-2x+3x^—4:x^ + 3x^-2x''+x^. 
 
 2. 1-4:X + 8.r3 - 14.c3 ^ 14.^*-8.r5 +o.x6 +6x'' +a;% 
 l_|.6:c + 15:c2 +20a^3 r l^x^ + (3.c^+x«. 
 
 3. 4a2-+??- + c4 +1 -4aZ/ -4af3 -4a + 2&c2 + 2/) + 2c3, 
 1 +a;2 + ,/2 ^23 _ 2^ 4- 2^ + 2z - 2;cy - 2xz+2y2, 
 kx^+y- + 362- - k^y + 6x2 - iyz, 
 
 4. a;6_2x-'^^ + 3x%3_4.,.3^34.3^3^4._2x7/5+;/6, 
 
 a2a;2 + 2ai;«3^.(2ac + &2)a;* + 2(rti + 6c)a;^4-(26(Z+r2)^8^ 
 
 2cc/.«^+(i3a;8. 8. 3{a^ +b^ +c^')-2{ab+bc+ca). 
 
 11. 4rt.-+i?>2a;2 ^__i^c3x- +4:d-x'^ - 2abx — acx + 8adx+ ^icx* — 
 2Wa;2-c(ia;3. 
 
 Exercise vii. 
 
 1. (&2_63)2. 2. ix4 + ?/4. 3. rt4 + 3fi253+454. 
 
 4. a;4_.^4. 5, ^2^ 6. IGr,?. 7. U. 
 8. 4a4-9&4_16c*4.24^2c2. 9. 5* -9c2 -4n'2+12ac, 
 Qc^-4.a^-b^+iab. 10. a;.8 -^/S. 11. ajS+x^^^+t/'. 
 
ANSWERS. "^ 
 
 12. a-'-a^b^+b^-l. 14. x*+.v* + iV'Z/'- 
 
 15. a;S+2.c«+3.K'» + 2.^2+1. 
 
 16. 4:a^x^ -Aa^xy + a^i/^ -a^x^ -2a'''.}^y+2ax^ +^ax^y -x^. 
 
 20. (a:2+i/2-2x//-z2)3. ■ 21. x»-.v«. 
 
 22. i-6«2 + 27a4. 23. (w+^)3-(w+^)2. 
 
 2i. 2.i;-+a;^+2x°-a;«-l. 25. 6*8-^,16. 
 
 F. 
 
 XERCISE VIU. 
 
 1. x^+ix^+Sx^-2x-l-2, a;-+i/3-2.r// + 8.c2-8i/2+1622. 
 
 2. x^ + 12x--i-49.t-2 4-78u;+i0, x''+6.«3 _a3_|_3fli_2/;^ 
 
 3. a3+8a6-10a4-104«3 + 105, a;^4-2.i;S -x^— 2. 
 
 4. a;44-5x3^^'- 12.^2//^ +5a;?/3+?/4. 
 
 a;2 1/2 2a; 2y 
 
 6. a;2»-2x"-a2-16fl-63, —+^+— + —-1. 
 
 (/- X- y ic 
 
 6. 7i2a;3_^2na;?/ + 7/2 + 10»a; + 102/ + 21. 
 
 7. (^. + a)3 + 27/(.i; + a)-3y/3. 8. u;4» + 2a;3» + a;2»(l-a-6) - 
 a.«(a+i)+a6. 9. ia-« -.C^/y^+i/* -a;^ + 22/a -8. 
 
 /I 1 \ 2 /I 1 > 5 
 
 11. a;4-8x3+19a;2— 12a;+2. 
 
 12. (u;+6)4-(a34-c3)(a;+^')^+»^<''^- 1^- «^+c<i. 
 
 Exercise ix. 
 1. 2(1+3j;4), ^xy^{dx^+x''y^). , 2. 96(a» + i2+a63), 
 
 b{21a^-21ab-i-lh^). 3. (a; + ?/)3. 4. Sa^. 5. Sx^. 
 6. 8a;3. 7. aS. 8. 27x3. 9. {2+x)^. 12. 8{x'+y-')^ 
 14. (a3 4-i3)(;x3 + 2/3). 15. 0. 16. 0. 
 
 Exercise x. 
 1. l-3a;+6x^-7x^ + Gx^-dx^+x^, a^ -h^-c^ -3a^{b-\-c) 
 + 3b^{a-c)-^Sc^{a-b)+6nbc, 1 - 6a:+21a;3 -oCx^ -h 
 lllx* - 174x-5 +219x« -204x^ +144x8 -64x'''. 
 
IV ANSWERS. 
 
 2. ~(x^ 4- iar« + 27.r' + 2f)x<' - lix^ - 36a:* + 5a;' - Sx^ - 2). . 
 
 5. 0. 6. ■i5x«+16Ssx*//^-432a;3^22a. 7. (^ax + bu + cz)K 
 
 ExERcisF, xi. 
 
 1. a;«4-6^^?/+15a;*//3+20a;^2/^+15a;3y4 + 6a;y'+2/^, 
 
 a;^+7a;G?/+21a;-5?/2 + 35a;4?/3 + 35x3.y4+21.i-27y«+7ic?/«+i/% 
 a;8 +8x^v/+28j.-6^2 + 56s;5y 3 4. qOx^yi + 56a;3yfi4-28a;2^6 + 
 
 8^//' +;//», a;i 2 4.12x1 i^+66a;iOi/2 + 220u;'^//^ + 495a;Sy*-h 
 792x-^^^+924^6;y«4-792^"//'+&c. 
 
 2. The signs will be alternately positive and negative. 
 
 3. a^ - 5a^h + IQaH^ - lOa^b^ + 5ab^ -b^, 
 
 a* — 8a3^-[.2ia-62 — 32tt/;3 4-iG/>^, same as last, terms in 
 inverse order. 4. l + Giii-{-lo)tt--\-'2{)in^ + 15m*-[-6m^ ■+■ 
 m^, )n^+5m'^ -f- lOm^ + lQm-+oin + 1, 6-i/M^ + 192;u--|- 
 240;«4 + 1G0m3 + GUw- + 12^ + 1. 5. 120. 
 
 6. x^ -4:X^y + &x'^r -ix^ij^+y^, a^ -10a462_f.40a3^4_ 
 
 160a9^9+240a6/;i--192a3i;i5^ti4^is^ 
 
 7. 49i5a8^*-792rt^i-'. 
 
 Exercise xiL 
 1. l+x3+a;*+-^•«+a;l^ 2. 1 +x + a:''+x3+a;4+.BG_|.a.7.|. 
 a;«+a;»+a;i«. 3. x4 + 2a;3-85a;2-86^+1680, 
 2^3 -3^6 +4x5+^4 4-^^ -2^- -x+2. 4. x^-57x^ + 
 266x2-1. 5. 18x8 4-21x' + 8a;6+a;5-f-63a;-^+96a;2 4. 
 43ic+6. 6. l-i^--ix-4. 7. 6a;i3_4^9_5^8_2a.7 + 
 9a;°-10x5+a;4-5x3+5x-2 4-:c-f-4. 8. x^ +dx- -irlQx+ll, 
 x4 + 3a;3. 10. x'^-'dx^. 11. a;4-f-8.c3_8a;. 
 12. (1), -1, (2), -1, (3) -4. 13. -1. 
 
VNSWERS. y 
 
 Exercise xiii. 
 1. Sa-S-2a;*-4r+2. 2. 5x* - 4x^+Bx^ -2x+l. 
 
 5. a3+8„2^+8.,:^.3 4..,.3. 6. 4:x'' -\-8x-\-7, -13a; -20. 
 
 7. 10^-3 + 5a:-+l, 10.^; + 10. 8. x^-xy + T/''. 
 
 9. a;2-«3. 10. x^ + {l~a)x^ + {l-a+b)x^+(l-a]x-\-l. 
 
 11. 3a;3+2.c3+a;+Ur3i(a;+l). 12. 5a;2+13.ry+12(/3. 
 
 13. 6x^-x^-x^+x^-x+Q, -1. 
 
 iJ. 2xi-3a;3+4a;2_5^ + 6. 15. a + b. 16. «+!/+*, 
 
 17. 10x3, 10(a;4-20). 18. wrc^ 4.^^.2 _}.«. 
 
 19. i+x-oix^-dx^+Ox^. '20. 33. 21. -4. 
 
 22. -2U. 23. 15^4. 24. 85x+8. 25. 755 
 
 Exercise xjv. 
 
 1. ?/3-2y2-4i/-9, if?/ = »-l. 
 
 2. ?/3 + 3?/+5, if ?/ = a:+l- 3. y^ + Sl, i: y = x-2. 
 
 4. ^^4^1,^3 _432/3+92y_67, if7/ = a;+2. 
 
 5. 3.v''+30y4 4-ll9v/3 + 238^3 _|. 249,7 + 106, if 2/ = a;-2. 
 
 7. V^-'-iy+W^ i!iy = x-h 
 
 8. (x-2?/-'')-3//(a;-2y)--18i/3(a;-2r/)-24yS. 
 
 9. (^-//)^-107/2(a._,y)3_207/3(a:-2/)'2-10^4(a;-_j/). 
 
 10. (2a: + i/)3 + 2//2(2a;+2/) + %^- 
 
 11. 5l2(/3-3//--ri/4, ifi/ = ia;-TV- 
 
 12. ■*-24(/-^+492/-28, if2/ = ^j;+2. 
 
 Exercise xv. 
 
 1. a^b, +ab^-\-n^c+b^c + bc^+ac'>, 
 
 (a-6)3 + (i-c)2+(c-a)3, a{b-c) + b(c~a) + c(a-b), 
 ab{x-c)+bc{x — a)-^ac{x-b), 
 abc{a^b-\-a^c + b-c + ab^ -tac^-^bc^), 
 
VI ANSWERS. 
 
 {a + h){c-a){c-b) + {b + c}{a-b){a - ^r ■■i-a){b-c){b - a). 
 
 (a+(;)--63 + (6+a)3 -c2+(c+&)^ -a^ 
 a(b+c)^+b{c + ay+c{a + b)^. 
 
 2. cibc-{-bcd-\- cda-\-dab, 
 
 a^{b+c+d)+b^{c + d + r,)+c^[d-[-a + b) + '!"(a + h+c), 
 {a- b) +(a-c) + {a-d) -\-[b -o)-\-{h - d) + {c-d), 
 a^{a-b) + b^{b-c)rho-{c-d)+d2{d-a). 
 
 13. X and y. 14. ax and i?/, x, y, z. 15. / and h, 
 16. a; and y, also ic and — z, and y and —2. * 
 
 n. a, ^ and — e. ^18. ic^, — ?/- and s-. 19. 6 and c, 
 
 20. a and c. 21. a and 6. 22. a^ and 2a&. 
 
 23. a-o and aic. 24. a^/^^ abc. 25. a;^, x*?/, and x'^y-\ 
 
 same ; a;^//, u;^?/-. 26. Not symmetrical. 
 
 28. uS u^o, a^bc, abed; u^, a'^u^. 29. a^, a^b. 
 
 Exercise xvi. 
 1. 4(a3 4./,2^c2). 2. 3(a24-62_|_c2)^2(ai+k+ca). 
 
 3. 4(a3 + fe2+c2+rZ3). 4. ^(a2+63+c3). 
 
 6. 4(x2+?/2_i.22+7t2). G. 2(a3+63 4-c3) + 62a3&-12a6c. 
 
 7. 14(a;3+2/3-fz3)+2(a;(/+?/3+2a;). 8. ^Ubcmnr. 
 9. 2ak(a+i+c). 10. a263+i2c2+c2a2. 
 
 Exercise xvii. 
 1. 115. 2. ?m»-3<7a2+3m-s. 3. 2. 4. -17-3533. 
 5. 1, 2(3a2+l). 6. or 27/", 2?/", 0. 7. 36. 
 
 8. -(i2+a2)3_(3&2)3. 9. -15a*. 10. StiSSa^b*, 
 11. a2fe2(« + ^,), 12. 0. 13. 2a3-3a6(a-6), 
 
 263-i-6«6(a+Z/), 2(«3 + i3). 
 
 Exercise xx. 
 
 1. 3. 2. 1. 3. -l±2v/-2. 4. 2. 5. 36. 6. 11. 
 7. -l^f 13. ;^=-?, 9 = 6. ■ 14. /'=-46, 5 = 14. 
 
ANSWERS. Vll 
 
 Exercise xxi. 
 
 1. ^=_3, r = 8?5, ^=-24. 2. c= -20i, f7=-13-^, ^=GOf. 
 
 8. /)= -3, (■= -10. 4. a = 3, /> = 0, ' = -57. 5. a=-2, 
 c = 24i, ^ = 0. 6. c=- 100^, (1 = 202^. 7. a = 200, 
 6= -810, c = 639. ^ 8. a = 4, c = — 27, d = 7, e = m. 
 
 9. 399. 10. x^-{p + -3)z^+{2p-{-g+3)x-{p + q-^r + l). 
 
 11. x3-(/?-3)a;2-(27J-g-3)a;-(^-g+r-l). 
 
 12. ra;3-(3r-g)u;2 + (3r-2g+7j)a;-(r-5+;>- 1). 
 
 13. x^-qx^+prx-r^. U. x^ -{p2-2q)x^ -t{(]' -2pr)x ~r-. 
 15. x^ — 2qx^+{pr+q^)x + r^—j)qr. 10. ra;^ — (y^y + 3r)a;2 + 
 
 (p3_2;>g + 3?-)a;-(/)?-r), 83. -i. 34. 1. 35. -1. 
 3G. 1. 87. -1. S8 -And Bd. a + b + c + d. 40.-1. 
 
 Exercise xxiii. 
 
 1. 5h*+15c^. 2. 6. 3. 3. 4.- {y}(?;+r+(0 + --- + --- + ...}. 
 5. 0. 6. 5i4_30fl/,3-f80rt3^a-5«36. 8.0. 9.0. 10.0. 11.1. 
 
 12. {a + b-{'r + d). 18. -1. 14. a + b+c + d. 
 
 15. (a+^>+c')(r/2 4-i3_^c"+rf& + ftc+m)+^//;6'. 
 
 16. {a + b+cy{a^+b^-{-c^) + 2abc{a-{-b + c). 17- a + 6 + r+(i. 
 18. (a + A + c+rf)2. 19. {a + b-{-c-{-d){{a-{-I) + c + d)^ - 
 
 (ab-{-ad+ac + bc + bd-\-cd)}+abcd. 20. a + 6H-c. 21. 3. 
 
 22. -1. 23. 0. 24. 0. 25. 0. 26. 1+lx-^x^ -{-T^t^x^. 
 
 27. l-ix-ix^-^'-sx"", 28. 1+x+x^+x^. 
 
 29. 1— 2a; + 3a;2-4a;3. 80. l+lx-^x^+^\x^. 
 
 Exercise xxiii. (a) 
 
 1. (p-pf+q)^ = {p + l){p2-pp'-q). 
 
 8. 9(p3 - .7)(r2 -g«) - (pr-t)^ = 9{3(^3 -^)(gr-p«) - 
 
 (j99-r)(pr-0}x{30)g-r)(r3-90-(??/--?;)(9r-;7/)}- 
 
 9. af{4:X^-[-Spx^ +Sqx+r) -r- (ic4+4i;a;3 + 6^a;2+4ra;+ij. 
 
VIU ANSWRRS. 
 
 10. -ip, (ip)^~2(Gq), -(4:p)'^-^B(ip)(eq)-^(ir\ 
 (4;^) 4 -4(4/>)--'(6y)+4(4p)(4r)+2(6^)3 -4^, 
 
 -(4/;)^ + 5(4p)3(6y)-o(4y>)2(4r)-5(4j9)(6?)2 + 5(4p)<+ 
 5{6q){4r), (4p)6-6(4;j)4(6^) + 6(4,v)3(4r) + 9(4;^)2(G(/)2 - 
 6{Ap)H-12{Ap){Qq){4r)-2{6q)^ + 6{6q)t+S{4r)K 
 
 11. .Sp.s^ -4si.s3+a'?^, .f^Sp -e.Sj.s^.+lo-Sj.s^ — lOs^, where So, -"i, 
 &c., are the coefficients of the terms (taken iu order) of 
 the quotient in No. 10. 
 
 12. a;»(4,x3 _2Sx + l) -^ (x^^ -Ux'^-\-x-S8) ; .s, =0. Pj^SS, s, 
 
 = -3, .54 = 544, .s'5=-70, 6-g=8G83; 2(a-/.j4 = 4526, 
 S(r/-6)6=2G4122. 
 
 Exercise xxiv. 
 1. (3m4-2)-', (o--l)2. 2. (y3 _,^3)2, 4,^?(2.r-f-7/)3. 
 
 3. (3ai + 2c)2, 4?/3(3«-.7/)2 4. (^a;^ -47/^)3, (^a^ _i^2c2)2 
 
 5. (., + /' +c■)^(3.^•4-l,/3)2. 6. (2-a;+2/)3,|(-) -(--]} , 
 
 7. (ar2-22)3. 8. {x-7j)K 9. (a + M-, (fi-'^—k-^)-. 
 
 10. (x-//)3. 11. 4(a:3 +7/2)2. 12. (x-+^)*. 
 
 15. («2_^3_c2)3. iG. (2rt-2c)3. 17. (2a2_3S + 4.)s. 
 
 Exercise xxv. 
 
 1. (7rr + 2/*)(7a-26). 2. {Sa+U){^a-U). 
 
 3. (3a-2i)(9.^3+4/;3)(;v, + 2&). 4. (10.r-6/y)(10.r+G.v). 
 
 5. 5i(a + 2x;/)(«-2x2/) 6. (3a;-^-4.y3)(3j534-4//2). 
 
 7. (ic + l)(fc-l). 8. (27/2-t:f2)(2.v= + §x^). 
 
 9. (3a-l)(3fl + l)(9r/3-f-l). 10. (rf-2h){a-\-2b){a2 + U^). 
 
 11. (a-Z-)(a + Z>)(a2+Z/3^(a4 + 64)(a8+&8^. 
 
ANSWERS. IX 
 
 12. (a^h-c)(a-b-\-e). 13. {a + 2b -•Sx + 4:y)(n-}-2b-'^x + 4:y). 
 
 14. (a;3 -7/3)3 15. (^x+y-\-2z){x+y~-2z). 16. XG(.r+l)(l -a;). 
 
 17. {x+y+z){x+y-z)(z—x+y){z + x-y). 
 
 18. ixy{x + y){x-i/). 19. (a;— 24-?/)(a;-2— i/)(.c+2+2/)(x+z-?/). 
 20. 4(a + c)(i+^). 21. 24a;(H-2a;2). 22. 8«i(a + 6)2. 
 
 23. (a + b+c-\-il}{a+c-b-d){a-h-~c + d){a + b-c — d). 
 
 24. (a;+j/+2)(a;-?/— z)(x+^-z)(.r-.V-l-2). 
 
 25. 8a3Z)3^aG_3a3?>3 + 66). 26. {a^ + bS){a^ -b^)». 
 
 27. (a;2+2/2+«2)(^2_|.^2a-22--2x^-2?/z-22x-). 
 
 28. {x+2z){x-2y). 29. (« + i-c)(rt-6+6-)(i/ + c+«)(i + c-«). 
 30. (a;-2/ + z)(a;+2/-2)(d; + 2/-f2)(a;-</-2). 
 
 Exercise xxvi. 
 
 1. (a--7)r.r+2), (a;-7)(a;-2). (a;+4)(a; + 3). 
 
 2. {x-3)(x~5}, (a;-7)(a;-12), (a;- 12)(.r+5). 
 
 3. 2(2a:-5)(a; + 2), 3(3a:-20)(a;- 10). 
 
 4. -^(a;+l2)(ia;-3), 5(x+l){5x + S), {Sx^ -4)i3xS - 5). 
 o. (ia;+4)(ia;+3;, 4(4a;-5)(a;+l). 
 
 6. (x-a){x+a)ix-b)(x+b), {2(a:+?/)- 11} {2(a;-!-y) + 9}. 
 
 7. (a;^ +7/2 -a2)(a;2 +7/2+^2). s. {a-tb-3c){a-^b+c). 
 9. (x+y)(l-\-x+2/){x+y+(x-y)^}. 
 
 10. (a-{-b){l-a-b){a+b + (a-b)S}. 
 
 11. (a;^+x//+y-+2a;+2/)x{.c2+a;//+2/3-(a;4-2y)}. 
 
 12. (a-86 + 3c)(a + &-c). 13. (x--'+y2 + aS)2 ^b^ =Scc. 
 U. (a;- - 10.-c-l2)(a;^ - 10a;+8). 
 
 le. (a;2-14a;+10)(a;-9)(a:-5). 16. {x^--y^-)2, 
 
 17. (z+l)i2-l)(z2-2), (.«^'-3)(ar3 + lj, 
 
ANSWEKS. 
 
 (3.c* + 5./y2)(3a:4-2?/2). 18, (L-'"-f 2)(c"' - 1), 
 
 19. {x"'-ufj{x"' + by"]. 
 
 JdjXEEciSE xxvii. 
 
 1. {x-hy){l>x-v). 2. 3(ic+2//)(2.c-?/). 
 
 3. A[Ux-5y){x-y). 4. 4(14.c4-5^)(a; - i/). 
 5. (14.c-^v}(x— 20//). 6. 4(7ic-5//)(2a;-2/). 
 7. 2(28x + 7/)(x-10?/). 8. A{14.x-5y){.r^y). 
 9. (8x-5^)(7x-4y). 10. (8x+5//)(7«-4?/). 
 
 11. 2(3x+?/)(a;-3?/). 12. (3x-2//)(2a;+3?/). 
 
 13. 2(28a;+?/)(^c+10i/). 14. 2(28.6- -5//)(;c-- 2y). 
 
 15. 2(28x+5.v((.«-2//). 16. {5Qx-5y)[x-4:y). 
 
 17. 2(4,.r-2/)(7u;-10i/). 18. ^{l4.x-\-y){;£-5y). 
 
 19. 3(3.i;+^)(4.c-5i/), 20. (8u;+.jy)(9x' -8^). 
 
 ExEKcisE xxviii. 
 
 1. (5:i;-7)(2a:+3). 2. (6:c + 3)(2x-7). 3. (5x--3)(2.c+7). 
 
 4. (2a;-5)(3x-ll). 5. (4a + l)(3a-2). 6. (dx-T)>yix-o). 
 7. (8a; + 7)(4.c+3). 8 (.5.,3 _46-2)(3a2 + 56^'). 
 
 9. (4a;+l)(3a;-l). 10. S^^^x— y/)(:J.c+2/y). 
 
 11. (2a; + 3!/)(2aT4-?/). 12. xH^b-^x){'lh--dx), 
 
 13. (3a;3 + 72/3)(2a;3-.5?/3). 14. (2a;3-9)(j;3 + 5). 
 
 15. (2a;+?/)(22;-j/)(a;-37/)(x+37/). 
 
 16. (2.c+4+y)(2x4-4-?/)(.c+2-3?/)(a;+2 + 3?/). 
 
 17. \mxy. 18. (19//2 + G0a;//-6a;3)(3oa;^-12x?/4-yU//3). 
 
 19. 2(4.i;y-3x3-32/3)(61a:2-49a://4-6l2/2). 
 
 20. 2(5^2 + 4.,-y + 10;/3)^x- + 10.cy + 2(/2). 
 
ANSWEKS. Xi 
 
 Exercise xxix. 
 1. {7x-^ey + S)(x-y-z). 2. (5a;-5i/-22)(4.r4-?/ + 4). 
 
 3. (3x2 +4y3 + 13)(x2 -2/3-1). 4. (4a-_|_52^)(5a;-4?/-«-7\ 
 5. (9x+8?/-20)(8a;-?/-l). 6. {x+Sy){x-i>/-5). 
 7. (4x + 3;/-z)(2x+3//+0). 8. (3a;-22/-2z)(2a;- S^z + Jz). 
 9. (3x2 — 27/2 +5z2)(2x3 _|_5y2 _ 5), 
 
 10. (15x-2+8//2 4-5z2)(a;a_2?/2+3a2). 
 
 11. (2a.-5i-7c)(2a + 3/i-f3c). 
 
 12. (rt - b-\-c){a + b- c){a + b + c){a-h - c). 
 
 EXEECISE XXX. 
 
 1. x-+-J+fi/5, 2a;2 + |±-||/5. 2. x^ +|^2-^|^/2 ^C ■ 
 j\(6x2-r5y^±y^vlS). 3. i(4a;2 + 6± v/13), 
 
 T-V{6(x+y)2 + 523±a2yi3}. 
 
 4. (a:2 + iy2)(a;2_|.6^^3), (.^2 +ii,/5)(a;2 + |^2). 
 
 5. (2.f'^+4i^2)(2^3 + iy2), if4(V(+/;)2+5ci>/13}. 
 
 6. ^V(6^^+5^2)(6x3+ll//2), (6x2 4-5)(6x2 + ll). 
 
 7. J(5x2+10 + 3v/10), i2a2^3-t.2j/2). 
 
 8. {2(x+yi2 + (3±2v/2).s^} ; 
 
 -H 10x2 + (10±3v/10)!/2}{10x'^+ (20- 61/10)2/2 }. 
 
 9. J(9.c^+7+v/13), 4{2x2 + (0.L--i/16)(2/+2;^}. 
 
 10. ^(2x2+6± a/6), 1(7x2 + 20±v/; 85). 
 
 11. i{4x2+(9±y23)//3}. 
 
 12. i{7(^.-fe)3 + Sr2+rV29}, i{-)a^±b-]/n}. 
 
 13. i{3x2 + (3±v/3)^2|^ ^{3(^a-^by + {S±^/S){a^hy}. 
 14 {7a--'+(G±v/14)6-}, (5»i3+9/i2)(5m--}-3,i3). 
 
Xii ANSWERS. 
 
 Exercise xxxi. 
 
 1. (x^±^xj/-\-^if9), (x^±xy-y''), {x^ +xj!-\-y^).' 
 
 2. (x^±2x]/ + 2y^), (4a;2±3x?/+?/2), (Xx2±xy+7/''). 
 
 5. (x"" +i/2x+l), {x''±VGxy+Sy^), {l±2y-Ay^). 
 4. {x^±Sx-\-l), x^±V6x + 3, ^,r2±2j-?/ + //3. 
 
 6. {2x^+y^±?ixy), (x^ +y^±ixyVS9), {2x- + l±2x). 
 
 7. (a:2'" + 8?/2'" + 4a;"'?/"'), x^'^ + 2y-^'"±:2x'"y"'), 
 (^a;2 _ |jy2 ^;;£c^/ V5). 
 
 8. (2d:2-l + 2a;), -(^icS _6?/2 +x?/) (^.7-2 -62/2 -a;y), 
 («- +a^y^ + axyi/2). 
 
 9. mig^—ny^±xyyp), a;2"«-f 2'"-»y2m^-2'"?/'». 
 
 10. 4x3 -3±a;, 2a;2-2±2a;^/2, 
 -(3x2 -2i/2+a;y)(3a:2 - 2;y2 -xy). 
 
 11. 2^2 ±4.1-7/ -3?/ 2, a;2 + 22- + 5. 
 
 12. 2{a2 + ab + h^)^, {2a^-\-a + l)2. 
 
 13. {(a:+2/)2 + 3(x+2/>+z'}{(ic + y)^-3(x4-?/)~4z'} 
 
 14. {a + b)^+^^dz^G^V5. 
 
 15. {4«3 + 6a(6-c) + 2(fe-c)2}{4a3-5a(6-c) + 2(6-c)2}. 
 
 16. 4(a2 + 5«i-2/>2)(62_|_5«i_2a2). 
 
 17. {ix^+y^-xy)^±S{x^+y»-xy){x-[-y) 
 + {x+y)^}. 
 
 19. (4a2 + 2a+l), a;2+7a;+4. 
 
 20 (a;2±9x// + 9//2), {l±:Sz+^z^). 
 
 21. 4(3a;2_2x+l;(a;2_2a;+3). 
 
 Exercise xxxii. 
 1. (a;2 + 3)(3: + .Sl(.r-l). 2. 2(.)-2-f 3)(r2 + .r -f?) 
 
 3. {x^ + A)(x+i){x-i). 4. (a;+2)(x-2j(3a;2+x+12). 
 
ANSWERS. Xm 
 
 5. (a;2-B)(5.f^ +4.^+15). 6. {x^ ■\-G)(10x^ + ^x-G0). 
 
 7. (ix2+TV)(i^^+-^0-«-To)- 8. (5^2 _i)(5^2_ 8^+1). 
 
 9. (5i;2-8)(7ia;3-6x-12). 
 
 10. (3x2-4)(21.k2_13u;-28). 11. {18x^-\-l){A5x^ +^x + ^). 
 
 12. (ll.'«2 + l)(22a;2-8x-2). 13. (^^^ -f)(^a;--^+ia;-f§). 
 
 14. 8(;r2-2?/2)(10x2-4a;?/ + 20y3). 
 
 15. (2x^ -5?/^)(12a;2-6:r(/ + 30//2). 
 
 16. (x^-16y^){2x'- + l^y + ^^y^}- 
 
 17. (a;2-|)(llx2+10a;+Y)- 18. 10(.t2+2)(4x2 + 3.^-8). 
 
 19. (a;2-6//)3(13x3-12a;y + 78y3). 
 
 20. («2+4?/2)(3.,.2+3a;y_i2y2). 
 
 21. (x2-3?/2)(5a;2 +4x^ + 15^2). 
 
 22. 2(x2-2//2)(2a:2 -7x?/+2//2). 
 
 23. (a;2 4-i.!/2)(ic2-t-80xt/-^//2). 24. (x^ -6y^)(2x^--xy ^12ij"-). 
 
 Exercise xxxiii. 
 
 1. (x2+3./: + 27)(:«2_9.^+27). 2. x2 +a;(l±,/3)+4. 
 
 3. {.j;2+l+i(l+ a/5)4. 4. x^+l-x(2±^/5). 
 
 5. 2x2 + 2-3x±a;\/23. 6. (x^ + lSx- 5)(.x-2_a;_5). 
 
 7. (la;2-2)(4x2-6x-2). 8. {x^-}-8x+4:)l,x-\-3x+^). 
 
 9. (x--^+7x-2)(x2-a;-2). 
 
 10. (a;3+5x// + 3i/2)(a;2-a;2/+3y2). 
 
 11. (x2+.10x^l)(a;2 + 2a;-l). 
 
 12. ix^+l.ry + y^){x^ -Sxy+y^). 
 
 13. 2x^ + xy-5y^±xyi/4.6. 14. [x^-^.'Jxy -y"-){x^ -xy-y^), 
 15. a;2 + 2?/2+3.r?/±a;yi/3, 
 
 10. (3x2+10a;?/-2?/2)(3x2 -4xy-27/2). 
 
 17. T-V{lla;^+22//'-' -h5x(/±ffx!/v/ll}- 
 
XIT ANSWERS. 
 
 Exercise xxxiv. 
 
 1. {y-z){T:^-y). 2. {hy + c){ax-^hy-c). 
 
 3. (z-+a){x-\-a){x-a). 4. (2a;-a)(a;- 2/*). 
 
 5. (.T+3a)(a; + 2^). 6. {x~b-^){x-a){x + a). 
 
 7. (a;-6)(a; + ?))(x-fl)(x2+^cG+a2). 8. (2a;-l-3«)(4.'c+5/;). 
 
 9. {a-\-hx){a-hx-\-cx'') 10. (a-&s)(« + ^^x+c.x-2). 
 
 11. (aa;-(^)(5a;3+rz-/). 12. {px-q^x"' -x-l). 
 
 13. („_6_c)(« + 2A + Br). 14. (ic + «)(x^ +x + 1). 
 
 15. (wx-n)(;9x2 4-(?a;-r). 16. (.'B-«)(.'c~/>)('a:- (■). 
 
 17. {x-\-<i)(x-h){x-c). 18 (xH-ri')(^4-/;)(>c-c). 
 
 10. {a^-{~z){x-ay){x,^ -y). 20. {abx+nly -c.fz)(ax+hy). ^ 
 21. ('/ic + c)(^'a;3-/«+c). 22. (x-y){x-i-y){nix-iiy-\-rz). 
 23. (i»x-w?/)(rta;+/'// + ''z)- 24. (;/'a^ + «)(«.i-'-6ra-4-'f). 
 
 25. (c2-a;3)(/>2-|/2)(«--a://). 26. (x^ -jwSa;^ -«)(.r3 -« + »-M, 
 
 27. {l + x-x^){l-ax + hx^—cx^y^ 
 
 28. (fla:-'/'/M^-«— ^.'A)("^+c?/). 20. (?7?.r+9)(;?a;4-")('"^-»- ")■• 
 30. {mx-\-Hij){:iix-ny}[p'x'^ +q-y-){x+l). 
 
 Exercise xxxv. 
 
 I (^a+x){a — b). 2. {ax-{-hy){bx ny). 
 
 3. {x-n){x+a){x- +ox + rr-), -L .r(n+x){a^ -^ax+x^). 
 
 5. (aa;-6)(ca; + f?). 6. (5a;2 - l)(5a;2 -.^ + 1). 
 
 7. (fl-/*)(rt + /^+.i--c). 8. (« 2 + 7,) („ + /;). 
 
 9. (x-?/)(a;+?/)^ 10. (a;-;y + l)(a;2 +^;_v-^-_y2^ 
 
 11. (fc-2.r)(2 + 6.v). 12. (;r-l)(a; + 2)2. 
 
 13. {p-q){p--2g^)- 14. (a-l)(a2 + 2a + 2). 
 
 15. (a/.2_l)(3a/;2 + l). 16. ^y-l)'-^y + 2). 
 
ANSWERS. X'V 
 
 17. (a^h)(2a2-nah-{.'lh^). 18. (Z^"'-l)(/>2'"4-2?>'"4-2). 
 19. (^" + z"j(y-'«_8v"z"+,2-"). 20. {u-b)(a^-i-ab-'db-), 
 ^1. (a"'-c")(a"'-2c''). 22. (aa;-6)(a;- -aa;-/>), 
 
 .28. (5a;"-3rt2)(7:c" + 3rt2). 24. (a6+6c-c«)(aft-ic + 01). 
 
 25. (w-6)(w + M(a-»t). 26. {^ -Sa-){l-Sa){l + 3a). 
 
 27. (a;-y-z)(x3-2.c^-}-//2+2). 28. {6m-7n){4:m^ + n^). 
 
 29. (.6-"' +/)(.*;" +y"). yO. (.c-+.c// + ^;x-+(/2)(a;-^+sr^ -ax-y/)^. 
 
 Exercise xxxvi. 
 
 1. {x~y){x-{-7/)(x^+x;/ + !j^){x^-xy+!/'), {x -l){x-' +x + l), 
 {x+2}{x^-2x + 4:), (2a--Sx){ia'+(iax-\-dx-), 
 
 {2 + ax){4:-2ax+a-^x^). 
 
 2. {x-a^){x'^+.r'^a-+x2a'^-\-xa^+a^), 
 (3f(-4)(9aa-[- 12^ + 16), {a^ -b-^){a^ + b-'){a<^ + b^), 
 (a;3 - 2//)(u;» + 2x-«//+4.i;4^iJ + .-:Jx-' //3 + 16//4). 
 
 3. (rt-/0. 4. a:+4.v. 3. (.c+//){ic- +2/3)(u;4 + //^) 
 7. i'(a; ?/)(i/-hl). 8- {x-a)[x^+ax+u-)[a r b). 
 
 11. {a^+bc){a^-^a^bc + 7b2c-2). 
 
 12. (x-a + ^){(.r-(0^-(^-«> + ^^}- 
 
 18. (a;2 _2x-// + 4]ry3)(.c+2?/+4.r,y). 
 
 !4. {2x+S,j){2x-3y)^. 15. (1 - 2.r)(l+4.t3). 
 
 LG. (a-^ -^ubc + b^c^){a+be){a^ -abc + b^c-'i). 
 
 Exercise xxxvii. 
 
 1. d(x-\-y){y + z)(z + xJ. 2. {a-b){b-c){a-c). 
 
 3. 3(aa-/>2)(>-!-6-3)(c2-(,3). 4. ^^_j.^)^^_|.^)(2_j.a;). 
 
XTl ANSWERS. 
 
 5. 8(r;f + /;)(/; + o)(c + a). 6. («+fc+c)(a-?v)(&-c)(c-a). 
 
 7. (a+//)(ft + c)(f+rt). 8. („3-Z»)Z.2_(.)(t.3_«). 
 
 9. (a + A)(/; + f)(c + a). 10. (a-6)(6-c)(c-a). 
 
 11. (.r3-f?/2)(//3 +23)^23 4.3,2). 
 
 12. («3 _f.52 .|. c^^ab~bc - ca)ia - b){b-c){c- a). 
 
 13. (a3 + />2^c2)(«4-& + c). 
 
 li. (C-i3)(a-c3)(6-a3). 15. (^2 _2/2)(y3 _22)(,c3 _2-2). 
 
 10. (x + ?/-f2:)(a;-//4-z)(.7-c+.'c)(z + 2/-«)- 
 
 17. (a-b){h-c){a-c). 18. 8(a + i4-c)3. 
 
 2i. (a-6)(6-c)(a-c)(rt2 + A3 + c2+<76-f-/;c+ca). 
 
 ExERcisK xxxviii. 
 1. (r,-2)(a2-7« + 2). 2. {x-2){x-S){x-'i). 
 
 3. (x-3)(a;-2)2. 4. (a;-2)3(.r + 4). 
 
 5. (x+l){x^ +2x-rS). 6. (a;2+2a; + 3)(x2+2x + 3). 
 
 7. (x+2)(a;-l)3. 8. {x^+2x+3)(x^ -2x+'d). 
 
 9. (»t-«)(m2-2wi»— 2?i2). 10. None. 
 
 11. (M-n)(»/-2u)3. . 12. (6+3e)(i2-26c + 13c2). 
 
 13. -(m-w)3(w2-wn + >(). 14. (rt+26)(fl— 26)(«3— 7ai+46-'; 
 
 15. (x-5)(.-c-3)3. 16. (.«+2)(x3-|-3.c+l). 
 
 17. (r*-l)(a2_2a-195). 18. {p + 2){j)—l){p+4:). 
 
 19. (a-l)2(«+2)(a+3). 20. (fl.3"-l)(a=''-2)(a2"-3). 
 
 21. rt2 + 463+7a&. 22. (a-i)2(a3+2a6 + 263). 
 
 23. {jj-2){2J^-2j>+-2). 24. (.c"-l)(a;'-» + ua;" +5). 
 
 25. (y/-2)(,)/5 -37/2 +2^+4). 26. None. 
 
 27. (fl-/^)('t*- + 2ai+362). 28. (a''+l)(2«3»_3rt» + 2). 
 
 29. (x-2)(x-3){x-6){x-7). 30. {x-t/){x-2y){x-3i/)^. 
 
ANSV/F.R3. XTU 
 
 Exercise xxxix. 
 
 1. 2(x-l){x"--9x-\-10), {x-2i/)^x-3y). 
 
 2. (ix + 3i/)(3x'-i-x!/ + y''), (x-l)(4c-2)(2x+3). 
 
 3. {x-5ci){3x^+a''), {2x + oy){x^-{-3xi/-y^). 
 
 4. {b+c){b-4:c){2h^~bc-\-c2}, (5a+4i;)(3a3+76i//-3&3). 
 
 5. {2p+q){2iJ + 3q){p^+q-^). 
 
 6. (lOx- 9v)(15^+ 16//}(x-3 - 5xtj-\-8y2). 
 
 7. (2./:-3//){2x--f3//){3a;+47/)(3a;-5?/). 
 
 8. (5x - 2z)<^2.cS - 3^2^ + B^^a _^ 12^3^. 
 
 Exercise xl. 
 
 4. (^+2^)(a;2+Hv-). 5. l-2x-+3a;2. 
 
 6. {a~x)ia-\-x)^. 7. a;^ +z/^ +s3 +ie^+7/2-2x. 
 
 8. [a + />){da+b}. 'J. (a;-2/)(2.c + 3^). 
 
 10. aa_/,2_f.^3. 11. 7a2-Sab+2b2. 
 
 12. a -7. 13. {a-h)(b~c){a-c). 
 
 14. (x-a)2-ft(.a-_^,)-f/.2. 15. a;2_j_^2_j_22 ^1, 
 
 16. x{x--ax+b). 17. rc3 + ^3. 
 
 18. {x-y){x'+y^). Id. a^-b^+c'^+l. 
 
 20. a3-63_c3, 21. 'i-i-x. 22. (c-i)(a-f-6 + r). 
 
 23. ab — ca — hr. 24:. x^+y^-\-l-xy'\-x+y. 
 
 25. (x3-2)(a; + l). 26. a^+5a-\-B, 
 
 27. (2.<;-t/)a2-(x- + //)a.7:+x3. 28. a(a;2+a;+l)-(a;4-l). 
 
 Exercise xli. 
 
 1. j:2-3. 2. x-\-o. 3. a;3-.T+l 4. aa;^'4 6a^4-c 
 
 5. None. 6. c^+c*. 7. (a-i)(.cf«). 8. 6(a;+3/). 
 
XVm ANSWERS. 
 
 9. (a-b){b-c)(c-a), 10. «.2'"+l. 
 
 12. 5(a-b){b-c)(G-a). 13. {y-l)(x-l). 
 
 15. (x+l)(x~ + l){x~l)^. 16. {x+l){x+2){x+'6)(x yi). 17.5 
 18. Same as given quantity. 25. (a — 6j((!>-c)(c-a). 
 29. a;^+a;2+2./;-M. 
 
 Exercise xlii. 
 
 1. (.r-l)--(a;2+4x- + 16), a;(3^-7)-^y{77/-3). 
 
 2. (^^-ax^a^)-^[x--a^), (x + 4)-^U- 1)^. 
 
 8. (a;-l)(,x+2)-^(a;2-f5.*; + 5), ix^ + 2xyd)~{x^--2x~3). 
 
 4. l-(6-2./:), l^(ar3_2.<;4-2). 
 
 5. 5a^[a + x)-ir-x{a'-'+ax-rX-^}, {4:X- +l^^{5x^ -i-x+1). 
 
 6. (a;-7/)-^(;c-L//). 7. [Sax^ -hl)~{ia^x'^-\-2ax^ -1), 
 [ax^bi/)-^{''X — bf/). 8. —1-T-abc. 
 
 9. -(.,+/'+')-H(a-6)(6-f)(c-«). 11. 5-7(»2 + a;2/+?/3) 
 
 Exercise xliii. 
 
 1. (4-.r)--(5-a-), [a^ +h-)^2ab. 
 
 2. X, 2a-(«--M). 3. r^(l+rO--(l+2r,+.3rt2), X. 
 4. b^-i-a^, {h + l)-^cib^. 5. («c--M)-=-(ac+^)c^), ^n-a. 
 
 6. l + 6x27/z(7/+2j--{y-3--s:3(v/ + 2!)3}. 7. (a^ +63 ^c2}-v-a6c. 
 
 8. 1. 9. -0/^+a^6^+o4)^rt6(a_6)2. 
 
 /I— a;\ 2 
 10. {a + b + cy-i-2bc. 11. |j^l , 4a2a;2H-(a2+a;2). 
 
 12. (a.-+2/)-j-{a;-y). ' 13. (a-6)S-=-(a + 6)3. 
 
 14. (a; + ?/)^(.<;-2/) 15. l^a;3, 
 
 m. 1--W. 1/. ±(l-i)-T-(l + o). 18. l-»-c 
 
ANSWEBS. XlX 
 
 Exercise xliv. 
 1. (x-a)-i-5. 2. a+b. 3. 16a^x^ (a* -a:"^)*. 
 
 4. 0. 5. 1h-(5:+2). (y. lH-(n4-a;4). 
 
 7. 12^?/^(9.r2-4y3). 8. (4^2 +2) --.-'•; 10x4-1). 
 
 9. l~-{x + l)(x+2){x + B). 10, 4(a;4 + 4^2y2+^4j_j_(.c4__;y4), 
 
 11. (^a-b)3-i-{x+a)2{x+b)". ' 12. 2,',- --a;. 
 
 13. mG-77x)^18{llx- 8). 14. l-^(a-6). 
 
 15. 15a(8a-«)-^(9a4-2a:)(« + 3a;). • 
 
 16, {10x~7)-^{x~l)['2x-5)-l-^{-2x-l',.--4:). 17. Z. 
 18. y^y^'-K"). 19. (fl-/>r"+2. 
 
 20. 0. 21. 4.c^~ix'2_i). 
 
 Exercise xlv. 
 
 1. a;-y. 2. a + h. 3. 0. 4. 0. 5. 0, 
 
 6.- {{a-\-h){c+a)x^-\.2iah + >' +ca)ax-%i''hc] ^ • 
 (fl + //)(rY+r;)(x + a)x(a; + />)i>+c.) 7. 1. 
 
 8. «+i+c. 9. 1. 10. a;3-.v3 11. Q. 
 
 12. (ff-fc)(/>-c)(a,-c)-i-(a+6)(6+c)(c+«). 
 
 18. a;2^(a;-fl)(a;-6)(a;-o). 14. 1. IC. 0. 
 
 16. {h{;x + a-b)-\-ax] ^ {rt64-(o-rt)(x— o)}. 
 
 Exercise xlvii. 
 
 1. («-i,)2+4c.2^0. 2 8. 8. 10. 4. rt2+i». 
 
 5. 7«. = 2, ?i=l. 6. 2a;2, or 5. 7. 7?7=-5. 7? = 6. 
 
 8. +12, 9. (a2+/;3l(c2+d2). H. _ 3?;C - 4^2 4./,2^2 _ 4^2. 
 
 12. (a;2-4a; + 3)(x2-4), also (a;^ -3x-4-2)(.r2 -« -6). 
 
 13. i(-l±i/5). 15. a-^c = d-^ ^e-, a^b^p ^e^. 
 
XX ANSWERS. 
 
 b^c = d^ ^p. 17. ac^ = b^d and 9ad = be. 
 
 19. ip^ + 27q = 0. 24. p = 27n^q±2mqV{m^-^l'}. 
 
 26. 4(jt)-3)=2. 
 
 Exercise xlix. 
 
 1. 5, 31, a, -3. 2. -4i, -a, 2, 10. 
 
 8. a + ^, c-rt, 6-c, 3. 4. -2,6, -5,12. 
 
 5. -14, a -3b, 2a- 3b, 5b -3a. 6. 7, 4, a, 6. 
 
 7. ic, 5-=- a, 0, 1. 8. -1, {(a + b)^-a}-i-b, a+b. 
 
 9. (/>-«), a + i. 10. 1-^a-h, l~(a-b}, l^(a-' + &£). 
 11. 2b, a. 12. a + 6, c-^(rt + 6), b-^{a-c). 
 
 13. (i-c)H-(«-6), i + c. 14. a+b, a^-^ab + b^. 
 
 15. «2_a&+/,2^ 1. 16. -1, (n-{-b)^{u-b). 
 
 17. (e + i)(e_/,), 2^15, 3---14. 18. -1-12, b~-ac, a-^b. 
 
 19. («-+;>2)-^rt-7>2)-^a2/ja, ^(63-|-c3J^5p, 
 
 20. 10, 12, 4, 4. 21. 1000, f, f. 22. 9/^, ab, bc^a. 
 
 23. h^^ac, c{n-\-b), b{a + b)^a. 
 
 24. a-f-i, {a-b)-^(a + b), -{a+b)^^{a-h)^. 
 
 25. -1, -1. 26. (a2_c2)--(a+6)2, 2, 3^. 
 
 27. ab, b^a, nc^b, 12. 28. 12, -ac-^6. 
 
 29. 9, 2. 30. 12, 1. 31. 3, 1. 82. (2fl -l)(2r, + 2), 0. 
 
 33. l-!-m. 34. 1. 35. {ab ^-bc-\-ca)---{a^ +bc+c-^). 
 
 36. {a^Jrh'+c-)^{ahJ^hc-\-ca). 37. a + b+c. 38. 1. 
 
 39, 1. 40. 1. 41. 1. 42. 15. 43. 16^. 44. 6. 
 
 45. 5. 46. {npqa+j)qb + qc + d)-^vmpq. 47. — ^. 
 
 48. 0. 49. -25H-136. 50. 1. 
 
answers. xxi 
 
 Exercise 1. 
 1. 2, 3. 2. *, h 3. +2, li. 4. 1, H. 5. ±f, 
 
 d=(a4-*), «• 6. 4, S, 2, 2i. 7. -8 or 2 ; 4, -3 ; 
 
 2i, -li. 8. 1; for-l; i 01-3. 9. -f or |, ^ or 6; 
 
 4 or -|. 10. -1, 2, -i, 1. 11. 0, -Z*, 3ft. 
 12. a, +av'-l. 13. 1; l( — l+v/5). 14. ±rt. 
 
 15. +6c, -(/)+c). 16. rt + 2Z>. 17. 6or+rt. 
 18. -2ai, iab{lztVl). 19. «, /;, - (a+i). 20. a, b. 
 
 21. aorl-rt. 22. -a, -b, a-2b. 
 .2a a, 6, /;(1 -/;)^(l+«.-6). 2-1. a;^ -6.c2 - 37.1- + 210. 
 
 25. a;4-4fla;3-13rt2a;2+64rt3a;-48a4. 
 
 26. a;(cc-l)(a;H-2)(a;-4)=0. 27. a;* -4A-3+a;2 + 6x-f-2 = 0. 
 
 Exercise li. 
 
 « 
 
 31 
 
 1. 4. 2. -7f 3. -107. 4. 8. 5. 3./. 6 
 
 7. 50|o, 17. 8. 22, 46^. 9. 7, 3. 10, 10, 10, 11. 
 
 11. or 11; 33. 12.3956-^3971. 13. i(15+s/190). 
 
 14. 3. 15. 3. 16. 4. 17. If. 18. l^-. 19. 3^. 
 
 20 4. 21. +3. 22. 11. 23. 2 aud -1±:|/ -3. 
 
 24. 2^. 25. 0. 26. 3a. 27. S- 28. \§. 29. 3, 
 
 80. 10. 31. 0, 1, or (-5±a/-23)-^8. 82. 102|. 
 
 33. (-ll±i/4681)-^20. 34. 2, i, |. 35. --4 
 
 86. or ±:y (fl2+62). 
 
 Exercise lii. 
 
 1. {l-a)^(l+r7), rt(m + l)^(/»,-l), h(m-^l)^a{m-l). 
 
 2. a -6, 0, 0. 3. h, ma^h, h-^ca. 4. 1, -1, 0. 
 5. -|or-l. 6. (c-/;)(ft2_f_c2):^2a6c.* 8. 14, 4^. 
 9. 2; 6^295. 10. 73 -=-210, {a -^b+c + d) ^(m+n). 
 
XXU ANSWERS. 
 
 11. h-^a. 12. h^a. 13. a or 0. , 
 
 14. ±:|/a2+l-j-2. 15. |. 16. ||. 17. or 4. 
 
 18. c-^ab. 19. 83|/(2a;-l) = 100A/(3x-3). 
 
 20. 75 --52. 21. 8. 22. 34^^?- 
 
 23. l^«(n-l). 24. ac-^{h-a). 25-. 4, 3i or 13^. 
 
 26. a2j2^(rt_6)2^ 3 27. 4a2^(l+a)2, 6(a + 6)2-7-(a-6)2. 
 
 28. (1+62) ^2a6. £9. ■,/(!-«) = 2 ^ (a + 1)3. 
 
 30. -a + aV{(l + & + i2)-j-26}. 
 
 U-1/ - U-1/ 
 
 5 
 
 ExEEcisE liii. 
 
 1. 8. 2. 0. 3. 3. 4. (v/m+|/n)». 
 
 5. ^;,^(1-2n/&). 6. 4-f-7. 7. l-^(a-2). 
 
 8. 18962^12393. 9. V^-^ ( v/^v + 2). 
 
 10. (c4-2/;c2)-T-(2c-2_26). 11. i. 12. 18a. 
 
 13. .r2 = 80-^81. 14. ±V°n/^t- 15. +Av/-ll. 
 
 16. 
 
 hs/ !«=- l^^-^^^^'U 17. 0. 18. ^. 
 
 19. (c-a-6)-''' = 27a?'c. 20. a;2 =a2(«-l)2 s- (2?i-l). 
 
 21. 16.r/y=(»-4x--2/)2. 22. 0, -ff 
 
 23. / ""^ -1) ^ 0. 24. 2s/(l -w2) -i-m ^(4 -m2). 
 
 \2rt — 2 / 
 
 25. (rt3_i)|,,2+2+ V(a^ + l)}H-a2.' 
 
 26. (m-an+c)2H-/;(n-l)2. 27. ±5. 
 
 28. 2V(3x2+*10) = (17i/17-3^3)---7. 29. ±5. 
 
 80. ±i/(36-2a). 81 |/|(a2_fe2). 
 
ANSWERS. XXm 
 
 32. {2i/ + 2z-2x)^-{-2Wx/jz^0. 33. |«-^/6. 
 
 34. a(^^2-4/t+_8) - (2«-4). 35. a^ +2a, 
 36. ±|/(3ft2+i2)^^3. 
 
 Exercise liv. 
 
 3. {(a-&)«2_2c(«2+a^ + 62)|^|„2_2t.(a3_/,3^j_ 
 
 4. —6. 5. a + 6-fc. 6. ab-r-{b — a). 
 
 7. a;2 — 3«a;-<*2=0, &c. 8. a. 9. K« + ^ + <') 10. I^a6c. 
 
 11. l3;(a + 6+f). 12. {a-b)[ac-2b)-^{a + byiG. 13. -c. 
 14. (ifa4-^/>)3. 15. ±2. 16. c^{a-b). 
 
 17. {a-b)^{a + b). 18. |a. 19. ±2. 20. ±2, &c. 
 
 21. i(a + c)-^(«-c). 22. a, {Sab-Bb'' -a)^{l-\-'da~db). 
 
 23. «. 24. a, b, 2b. 25. a, {c- .+ Qab)^Gb. 26. i(c + 6rt). 
 
 27." ia. 28. r( + 6. 29. (a6 + k;+ca)H-(« + 64-c). 
 
 30. ±b, ±a. 31. |/{l-=-(a 1)}. 
 
 32. {6{a-b)-4:c{c~b)}-^{4:C-db-a). 
 
 33. (c-2-a/.)^(a + 6-2(;) 34. -^(- 29+ v/37). 
 
 35. (a; + a)2=2i3-a3. 36. v/(^^2._|«i), 37. ^(&- a). 
 
 38. 8h I- 89. x--6x = a. 40. 1±]/19. 41. Z*, b~a. 
 
 42. , («3 + 63)^(a + 6). 43. a;=-5-^2. 44. -1(5+ a/3). 
 45. -2a, -^a, |a. 46. -3a. 
 
 Exercise Iv. 
 1. bc-i-(a + c). 2. (a3 + 6_2a5)-j.(a + ^>2). 
 
 3. (arf-66)-^(«-i). 6. -^;X7- C- ^- 7. ^(a+6) 
 
 8. a + 6. 9. 0. 10. 0. 11. abc. 
 
 12. («2+62+c3)--(a + fc + c). 13. (a + & + c)-(a2 + 63+c2). 
 
AXi^' ANSWEES. 
 
 a^b ,^ ah 
 
 14. {a^^h'-^-\-c^)M'tb + bc + ca) 15.^3^. 16. ^^• 
 
 17. 4*. 19. 4. 20. -140. 21. 17. 22. 10. 23. a. 
 24. ^ !!^'7^l, . 25. 3i,0. 26.3^. 27. («6-6-2)--(« + Z>). 
 
 r^< (lb 
 
 28. -i, «, 29. 0, 0. 30. U'l-^b-c). 31. —-,. 
 
 32. d. 33. fl6--(a3-62). 84. -3|. 35. f. 36. -3§. 
 
 37. Infinity. 38. 10. 39. abc-^{ah + hc + m). 
 
 40. {ab-\-hc + ca-ad — bd-cd)^(a+b-{-c-Bd). 
 
 41. a{b + c)^-^(b^ + rJ-ab-]-bc-ca). 
 
 42. Z*c{f/-«) + (a-6)(&-c)(c-rf)-=-(«tH-ic4f'?-'^'d-&---c2)- 
 
 43. bc^—b^c-ac'^-\-b^d-abd+cicd^{ah-^bc-ac-h^). 
 
 44. _(fl_j_&.4.(.). 45. rt + /) + c. 46. {ab + bc + ca)-^abc. 
 47. _i(i4.c). 48. («^ + c)--2«. 49.9. 50.2. 51.7. 
 52. 4. 53. r,S(5±^/7S5). 54. 4, (am.-nb)-^{n-'m + a-~b). 
 56. ^Ij, b{a + o)-^{cr--]-ab-\-b^). 56. 0, -|, *. 57. 10 
 
 n(i{ap-\-i)ih) — wp{cq-\-nd) 
 
 58. («prig - cmjpq)-^{apn^ +cqn,2); ^^^^^^2^,nnH ^^n^f^^^^m^d' 
 
 59. rt6-(6-c), c{a2 + (6-0«-ic} + a(62_f2)^ . 
 ((j2^i2_c24.rtJ_5c — rtc). 60. b-^{u + b), 
 
 61. mpcq-]-apnq-~{Cjtn^ —cqrn^). 
 
 02. {&m(a— c)4-CT«(6— rO + f(/)(c-5)}-^ 
 
 {m((a-c)+n(i-a)+y>(6--i)}. 68. (a2-|-,62)^,,6, 0, ^. 
 
 d{u-q)-q{h-d) 
 64. (a;.-cm)H-(«7i-im), ^,, _ ^) _ ,,,(^ _ ^y ^ 55. i, 3. 
 
 66. 100. 67. 13, 111. 68. 11, 7. 
 
 a-2 +2ac+«rf+26c + 2«fe 
 69. (rt + ft-m-n). 70. ^3-^ ; • 
 
ANSWERS. XXV 
 
 71. ^^^h) + i/{^(a-hy-^c^\,aoYb. 72.0. 73. rt + 6 + c. 
 
 74. ^^1!+!!^+^^ 75. a + fc + c. 76. a + b + c. 
 
 abc 
 
 77. (a/* + /;c+r'^)H-(a4-?' + c). 78. b^+a^-c^. 
 
 79. c-a-6. 80. 0. 81. or 11. 
 
 ExEECiSE Ivi. 
 1. ^ = 0, orB = 0. 2. A = 0, or 5 = 0, or C'=sO. 
 
 3. x-0, or(i—b = 0. 4. x = 0, or;y = 0. 
 
 5. In the first case either .r — 5?/ = 0, or x— 4y + 3 = 0, in the 
 second case both conditions hold. 6. x = 0, or x~a. 
 
 7. x = 0, orx=—b. 8. x = a, ora;-- c^-&. 
 
 9. a? = 0, ora: = 3. 10, x = 0, or a- = a + h. 
 
 U. 3; = 0, or «=+«. / 12. x = 0, or x^b^-^ a. 
 
 13. :c=:0, orx = a. 14. x = 0, or a. 
 
 15. x = 0, or ic=rt + &. 16. a; = 0, ora + i^. 
 
 17. -{2ab)-^{<i + h). . 18. a: = rt, or 6. 
 
 19. « = a, or6. ore. 20. 5. 21. 1. 22. 21. 
 
 23. x = \, « = 3. 24. a; = 9, .r = 4. 
 
 25. x=l, orB. 26. (a/O ^ («' + ^'). 
 
 27. x = a,orb. 28. a; = (a^ +/,2) h- («+i), x = /v+a. 
 
 'J.9. (2r/.ft)-i-(rt + '0- ^^- « = «, or^. 
 
 1. a;=i, or (l+a)-f-(l -a). 32. x = a. 
 
 34. a; = a - 6, or i(54-c). 35. x = «+??, or i(rt + c). 
 
 36. X = — " — , orl. 37. a-^b-c, 
 
 a^rb-\-c, 
 
 38. x = rt, or ^(46-a). 39. x= -c, or a^h + c. 
 
XXVI 
 
 ANSWERS. 
 
 40. x = lf or 
 
 m—n 
 
 41. X 
 
 _ nc—pb 
 
 n—p 
 
 42. P{'i —^)-<^{m-'>^) 
 )n{c ~ b) — a[n — p) 
 
 •1:4. x = 2a — b, or 36 — 2a. 
 
 17(7-76 
 
 46. « = « + /', or 
 
 mc — ap 
 43. a; = i(a+6), or ^(6-4 
 
 45. x = a-\-c — b, ora; = 
 4a+4c-26 
 
 47. x = 4:a + h, ov a + b. 
 
 48. a; 
 
 8 
 
 • ' a 
 
 - — , or_ 
 b — c 
 
 49. {a - h){b - c]x^ - (^2 4-^3 ji^c- -ab-bc- ca)x+ 
 {n-c){a-h)z=Q. 
 
 50. x=±d, or ±2. 51. a;=±G, or +2. 
 
 a-\-b 
 
 52. a; = 3, ori. 53 a; = ^^, or ^ 
 
 a 
 
 r. 54. x — -r-, or 
 
 a 
 
 55. x = b — 2a, or a — 26. 
 
 • 2rt + 36 Sa + 26 
 56. a; = — ^ , or 
 
 5 ' "^ 5 
 
 57. x = {tnb+na)-i-{m + n), or (wa-w6)-7-(m+n). 
 
 58. a;=v/{(»i + 2/i)- A/(w-2«)}^l/{(m-|-2;i)+ i/(»i-2n)}. 
 
 59. a 
 
 [v/(c+i)+ A/rc-1)) 
 
 60. a{y(3c-2) + i/(2-c)}H-{i/(3c-2)-|/(2-c)}. 
 
 61. a{v/(2c-l) + l}-=-{l-'/(2c-l)}. 
 
 62. H«+26). 63. i(a+6)-(a-6)v'(7n-2w)-=-y(w+2«) 
 
 64. 
 
 66. « 
 
 26 
 
 , or — 2 • ^^' 2a6-^(a+6). 
 
 3a 4- 56 86— 5a 
 
 8 
 
 -, or 
 
 8 
 
 67. « = 2a{v'(c+4)-i/(c-4)}--{v/(c + 4)+ \/(c-4)}. 
 
ANSWERS, XXVll 
 
 68. ir = 4, 01-3. 
 
 69. ^{«±:|/a3-4w) where m = «2+^\/(''+«*); 3 or 1. 
 
 70. a, b. 71. x = ^[a + b±i/{{a + b)2—4.{ab-\-t)}], 
 
 72. a; = 0, or a, or ■^rt(ldr-i/-3) ; a; = 4, or 2. 
 
 73. x = 0, or «+&, or ^{{a+b)±lV{a-b)^-4:ab}. 
 
 74. a;2-(a-6)x+a6 = &c. 75. x= ^(3a-fc). or ^(3fe-a). 
 
 76. a; = 3a— 2&, or 3i-2a. 
 
 77. y^ -m^ =0, where y — m—x and 2m = a+b. See Key. 
 
 78. 2/^-»*^=0- 79. ^2_^2=o. 80. y^-m^ = 0. 
 81. i/^-m3=0. 82. t/3-m3=o. 83. ?/3-w»=0. 
 
 84. (?/2-^3)(52/2^.7;^.2)^0, (where also k=i(a-b), 
 
 85. k^-y^ = c. 86. A.- + 10^3y2^5^^4 = c(^4_y4)&c. 
 87. s?/ihA:N/(A; — 3c±r) = 0, where s^^g^ + c, and r2 = (A;— 3c)* 
 
 4-(^_c)(3/.-+6-). 88. -3±:a/(9 + 12/24). 
 
 89 — 102. Work with a variable w such that wx = x^+l. 
 
 89. w = {a±s)^b, where s = a^ + 262. 
 
 90. «j = (3a+26±i)-=-2(a-6) where s= ±:6/(a3+2a6+462). 
 
 91. M; = (3±:s)^(l±i>) where s = (6 — 4a) -^6. 
 
 92. (M; + l)3=a-^(«-i). 93. Mj3==2a-^-(Z»-«). 
 
 94. {x+l)~{x-l) = a^{a-%h). 
 
 95. (w + 2)-^(w-2)=^(l + sJ where s=(16rt+6)H-6. 
 
 96. «;3(4a-6)-^-(a-6). 97. ■«;3 = (4a-36)H-(a-fc). 
 
 98. M) = (6±s)H-2a where s^ = 63 -(-16a2. 
 
 99. u; = (a + i»±s)-i-2(a — 6) where s = (a + 6)3 +8(^a -6).. 
 
•XXVlll A>'SWERS. 
 
 100. Wz^(a-\-b±s)-^2{a-b) where s^-^{a-b)^ = 
 
 101. (it;-f2)it(t«-2)=+s-=-(4+3s) wheres2 = 2a^(a + 6). 
 
 102. {w-{-2)-i-iv=±]/{5a-i-{a+U)}. 
 
 103. i{2a + b), i(a + 26j. 104. 2a-6, |(a+6), &c. 
 105. 1, 2, 4, 5. • 106. ±1, 2, 4. 107. 1, 2, 3, 4. 
 108. -^, -1 1, f. 109. -1, 3, 4. 110. -a, o^t, 5a, 
 111. 15, 20. 112. 2ia. 113. 4, -1. 
 114. 7, —1. 115. ^(6c-^-a + 6•a-^ft + rt/;^c). 116. ±a^m, &c. 
 
 117. 26(s-a)(s-6)(«-(-)^V^{s'2_a2j(s'3-6)(.s-'2_c2j} where 
 
 2s = a + />+c, 26i=a3+i2 4-c; 
 
 118. (2«6+2ac2 + 26c8-a3-6'--c4)H-4c2. ll'j. a, b, ^{a + b). 
 120. ±a or ±JaV^' 121. a, 6. i(ct+6). 
 
 Exercise ivii. 
 
 2. a;, 2; ?/, 1. 3. x, 8 ; y, 1. 
 
 5. X, — lOi; «/, 51 6. a;, -2; t/, ^. 
 
 8. x, -2; 2/, -3. 9. X, -f; 2/, i. 
 
 11. aj, 12; y,S. 12. x, 8 ; ^, -9. 
 
 14. X, 12; t/, 15. 15. x, 18; 2/, 13. 
 
 17. x,l\ y, 9. 18. x,l; y, -3. 
 
 20. a;, 2 ; y, 3. 21. «, 3 ; y, 4. 
 
 23. X. -3; y,\. 24. «, 12 ; //, 15 
 
 26. X, 8 ; y, 9. 27. a;, 3 ; y, 1. 
 
 29. a;, 11; ;/, 7. 30. x, 17; y, 13. 
 
 32. a;, -4^; ^, -|«. 33. a:, 13 ; y, 10. 
 
 84. a;, 4|;y, 3tV ^S- x,ll',y,Q. 36. x, 7 ; y, 5. 
 
 1. 
 
 X, 7 ; y, 9. 
 
 4. 
 
 x, 9 ; y, 5. 
 
 7. 
 
 «, -1; 2/, 1- 
 
 10. 
 
 X, -i; ^, f. 
 
 13. 
 
 «, 10 ; y, 12. 
 
 16. 
 
 X. -3 ; y, -2. 
 
 19. 
 
 a;, 7 : y, 3. 
 
 22. 
 
 «. r ; 3'» ^• 
 
 25. 
 
 »' Tff ; 2/' A- 
 
 28. 
 
 a;, 7 ; ?/, 8. 
 
 31. 
 
 a;, 5; y, -4. 
 
ANSWERS. XXIX 
 
 37. x,2; y, 3. 38. a;, 5 ; y, 3. 89. Equations 
 
 40. x,d\ y, 1. 41. x,l;y,5. not independent. 
 
 42. x = = y = 0. 43. 0,0. 44. a; = Oor 13 ; y/ = Oor ^f. 
 
 45. X, 17; y, 20; z, 5. 46. a.-, ^„ y, \%^, z, f|^. 
 
 47. 11,7,9. 48. 21,22,23. 49. -15,-0,-8. 
 
 50. 3, 4, 5. 51. 12, 15, 10„ 62. 5, 3, 1. 
 
 53. f, 1^, |. 54, 3, 5, 7. 55. 11, 13, 17. 
 
 56. 5, 3, 1. 67. 9, 7, 3. 58. Ik, 8i, 9^. 
 
 59. 3f, 2|, 1|. 60. 2-3, 3-4, 4-5. 61. 30, 20, 70. 
 
 62. 88 H- 59, 1098 h- 59, 1004 ^ 69. 63. 30, 12, 70. 
 
 64. 6, 12, 20. 65. 5, 2, 0. 66. 1, 1, 1. 
 
 67. 11, 9, 7. 68. 6, 3, 1. 69. 2, 3, 1. 
 
 70. 3, 4, 5. 71. h h i- 72. 5, 4, 3. 
 
 73. 7, 3, 1. 74. 2, 3, 1. 75. 1, 3, 6. 
 
 76. 0,1,2. 77. 1755-698, 360 -f 319, -15705^698. 
 
 78. ^,i, 1. 79. 5,4,1,3. 80. 4f , 3^*, 2^, U- 
 
 81. 31, 41, 51, 21. 82. 7, 4^ 4, 8i. 
 
 83. 20,10,0,30. 84. 11 --24, J-, 1 ^24, ^. 
 
 85. 270 -- 117, -52-- 117, 15-^-117, -126--117. 
 
 86. Eaok 210. 
 
 Exercise Iviii. 
 
 1. [a'c—ac')-i-{a'b — ah'). 2. b(cn- dm)-^(ad—bc). 
 
 3. b{d-c){d-a)^d{b-c)(b-a), c{d-a){d-b)-^d{c-a){c~b). 
 
 4. ?/ = cz+f^'^ + ^M^ + ^aJ) 2 = rfM-j-eif'-|-aa;+«t/, 
 u = ew+ax-\-by + cz, iv = ax + by-{-cz-{-du. 
 
 5. a; = ^m(a — ^4-c), &c. 6. x= {j:>(a3 — 6) — ?n(aA — l)-j- 
 
XXX ANSWERS. 
 
 n(b^ -a)'^{nS + b^ — Sah + l}, &c. 
 
 7. x={l — am + abn — abc27-\-abcdr)-^{l-\-abcdf), &e. 
 
 8. l=a-^{l-i-a) + b-rr{l-{-b)-{-c-i-[i+c). 
 
 9. l=-ab + bc + ca-\-2abc, 
 
 ExEEasE lix. 
 
 1. (nc — bd)^(na — bm), {'mc — ad)-~-{mb — na). 
 
 2. (??a+6d)-r(aw + 6?n), (mc — acZ)^(6w+(m). 
 
 3. c{n — b)-i-{an — mb), c{jin — a)^{hm — am). 
 
 4. (6 — c)a-^(/>— «), Z/(a — c)-i-(rt — i). 5. a5-T-(a+5), t/, same. 
 6. a62-^(a2 + '^3)^ a,n^{a'^+b^). 7. ac-^(a4-6j, 6c-r;a + /:') 
 
 8. {a- —b')-T-{am — bnfy {b^ —a^)-T-{bm—an). 
 
 9. « + 6 — c, c+a— &. 10. a + c. b+c. 
 
 11. a(c»— <f?n)-j-(ii — ac), b{cn — dm)-T-{ad — bc). 
 
 12. t/={93(a2_c2)_o(6+2a)}^{(a-Z>)'^-i-s + 4ic}. 
 
 13. a + 6-c. a-b + c. 14. a+6-6-, c+a — 6. 
 15. (TO-a)(?i — a)-5-(6-a), &c. 16. l-*-(a-^)(a-c), &c. 
 
 17. (Hi-ic)(?--«)-^(c— a)(a — 6), &c. 
 
 18. x=p-i-{pl+mq-r'iir)+a, BO y and g, 
 
 19. p{l — (?rt4-w6 + wc)}-^jp^+m3+nr)+«, &0. 
 
 20. {m^ + '2a^ -b- -c^)-^Sa, &c. 21. y = a — 6-fc, &c. 
 
 22. x= {ab+bc-\-ca){b+c-2.a){2b-a~ c)-i-{{a- c){b +c ^ 2a)-{- 
 (^_c)(2&-a — c)}. Corrected equation, a; = i(i4-c), &c. 
 
 23. nui-^{a-\-b+c), &C, 24. npr-^{anp'^bmp-^cmq). 
 25. l-=-(6-c), &c. 26. -^(i+c— a), &c. 
 
 27. a«H-rf, &c 28. «=l-^(«+6-c). 
 
 29. a+fe, &c 80. l-^2a, &c. 
 
ANSWERS. IXXl 
 
 31. (mS+nS-ZSI-^-Smw, &c. S2. i{a + c-b), &G, 
 
 83. l{m'^ +n^)-i-'imn, &G. 84. l-=-(ft + c- a), &c., 
 
 85. bc-^[b-\-c), &c. 36. /> + c-a, &c.^ 
 
 37. a, 6, c. 38. ^>^-c3,&c. 
 
 89. i(a4-26-c + 3fZ), &c. 40. i(4a+^ + 3c-2(/ + 5«). 
 
 Exercise li. 
 1. ,, + ?,. a-h. 2. i>.2+;,), ^(^2-6). 
 
 3. {8;^2+m2)-^-5m, (2^2 -w3)^5?n. 
 
 4, a_^(a_&), />-^(rt4.6). 5. l~{a~h), l-4-(a + 6), 
 8. a-f-6-c, a-h+c. 7. a + 6-c, o-fe + c 
 
 8. (a2+fi/j+&^)-^(« + &), (a2-a/j+6-')-4-(a-&). 
 
 9. (a&-l)^(«-l)f^-l), {a-Z;)--(a-l)(6-l}. 
 
 10. (l+a)--(aft-l), {l+h)-^{ab-l). 
 
 11. (« + l)(^' + l)-5-(«^-l). {a-b)-^{ab-l). 
 
 12. a(a+fc), b(a-b). 13. a{6(a4-o)-c(a-c)}-4-(a3 -6c), 
 a{6(a-6) + c(a+c)}-j-(a2_6c). 14. -(a + 6). at. 
 
 15. i(^+c), &c. 16. (a-26+3c)-f-38, &c. 
 
 17. 2-T-(/>+c), &o. 18. fl + 6, &c. (by symmetry). 
 
 19. b^-c^, &c. 20. b-2-c^, &e. 
 
 21. ia&c, (l-a)(l~^)(l-c), (2~«)(2-6)(2-cy, 
 
 22. 2a&c-4-(a&+fec — ca). 28. 1, 1, 1. 
 
 24. ar-{ma + 7ib+pc-\-qd), &c. 
 
 Id \ Id^ dn 
 
 25. w = 0, or(— -l)-(^-^|- 
 
 26. (5-|.c-fl)-f-(rT + 6+c), 2/ = (6-c-a)-«-(a-*-«). 
 
 27. ^(a— 6+w-w), &0, 
 
XXZll ANSWEB9. 
 
 28. (4:a + 2c—d—Bb), y+z by symmetry- 
 
 29. —{a-b-\-c-{-d}, {nb-\-bc, &.c.), -{abd-\-&c.), abed. 
 80. ^(a — 6 + e — rf-}-e), others by symmetry. 
 
 31. x = (a — l.b-\- Imc — Imnd + Imnpc) — (1 + Imnpq), the others bj* 
 symmetry. ' 32. a; = 6 + c— <?, &c. 
 
 34. j^ = (a+56 + 3c-7rf+9e)H-22, &c. 
 
 35. z = ^(a+c), then symmetry. 36. 2 = -J+^ + e, &c. 
 • 87. a5 = a— 2fe+3c — 2(i+e, then by symmetry. 
 
 Exercise Ixi. 
 
 1. a;=(2a^ + a+fe+r)-=-2(a-/») where r = 4rt(&2-j.i + i) _j. 
 {da-h){U-a\ 
 
 2. x=(ar + l)^(fflr-l) where r« = (J2 - 1) H-3(a3 -62) 
 
 3. a;={v/(l + a)(l+6)-i/(l-«)(l-6)}-5- 
 
 {l/(l+a)(l + ?.)+v'(l-a)(l-fe)}. 
 
 {^/(a + 6 + c)(« + ^>-c)-i/...[. 
 6. {a-\-b)^{l-ah). 7. x=(a^-ai)s-(a^+6a) 
 
 8. / ^+^ \ * _ («+»0 (^>-f-n) ^ 
 \a; — 1/ " {a-~ni)[b — n)' ' 
 
 9. a;={aT/(l-6^)-6v/(l-a3)-^ ^(^S-js). 
 
 10. x={b + c-a) ~ {(Vb-^c-a)(c-\-a-b){a + b-c).] 
 
 11. x = {b + c)^^(a + b){b-^c){c+a). 
 
 12. aj= VC^+t + c) -j- rt, &c. 
 
 13. a; = {62.^.^2 _^(t+c)} - i/2(a3 + 6»-f-c» - 8«5c). 
 
 14. (fc+^-a), &n. 16. a or (a^— 6j -j- (i_ai). 
 
ANSWERS. ZXXIU 
 
 x-y= V(a-i-b)(a — 2b) -^ V{a — h), &c. 
 
 17. ix + ,jr^ = l[3b- ^^, {x-yy- = il^^ - b'j . 
 
 18. (x ■}-]/) H- (x — 7/) = -/(«+ 36) -j- i/{a — b) = m suppose. 
 
 19. y^ = m -j- iam^ — 9??.+ 1) where m = 
 
 20. a, 6. 21. a;={^(a-c) + |/c[y-- {i/(a-c)-y/c}. 
 
 22. «= (a-|-c)?/ -;- (a — c), &c. • 
 
 23. x+y = {ab-l)^{a-b), &,G. 
 
 24. a;2/={v/(o + 2)-i/(Z»-2)}^-,/{(?<4-2)-i/(&-2)} = 
 p suppose, x-^y= {]/«4-2) + y («-2)}-j- 
 {s/(a+2)-T,/(a-2)}. 
 
 25. x'1j^z'^ = \{a + h-c)[h-\-c-n){c^a-b), &c. 
 
 26. x= (ai — 6c — ca)-^2s/a?y(;. 27. a — x'^ = ±m, inhere w is 
 the value of t; in the equation 4m — 4(r +</)»-+ ^f* =» 
 
 (ca - ai - 6c) 2 4- 46(ca -ab- bc)v + 462^3 . 
 
 /I 1 \ 
 28. a; = i|/(<5i6c) \-t--\ I , y and z by symmetry, 
 
 29- a(624.c^)^(52_,_c2)_^(J3_}_c2)^ ^c. 
 
 30. c{V{a-\-b) + i/{a-b)}^Via + b). 
 
 81. or a ib + c)^2bc, &c. 
 
 32. »= -1 or ff/(a«-l)-^v''(ft'« — 6'), &e. 
 
xxxiv answers. 
 
 -Examination Papers. 
 
 I. 
 
 1. (a + h + r)(x + y+z). 
 
 3. x^-6x^ + ldx^-12x+i, x^+dx-^ + d, 
 
 X' 
 
 nln- 
 
 • 1) J.;^"!"— 3)_l.a;«(»— 3) 
 
 4. 2aj2'"-ix3m A_ — _ — . 5. -jhl4|/ - 19-^-27, or ^. 
 
 '* c a ' 
 
 6. 16, 7. 199, 8, AB 37, CA 52, BG 45. 
 
 II. 
 
 1. {a + h]^. 2. (a+h + c)(a^ + J,S-^c^ -^ahc). 8. l-^3;3. 
 
 4. (a2'"4-2rt'" + 2)^(a"' + 2), (<(+i-|-c)-^(rt-6-c)• 
 7. -I, 4 or 6. 8. p'^q^{r-pq). 
 
 9. 584. 10. (am + b)i)^ {a -\-b), {by common rule). 
 
 III. 
 
 2. ,/4-fca;+ca;2, 3 — 4a;-h7a;- -10a;3 
 
 3. 2i ; 6, 9, 12. 4. 160 eggs. 
 
 5. 40, 35. 6. /|(l+«)+i/Kl-«). 
 
 7. 5 or J; 4. 9. (rt+6+c)(a— 6)(6-c)(a-c. 
 
 IV. 
 1. (a + h + c)9. 2. a-;-6. 3. (4a;2 -92/2)(4a;«— 4?/5). 1 + Va;. 
 
 4. L^3-|v^6, 17 5.4. 6.1. 7. Cor 4; 8,1. 
 
 8. 7i, 12. 9. 4 or 6. 10. i(-3± ^ -39). 
 
 V. 
 1. 8x3 + 1^125. 3. IGa^b^. 4. b-^8a. 5. 15, 12. 
 
 6. ia, ia; 4 or -9, 7. 6, 7, 8, or -6, -7, -*8. 
 
ANSWERS. XXXV 
 
 Vi. 
 
 1. (r,3-f-ft2). 2. a^ -\-b^ + e* + ah + ac -he. 
 
 3. .c'+r±%,xy, {lx+6y-9){x-y-{-4:). 4. -20, 0. 
 5. borl^i. 7. x- = 4 01-9, i/ = 9 or4; 1, 2, 3; 
 
 a; = i(_7 + y^33). 8. -1, 0, 1, or 5, 6, 7, or -V. -4. ?• 
 9. xix'^d)[x^-tix-l) = 0. 10. x-^//-3^4. 
 
 vn. 
 
 2. --01. 4. a;2+x-l. 
 
 5. ^36x--^+l8a:+9)--fl6a;4-81), (u;3 -a/yj--(x^ +«6). 
 
 6. 9; 3i .t; = -^(a-Z*). 
 
 VIII. 
 
 3. (»3_y3)2(a;_y); a=y;4.ry, i=i?g. 
 
 5. <'.6-3 + ^/x + ca;-^ ; ^a;* + i^" - i« 
 
 6. x^+px+p''; 50{x+o){x-4:}{.c-6). 
 
 7. a=0 = ior«.= l, ^1 = 2. 
 
 8. 3 or — 43-J-7 satisfies the equation 2 - y . . &c. 
 
 IX. 
 
 4. axij+b. 7. +^/a6; A/(a4-6)-f-i/(«-6), andt/ = reciprocal 
 of this. 8. P(22a-216)--20a(a-6). 9. 7, 15, 48. 
 
 X. 
 1. aa;3 4-%3+2ca;?/. B. x-2. 4. 24. . 
 
 .5. 5x-^-Qux+ia^;y+-^-Vi- 7. 3;*. 8.4,6. 
 
 XI. 
 1. 2. 2.6,8. 8. (4x*-9i/V,TV^^+-5%*. 
 
 4_ ^3_^3 4.9a:-S x'^ -{a-\-b)x-c. 6. 0. ^ 
 
 7. Oor -2i; i/ = di3or ±|/-9, &c. 
 
XXXVi ANSWERS. 
 
 XII. 
 
 « «• 
 
 6. m^ —vi=p, q = 0. 
 
 7. a;2=::(a_2)-^(«4-4), y/iV= {b{n~ I) - cn\ -i- y'a[n-y} 
 a a6-{(a-l))i-l)-l}. 9. 87. ' 
 
 XIII. 
 1, 2. 3. 14-rja;+Coa;^4-&c., where c^, Cg. <^c., repveseut tbt 
 
 combinations of rt|, (''21 • ■ ■ taken one, two, &c.. at a time, 
 4. §4000. 5. {m{b'c — bG')—n{ca'~c'a)}-i-{ah' - a'b), 
 
 V=P""c^' 7 (« + /3f+(ar-/i)'-2/ = 0. 9.66. 
 
 XIV. 
 
 1. 12rtie. i. Smiles. 5. «tm-i-/i; a;+^ = ±5 or ±1, 
 X - ^ = ± 1 or ± 5. 6. ■ 20. 
 
 XV. 
 
 2. A. 3. c2(r2-6d!) + ci!2(62_^,c)+a/r(rt(i-ftc). 4. 2^(OT-n). 
 If m— w is negative u; is neg. -wbicli shows that they were 
 together before noon. If m-n = Q, x is infinite, i.e., they 
 are never together. 
 
 5. x^(x^ -r,2)(x-2a) ; (x^ - a^)(;£^ ~ y^} . 
 
 6. (rt+6+c+3d)-=-(a-}-6 + t-l-c/): {x + y-^z)-^{x-y^z); 
 
 3a-T-(a + i). 
 
 XVI. 
 
 4. 2x* -3a; 3+ 4a; +3. 5. See paper XIX., prob,, 4. 
 
 1 i 1 
 
 & i{l±l/(«" + 4//)H-\/4ft". 8. (i,Ci-^C2)--(ai63-fl2&j) 
 
AJ^SWEES. 
 
 xxxvu 
 
 1. 
 
 8. 
 
 1. 
 7. 
 
 1. 
 
 4. 
 6. 
 
 4. 
 
 5. 
 
 7. 
 8. 
 
 1. 
 
 4. 
 
 XVII. 
 
 8a^-2ab-10ac-Sb^+2c^ + 5bc. 6. ?/-»i^ = 4. 
 (w. + n){bq—pG)^niq -pn), {p-\-q)(j)ic - bn)-i-{mq —pn). 
 
 XVIII. 
 
 l-aj»«. 2.1. 4:. x-a-b. 6. {x- y){x-z). 
 
 1. 8. 9a; ±:1-=-a/2 or ±1^6. 9. 10,11. 
 
 XIX. 
 
 i(4a;-« + l)8. 3. 5;c2_i. (3.3+^^^4.^2)3. 
 
 (u + 2)-'^4(a2+a> 7. (76'-^ -a-')^12fl. 8. 18, 22, 50. 
 
 XX. 
 {l+w.)x+(l-n)y. 3. a,-2_i. 
 
 {a^b-2ab^ - a^ -^ab^ + b^)-r-{a^ -b^). 
 
 1. 6. a>6'-6c') + /*^(ac'-a'6-)4-c"(^'6-«ft') = 0. 
 
 l/{^*^ 
 
 3m 
 
 ; 2, 3, 4. 
 
 10- 
 
 a{b-c)-T-{b-a}, b{c—a)^{b-a). 9. 3. 10. 2000. 
 
 XXI. 
 8. 2. 24a&c; 2a^ +4:Sa^x^ + Q2a^x'^+A4:a^x^ + lSx». 
 
 0; rj7+16. 5. 16; x+^Jx^ -2a. 7. 3377 oz. of gold, 
 783 oz. of silver. 9. - (8±4-i/3)h-(3±2 \/3 ; 
 a;==fc2 or %/— 1, y= +1 or q:2|/-l ; 8, 4. 
 y = cost of 2iid bale = 60 ± 20 \/ 7. 
 
XXXVUl 
 
 ANSWEBS. 
 
 XXII. 
 
 1. -02997, a^+aq+p^. 2. ab^'~h^+c = 0, 
 
 {a).{a-b){8a-'db). {b).x{x- l){u—b){b - c){(( - c). 
 
 8. b^=A'ia. (a), (a + b)^. 5. (a) {ax-bi/)^{i:x + by) 
 
 7. (a) U(^ + b + c), {b), f, -I, 2. 
 
 (c). b-c,c-a,a-b, (d). -1±V2. 
 
 XXIII. 
 
 1. ^x' + 4.r*-.T3 + ia;3 + ^^3^ + (95a;3+21a;3 - 40a:+42)-- 
 
 (3x4-21a;3-j-9a;-6); -382. -4. x = a + 'Ic, y = b-[-Sc. 
 6. (2). ^. 7. (1). a+i + c. (2), 1, 2, 3, 4. (3j, or ^. 
 
 8 5'j(^2_i(3yi-|-2). 11. 
 
 ayb^c^^d-^' 
 
 a^b.^c.-^d.^ 
 
 a^b„c^,l., 
 
 = 0. 
 
 12. Coll. to Newmarket 63 miles. 
 
 XXIV. 
 
 1. i^^x-y){y-z){z-x). 2. -d-rti—n; 0. 3. 5, 3, 4^. 
 
 4. (1). A = bc^2a, £ = ac~2b, C = ab-ir-2c; (2). a^+b-^=c^. 
 11^ y=+2, x=±5,&c.; x'-'-lOx=-lQov -16, 
 
 1 i 4pq 
 
 :. a; = 8or 3, &c.; (6 ' ±a )^ 
 
 XXV. 
 
 1 % 
 
 1. (a2-3x)-^(a2-x2); 1. 2. — - — -i2(a;+a)H-a=a;3. 
 
 5. ^•2-2.<; = 2, a; = 3, y/ = 2; z/ = 53--24, &c. 6. 4 miles, 3 do. 
 
 7. (a;»-l)H-(x3-l).*;' 
 
 .»-i 
 
 8. fl-IP-fl+f- '■>■■'■ 
 
ANSWERS. 
 
 xxxix 
 
 1 1 1 
 
 10 1+ -j-+Y-2~^r'2^'^^^-~^''^'^^'^^ approximately; 
 
 {(5x)>3'"^^}{l-6-ll ... (5r-4)}^|jl. 11. f and -^. 
 
 XXVI. 
 
 i. a-^{nh-a^)', 4. 3. 2, i, or ^(-8± '/S) ; x-\-y-^z— 
 v/(a3+2^*2), .■.2 = o+i/(rt3+2?>2), &c.;|(_4±i/76). ■ 
 
 4. 3. 7. )t>3H-4+r3-f-27 = 0. 10. (l+a:)-^(l-a;)*- 
 a;"-l{3.«-l+2«(l-a;)}-^-(l-a;)2 ; n-^(15u+9j, yV- 
 
 xxvn. 
 
 1. {Tm-nltY-. ' 2. (a + t+^)(rt-ft)(6-c)(a-c). " 
 
 3. ia-b){h~c)-\-{b~c)(c-a)-^(c-a){a-b); {a-b)^ + 
 (6-c)2 + (c-a)2. 4. a; = a^(«2+62 4.c2), &c. 
 
 5. N/a6; 18or-2; i + Sj. 7. i(« + />)&c.; 
 
 a;={l+rt2_63 + i/(l-a-6)(l-a + ?>)(l + a-i)(l+a+i)} 
 -i-2a; x-i-y = {l+n){l — b)-i-{l — ac), &c. 
 
 8. —8; (^4-4;j2^+87?r)-^(yj^-4g). 
 
 9. (m-}-n)'i-2>inn, (?i — mJ-5-2m» 
 
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 The Shorter English Grammar. 
 
 is intended for learners who have but a limited amount of time at their dis- ' 
 posal for English studies ; but the expsrience of schools in which it iias 
 been the only English Grammar used, has shown that, when well mastered, 
 this work also is sufficient for the London Matriculation Examination. 
 . — — ^,^ — . — . — ^— 
 
cB. J. Sage ^ (E.o's. |leiu" educational Mlorks. 
 
 HAMBLIN SMITH'S MATHEMATICAL WORKS. 
 
 Authorized for use, and now used in nearly all the principal Schools of 
 Ontario, Quebec, Nova Scotia and Manitoba. 
 
 Hamblin Smith's Arithmetic, 
 
 An Advanced treatise, on the Unitary System, by J. Hambltn Smith, 
 M. A., of Gonville and Caius Colleges, and late lecturer of St. Peters Col- 
 lege, Cambridge. Adapted' to Canadian Schools, by Thomas Kirkland, 
 M. A., Science Master, Normal School, Toronto, and William Scott, B. A., 
 Head Master Model School for Ontario. 
 
 12th Edition, Price, 75 Cents. 
 
 KE Y.^ — A complete Key to the atjoy* Arithmetic, by the Authors. 
 Price, $2.00. 
 
 Hamblin Smith's Algebra. 
 
 An Elementary Algebra, by J. H.\mblix Smith, M. A., with Appendix 
 by Alfred Baker, B. A., Mathematical Tutor, University College, Toronto. 
 
 8tli Edison Price, 90 Cents. 
 
 KEY. — A complete Key to Hamblin Smith's Algebra. 
 
 Price, $2.75. 
 
 Hamblin Smith's Elements of Geometry. 
 
 Containing Books I. to VI., and portions of Books XI. and XII., of Euclid, 
 with E.xercises and Notes, ijy J. Hamblin Smith, M. A.. &c., and Examina- 
 tion Papers, from the Toronto and McGill Universities, and Normal School, 
 Toronto. 
 Price, ■ 90 Cents. 
 
 Hamblin Smith's Geometry Books, i and 2. 
 Price, 30 Cents. 
 
 Hamblin Smith's Statics. 
 
 By J. Hameli.n" Smith, M \ , with .\ppendi,x by Thomas Kirkland, M. A., 
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 Price, ~ 90 Cijnts. 
 
 Hamblin Smith's Hydrostatics. 75 Cents. 
 
 KEY. — Statics and Hydrostatics, in one volume. $2.00. 
 
 Hamblin Smith's Trigonometry. $1.25. 
 
 KEY.— To the above. $2.50. 
 
m. J. ©age & Clo'e. ^etu ^ixtratioital gEorks. 
 
 BOOKS FOR TEACHERS AND STUDENTS, BY DR. McLELLAN. 
 
 Examination Papers in Arithmetic. 
 
 By J. A. McLellan, M. A., LL. D., Inspector of Ili^h Schools, Out., and 
 Thomas Kirkland, M. A., Science Master, Nonnal School, Toronto. 
 
 " In our opinion the best Collection of Problems on the American Con- 
 tinent."— National Teachers' Monthly, N. Y. 
 
 Seventh Complete Edition, - - Price. $1.00. 
 
 Examination Papers in Arithmetic. ---Part I. 
 
 By J. A. McLellan, M. A., LL. D., and Thos. Kirkland, M. A. 
 Price, - . . . . 5o Cents. 
 
 This Edition has been issued at the request of a large number of Public 
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 Hints and Answers to Examination Papers 
 in Arithmetic. '' 
 
 By J. A. McLellan, M. A., LL. D., and Thos. Kirkland, M. A. 
 Fourth Edition, - - - - - $1 o6. 
 
 McLellan's Mental Arithmetic,---Part I. 
 
 Containing the Fundamental Rules, Fractions and Analysis. 
 
 By J. A. McLellan, M. A., LL. D., Inspector High Schools, Ontario. 
 
 Third Edition, - - - - SO Cents. 
 
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 McLellan's Mental Arithmetic. ---Part II. 
 
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WORKS FOR TEACHERS AND STUDENTS, BY JAS. L. HUGHES. 
 
 Examination Primer in Canadian History. 
 
 On the Topical Method. By Ja.s. L. Hiohes, Iiis[K3ctor of Schools, To. 
 rente. A Primer for Students prejiaring for Examination. Price, 25c 
 
 Mistakes in Teaching. 
 
 By Jaj. Laughmn" Hughes. Second edition. Price, 50c. 
 
 M)0FTEI> BY 8TATB nNIVKllSITT OF IOWA, A3 AS 8LRMBNTART WORK FOR USE 
 
 OF TEACHERS. 
 
 This work discusses in a terse manner over one hundred of the mistakes 
 commonly made by untrained or inexperienced Teachers. It is desifjned to 
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 be preventing their hiyher success. 
 
 The mistakes are arranged under the following heads : 
 
 1. Mistakes in Management. 2. Mistakes in Discipline. S. Mistakes In 
 Methods. 4. Mistakes in Manner. 
 
 How to Secure and Retain Attention. 
 
 By Jas. Lauqhlin Hughes. Price, 25 Cents. 
 
 Comprising Kinds of Attention. Characteristics of Positive AttentionI 
 Characteristics of The Teacher. How to Control a Class. Developing Men 
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 (From The School and University Maoazinr, London, E^-g.) 
 "Replete with valuable hints and practical suggestions which are evident- 
 ly the result of wide experience in the scholastic profession." 
 
 Manual of Drill and Calisthenics for use in 
 Schools. 
 
 By J. L. Hughes, Public School Inapector, Toronto, Graduate of Military 
 School, H. M. 29th Regiment. Price, 40 Cents. 
 
 The work contains : The Squad Drill prescribed for Public Schools in On- 
 tario, with full and explicit directions for teaching it. Free Gymnastic Ex- 
 ercises, carefully selected from the best German and American systems, 
 and arranged in proper classes. German Calisthenic Exercises, as taught 
 by the late Colonel Goodvon in Toronto Normal School, and in England. 
 Several of the best Kindergarten Games, and a few choice Exercise Song^s. 
 The instructions throughout the book are divested, as far as possible, of 
 unnecessary technicalities. 
 
 "A most valuable book for every teacher, jnrticularly In country places- 
 It embraces all that a school teacher should teach his pupils on this subject. 
 Any teacher can use the easy drill lessons, and by doing so he will be con- 
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 i Life Guards, Diill Instructor Xormal and Model Schools, Toronto. 
 
Authorized for use in the Schools of Ontario. 
 The Epoch Primer of English History. 
 
 By Rkv. M. Creigiito.v, .M. A., Late Fellow and Tutor of Meiton College, 
 
 Dxloid. 
 
 Sixth Edition, 
 Most thorough. 
 
 Price, SO Cents. 
 
 AEERnF.rS' JOfRNAL. 
 
 This volume, taken with the eight small volumes cciitaining; the ac- 
 counts of the different epochs, presents what may be regarded as the most 
 thorough co\u-se of elementary English History ever published. 
 
 What was needed. Toronto Daily Globe. 
 
 It is just such a manual as is needed by pul))ic school pupils who are 
 going up for a High School lourse. 
 
 Used in separate schools. M. Stafford, Prik.st. 
 
 We are using this History in our Convent and Separate Schools in Lind- 
 say. 
 
 Very concise. Hamilton- Ti.mes. 
 
 A very concise little hook that should be used in the Schools. In its 
 pages will be found incidents of English History from A. D. 43 to 1870, in' 
 tercsting alike to young and old. 
 
 A favorite. London Advertisf.r. 
 
 The book will prove a favorite with teachers preparing pupils for the 
 entrance examinations to the High Schools. 
 
 Very attractive. British Whig, Kingston. 
 
 This little book, of one hundred and forty pages, presents history in a 
 very attractive shape. 
 
 Wisely arrang'ed. Canada Presbtterian. 
 
 The epochs chosen for the division of English History are well marked 
 
 — no mere artificial milestones, arbitrarilj' erected b}- the author, but roaj 
 
 natural landmarks, consisting of great and important events or remarkable 
 
 changes. 
 
 Interesting. Yarmoi'th TRiBrxF., Nova Scotia. 
 
 With a perfect freedom from all looseness of style the intei'est is so well 
 sustained throuKhoirt the narrative that those who commence to it.. .</ 
 will liiid it difficult to leave off with its perusal incomplete. 
 
 Comprehensive. Literary World. 
 
 The special value of this historical outline is that it gives the reader a 
 comprehensive view of the course of memorable events and epochs. 
 
m. J. (gage S: Co's. Jlcto eiurational moxks. 
 
 THE BEST ELEMENTARY TEXT-BOOK OF THE YEAR. 
 
 Gage's Practical Speller. 
 
 A MANUAL OF SPELLING AND DICTATION. 
 Price, 
 
 SO Cents. 
 
 Sixty copies ordered. Mount Forest Advucate. 
 
 After careful inspect on we unhesitatingly pronounce it tiie best spell- 
 ing book ever in use in our' public schools. The Practical Speller secures 
 an easy access to its contents by the very systematic arrangements of the 
 words *in topical classes ; a permanent impression on the memory by the 
 frequent review of difficult words ; and a saving of time and effort by the 
 selection of only such words as are difficult and of connuon occurrence^ 
 Mr. Reid, H. S. Master heartily recommends the work, and ordered some 
 sixty copies. It is a book that should be on every business man's table as 
 well as in the school room. 
 
 Is a necessity. Phesb. Wit.ness, Halifax. 
 
 We have already had repeated occasion to speak highly of the Educa- 
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 before us. 
 
 Grood print. Bow.MAiNviLLE Observer- 
 
 The " Practical Speller" is a credit to the publishers in its general get 
 up, classification of subjects, and clearness of treatment. The child wh« 
 uses this book will not have damaged eyesight through bad print. 
 
 What it is. Stk.miiroy Aoe. 
 
 It is a scries of graded lessonSj .containing the words in general use, 
 with abbreviations, etc. ; words of similar pronunciation and different spell- 
 ing- a collection of the most difficult words in the language, and a number 
 of literary selections which may be used for dictation lessons, and commit" 
 ted to memory by the pupils. 
 
 Ex«ry teacher should introduce it. 
 It is an improvement on the old spelling 
 introduce it into his classes 
 
 C'ANADIA.V STATESMA.N. j 
 
 book. Every teacher should | 
 
 The best yet seen. Colchester Sun, Nova Scotia. 
 
 Itis away ahead of any"speller"that we have heretofore seen. Our public 
 schools want a good spelling hook. The publication before us is the best 
 we ha\ e yet seen. 
 
. J. (iagc S: aro'0. JlcU) @^urationnl ^orks. 
 
 The Canada School Journal 
 
 HAS RRCEIVED AX IIONOIiARLE MKNTION AT PARIS EXHIBITION, 1878. 
 
 Adopted by nearly every County in Caiia«ia. 
 Recommended by the Ministei'of Education, Ontario. 
 Reconimeiided b.v the Council of Pul>lio Instnictioii, (Jftelioe. 
 Recommended by Chief Sujit if Education, New Brunswick. 
 Recomniciidec liy Chief Supt. of Education, Nova Scotia. 
 Recommended by Chief Supt. of Education, Uriti.sh t:o!umbia. 
 Recommended by Chief Supt. of Education, Manitoba. 
 
 IT I.S EDITED BY 
 
 A Committee of sotnc of the Leading Educationists in Ontario, assisted 
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 Brunswick, Prince Edward Island, Manitoba, and T5ritish Columbia, thus 
 having' each section of Vm Dominion fully represented. 
 
 CONTAINS TWENTY-FOCR PAGES OF READINO MATTER. 
 
 Live Editorials ; Contributions on important Educational topics ; Selec- 
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 Practical Department will always contain useful hints on methods of 
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 .Mathe.matical Department gives salutions to diillcult proljlems also on 
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 Read THE Followino Lettkr from John Greenleaf Whittier, the Fa- 
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 I have also received a No. of the " Canada School Journal," which seems 
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 truly tliy friend, John Greenleaf Whittier. 
 
 A Club of 1,000 Sifoscribei's from Nova Scotia. 
 (Copj) Em-CATiox Office, Halifax, N. S., Nov. 17, 1878. 
 
 Messr.s. Adam Miller & Co., Toronto, Ont. 
 i Dear Sirs,— In order to meet the wishes of oxir teachers in \arious part^ 
 I of the Province, and to secure for them the advantage of your excellent 
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 , at dub i-ates mentioned in your recent esteemed favor. Subscriptions wiij 
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 , days. Yours truly, 
 
 1 David Allison, Chief Supt. of Education. 
 
 I Address, W. J. GAGE &CO., Toronto. Canada 
 
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