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In eleven numbers oaoh 1 00 -In )efl- )aoh 07 CA CO S.- ld2, 15 THE I.IHRAKY T] TNIM RSI IT OF WESTERN ONTARIO /^'^"%^ CiY- 'rin- l to ad 4 015 I sntB, "^ pfe. A a by I Neu bia, El THK J. 1). BAKN'KTT TEXT-HOOK COLLECTION ' Aasiated >a Scotia th Colunt iter. Educational topics; Selectiong-Readlngs for the School' BiooS^f^and No*e8 and News from each ProvincA. ' I on pro- r be University of Western Ontario LIBRARY LONDON . CANADA Class LX\00\ I ras ( ENGLISH. ICASOirS ADVANCED GRAMMAB-By, 0. P. UMoa. B.A., F.C.P., Vellovr of Univeraity College, London. S7Ui edition t II MASON'S ENGLISH GRAMMAR.-(Comraon Sohool ^_ edltiou) with copious and oaretully gruUed exeroiBes, 248 pagM 60 MASONS OUTLINES OF ENGLISH GRAMMAR, for ^ _ the aie of Junior classes OflO ENGLISH GRAMMAR EXERCISES.-Mason's. (Be- _ printed from Com. Soli. Kditiou W MILLER'S SWINTON'S LANGUAGE LESSONS, (revised edition), adapted as an iutroduotory text-l>ook to ; Mason's GramTnar, bv J. A. Mocmillnn, B.A., OttawcN^cllegi- ate Institute. 6th edition. 40tb tbooeand ... IB NXW ENGLISH GRAMMAR.— In three parts: Etymology Syntax and Analysis. By William Swiuton, A. M. Keyised by J. B. Calkin, M. A., Principal of the Normal School, Trnro, N. 8 10 DAYIES' INTRODUCTORY ENGLISH GRAMJAAB Oil MILLER'S ANALYTICAL AND PRACTICAL GRAM- •'*■•• ' IS Lk LIBRARIES THE UNIVERSITY OF WESTERN ONTARIO LONDON CANADA LS-60135 't I y M!. |. 6a0c & Co's Blatbcmafuiil Strics. TO THE TEACHER'S HAND-BOOK OF ALGEBRA BY J. A. McLELLAN, MA., LL.D. 2n'I) EPrTTOX—KEVTSKP. TORONTO : W. J. GAGE & COMPANY, 1880. Kn^iered accordiii},' to Act of Piirliament uf < 'itiiiida. in the yoar 1880 by W. J Gage & < 'ompant, in the office of the Minister of A.griculture. il2> PREFACE. At the request of many teacViers who wishccl to rse the Hand- Book as a practice-book for their pupils, no answers and but few hints were given in the book, but it has been thought desirable to prepare this " Companion " which will fully meet the wants of both Teacher and Private Student, as expressed in numerous sug- gestions from Teachers and Students throughout Ontario. Full solutions are given of all the difficult problems, and in every case, the hints and steps are such as to meet the requirements of the Student. It is believed that the Key will prove useful not only in lessening the labour of Teachers, but also in assist- ing that lar^ class of Students who are endeavouring, without the aid of a Teacher, to obtain a thorough grounding in Element- ary Algebra. To J. Ryerson, M.A., of Barrie Collegiate Institute, J. C. Har- stone, M.A., of Port Hope High School, and Mr. Jas. Miller, of Bowman ville High School, my thanks are due lor valuable as- sistance in the preparation of the Key. J. A. M. CHAPTER I. 1. . ^Exe-cise i.. pa^'e 1. 1. 0; -en; 1 ; 0; 1206; -25); 1|. 2. -1<;0; 100; 41 ; 108. '• Tff' 38» 25, 125; tV. -»1. -4V, : 0, - 1. 4. 9, 8 ; 7. - ,V. 6. 176; 82, 254^ «; -87-7^3. 6. 18 each. 7. 146; 14; -72; -270; 896. 8. Each = 0. Exercise ii., page 7. 1- -1; 2. -100542; 4. -2967511; 5. 008; «• -8; 9. 0; 10. -20 12. 3. 100. 6. -102 7. 10 11. 700440254900. !l + 3 -13 -88 +3 8 + 10, 75329 + 31.8048 .6 •5 + 3,29222 + 5.3007 8 8 .52075 .8481 4 4 26.34 424 4 4 203 42 8 8 20 8 l'fO-58448 + 10.0014; -8 1 1+3 -13 -38 .7 -8.77981 + 2.3379 + 30.165 7 .5455 7.038 9 545 .704 8 69 90 1 2 8 «M— Mi - .77931 -10.055 ; A Am. CBAFTIiR ONE. -2 .8 5 1 2 1+8 2.80512 -18 - .:i808 .1550 10 -88 -1- 27.0084 + 10.8874 ()77 18 8 1+ .10488 -18.54r.7; A ins. 18. lor each. Exercise iii., page 7. 1. 0, 16a*; 2. tt, «i/8; 8. 2a, 0; 4. 20a«,-2C^»«; 5. 0; 6. la^ ; 9. c] 10. 0; 7. 6a*; a 8 s . a + o ' <(''(• 12. ya (iT2r); (18). «2+/,2+r2; 14. 0. 16. From tlie value of x, the cuhf \. the given expression becomes o^— - 1 ; also a; + 2a + A = - - ^— , and a; - a - 2ft = - ~ »2 — *» •• ^^^® second fraction in the given expression = -1, and the result is j .^^^ — ^ij + 1 = (12^'^/' - 24o/'- + 286S) - (36 - a)*. IG. 0; 17.0; IS. - ^Sc ; 19. 20, 21, 22, each = 0. 23. The square of the difference of two quantities is equal to the sum of their squares diminished by twice their product. 24. (a + 6)(a-/>) = a-'-/>a; 25. 2(i+^)^, 4a;«. 20. The product of two binomials which have one common term, is equal to the square of the common term, plus the sum of the unlike terms multiplied by the common term, plus the product of the unlike terms. 27. (a + t)2-2a/> = a3 4-i3. 28. The sum of the cubes of two quantities is equal to the sum of the cubes of the quantities, increased by three times their pro- duct multiplied by their sum. 4 OaAFTER ONE. 1. € i 29. (T~y)^=.r^-y^-Xri,(x-y.) 80. ^^- f' == (a- ---»/)' + .77/ SI. Tlif (1 iff. 'ion CO of the cubes of two (junntities divided by the diffcniDco of the (lUiintitioH, is cr|ual to the square of tlie snru of the quantities, dimini.sJiod by their product. n^^. 'r' = '.\n. 88. / = v(:l'/»). Hi. Aroji = }, ^ (2///2) X v'(2/2), xshere /, w, are tlie sides of rect. 8r,. Tr-. w(/-'- r"'i) = nr{r-\-r')(r-r'), where /•, ;' are tlie indi:. 8(). ir/^'A. 'wr^/i, f^TrrVi. Exercise iv., juige 10. 1. 2(/u' 4- '•//). 2. 8r<^r-%). 6. 'Z[x-^!,-{-x)Or-i-{.{/'i+c'-i-ah-nr-l,r). 7. Q. 8. 2{aj:-\-ri/^/,z), 9^ „2^/,a^,,2. 10. 2a:\./-2/>j. 11. {a-\-b-v) Exercise v., \m{^o. 12. 1. 2(a;2 4.0//4), l.,-6a. 4. 4(a2-68)3. a .r2 - 0r»4-9a** +2x1/- (Sxy'-^ - (b'=//-f IHr^v'-'-f ,/2 -fiy-^ 4-9^4 10. 1 /.2 1 R ,W 0. 4x7/(^2 -y/2)^ 2(14- 12.f;2 + l-, 8f//*^^-f A|2. 18. |ir^4 lv/^ + j2^4-.l(.'7/4-//2+3a;) 16 4(./v/-t-/y2 + 2:c) - 2(0-2 -4- ^2 ^2- j, 18. («-'+i!/.-'-2f2j3. 20. - 4r/6. 28. 4{l+x^+x^+x'''). JIiNTs AND Solutions. 1, 2, 8. See Ex. 1. 4. The quantity iu first brackets is seen to be {a + ou + u b)~ -i^u il)', iiow multiply thi.s by (a-b)^. 12. 2(/r- c){h-d), 15. (l+.c2)3. 17. x^. 19. iaa,'-'y3. 21. 4(a + 64-r)2. 24. (a2a,-2 +^2^2)8. CHAPTER ONE. 5. ActuRl expaiision, or use formula [4]. 7. Actual expn. of given expression, or by symmetry. 9. See Ex. 1. 10. The expression is seen to be {2a ~ b-^2h ~c^2r-a)* = {a-\-b+c)l = &G. 12. Actual expansion, or by formula [i] {a -f 2b+<'){a - r.) - (r-{.2'!-{-a)(a ~ c) = &c. 13. Expression is seen to be {^x — >/+Si/ — z + lz — x)^ = Ac. 14. Trans])n'^c left hantl member, and expression becomes {x~ i/ + 7j — z-\-z~x)^ =0, Sec. 15. Expn. = {l-|-.c)4 -2(l+r2)(H-a;)24-(l + a:2)2 + (i-.6-2)-' = {(l + ^)3-;H-a:2)i3 + (l-. a;2)2, &c. See Ex. 4. 17. { {x - 2ij + H2) - (;:2 2/y) j 2 := &c. 18. Expn. = }(f/2 4-/;3_c2j ~(r2-/y2)l2=&c. 19. Expn.= {(.f4-//)--(aj-//)2p = &c. 21. Given expn. = {Sa - b+3h - c + 'dc - a)^ =: &c. 22. Expn. = (2x3-^2^2?/2-c24-222_ic2)t'^ a;3+//24.23)a = &c. 28. Expn. = 2{(l-^a;'<)2-+(] -x^)-2-^{x-\.x'^)^ ■^^[x-x^)^ = itc. See Ex. 2. 24. Expn. = ((Ta-+%)4-2(./2x-2+/>37/2)(,,:r + ^;y)a • =i(^a'H-%)2-(./2^.2^/,2_,y2)|24.(,^3^2 . i2_,/2)8 = 4a262u;2//--' + (,r-.r2-//-'^2)2=&y. See Ex. 4 Exeicise vi. ^ w 1. l--4a-+10;^2_20^3.|.2.5^;4_2b;'>-fvlO./;« ; 1 - 2x-{-3x^ - 4.j;3 4-3a)* - 2^;^ +x^. l + Qx+Ux'^ -t20x'^-\-15x^-Jr&x>+x^. x- CHAPTER ONE. 1 +a;2 ^y2 ^.2,2 _ 2x + 2// + 2z - 2x>/ - 2xz + 2//Z ; u3.<;3-|.2^a;»+(2r/6-4-/>iJj^'t-f 2(a.r/ + />r)x'' + ^2/^/4- 6'2)^g 4-2c'(/.t;'+t/-'^«. 5. By actual expansion, 0/ left liaad member is seen to contain [ay — bx)^ :. by symmetry, etc. 6. Expression = { (ax + bi/ + cz) + (/>^ +(\y -\-az)\ { (a.i- + ^>Z/ + '-^^) - (^'^+ ^2/ + ''2) } = (a.r + bif +('2)2— (/u- + ^?/ + (iz) ^ 7. Multiplying by 2, we have to show {a-b)^ -j-{h- cy^ -{.{a-cy -2{a - h){a -c)-\-2{a ~ b){b-c) -2{a-c){b-c)=zO, i.e., {{a- b) + {b - c) - {a - c)}^ =0. a S{a^-\-b^-\-e'^)-2(ab-\-bc + ra). 9. Expand the given expression, and result can be put in the required form. Or, expression is symmotiical with respect to b and a,, which may therefore be interchanged, giving the re- quired result. 10. l^ut n — b — x, b -c=:y, and :. a — c — x+y', then we have to show (xy -xy ~!/^ - x^ - xy)^ =x-y^ + y^{x-\-yy- i-x-ix+y)^, i.e., («2_f-;c//4-,'/^)" =&c., then see Ex. 4. 11. 4.a'+lh-x^+T^^('^x^-^4:ff^x^-2abx-acx-{-Sadx -\-lbcx^-2b>U^—c)2^6-c)2 + 2(/>-r)2(r-rt)24-2(r-«)-'(a-6)3 = 2(a:2//2 4-//222_|.23^2); but (a:2-v/2j24.&c., = 2(u;4^V/*-f-24)- 2{x^y^-\-y^z''^ +z''^x'^)=x'^-\-y'^'\-z^, by subytituting from the first result. 13. The given expression reduces to 2{a^b^ + b^c~ -\- c^a^) + 4fl6c(rt + i+c), which by Ex. 2, gives the result. 6 CHAPTER ONE. Exercise vii., page 15. 2. ].f *+.'/*; 8. rt4-f-3'7 3/>2 4.46*. 7. 0. 1. (a2_/,2)2; 4. x'4-//* ; 5. .r2 ; 0. l(k-3 ; 8. {2r/2H-(8A2-4.-2)l{2'/a-(8/>--4r3)} i _3,..L(2,, -/0|- 1 -3r -(2.1-6)} =*>- -4r/3 _/A' .-u-laft. 10. x'^-pK 11. ;i;«+x-4//H//'- 12. Expres.sion = (.^ + /))2-('f/>-f-l)3=^/2 _,,2/,2 ^ ^3 _ i. 13. First two factors are {h^+r" -(i^){l>'^ -\-c"-i'U'i) =:26-c2 by given condition. Tiie soeond pair of factors gives \a'^-\-ib'^ -'"^)} X .[rtS _ (/,2 _ ^.2 j} = rt^ - (62 - r2)2 = rt4 _. (/>4_|_p/i _ iih2c2) ^ 2h"r^ bj the given condition ; .-. le suit is 262<,'2 y/ 262(_.:j _ 4./j4f.4_ 14. ..•4 + //^ + /,.^-//'^; 15. {x^ + 'i^x^ + l-Cix^+'Ix)} X {a;'*+B.c«-'rV-f-(2a;» + 2.7;)} = a; 8 + 2a; " 4- 3.^4 |. 2.1; -' + 1 . U). 4'(4a;2-4.^-ia;// + .<4//2-3 _(.,.2_{_(<^ bc4-(.'a — >i'-^-((h—(i('-\-hr-\h-{-c)^; .'. product is {(« + '■)* + (a+M2; .; i^o+ry^ = ^a+cy^(b+cy' i-iu+by-iib + cy-i = &c. 10. By formula [4J expression = 2(^///H-c(f) + a2 4./;2_ca -^/s multiplied by 2(^/6 -f v-(« + Y) [■ = ('" + /')- -("+'7)2 = &c. 2t. First pair of factors ^'ives (x^ +y, -l)(x^ -h''-\-l)=x*-\-2x^ + r'^ - 1. Tiie second pair gives (l+x- -x){l-{x- -x)\ = 1 - .r^ -t2j:^ ~x'3. Tlie only term exactly the same in these two results is 2x^, we have .•. (2;t3 +,.4 +.,.3 _._ 1 ) |2.,3 _ (.,4 .^.,2 _ 1); ^ 2.>--' +.,'4 ^2x^ - x' ~ 1. 25. Expression = (a;* +.>;5Jy'- +//4)2 __ (a;2,/2)2 =, by formula [4] (i«4+x3//a-f_,^4 4.;,;2_,/2)^.,.4 4.^.2_^/2_f_.,4_^2,^2^ =:(a;2.X.,,2)2(^.4_|._,/4)^^.4(.,.2 +,^3)2^^4,^.2 +^3^8 = (.f4-f-.c'3./2^3+(^4+^.2^3)2. See Ex. 4. Exercise viii., page 16. 1. X*-\-ix^ iS.r'i -2x-12; .•2-|-./3_2.,vy^_Q.^,^_8,/2f 15z3. 2. x*-^l2x'^ + i\)x^-\-78x + i0; see Ex. 2; x'-+bx^- a^ 8. ««+S.^«-ir)rt4_l01«2+105; .f« + 2a;c -u;-'-2. 4. ;,;4^.5^.3^1.> - 12.l-22^'3 +5.^//3 + z/*. 2 .f2 f). a;^"-2uy' .,2-16.,-63; ~ + ~ ■{■ -- + — -1 ;'/*• x^ y Hi 6. w3».2 4. 2nx!/-}-y^ + 10nx+ 10// + 21. 7. (a;^r02 + 2//(.'c+f/)-3^3. 8. .1-4'' f 2.i;3" + -^.2n^l_,,_/,) x'{o+h)+ab. 9. -Ix^ -x^!j-^-\-!/*~xA + 2f,^-~8. 5 10. 1 1 > 2 n 1 . 11. First and third factors give {x~2)^ —2 = x^ -4x+2 • the second and fourth give (;«-2)2 -8 = 3-2 _4a; + l, ^ua (:e2 _4j_|.2) (.1-2 _4a;4- 1) = (3?2 _ 43.)2 .^ 3^^.3 _4^.j +2 = ic4 - 8.6-3 + 19u;2-12a; + 2. 12. {x + b+a){x + h-n):^{x-{-b)^ -a^ ; and (x-hh - r)(x-^h-\.r) - {x-{-b)^ - cs ; product of these = (x-j-b)"^ - {a^-\-c2 )(x-\-b)2-{-a2p2^ ^1 8 CHAPTER ONE. 13. Expression = (fli4-M^ + 0^ + 6)(r + 'fH^^ '\-{c+d)^ + {c-\-d){a + h)+ab 14. Apply the formula. Or by symmetry aa in Ex. 7, pa»e 86. Type-terms are a^ ami ab ; we see that there is no a^ and 9ab :. &c. Exercise ix., page 17. 2. 96('<2-f.62+^7/;2); h[27a^-2,lah + 7b^) 3. (x+y)^. 4. 8a\ 5. 8./;3. 6. Sx^ 7. a^. a 27a:3. 9. (2-|-a;}s. 12. 8(a;2-fi/2j». 11. (u^ ■^b^){x^-^y^). 15. 0. 16. 0. Hints and Solutions. 1 and 2. See example 1. 3. -{x-^y-z-^-z)'^. See Ex. 2. 4. Sum of the two cubps is (Ex. 1.) 2>((a^ -{-^h^), :. &c. 6. Last term may be written +i^{x — y){x+t/)^ .*. expression = {x-y + x-\-y)^ = {'2xy' = &(i. 6. Expn.= {(l+.«;+^-2^-(l-a; + a;2)}3 = (2^)3. g^e Ex. 3. 7. Expn. = (^/-6-c+/» + r)3. See Ex. 2. 8. Expu.= {(3x--4//+5s)-(o2-l//)j3 = &c. 9. By formula [Gj tliis is {[^l^x-^x^)-\-{l-x)}^ = &o. 10. First term of given expression is a* — Gn^b + 12((^b^ —Snb^ ; in this write a for h and b for a, and 2ud term is b^ — 6b'^(i-\- 12/>-rt2 _ sha^, :. difference is «4 _ hi_Qab{a^ - b^)+Sab{a^ -b^) = a*-bi-\-2'ib{u^-b--^) = {a^-b^){»^'-\-b'^-\-2ab) = (rt - b){a + 6) 3. Or, expand boUi sides ol identity. 11. See last solution. 12. By formula ('jj thi •fee. {(x^+xy+y^)-\^{x^-xy + y^)] CHAPTER OITE. 9 ^' 7, paare 86. «^ and 9ab 3. (x-}-y)s, 9. (2-ha;j3. 6. 0. expression 5 Ex. 3. 18. Treating left-hancl member an in 10 al)ove, there results = (rt6 4./,«)2^.i4,,3/,3^,t6_^/,6^+4(j^6^c which, formula [11 X 14. Expand tlie first cube by formula [6], and last term of given txpressiou cancels out, .•. &c. 15. From the given condition c=-(a-{-h)', substituting this value of c in the given expression, it becomes a^ +6-'' - (a-fi)-"* ^ +3ah{a+b) = {a-i-b)^ -{a-{-b)3, by ibrmula [«] . 16. As in last solution ; y-=x^ +z», substitute in given expves- , sion this value of ?/-, &c. Exercise x., page 19. 1. 1- Sx+Q.r2-7x^+Gx^-Sx'+x''; 1 -6a; + 21./;2_ 5(U'3 4-11U.4_ 174^5 ^219a;«-204u;' +144a;«- -64a;^ 2. - (x^+lS:c^ +2735' +29.r« -24x' -SGx^ -\.5x^ - Sa:^ -2.) 6.0. ^. iox^+lGS:>:x^i/^^-iS2.c^u-hK 7. {ux + by + cz)K Hints and SoLunoNS. 1. Cube first two by [7] ; last prob. may be treated as a bino- mial, and cubed by formula [GJ ; or see Ex. 4, p. 36. 3. Equating the right-hand members of formulas [8] and [9] the identity is at once derived. 4. See hint on Q. 14, Ex. 9 6. By formula [8] the first four terms of the given expression is seen to be {x-2yi-y-2z-{-z~2xy'' =:-{x+y+z)^, hence, &c. 6. By actual expansion. But more easily by symmetry, as in Ex. 7, p. 36. 7. By formula [8] the given expression is seen to be {2ax-by4'2by-cz-\'2cz-axy^=&c. t 1 1 i i' I 10 CHAriER ONE. 8. Ciibin? by forniulfl ^0] , expressiou l)ec;)n-ie8 + Sx!/^{x+!/)'^\ ; now the (inaiitity in tbe la&t brackets = (x^ -yS)2yfix!/(xhj-\-x^-xy^ +x^y + 'Sx^y'' +nu:u^+!J*) = {x^ -y^y +3xy{x'' -^xy-^- y'^y = [x'' +xy + y'')- {{x- y)'' +Sxy} = {x^-^xy+y-)^; rind{x^-yh^ = {x~y)^x-'+xy+y'')^ :. &c. 9. From formula [9] (,/-|-A4-<.')3 -3 2a2i = a3+63+c3 + 6a6c. Add lS((hc to both sides, aud this becomes ^ _3.;„(/,_,.)'2 + + ...i_2-k,.j its value from lo above, = {['i-h)-^-[b-c)^-Jr{c-a)-\(n + /; + c) + C{«(/>-c)-+%-t<)" -\-c[a — b\-\ =, by addition, the required result. 19. In problem 8 of this exercise (derived as in IG) put 2a -b for .i;, 'Ih-c for //, and 2c— a lor z, and the identity appears at once. 20. From the last //uv/t relation 'I.c/jz = 'labc ; add this to the three other given rehitions, then ^'^{ll+z) + if^{z + x)+z-{A-\-ii)-\-%c;iz -= 'labc. But by equating the right-liaud members of formulas [7] and [8J, we see at once that the /t/Miand member ot the above Note. — The above identities can be proved as in Art. XXVi. Exercise xi. page 21. 1. a:6 4-C.ir'''//-f 15./;-i//-^+20a;3//3 + 15.6-=^//4+rxc//''' + v'' ; x'' +7.>:«// + 2U;\//--^ -f-3ou;4//2 + 35./;^//^ -h 21^-//^ 4-7x7/^4-^^ , a;i2_^12.r"//+G(3u3iO//2^220.6-«^3 + 4y^^s^4_j_7(j2^-^.''^ + 924.c«/y6+792a;-^^/7+&c. 2. The signs will be alternately positive and negative. --32a63^1(j/^4 ; same as last, terms in inverse order. 4. l-fO/«f 15///2 4-2Um3 + i5„<4^.0,y,-+m« ; /yr^ +5w* +- 10m 3 •f 107/<.^ + 5m-|-l ; 04//t*^-|-iy2//r'-{-2'10m4-f 100«i3^.60wi=* -t-12?/i + l. n CHAPTER OXE. 5. 8. Given expression = 5;/-4// + 10a;^//3 4-5a;'-'y/3-f 5a;v/« 9. Miiltiplv identity iu 8 by 2, then 2{(^+i/)' -a;''-//'M =5'''//(-«+//)(2^' + 2//2 + 2a;//) In til is identity put a - z for £yC-h ior y, and Uierefore a-hioi x+]h tlien -|- . . . } . Seo cxtimples ptige 13. Exercise xii. ANSWERS. 1. 1+.r^+.7•4+.r«4-./•'^ 2. 1 +x-{-y^ -\-,r^ +.ri +.»;'• +x\-\-x'' -\-x^ +x^ ^. 3. xA + %''^ - 85.<;2 _ 8().r4. 1080 ; 2.r^' - a./;« +4a;''H-a:4 + .t3 - 2a;2 _a;4.2. 4. a;«-57.«4+20r).c-2-l. 5. 18.c« +21.6'" +ii.c^4-JJ^'+G3a:3 + 96a;2 +43.^-1-6. 6. l-lx'-lx^, 7. 6.ri 2 _ 4a;9 _ n^^s - 2a;^ + 9a;« - 10a;» +»* - 5a;3 4- oic^ 4 .r+4. 8. .f3 + 9.c'-' + Uk-+ll. 9. a;4 + 3a;S. 10. x^-'6x^. 11. a:44-8a:3-8.^. 12. (1), -1; (2), -1; (8), -4. 18. -1. Hints and Solutions. Nos. 1 — 11 are intended as simple exercises in Horner's mul- tiplication, but the student who has read the first three Chapters will notice short methods of solving several of them. 1. Omit x^ from the second member (l+^+a;2 +a;3 +a;*)(l -ic-a;^ +a;»_-.T;^ 2 ^a;i 3) = {\+x^x''-^x^'^x^){\-x)[l-x'' -y.^^) ^(l-..-^){l-X'll4-^5)}^l_^5_^7(l_^IO). Now add the omitted (l+x'+x-^ + x'^ '\-x^)x^. 1 xy). ) e a — b for 8. -1. 's mul- aapters cnAPTr.n one. |g ,. [Tho above is the solution of tlio p^oWem as it appears in tho - HaT,d-i3ook ; H misprmt, however, occurred, the problem should #of which tiic pi-oiliict is ] -f-;c+.>'3-f ^^.4_{. ,.i 7^ l+a;+a;2 4-x--i+^i +^.5^,pfi ^ ;,;7_|.^s ^^9 _x^\ -a;!^), 1 (corrortod) ami 2 :ro ]>nrr,icnl:ir cases of the ^^eneral theorem 6. i/(l+u;) = l4-U-— ^.,--' + &c. •': ^(|-^-) = ^ --'^ -^-^*-'-&o., (by v^vWmcr-x for .r), and v/(l --x-)^l-ix- _ i^^ -.tc, (by wnting-;^- for ..). 12. AiTanjjfin.'^r tho toi-ins of tlio sum in order x^^+x''+x'^-{. +a;2+.,= _i, or,a:^«+^'.^-f-.,;i4.^. +..-2 4-^. + i =0. («) Multiply both members by x aud add 1 to each product. ^''+-^''+^'' + -\-x^+x-'^x+l = i (b) ih)-(a) .-. x^^ ^j ^ ^ The product of tlie first pai)- of factors is '^^^"+x^'''+x + x- +x^-]-x'^-{-x' +x^+x^' +.c'^ = -I. Similarly the other products may be found. Instead of writing [the two Imes as above, the student will find it convenient to write only the latter line ; this may be done by subtracting 17 at once from any index that rises above that number. Detached co(>ffi- cientrf should also be used. 13. Arranging the te)-ms of the sum, x'''A-x''-^x'^ + ^x^-+x=-l. 14 CHArXKU ONE. if ,( !.'l , I I ' ^ Multiply bofb mniiib'^rs by .r. liuii raid 1 to grcIi pvocluci ;ci3_|_,,.i.'4.,.n^ Multiply tlie given factors, subtniotiug 13 from auy index greater tlian that uumbcr, the product is The following are other examples of this class of problem".. la eacii case tiie aum of the factors is assumed to bo otiual to —1. 1. [x + u'^)[.r:--\-X'')=-U I. {.>:-{■ x'> ; ix- -h x"" }{x^ 4- r 4) = 1 . 3. (,/■ + ./•- +.6'4)(./;3+.,-^'-i-.,;«) = 2. 5. {x+x-^ +x^ i-x-'-tx^^ +u;' -•) [X-' -i-x'' f u;" -h^-^ +^8 +^i ») , (j;i I x'' f A'^-f u;> *^ +.f> a +^-' ■'^) = 7. {x-'+x"^-hx^+x'^-^x''-^x'-^+x'^-i-x^^+x'>') = 5. Exercise xiii. I. Sx^-2x^-ix + 2. 2. 5^'4-4a;3+3a;2-2.r+l. 3. a^ -f 2(/3 4.8«.3 r 1" +5. 4. x^-i-2x''u + irx>/'^ +4y/3. 7. lOx-s+S^-i-fl; 10^-flO. 8. x^~x>j-{-yK 9. .B2-(t2. 10. x^-\-[l-a)x^ + {l-„ + h)x''+(l-a)x-\-l. II. 3a;3 + 2a;'-+a.- + U; 3,l(.f + l). 12. H^*^ +13a;// + 12//2. 13. (jx^-x^-x'^+x^-x + (j\ -1. 14 2x^-nx^-^4:X'i-5x-{-Q. 15- «+&. 16. a; + //+2. 17. lOx^ ; 10(a;4--20). 18. )/Kf3^_„^2^,,. IQ l + .i._5;^2_3^3 4.9;c4. 20. 33. 21. -4. 22 -20. 23. Uy\ 24. 85a;+8. 25. 755. U Kluct. = 1, Qdex greater »bloim. Iq il to -l. CHAPTER ONE. 5. H-1. :-20. -5a:+6. 3 + 9;c4. 15 IT o Hints and Solutions. n.oia:'""' "' ""^ ""•^'■^" ^"'"-'" "« -•'^^ »»• Horner's Powersoft. ^^ "^ ""'• ""■•■'"«« i" ««ceii.li„ ,i^ -^^ « 17 -5 -6 3 is merely a variation of 2. 6. («a-X-3)3^(«_^)3 = ^^^^^.)S^ 7. 9 10 10 5 6 -!J0 __yo -44 __45 10 8. «« - V 6 10 1 10. 11. 12. -1 i ~1 ■ 2 ir 12^ 6 8 60 7 -3 2 " 91 65 7 6 -2 -1 -3 -2 1 1 ' • 6 3f 169 -25 -91 156 -05 3 3^. 00 -60 5 13 127" This isaparticnlar case of {(.,2_,,3j^.,^^,„.^^^ -, verif, In'tlietale^r^t.^^^^^^^^^^^ -ude.t in .' • In! I ■ ■^ ? <. 14. OHAPTFR ONE. 6 -1 2 -2 2 10 6 -4 -8 12 -Ki 20 -24 1 2 -8 -2 8 -4 5 -rt 8 4 -6 Work also iu asceiidiug powers of x, 15. a4-i-2 + a,^« TT'lili" 2 -1' 8 3 8 -8 1 -8 -3 1 1 1 10. {(.r + //)4-2}S4 {Or-|-//)+2}». VJ. Fiit'tor the divisor lirst. ^(2 + 18a;-8(;.f3) = A(l + ().r)(2H-.r-nr»). i(.i- IJU.2 +G.r3) = i(2-u:)(2+u;-Gu;2). .-. quotient = i(2+u;-0u;2)3. 23. {iJU-)*-(2//)4 + K,^4}^.^a^.)_(2^)j. Exercise xiv., page 81. All these are done as iu Exs, 1 and 2, Art. VIU. 1. y^-2>r--ii,-\), iiy = x-S. 2. ys^mz + h, if //-.(•+ 1. 8. 2/* + 81, if y=x-2. 4. yi-i^i,!^-. .iHij'-\-d^2y-{M, if // = .r4-2. 5. 8.v»+30//4 + ll})//S + 288//34-24l)/y + 106, if t/ = aj-2. /;■* _ ft S» i/'J _ '.» 1 ,, 4_ H 4 .'•, :f ., _ ,. 1 :< ! ' "^^K CnWTT.n ONE. H. (x - 2.v'«) - H;fi.r - 2// ) '^ - 1 H>,'i(x - 2//) - 2 ly ^ ». (.*-• -//)'- l<>//-\;c-//)»-2l».v\^--^)-' -l()^t(^_^)^ 10. (2.r + //)3-f2//-'(2.r+/y)-f.w/S. 11. r,i2.v«-;{v ,';••.. il>=,i.r ,'«. 12. //4-2|//'^-f- tt)// 2S. it ,y^.,.4_2. 17 8i)J. of 9. 4-1 -4 -4 6 -4 -1 +1 4 + 1 _.4 -8 ~4 -7 + 1; -u + 1 ' - 7 - 2 -11; - 9 10 2 + 1 -1; + 1 - 1 -10 -20 1 fl 0^ ! I 'i! 1i It CHAPTER II. Exercise xv., page 33. -n\2 a(0-c) + b{c -a)-\-c{a-b) ; ab{x-r) -\-hr{x-a) + ac{x-h) ,) > {„4_/,)((.__,,)(,._/,)+(/,-f.r)(^/-/;)(rt-6')4-(^+«)(*-c)(*-'0 ; + (^'-c) + (6~^/); a^{a-b)-^b^{b-c)-\-c^{c-(l) + tV^{d-a). 3. Write a for b aucl /> for a, iiud there results (ic+'')(^+'^')(^+^')+^'''<^> which is identical with given expn. 4. Interchauge « and />, tlieii [h ^ a)'^ -\- [b — a)^ ^ which is iden- tical with the sjiven expn., it being remembered that (6- a)3 = Jrt — 6)2; intercliange ^< and -6, then (a — b)^ -{-[ — b — a)^ = (« + 6)3+(a-6)2, /. &c. 6. This is at once evident. 6. Change a into b, b into c, and c into a, then 62(c— x 2{a + r){h+r) x 4r{»-^b) Winch IS plamlv symmetrical with respect to a, h, a. 12. In the given expression Xnow into.cliaugujg al,, he, ,«, tberc results 13. a: and y* 14. „;, and /,,/ ; x, ,,, and 2. 15. /■ and A. 10. ^ ^ud ,,, also .■ aud -., and , and -. 10, b aud 6'. on r. o^;i „., , „ , - ^"- " ""'^ "• 21. a. and J. ^, ' ^'^- "'^ ^'^^- 24. «'^6, aic. ^'^^*,^4v. ^-V^ Same. x\if, x^i/^ ; 26. NoL symmetrical, .-. none. Same. 27. .4, ,3,, ,.y,, ,.^,, 28. a4, «S6, «2^^.^ ,,^,.^^ a4,«263. 29. a^, aH. ..t I 20 CHAPTER TWO. Exercise xvi., pasfo R7. 7. U{x^+y--\-Z'') + '2,[xy+yz+zx). 8. 24f//'rm>m.^a^nb in second and third, .-. this term vanishes ; Qabcmnr in each of the fom* terms, .•. &c. 9. Type-terms are a^bc, ab^ ; ab^ iu first, -ab^ in second, .•. this term vanishes ; a^bc in first and second, .♦. 'la'^bc, &c. 10. Type-terms are a'^b'^, a^bc ; 2a^bc in first term, -2a26ciu second, .-. this vanishes ; a^b^ in first term, .-. &c. ii ) CnAPTEH TWO. 21 1 fourth, m lust, Aiiee^ .'. 'st term, -6xy in fourth, each of 11. Type-tsrms are a^x'\ ahxy \ a^x^ occurs twice, and 2.//).r// occurs twice, .-. 2(a2 +/,2 ^^2)^.^ ^ud 4(a6 + ^r+m;;.-^, aiid tlie same lor y^,z^, and //e, zx, .-. c^c. . 12. Type-terms are a^ a3/,, ,,2/,.^ ^3/,^ . t^ese are obtained bv taking terms of the expansion ; tliiis a4 ill eacli term ; W^b in first and last terms, --iuH^ in second and fourth, .-. this term vanishes; 6^2^,2 i^i each of the teims = 24rt362 ; l%,2hc occurs in first two terms, and —VZa'^bc in last two, .-. this term vanishes ; hence result. = a^+-ia^b-j-^a'^hii + + 4^^,,,3 + y,,2/,_^. ..) + ... Hence the type-terms are «4, 4,^3/,^ 6a^b^, Vlabc, &c. 14. {>^a)i = {a+b + c+d+..:)i=:{{a+b+c)+d-r...}^ ^ Tlie first of these powers gives the type-terms found in last Ex., and the second gives the only other possible type-term, abed, with coefficient 24 ; .-. &c. 15. (a2+/')2_{_,.2)3^ v,,r,4.3va4/,2_Ln,,?/,2,.2. -Sia^ +h2+c^){ab+bc + cay^ = -32a^b-' -6Za^b'2c^g^fil,2c2 Now add, and the given expression - S./G4_2s,/3/,3 _osrt46c + 9rt262c-2 = (Sa^-Sabc)^ =^[u^-{-b-'i+cf^~-3abcy^. A shorter solution is as follows : By formula [0] we prove at once a^-i-h^+c^~3abc=i{{,c~b)^i-(b~^)2 + ^,_^^,,^^_^^_^^^ (See Exercise X., q. 15). Now the given expn. is same form as left-hand member of tins expn. ; for it is {ab-\-bc-\-ca)'^ = h,{{u-^ +b2+c^ -ab -bc~ta)2 -\-0-i- ■ '^..=..^=,ef;irti;r' -• '="'"■"'""' -- :-='';'■ ^ "' in first an eeS "'f " T <"• "^''^ «»*^ «»*e, "■■'"• fi-' and third, -o, " in^c;: T w"' '"^'' ^°-">' =0 m second and fourth = 0; 10a »6» in CHATTER TWO. 23 first and second, -lOaH^ in tlnrd and fourth = ; 2a 3/.. in all the terms, .-. ve^^ih=^ ^,{8[)Za'^bc) = lMc{a^^ ^b'^ ^c^). Hhmhvly first facto:- of ri-ht-hand member may be proved = 8a/>., and the second = 2fa2-{-/.3 4-.-^), .-. &,. The problem may be more easily solved as m Ex. 4, p. 48. Exercise xvii., page 40. 1. U5. 2. pn^^3qrr-i+Hra~s. 8. 2. 5. 1, 2(:i/2 + .l). G. or 2//", 2y/", 0. 11. a^b'^[n-^h). 12. 0. IS. 2a^~8ah{a-b); 2h3+6ah{a+h) ; 2{u^ + bS) Hints and Solutions. 1. Find remainder by Horner's division. 2. Write a for x in given expression. 8. Find remainder by Horner's division. 4. -17-3588. 7. 3G. 10. 8888a4/;4. 11 id 10 -20 -10 11 0.9 -•89 21-89 -8-9 -25058 + 20 -87-3588 1 -.9 - 1-99" -2-278 -^8-8958; -17^588 Observe that the zero coefficients in the dividend need not be mserted if the zero coefficients of divisor be not considered. 5. (-.l + l)5_._i)5^i. ^j.jting -2for a;, we have (a-hl)^ 6- (-.'/)" + //" = or 2//" as n is odd or even ; (-2/)2"_j_^3« 8. Expu. = {^^-,r^)3-f(a,2_262)3,in which write-/)2 for ^^ 9. In the first pair of factors write -2./ 2 for :c3 and we get (ax-a2X -ax-a2) = _,2.,2 +,4 = 3,4 (1,3, substituting for x^). In the second pair of factors x^^ + 'Ia^ may be at once struck out, giving -{-Sax)xSax^ikc^x^^-18a^^hy substituting for X''). ° 24 CHAPTER TWO. Li t ^i ;^ 10. Divisor =0^ive.s Oa^+U^ = 12ah, whence Bl^^+IGM^ 72»<2/,2 ; snbstitiitiu^' those values in f?iven expression we have 18^//>x(h//)x8G«2/,3^ 3888^4^4 = 768/>3. But observe that this is the remainder en dividing by Sa — 2h. To get the true re- mainder on dividing by 9a^ — l'2ah-^4:h^, we must divide the given expression ( = 0501a^ + 1290 <464+2oG/j**), we may use Horner's division, thus : 65G1 1296 256 + 12 - 4. 9 729, 972, 972, 804, 804, 768, -640; 4608 -2304 where the coefficients of quotient are all positive, ancl the remainder is 4008^//;^ —23046*. If in this remainder we substitute the value of i^a ( ='ih) we get 708^*, the rem. on dividing bv Sa — 26, 11. Put a + h for X : .-. a'\'(-{-h -a)^ + lj^(a-^b-b)^ =&g. 12. Given expression (cubing by formula [6] ) = (a3 + />3)f;(.3_}.;^3 )^ of whicli one factor is exactly divisible by a-^b, and the other by x-\-)/. 13. Put (/-6 for u;, then {a-h)3J^a^-^b^-'d'ibia-b) = „-i - h-,x ^ nlJ) = {ln + n)2J-\- {}/l +)i)x= {)n~\-n)(x-ty). 8 I)ivideud={(,c- + v/)3|3 + |(^_^p|3„-.{(^^,^y>|3_ {-(^-//)^}^, winch is divisible by [x+f/)'^ -\ -(x~u)n =:2{x^+y2y 9. Divideiid = (..-2+.^y4.y2)3_(_3...>^^,^_,^2)3, .vhichis divi- sible by a;3 +^-// + ^ -' _ ( _ ^2 ^ ^,^,j -^2^ = 2( u;-' + //^). 10 Dividena = {(r.-h/.)3|3__{(,,_6^3p, ,^1,,^!, i, divisible by 11. Dividend = i.c.:^4-o/..«-f-6-')7-(-^r2+/,„_7,o)7^ ^l^i^j^ js divisible by ^- i- o/^x- + 6'^ - ( -^2 ^/,^_/,2 • ^ o..-' + 46.6-+ 2//-' = 12. Dividends {(., + /02}2n+,_|_(^_/^^,,,„^,^^,^^^^j^.^j^^,^_ sible by («+/>)3-{-(,7-/,)a}=2('). 13. Dividend = {^-3 J^';)xy{x~y) -^3} 3 -{-u;3 4.9^,V..^-,y,+^3|3, ,vhich is divisible hy x^+^x,i(x-y) -y^-{-x^-^{)xy{x-~-y)+y^]^2x^~{'yxy[x~y)-%,^ = ^l'x->ji\ ^14. By Ex. 14, Exercise XVli., the remaindci=3 -0 + 4 - 2 15. Let tlif polynome be r^r" +/>,,"-i-f j^Ji^^jf, Then a + /. + + A+/c = (J ; subtract, term by term /. a(..«-l)+6,."--.l) + /.(^-i;, ,vhich is * exact! V divisible by.t;-l. 16. Let the polynome bo ax''-\-hx"-^-\-rjf'~^^ +hx-\-k U this be divisible by x + l, it must vanisli foi' x=.~l i e «(-l," + 6(-l)-^ + r(^_ly-i4. 4./,(._i^lo;' if n is even we have /. (,-h-\-c— _ /^ ^ j^ _ q' I.e., rt+c4- 4-^- = 6 + &c., and similarly when n is odd. 26 CHAPTER TWO. ¥'% Exercise xix , p'tgo 43. 1. Put (f = iu the C'xpu. aud it becomes -/-^ //^ -^f,^r^ - h^x^ -\- h'\: " =0 ; .-. it is divisible by '^ and .'. by A (by sy mine try) ; put a; = 0, expn. becomes -b^i/'^~ for a, ... / _ hx - hij ) ^ 4- (6.<'4- /'// )^ -^(1-^(1; , •. ,/ -^h is a factor. Lastly, put X = // ; .-. {(i>j - hij)'^ + {hij - aij)'^ +0 = (», .•..?.•-?/ is a factor. 2. Substitute ax for 6, .-. a.c^ --{(i- +'ixuv- -\-(1-j',^ = 0. 8. Sib>ti^!ite -//for.*;, .-. {I>>/ -xij)'^ -{a-h\(z-y)[hy -ay)^ {ii - /')=^//2 = {((-l>)^\ i/^ -\->iz - if- — !iz) = : (-'^V-''//'^ -("+^'1(2 - //»( -it[/-l'U)-['^+'^)'i/Z = \it -\-br [y'^ + ijz - y- - ijz\ ^0. 4. Putting ii = 'l'(.r m given expression, it becomes 6r/33._4rt^3 b. DiviJeud = .c-u-'"' +//) -^//■-(x-" + //). whicliis divisible byu;" +y. 7. Dividend = '/ -{c - / 11 \ 8. Put ?/ for a;, then // (—;.'/) +V\- j^//) 9. Patiforrt, then A(A + 2/>)3-/-(/; + 2A, 3=0; also put -6 for n, then -b{-h + -lh)^ .. h^i, ^o/,ys ^ ^/j^b)^ -b[-b}^ =0. 10. Put b for //, expression vanishes. ;. a—b is a divisor; put -a forx, &c. (h; dividend = rt3(,t-' _ o,,/, ^ /;2)+x-3(a3 - 2rt6 + 6-) = («•"+. f^)(a-Aj2^ which vanishes when a — b does, and also when ii-\-x does. 11. Put rt = Ain the expression, then 6( /)-(?) 3 ^b(r—h)^ + :{b~b)^ mmetrv 11 \ * -— „ =0. = b{b b — f, <: — a -b{b are d by sy; ivisors. 12. Put a = b, then b^{b - r)-Lb-^ - :,)J^ci[h -b) = b^{b - c) — b^{b — c] = 0, :. a — 6 is a divisor, and by symmetry 6 — c, c — a are divisors. 13. Put 6 for rr, then bi{h -r)-{-b'i'{c-b)+r'i^{h -b) = b*{b-c)- b\b-c)=0, a 6. b- c. c — a are div isors. CHAPTER TWn. M 14. Pnta = ft, then 04-(/^-r)2(r/_/,)a „(,/_/,), ^-;,._,,^,^Q^ ^^^ 30 for the other divisors. 16. Dividend is symmetrical with respect to a, b, -nd c ; pnt for a, then {(6-c)'i4-(^'-*)2}{(/>-r)2//-^^(V;_/.v.7,iJ • _ 6-{(A-c)a + (c-6)2;.3 = 0; .-. .^-6 is a divisor, and by symmetry 6 — c, c^a^ also &c. 16. Put -zfoYx^y, -X for y+z, -?/ for ..+ar, and expn. be- comes -a;y2+^.-^^ which = 0. 17. Substitute -a for h+c, -h for c-f «, and -c for a + />, -rt^. 11. 7. -1.^^. a -2. 9. 3. 10. 3. 11. -0. 12. i2 13. ?>=-^, .y = 6. 14.;.= -40, ^^ = U. Hints and Solutions. 1. Find remairder as in Ex. 1 on dividing bv x'- -x±l. 2. Find remainder on dividing by x'^ -2x-l. 8. On dividing by o^!— 2x-f3, as in Ex. 1, reraain^er is found to be -2:c-hl;but(;c-l)^=-2, .-. x - 1= ±^( -2) and a:=l-j- v'(-2), substitute in remainder. 4. Dividend i^^x'^ + nx-- -{-W^' + 7^^ -^.^u^r^i, divisor x^ ~-'^ — 2j;-f-5 ; remainder is 2. •o^^ ( I'' ! ,1 I' > ! I -1 29 CH^VPTT^V. TWO '">. -1 3 6 -16 - 2 -3 + 8 »> 3 u -12 -6 60 4-6 -fl2 -21 - 6; remembering;; to " skip " tavo places on account ol the two zero coefiicients of the divisor. 6. I 1 -11 ■fl3 -11 + 26 -22 + 4 +52 -14 + 8; + 8c -88 """ 8c - 88, •1 +2 which remainder mu!-t = 0, .'. c = ll. 7. Since 3^+7 = 0, .-. x= —^ substitute this vahie of x in the term 2cx (as in second Daethod, page 45), and muUiply (as on Dage 28) to avoid fractions, thua 1 _.2 -9 +0 ~i^'' -14 _1 3 9 27 81 81 ~1 -7 6 -0 378c 1134 1 -13 +10 -70; which remainder = 0, .*. c= - IJf. 8. + 1 +6 1 -4 -1 +10 +6c: 1-3 2 6 -1 8 12 1 -3 +2; -378c -644 Divide by a;* — «— 6, 2 -10 +3 6 18 -3 -6-1 -J 6 6c + 12 .-. c=|-l. 0. 2 -10 4c 6; dividebya;2-3a;+3, 6 18- 6 47"-~12 .-. c = 3. 10. 2a; =5. .-. x=^ and 11a? = 27^ ; also -7cx^ = — 43fo; then substiluting for -Tea?- and ll.«. the dividend becomes 2 10 — 43|o-i-37i, the multiplying to avoid fractions, as in 7 above, this becomes 12 1 -700c 600 5 1 10 -700c 60 300 1500 2 12 UU 3U0; 700c -210J, .-. c = 3. CHAPTKR TWO. 20 11. From the given condition, rx'^ = 2irx~7lr, ... tllvidoncl br- comes 4 (110+2k) -T^.-lUo; and multiplying as in last example. 4 (88()-f2()r) -120r -lOSO 20 100 -100 111 20 ^- (iOO 4- (100 4 20 -20; ■ 180 + t20r, - I'.^0r-1T)H0, .-. rr. - (). S.^e pa-e 28. Or omitting c. as snggested in second method, page 45: — + 5 -ao 5 -80 880 20 100 -100 -120 -000 -1(580 + 000 4 20 - 20l Tho; TiMT in which remainder we have to insert 8x2^^ with 180, and -10 X 7^n with -1080, getting the same results as before.' 12. x^ = 'Sx-4,,-.x^=:n.r^-4x = <.)x-12~4x = l)x-12, .-. o-3 ==5cx-12c; 80 -5ic3 = 15u; -20; substituting values in the divi- dend it becomes 8 -16 5C-129 -12c +220 9 -21 -99 -218 -12 +28 4-132 +284 + 3 -4 3-7-83 -71; 5.-210, -126- TSCH .-. r = 42. 18. x^ = Sx-3, .•- 10a,-^ = 80a;-80, substituting this value in dividend we get 1 2 -0^+30) ,/4-30 8 15 30 - '^ -15 -30 + 3 -8 1 5 12; -JO -9, ^-0 .'. -/>-9 = 0, //-e = 0,or/>=-9,<7 = G. 14. The divisor is ^*4_ 5^2 +7^ and "skipping" for the zero coefficients in dividend and divisor, the work will stand : 9 + 5 -7 1 -5 10 -15 29 -p 6 16 40 - 7 -21 -50 .••7? = 40, 2 = 56. 8; 40 -jD, 2 -56 i I ll ' f i \> ll i , ■ f 80 rnAPTKR TWO. Exercise xxi.. pa^o HI 1. ft= -3, c = R». J= -21. 2. r= -20[, ,/= -13.\, ^ = 60J. M. A=-8, r=-l(). 4. rt = 8, i = (), c= -57. 5. a=-2, c = 24j,, .' = 0. ('.. c== -m\\, (i = 2U2i. 7. (/ = 2()(), />=-B10, r = (V.V.). H. a = 4, <'=-27, '/-7. ^' = m). 0. 890. 1(1. j;3._,/,^n).^2 4.(0^,4.,^ + H).c_(y,-(.,/ + ,--f 1). 1 I. r« - (/* - :-3)x'- - (2/; - r/ - ;•}),»; - (p-(l + /■ - 1 ). 1 2. rx^ -{Hr - (/)x'' +{^r - 2r/+/^).« - (r - 7 +/> - 1 ). 1 ."). x^ — 2ijX' -f (/"'+7 - )JJ -h /"^ — /'7-f3/-).c-+(/>3-.2/;y + ;-i/>; -(/"/- /•)• 83. -1. 84. 1. 8r,. -1. 80. 1. 87. -1. 88. ff-f-ft + r+r/. 80. « + />+'■+'/• 40. -16. 42. Let II aud c be the lengths of the ptirallel sides, aud h and k be the lougtlis of the diagonals, the sqiiaio of the area = T c ('* + A* + ^f + r) (/i + k - (1 - (•) {a ^c-{-h-k){a+r~h-}- k) . 43. Let a, 6, c aud d be the lengths of the sides, the required l)olynomc will be ^\{a+b + c-d){a-^h-c-\-d)( »-b-\-c + tJ){-a+h+c + ]){x-4:h){x-{-r). Equate the coefficients of ic^, .-. 0= —1:^+8 — 4^4- r, .*. r = S. For /- sub- stitute this value and expand. (The factor x-\-r must be intro- duced into the right-hand member to raise it to the same dimen- sions as the left-hand member. In the whole of this Exercise the sign = must be understood as the symbol of identity.) 8. x^-\-bx' +ix-[-'li = {x-2){x-\-S){x-r). Equating terms in- dependent of aj, 24 = 0/', .-. r = 4. CTTAPTRR TWO. 81 ••. ^/ = 8. ' *»; = a(-U+.i^u+r) ••• 0= - li+t _2i + , ... , = ,,. AIgo« = <,(-l.J)(-l)(-2.i),- = 0: -80-«(-lj)(4)( -2,i) = 15 7 ..^^+^.^^-1..^^ -81 =.(^-3)(,,_ |)(^_a)(^+,). Terms in Also 81 = 3.^. a. Y> .-. a = 200. 8. '^^'^■^<'x^-'h'/x+r = a(x-2){x-l^){x + l)(x^r) .'. = a(,-2-li4-l4-/-) .-. /• = 2i .-. U = a(l-2)(1-H)(14.1)(14.2.A) = 3^,, .-. a = 4. .-. = a(-li-2j + /-) .-. r = 4. .•. 49-«(3-l^)(8-.2f)(34-4) = f||« .-. ,/ = l6 .-. value = 16(-3-U)(-3-2^)(-3 + 4j = a01. 10. .*. x~l =a or b or c. .-. (»-l)3-/>(a;-l)2+^(a;-l)-r. ~1 -/?-l i ^1 i ( > ) If f 32 CHAPTER TWO. P r r r; r q-r p-q-\-r', q—'lr\ i) — 'lq-\-'dr q - 'Sr 1— ji; + 7 — r. rx^+{q-Sr).>'^-{'(p-2q + Sr)x+(l-p-\-q-r). We have solved this problem at length from the data ^nd by the methods used in the jn-ecediug problems, but the work might have been much shortened by taking for granted the following theorems which the student should note : If x^ —px^ +qx — r vanish for x = a, or h or c x^ +px'^ -{-qx-^r will vanish for a;= — a, or —6 or — r)x-r^. CHAPTEK TWO. dd ta und by 3rk might following -c ►r — once the x^-^ 15. p = a-i-b-\-r,, qr^ah-^bc-\-ca, f' = abc. .'. a{b-{-c) = q^ba = q- — X-ff 1 ~z~ = or r a \-n\ 3 1 l [x - q 1 —-or - — . C .*. r x + l\ ? /^•+1\2 , /X-^LX " \ ''I "^^1T~) +^( 7~) + l- See note to No. 12, above. •■• i^-q)^+q(x~-qy+pr{x-q)-{-r^. IG . ^-I^^- 1 1 1 /iC-l-l \3 ' ^^ 1^. (.7:- 1) 1 2 _,,o ^ ^,.0 _ 2,,_^ ^), _ ^, = MilIt. of {(;^;2_2,;+i)+,| o;uj3_^+i .-. (•^-l)^^-^« + ;..^--.«-i) = Mult. (^^-.^4-1). Also(^'-l)'2_^G_^^^2_,^.^l^, vanishes for x-1 anl .-. = Mnlt. {x-1). But x^ -x + i does not vanish for a? = 1, ami .-. it is not a Mult, (x-1) .-. (^-l)'2-./;« + ;x-=^-x- + l)'^-Mult. (x^-x+l)(x-l) OTx^-2x'^-\-2x~l. 18. f.^;- 1) 1 _^s ^ ^ ,o _ 2,r+l)8 -a;s = Mult. {{x^--2x + l)-x} &c. 19. (^-2)i«(2.^-5)'o._;cio = (2,r3_0.z;-f-10)'o_a;i« = Mult. {(2x^--dx-hlQ)~x} or2{x^'-4x+5} = Mult. (a;2. 'x-{-r>). Also, 2io(u;3 -4.i;-f-o)"'-a;i<> = 4'^(:cS_4.^_^.5V5_(^^3)5 = MuIt.{4(.«2 -^^^5)-x^} or 3.<;-^ - 1G. + 20, which = (x--2)(8/;-10) 1 1 } ! I :» •« r ; * |. ' ; f.' f i i t tf4 CHAPTER TWO. .-. 2>o(.r-4a;-f5)*-a;^*' = Mnlt. {x-2) .'. (^-2)i«(2^-5)io-.ci" + 2i"(a;-4aj+5)«= Mult. (^-2). [Instead of the part from Also, we might have proceeded by enquiring whether the expression would vanish for x — 2~0 or 2a; — 5 = 0, the factors of the first term successive!}' equated to zero.] x^—ix+o and a:— 2 are prime to each other, .*. since aach measures the given expre?.sion, their product will do so or (a; - 2)io('2;^_5)io ...a;io ^2(d;3 -4.r+5)« = Mult.{x -2){x^ -4x' + 5) or x-^ -dx'^ i-Vdx - 10. 20. (^24.4a;+3)i8_a;i8 = Mult.{(a;2+.4a;+3)4-4 ovx^ + Bx-^-S .-. (u;3+4a;-|-3)i*-.ri«-a;'^-5a:-8 = Malf.(^3 + 5a;+3). The factors of x'^+ix+3 are x+1 and.c+3. Trying these we find x^»+x^ + 5x+S = 0Ux-{-l:=0, .'. {x^+ix+3)'»-x'^-x'^-6x-'d = Ualt.ix+l){x^+5x-\-S), 21. (9a;-4)2i(a;-l)2i_a;2i = Malt. {(9a;-4)(aj-l)-a;} or 9a;3 -14a;+4. = Mult. (9^3_.i4^_|.4^. Alsoa;3i+(9a;3-14a; + 4)3i = Mult. {x+{dx^-Ux-{-4:)} or (9.r-4)(aj-l) .-. (9a;-4)3i(a;-l)2i-u;3i-(9a;3_14a;+4)-=5i «Mult. (9ic-4)(a;-l). Again (9a;-4)(a;-l)2i-(9a;2-14a;+4)2i = Mult. {{dx-^^){x-l)-{dx^ -14^4.4)^ or a? .-. (9a;--4)2i(^-i)2i_^2i_(9^2_i4^_^4)ai=:]^ijlfc.a, .-. (9.c-4)2.(^,._i)3i_^2i_(9^2_i4^^4^ai = Mult. 4a; - l)(9u;-4)(9a;a - 14uj + 4). CHAPTER TWO. 85 ^^"It. (^-2). . since aacli 30 or ying these f 5a;-j.3). .as "J 22. {6(.i--l)}i3_(2:^2.i_o^__4)f3 = Unlt.{(i{x~l)~(2x^ + o^c-i)} or -(2x^-Sx + 2)', «Mult. {i^{x-l)Jr{'2x'--3x-h2}} or 2a;24-33j-4; (2.ci^-f 3.^-4)1 3 _ (2,*,-3 -3;«-}-2)l 3 = Muli. {(2aj3 + 3.c-4) - (2a;2 --8a;+2)} or 6{x-l) ; ,-. given polynome = Mult.ofG(.-c-l)(2a;2 + 3;c-4)(2x2_3.r-f 2) 23. As in 22, 1st and 2ncl taken together are found to be mult, of 3rd; 2nd and 4rd taken together are found to be a mult, of 1st. ; and 3rd and Ist are found to be mult, of 2nd. 21. (2:^;8+3a;-4)'«-{0(a;~l)}i8 = Mult. {{2x^+-Sx-4.)-G{x-l)} or(2aj2-3a:+2) (2.t-2_3^+2)'«-{6(a;-l)}i6 = Mult. {{2x-^-'Sx + 2) + Q[x-l)\ or (2a;2+3:»--4) • and the given polynome vanishes if a;- 1 = 0, .-. &c. [Problems 17 to 25 are merely cases of the following simple theorems 20. Ifa;2-fa;-hl =0, (See Example 10, p. 49, Hand-Bookj. .-. .rSrrl, x^=x, x^^x^ U, x^ +.^-4 + 1 = if a'3 -4- .r-t- 1 = ... it'«+a;4 + l^Mult. (a;2+:c+l), 27. If a; -^+3://+// 2=0, .-. u;3 + :c3^-{-^^3i3 0, ••• ^'^-\-x''ij+xy^+y'^ = iii, .'. x^+y{x^+x!/ + !/^) = y9^ .'. .6-3=2/3, .-. x'^ = 7f\ .-. a;^"=ic?/» iindx''=x-y''i .*. x^i/'=x''-ij^ .-. ic^o+a;'\y^ + ?yio=.c^9+a;2y8 ^^10 ifa;'^+.c//+,73 = Oora;io+x-''2/^+//'«=Mult. (^^^.^.^^.^.^g)^ ! I I'' i r. I ; \k p,(] cH\rx':R TWO. x^=x .-. x'-' +x:' +x'^ ^x^ ^l^x'^ +x^+x-^-x^ -\-l = 0, 29. By last exercise ; x^ ''=x, x^-=x^,x^=x^ and ^., a;i«+.*;^24.;^8_|_.,.4^1=^ + ^.3_^x.3.f^.4 4-1 = if 30. If .r'+x-^//+.r//2 + ,/3 =0 .-. x'^ = !i^ .-. x^^:=:x^y^^, 31. May be written x^'' - x^ -\-x^ -\-x^ +x-2-t-a;+ 1 = Mult. (.t'4 +-.f3+.i'2-f X + 1). iC *^17 -r>r2=ryr3/Kia- a; a;3(a;i«-l) =: Mult, (.t"'^ - 1) = Mult. (.i;4 + uj3 4.^2 4 ic+i). =:Mult.(l+a;+x'3+.T;3_{_^4+a;.5+.7,«). x'-^- x^ =:;«4(a;49— l) = ]\rult.U-7_l) = Mult. (x'« +X'-' +^1 +.^3 +.T24-CC4.I). [31 and 32 may also be solved by the method employed above to solve Nos. 26 to 30. See also the note to the solution of Prob. 2, page 13. Probs. 2G to 32 may also be solved by the methods illustrated in Art. XII. of the Hand-Book. The solutions of 28 and 29 by the second of these methods will be ;pl2_|_^9^^ 29, -^-ZJl -Vx'-^r^^x^-^X _ (a;'o-l)(.c-l) u;* + x-'+a;-+a:4-l "" -l)(a;is-l)a; ^^^_l)(^4_l) '.Si' (««-!)(*« ^ 1) IT1-] CF>PTEE TWD. 87 88-37 and 40. If 6 — c = 0, these polynoraes vanish, hence they are multiples of h — c, and ;. by symmetry, of 7). But they are only of the same dimensions as /), viz., 6, .'. they are numer- ical multiples of D. The numerical multipliers may be deter- mined, as is done in Ex. 9, page 48, Hand-Book, by assuming particular values for the letters, or it may be determined by div- iding the terms involving a^b-c in the several polynomes by — a^b^c, which is the term in J) involving these letters to these powers. Thus in 83 put d--0 (since d is not involved in a'^b^c) and we obtain at once -ha^b^c + &G ; and -^a^b'^c- — (i^b^c= —1. 88. This polynome is symmetrical with respect to all the letters it involves, and it vanishes if 6— r = 0, hence it either vanishes identically or it is a numerical multiple of {a-\-h-\-c-\-d)D, for a numerical multiple of [a -^ b -\- c -{- d) is the only linear symmetri- cal function of a, b, c and d, and the polynome is but one degree higher than D. Assume values of a, b, c, d that will not make (a-f/^+c+f/)/^ vanish, and the numerical multiplier will be found to be 1. Or divide the term involving a^b^d in the polynome by the corresponding term in {ii-^b-\-c + d)D, 30. The above solution of 38 will apply, word for word, to 39. 1 ^ 1 ¥ 'n by the sum of their products two by two, increased &c. 41. The sum of the fractions — , 2 increased = (^ + T)(i + i)(^ + l) (i+|)-iwhich w-fl - lz=(/<-f-1)-l = n. ~ T ' "2" * ¥ n 42. The area will vanish if the parallel sidos lie in the same straight line, in which case /<+/.—« — 6* = 0, or a-\-c-\-h — k^ or a+c — /t+A;=0. The area will also vanish if the diagonals (and consequently the sides) vanish simultaneously, i.e.^ if h+k-^-a + c = 0. Hence the area, and therefore its square or any other power, will involve /t-1- A- -t- a + c, h-\-k — a-c, a-\-c+h — k, a-{-c~h'^k as factors. But it is given that the square of the area is a polynome h SN 1^ I > II- ■ 88 CH\PTER TWO. of four dimensions, hence it can only be a numerical multiple of tliese four factors. Write s- for the square of tho urea anci m for a numerical constant s2 = m{h + k -t a + (){h -\-k-a- c){a + c+li - k){a-{-c -h + Ic), To determine m, let c = in which case the trapezium becomes a triangle with sides a, h, k and by Ex. 6, p. 47, Hand-Book. 6'2 = ^^(^a^k-\-a){Ji-\-k-a){a'irh-k)\a - h-^k). 711 = _ 1 T^' Or m may be determined by taking as a particular case a rectangle with adjacent sides of iv^ngths 3 and 4, in which case a = c = i,h — k = o, and s = 12, from which it will be found m = ^q, [The formula for the area of any quadrilateral is in which s is the area, h and k the lengths of the diagonals and a, b, c, d the lengths of the sides taken in order. This formula is due to Gauss.] ^ 43. The only way in which the area of a quadrilateral inscribed in a circle can vanish is by three of the sides becoming equal to the fourth, i.e., a-^-b-^c-d = 0, or b + c-hd — a = 0, or d-\-a-i'/> — c = 0. If all the sides vanish simultaneously, it will be found to be only a particular case if one of these four, or rather of all of them simultaneously. .-. s^ = w{a-{-b+c-d){a+b-c-j-d.){a-b-\-c-\'d){—a + b-\-c+d). To determine /« tnke the case of the square .-. a=^b = c = d, .'. a* = m2rt.2a.2a.2«. ?n = _ 1 T^' Exercise xxii., page 53. 1. Transfer the right-hand member and the resulting polynome on the left vanishes for a; = or -}-6 or —b, i.e., for three different values although it involves only the second power of x, hence it vanishes identically. 1 CHAPTER TWO. 31) 2 and 8 may be proved in like manner. In No. 8 divide through by z^ and replace zhyc\ the formula will then be seen to be symmetrical with respect \o a, b aud c. j. These theorems are merely cases of {x—a)(y — a){z ~a) (h — a){c ~ a) (x-a)(y - a) {h — a){c - a) {x-n){y~a) -+ two similar iovms-x+y+z a-b — c, + " " " =1, and -+ three '* " =0 (h-a){c -a){d -a) respectively. They may be proved like Example 1., Page 58 Haud-Book, or by the method apphed to Probs. 8-25, Exercise xxiii. For the general theorem see the solutions of those prob- lems.] 4. Change the signs of a, b, c, in Example 1., page 53, Hand- Book. 6. Holds for rt = 0, or -6 or -c or -(6-f c) .-. &c. Or thus bc(b^-c^)-\-cn{c^~a^)-\.ah{a^~h^) = {f>-\-e)bc(h-a)-{.[c+a)ca{c-a)-{-{-,-\-b)nh(a-b) A. kho = abc{b~c)+ bca{c~a)-^ c ab{a—b) B. then A-\-B gives, &c. 6. Treat as a polynome in a. This identity is a case of 7. a^{b-c)+b^{c-a) + c^a-b)=-{a-b){b-c){c-a) .-. a*{b'^ -c^)^b'-^{c^-a^-)-^c'^{a^ -b'-')^ _(a2_62)(63_c2)(c'2^rt3) and {ni-h^c)^-a^-b^-c^ = 3^ni.b){b-{.r){c + a, by [8]. 8. See Prob. 2, Exercise V. (a^-}-h^)(c^-}.,r^){e2+n)={(nd+bc)2+{hd-ac)^}(e^+p) ^{(aJ + bc)f-j-{bd-ac)e\^-t{ad+bc)e-{bd-acy}'^. ; I'lii 40 CHAPTER TWO. lilt ' 1 It > pi {). See Pro!). 0, Exercise XI. Or thus If x^ ■4-.'7/ + .'/- ^ X-* = //^ and x (x+i/) =-ij'^ .'. x-"" {x + ;/)■"' ==-!/^''=-x'' If .-. {x + u)^=-x*i/ .-. (x+ij)'' -x'' = -xAy -X'' -x^i/'' = -j-'^x- -\-x>/ + fj'') = Q. . '. (x+i/) -^ - x-^ - ^« = if .i;2 -|_ jv/ + //3 = .-. X'+xif-i-i/^ is a factor of [x+i/)^ -x'^ — y'^. Also (x'f z/)*-./;^— .//'^=0 if x = or // = or .c4-*/ = ••• (//+//)"—/'"-//' -/'•^•//(•'^•+//)(-^'^+'^7/+//') = U-Mx+i/){{x-\- yy^-\-x'' +!/^, Determinincr k it will be found = 5. Substitute rt — 6 for x, h- <• for y, and .'. -(c — a) for x-^y = ?iOi-U){b-r)[c-a}{{a-hy+{h-cy-\-{n-a)2}. 10. The loft-hand side vanishes for .c = 0, or ?/ = 0, or 2 = 0, and it is of but three dimensions, . •. it = kxyz. k may now be deter- mined. Or, transfer the left-hand member, and the resulting polynome on the left will vanish for a; = 0, or // — 2, or — y-^-z^ or ~-y -z, i.e., ioM four values, hence it vanishes identically. 11. By Ex. 2, p. 18, Hand-Book, and.-. (.t'-^-/.2^3-4.//,3_^.2)3^(,.2_^jjs 12. a;2(v/4-2)3 4-^/yz(//+z) = x{;y-\-z){xy-\-yz-\-zx) y^[z-\-x)- -\-xi,z[z + x)= y{z+x) \xy + y^^ zx) .'.by addition, &c. Exercise xxiii., page 57. ANSWERS. 1. 6i4 + 15r4. 2. 6. 3. 3. 6. 0. 0. 5/^4_:30a/>3-i-b0a262-5a3^. B. 0. 9. 0. CHAPTER TWO. 41 J/^) = 0. X+y y = 0, and be deter- ges ulting 1/+^, or 22. -1. 25. 0. 27. l-^^x-lx^^J\xK 29. l-2x-\-dx^-ix^. 10.^- n. 1. 12. a+/>+c+^^. 18.-1. 14. a-\.h~[.c-{.,(. 15. (^^+6+c)(a2 4.62+r« + fli+^,c+ca)+«ic. 16. (« + 6+c)2(a2+^3+c2)-|.2.,^« + /, + ,.j. 20. a + b-^c, 21. 3. 23. 0. 24. 0. 26. l+^a;--Ja:2 + ,igajS. 28. l+aj-ficS-hajS. 30. l4-^a;-^^2_^/Ta;3. Hints and Solutions. 7. Let (l + rta;+ta;3-fra;3+&c.)(l-rta;+6a;2-£.a;3+&c.) = Change x into —a;. ^ .-. {l-cix + bx^-cx^-\-&c.){l'^ax+hx^-{-cx^-\-&c.) = l-Ax+f3x^-Cx^-^Dx^-Ex^+&c. .-. l + .4a;+i5.i'2 + Ca;3 4.&c. = l-J;c+l?a;2~Cx3 + &e. .-. Ax+Cx^^Dx^-\.&c.=^0. Now this is to hold for any value whatsoever that may be given to a?; .-.^4=0, C=0, D=0, &c. Of. 9. 0. 8. AssiUne {a;-a){x - b){x - c){x—d) .+ B + .+ D (a) x~a~x-b x—c~x-d Multiply by (x - a){x - b){x - c){x - d) .-. l = (^+/^ + 6'+Z))a;3-&c. Multiply both sides of {a) hy x-a and in the result put a: = a 1 *• •*• (a-6)(«-c)(a~~rf)=^' .•. by symmetry, &c. ! !' I I fi •I I * 42 OFAPTKR TWO. 10. Example 1, pai^e 55, TIand-Book. x^ A 11. Assume ; tt fc-, tt 7s = +&c. {x - (f){x — (>){x - c){;x— a) x-~a x'4 A 12. Assume -, w ,w ^,7 7\='''^ +&c. .'. x^=x^-{ii-{'h-\-v-\-il-A-H-C-D)x'^-\-kQi. ... A-\-n-\-0-\-D = a^h-\-c^-d. 1 .1 13. Assume , w tw-. r/^. — yw \= — -f-&c. Also ^ = -, B^ka, {a—b){a - c){a - d)(a — e) . 1 _- _ . + 1 Let ^ = 0, and multiply by ahrd ; bed ___ Or thus: Prove by the method employed ?u Ex. 1, p. 58, Hand-Book, {y-h)(y-(;)(y-d) _ {a-b){a-c){a-d) "^ ~ ^' Then take the case ?/ = 0. 14. is( Solution. Fmd a polynome in x that will become ^<(a+6)(a+c), />(i+6')(6-f-a), or c(c4-rt)(c+6) according aaajsa or b or c. 6(6+c)(6+a) = i(^4-«)(6+6)(6+c) c{c+a)(c+6) = J(c+«)(c+6)(c+c) .•. requii'ed polynome = ^ (a; + «)(a;+&)(.r 4- c). On^PTRR TWO. 48 (a) .-. 2(a-f-/> + r) = /i + « + c'. Multiply (a) by «;-« and let x = a. 7a-6j(,,_-.) - = ^. values of e aud C hy symmetry. ^^^Add together the values oi A, B aud (7 thus obtained and take Snd SoluUon, Assume (x-n)(x~h)(x^c) = x^+f,^^^^^^^,^ .-. ?>=-.- (,,+/, 4.,.)^ q=.ah + hr + ca, r=-abc. Express a(rt-f-/>)('r«-f-(') in terms of a, p, q, ,-. Substitute X for a and use the resulting polynome for a numerator. » r j ^^^ im a _--pa;2— r A .'. A-^B+C^~p=.a-}.h + c. Determine A, B and C as before. [Either of these solutions is easily derived from the other for any multiple of {x^a)ix-h){x-r) may be added to or taken trom any polynome satisfying the conditions stated in the first BO ution, and the resulting polynome will still satisfv those con-' ditions. Now U^ + ci){x-\-h){x+c)~l(x-.a){x-b){x-c)= ~px^ -r showing how the second solution may be derived from the 'first.] 15. a^{a-{'b){a + c) = -pa ^-ra^ — px^ — rx' Assume ^^^---~-^-___^ .^^+^,3 + _^^^^^ .-. ■~Px*-rx^=-px^ + {p^~pci+A-{.B+C)x'^~&c, .-. -r=:pS-.pq-^A + Bi-C .-. A + B-^G=^p{p^^q)^r. I- > ll I* Si I' I t .! ■. I ■■ -I I ; 44 CHAPTER TWO. IG. rt4(a4-6)((/»c)= — pa*— rrt* Assume — iior.^ {.r^ii){x--h){ir. — c) ' ' 1 ^11 ' »— a .'. A-\- /i + (' = i^ -'11/^^1 + 'Ipr. 17, 18, lii, a[u-i-h){,f-br)Oi-i-,l) = a{-p^—f)).>:—r){x—il) = x'^+px''^+>]x'^-\-rx+s so that p=z — [a -\-b ■{•€-{- d), q = ah-{-hc-\-cd-\-da-]-ac + bd r= — [((hc + hcd -\- cda-^ddh), s^mbcd. —px^ — rz A {x—a){x—b){x-~c){x — d)~x-a For 17, assume giving A-\-B-{-C-\-L)= —p, _ -px'*'-rx^ A For 18, assume / — w rr? r, T\= -/'-h +&c« ' {x~ =-;>2 , -px^'-rx'^ A For 19, assume r'~~u i\/ u n= -px+p^ + +&c. {x-n){x-h){x-c){x-d) ' ^^ a;-a giving A+B-{-C-\- D= -p^+pq-r. [Divide -^;a;^—?'a;3 by x^+px- -\-qx-^r and compare the coeffi- cients of the powers of x in the quotient with the values of the expression in 14-19] . 20. 1st Solution. het{x-((){x-b){x-c) = x^'\-px^+qx-hr h-\-^—a= —{a-\-p) — a ... bc(h+c-abc)=~{{a-{-p)-^a}{{a+p) + b}{{a-\-p)'{-c] = -{a-p)^ +p{a +p) 2 - ^(a+i?) +r ... -.bc(b + c) = {a4-2')^ -p{a-{-p)^ +q{a-\'p). THAPTKR TWO. 45 SnT>stitnf,fi x for n in tlic rip;lit-lian(l menihor and expand (See K.X.. ij, p. 47, Ifriiul-I^ook), and wo obtain iy^iY numerator Assume ^^^l^_^{l^±l}x^m _ ^ A^ ^^ ^^^ .-. /l + /> + C = />»=-(aH-/^ + 6'). Mult, {a) by x — a and then put u; = a. (a — /m(/<"c) ^mi Solution. 6 + ^= —()—) "6 a+ft+c = (a — /J) — a .•.(a+6+c)(2rt-f6)(2^/ + c) = (a-p)3+/;(./-p)2+7(a-p)+r Assume ^_« -2,.^ + (p^+7)^-(;.^^.-) ^ ^ ^^^ (a;-a)(;tf -/>}(a;-c) a;— a Determine A-^B-^-O^ and ^, i?, (7. (It will be ftjund to be necessary to divide through by a-\-0-[-c, which was introduced as a factor.) J2nd Solution. {2a+b)(2a-j'c) = 4:a^+2a{h-\-c) + bc = 3a^+a{a + b-{-c) + ab-{-bc-\-ca = 3a'^ —pa-\-q. 'II 'in •i mm i m n « 1 I ' p. V I III ll 'If ■'i ) 3 '' i I 46 f I It'- :i * • I ' .'. Assume ,—' CHAPTER TWO. 8./' 2 — px-\-q {x-a){x-h){x-(') x-a 22. a{b-{-c) = -a{a~(a + h + e)} = .-a{n,-j-p). + &c. Assume — x'^ ~l X + &c. + &o. {x-((){x-b)[x-c) x — a ... j + /y+6'.= -l. 23. A +6' -ft/ =-(/?+/;) . —X — p A Assume ,--- — -7 ^ = {x—a){x — b){x — c){_x — a) x-a ... A-\-mC \'D = {). Or, (^< 4- /j + 6' + r?)( Problem 8) -(Problem 9). 21. bc-\-cd-\-iU)= -a{h + c ^(l)-\-q--^a^ ^pa-\-q, x^{;x^-^p,c-{-q) A Assume / — — r, ;w _.^_j_^- 4.&C. {x — a){x — h){x — r)[x—(i) x—a .'. x'> -{-px^-^-iyx^ =x" -\-px^ + [^q-\- 1 + B + (J-\.D)x^ -\-&Q. .'. A + B+C-^B = (). Or, a^(Jx'-{-cd-\-)2(6(/)-i-4{4p)(4>) + 2(0r7)2 -4t i>; it- II ^Pl—! |<;t '< w ,!; M hi t » .j'l;: !l? 48 CHAPTER TWO. _(4;.)«-f5(4;))S(r)7)-5(4;))2(4;-)-5(4;>)(67)2+6(4p)« + 5(69)(4r); (4;>)6 6(47))4(G7) + 0(4/0^(4' +0(4/^)2(67)3 -G(4/>)3i - 12(4/0(6(/)(4/-) - 2(0^7)3 + G(G and .-. every C.F. is of the form mx^+nx^m which may be written {<^^'-¥hx+c)^{x''-+kx-\'l)=.ax^-akx^^-ir\owQV terms. (c+bx+ax'^)~{l+kx-^x''):=c + {b--ck)x+ higher terms. Assume C7=(it:2^^^^ i)|^^^3_rt/£^2^(i_^^^^_^^| i.e. ax^ -{-bx + c = ax'' -[ak^+ek-a~b)x'i ~{ck^-^ak-^bk-c)x^-\~bx+c, r. ak^-\-ck-ia+b) = ck^-{.{a~b)k-c=:0. These equations must both hold for one and the same value of k. they must .-. have a common factor, .-. by Ex. i, Art. XL, Hand-Book, 62c3 = (c^-a^ -ab){c2 - (a'^-b^)}, i. hi Solutiun. {{ax+c)V~{cx+a)U}-rx = (c3 - a^).f4 +abx^ - abx - (c^ - « 2 ) « (cx^-a''){x*-l)-{.ab{x'^~l)x = {{'''' ~a^)(x^ + l}-i.,ibx\{x^ -1)=W. m 1 I Li ffl li w ; it i 1 ■f ' 1 1- i^' i'i|. * ;-i 50 CHAPTER TWO. Also {ax^+c)V-(rx^+n)U}^x^=: abx* -f (c3 - rt 2 - k)^3 _ (c2 - a 2 - ftr)a; - ah ■ ah(x*-l)-{-{c^ -a'^ -hr){x^ -l)x = {ab{x^ + i)-^{c'^-a^ - h<')x}{x^ -l)==W' ,'. W and ^F' cau differ only by a numerical factor, c^ - «3 ab ab c^ — a- — be 2nd Sol. {ftx'^-{-bx^-\-c) = {x^-\-h-x-\'l){x^-akx^-ck-^l)zM ax^ — {ak^ +<'k —a ix'^ — {ck^ -f-a/f— t')a;2 +c ,'. «/i:2 -f- rk - ^^ = and — (r/i;^ 4- a^ —o)=b i.e. ck^ +• rt/i; -\-J) - c = 5. ax^-\-hx^ -\.rx+d= (x-2 -f.A;.^-f 1) {,,:;ji>4.(/; _ a/^aj + f?} » and {a-d)k-[b-F) = ,'. dk^-ck-{n-\-d] = .-. (a -(/)3(a+f/)2=(/>,7_rtr){/^(a+./)-c(a+r/} .-. (a-<(Y{a+'i) = {bd-ac){b-c). 6. (a;3+^?a:2 +(y;>;-f-r) -^(a;2+r/a; + /')ai «— (a — /)) +linear remainder. For exact division this remainder mnat vanish, and .*. x-{a — p) must be a factor of x^+px^ +qx-\-t', which will .*. = if in it a-phQ substituted for x. .'. {a-p)^+p{a-p)^-{-q{n-p)+r = 0. Expand by Horner's method, Hand-Book, Art. VIII. {x^-{-px^ +qx-\-}-) -i- {x- +ax-^h) = ix + —\ + ic (linear polynome) CHAPTER TWO. 51 ,». 0*4- -- must be ^ ftiotor oi x^+2^x- ^qx+r, which will .-. 6 r 8 •'• " 6^ + ^ -^ - ^ T + '• = ^• 7. (^*+/7a;+9)-^(a;3+flra; + 6) = a;2 — aa; -{-{a^ ~ b) -[• linear remainder ; .*. (x* -hpx+q) = {x^ +ax + b){x^ - ax + a"^ -h) x^ + {a^-2ab)x-\-{a^-h)b. :. a^ -2ab=p. [a^—b)b = q . Also a^b{cC^b'-''lb^)^=pn^ ••• {q + b^){q-b^Y=p''h^, {2a^ -2aby2ab = ia^q I. II. 8. a a Ap Gq a a^ A-4iap 4,. .%^-{-4:a^p + 6aq t T ,t^-\-4:a:^p-\-() + Gq\ 4a 3 -I- 1 2tt -> -h 1 2aq + 4 /• .-. 4a3 + 12a3^j4-12rt^ + 4/- = 0. .». a;3_j.3yjaj2_^3^^_^^. js divisible by a? — a and .*. has a C. F. with px^ ■ r'^qx,^ +^^'x-\-U 9. x — a ] 1 1 x-b x — c x—d 4a;3 -3(a + & + c + f^)aj2 +2 2 nbx — {ahc - f bed + ctfrt + Jry&) ~ ^H-^>^M^6^^'+4r^'+« Multiply this by x"". 10. -Ap ~Gq — 4r 12/> 12q ir ■IGp 16/^2 _ G4:p" + lSpq 25Gp^ - 288^^7 -I- 48pr -24y 24;;r -90/>2<^+72^3 — 16r 16/?r -4i 4, -4y>, 16^2-12^, -G4i>3+72/,5'_12r, Continue the division to three more terms. i 1 if- ^*£ # :«.: l! til li fit WL , ■ V V" y i . •;■,■ li ft ■A t ?" -1 •1. i t. 52 CHAPTEI7 TWO. = 4a;* — 4sia;^ H-G.s'gaj^ — 4.s'3x-|-.s'4. Put a; = a, =6, =(;, =(/ in succession, (a-c)4 +(/'-c)* + (('•- (h-^ =So^'*— 4s J c3+ 6.9.36'=^ - 4.s36-hs4 ... 2{(a-6)*-f(a-c)*+ . .. +0'-^O*} = SqS^ - 4S1S3 +6s2i'3 -4S3S1 +4.S4 .-. 2(rt-6)4=s„s4-4sjS3-i-39*;|. Similarly; (a;-a)e + (a;-6)6 + (a; - c)6 + ra;-(7)« = «oaj^ —Gsja;* + 15.5233* — 206-30;** -f-lui'^a;* --6s.x f 5g. Put a; = a, =6,, =c, =dm succession and proceed as before. N.B. s,j, Sj, S3, &c., are the coefficients of the terms taken in order of the quotient in No. 10 ; they are .•. known polynomes in p, q, r and t. 12. Substituting in Ex. 9, p = 0, 6^= -14, 4r=l, «= -88, the result there obtained becomes a;"(4a;3-28a;+l) Dividing as in Ex. 10, omitting the line for p. 4 -28 1 14 56 392 -42 7616 -1 -4 -28 3 38 152 1064 4 28 -3 544 -70 8683 ...So = 4, Si=0, 6-g = 28,S8=-3,S4 = 544,.s- = -70, Se=8683. Now substitute these values in expanded and reduced expres- sions for 2 (a -6)* and 2 (a- 6)^. X^": CHAPTER III. Exercise xxiv., page 62. 1. (8;» + 2)2; (c'^-ljs. 2. {y^~z^r-; 4//3(2:c+y)3. 3. (3a6+2c)'-^ ; 4?/-(8a;-y)3. 4. (^ic2 -42/2)2; (1^,2,1626-3)2. 5. (a+6+c)2 ; (aa;4-i?/2)2. M.-...)'.|f^)"_(±}"r. 9. (a+64.c-r)2 = (nr+6)2; (i/>3_ 4^3)2. 10. (3a;-4.v-2^4-3^)2 = (x-7/)2. 11. (aJ^-a;y + y--f-ic2+a;//f ?/-^)2=4(a;^^+7/2j2. = (a;2 + 2.^;«/+^/3)2 = (.^:-|.2/)4. -f(T)"-(T)T- -(-+^)^ 15. (a2_62_t;)a. 10. (a-6)2 + (i_c)2 + (c-a)2+2(rt-6)(6-c) --2((t-6)(c-a)-2(?>-c)(c-a) = {a-6+6-c -(c-a)}2 = (2a -2c)2. 17. (2a2-3/>+4c)2. Solve last three as in Example 4. Exercise xxv. 1. (7a+2/>)(7rt-26). 2. (3a + l5)(3a--l7,). 8. (9a2 -462)(9a2 -f-462) = (3a+26)(3a - U){ir- +4^2). 4. {lOx~Qij){10x+Qy), 6. 6b{a-\.2xy){a-^lxy)^ 6. (3aj3-4«/2)(3;c3 + 4^2). 7^ (|c+l)(|<,-i). !i(i I. .'*; ¥ I B M i •f '1'; t OHAPTJtR THBEl. (). ^<),,2_ij^0a2 + l) = (8«-l)(3rt-»-l)(9rt« + lV 10. (rf3-4/^2)^,,3+4/;2)(rt-2/0(a+2i)(a2+46a). 11. ai« -/>!« = («'* -/>«)(rt»+i8) = 12. a--(A-r)2=(^< + ^>-'')('*-^>+^)' 13. (rt + 2/; + 3a;-4//)(a + 2/>-3.c+47/). U. (a;2 + 2.n/+//3)(.c2 _2.ij//+2/)*^ = (^' +//)'(« -y)'' or(a;«-2/3)« 16. {x + y+'^z){x+!/-2z). 10. (S.*; + H)(2 - L>.r) = 16(u^+ 1))1 - a-). 17. (2.C// + .c3+,y3_2'-')(2.«_;y-a;2 -7/3+^3)* {x+!/ + z){x i-y-z){z-x + i/){z-{-X''y) {a;3.-ha;^+i/^ -(x-^ -^i/4-^-)} =(2a;' -2/y') X2anf = 4a^?/(ic+2/)(a;-2/). 20. 2(a + c)x2(/^ + ^i) = 4{a + c)(6 + t/.). 21. 4(1+2^:2) x6x = 24a!(l + 2a;2). 22. 2(a^ + 2^^/^ + 62)x4rt/> = 8^<6(a+ft)3. 23. (^3-62 ^.^2 _,/3 4-2rtc-26tZ)(ri2 _?,2 4.c8 -d^ ^^ac+2hd) (^^^_^lJ^(•^d){a + c—b-d){a-b-c + d) x{a + b-c — (l). 24. (.c2 -//2 - z"^ -%jz){x'' -?y2 -«3 +2//2) 25. 2(^«c-3a3/>3+6^)x4a3/>3=8a3/>3(aC-3«) + G.e3A3(,,6 „ /,«^ _ 8a363(a« - ftfl) =.(a«-/;3)(,,3^.6S)(,,3_^3^3^^^3^^3^^3_ 27. (^^+//^+.^)(^.-> + ,.+,o -2^^-2.y.-2^..). 28. 0^'-//+^)-'-(//+2)2 = (^-f.22)(^_2^). 2y. -(«^+/5>'t+^4,2a^/.2,2a3c3 -2/^3,2^ 30. .«*+//*+24_f.2.c2,^3_2.c'^22_2^222 Exercise xxvi., i)age G7. 1. (ar-7)(:r + 2), (a;~7)(^-2); (a;+4)(:r+3). 2. (.r-3)(a;-5); {x-~7)(x~12) ; (x-12){x-h5). *^. (2.^-5)(2.^;+4) = 2(2.^-5)(^+2); (3i; - 20)(3^ - 30) -3(3.c - 20)(;c - 10). 4. (U^+12)(i:«-3); (5.t' + 5)(5a;+3) = 5(a;+l)(5;c4-3) ; (B.c-''-4)(3^3_5), (4.^— 5)(4.«-3 +4) = 4(4a;- 5)(a;+l). (a:-rt)fa; + rt)(a;_6)(a: + />) ; {2(;c+^)-ll}{2(;r+y)4.0} 8. (a+/.-3o)(.+/,+.). 9. {^ + .+(^^,)3;.;. + , + (^_^)2. 65 % ^!!!*J I)/ 1 !i'' riit l-S :i. 56 CHAPTEU THREE. 10. {a + ft-(a + 6)a}{a + 6 + (a-fc)3}- 11. Last three terms = — (2.c + //)(u; + 2//).'. the required factors ai-e {x^ -\'X;/+7/^-{-2x+!/) x {^'-^ •fx7/4-//2 - (« + 2//) [ 12. {.<-3(/>-c)}{^f+(/>-6'} = 0/-8/> + 8r)(a+6-c). 18. [(.r3 + v3)34.2a3(a;3_f.y3j,i.a«J-6«a 14. (a;2-l().^-12)(a;2-10j; + 8). 15. (a;2- 14.^4-40 -30)(.*;3-14j^+40-f-5) = (a;2_l4a;-f-10)(a;2-14a; + 45) = (a'2-14a;+10)(.c-9)(a;-5). 16. /(.c2_a;y+y3)+;U7y[(.ir=J_a;y-f-^2-ar^) 17. ('22_l)(g3_2) = (2+l)(2-l)(z2_2); (a;2 - 3)(u:'-^ + l) ; {Sx^-\-oy''){3x^ - '2.y»). 18. (c"»+2)(6''«-l); (^••■i-2)(.c3 + l) 19. (a;"'-a^")(a;'"4-62/"»). Exercise xxvii. 1. (x — hi/)(hx — y). 2. (3a;+/v/)(2a;-i/) = 3(3r+2?/)(2aj-y). 3. (5G.c-2()//)(a;-?/) = 4(14aj-5//)(a;-y) 4. {5Cyx-207/){x-y) = 4:{Ux-5y){x-y). 5. (56a;-4)(a;-20?/). 6. {2Sx-20>/){2x-y) = mx-5y){2x-y). 7. (28.A:+//)(2a;- 20//) = 2(28a;+2/)(a;- lOy). 8. (28rc-10?/)(2a; + 2^) = 4(14aj-5:v)(a;+?/). 9. (8.r-5?/)(7a;-4?/). 10. (8.tT4-5,vf7aT-4//). 11. (3a;+y)(2;c-%) = 2(3:i;+?/)(a;-3i/) 12. Cdx-2y){2x+3y). 13. (28.c + v/)(2a;4-20z/) = 2(28.c+»/)(a;+10?/). CHAPTER THKUiS. 57 U. (56.r - 10ij)(x~2}f) = 2(2Hx - r>>/){x - 2//). 15. (28.i;+r)/yj(2-r-4//) ^ 2(2y.c + 0^j(u; - 2v/). 10. {i)iyx — 6y)[x — ii/). 17. (H.c-2//)(7a;-10//) = 2(-la;-2/)(7a;- 10//), 18. {r){ix+hj){x~5i/)=4:{nx-^i/){x-oy). 19. (y./; + a//)('4uj-r)/y) = 3(au;4-//j(lx-o//). 20. (tiu;4-5i/)(yu;— 8//). Exercise xxviii. 1. (5x-7){2x+S). 8. (6.B-3)(2.c4-7). 5. (4r^ + l)(3a-2). 7. (8u;-H7)(4a;+3). 2. (5a;-f-3)f2.r -7). 4. (2j;-5)(8.« -11). G. (3a;-7)(4.i;-8). 8. (oa3-4//J)(8r<3+,y,3). 9. (4^+l)(a.«-l). 11. (2aj-f-i3//)(2.f-f//). 13. {3x^+7y^){'2x'i-5y''). 8. (5rt»-4//3)(3a3 + r>68). 10. 8/y2(^--?/)(3x + 2//). 12. a-2(36 + a;)(26-3.c). 14. (2a;»-9)(a;3+5). 15. (4a;3-?/2)(.fa-97/«) =. (2a; + ^/)(2;c - 2/)(.«- 3^)(a; + 3y). IG. Here we have exactly the 15th with x+2 for aj. .'. the factors will be found by changing x into a; +2 in the factors of the 15th. {2x-{-4:-\'ij){2x+4:-7/){x-{-2-Sy){x-{-2-{-3i/), 17. Let2 = (2a;+8y) and a = (3a;-2?/). It becomes Gg^ + s^z-Ga- = (3a-2a)(2«4-?a). ;. Ans. (6x-{-dy — 6x-{-4:y){ix+Qy+dx-iJy) = 13//xl3a;=169a;?/. 18. Making same substitutions as in the 17th, ez*-{-rjz^a^~6a* = (3z2 -2.<2)(2z2 4.3,^3) = (12u;2 +36.r^+27//2 -18x'^ +24.C?/ - 8//2) X (8a;2 +24a;y +18i/3+27a;3 - 3Qxy+12y^) = (19//3+G0^-6a;2)(35u;2 -12xy+S0y^), m '(\ 18 rnNPTFR TIIRKR. V 1^' f (J y I 19. Leta = (a;-4-.'/^4-JV/). h = (x^ -xif ■]-!/''), :. 6a« + 13a/> - HH6/>a = [a 7/>)(5J ) = (7^J + H6)(3rt - 26) = (10^3^.8x//+20v/3)(^3 + 10.<;// + 2//2) Exercise xxix., page 73. 1. (7^4.6,/+8)(u;-//-2). 2. (5j;-5»/-22)(4a;+y+4). 8. (3.c»-}-l//3 + l3)(.«3-//3-l.)4. (4^+5//)(5.K-42/ + 7). 6. (9.c+8//-20)(8a;-//-l). 0. (.c+3i/)(.c-47/-5). 7. (4.c+3//-2)(2.c+3//4-2). B, (3a;-2^-2/- t-5«3)(2^2_|.5,^3^5). 10. (15u;='4-8?/3 j.5z3)(^2_2^3+322). 11. {2a-5b-lc){'la-^Sb+3c). 12. Expression = 0*3 _/,3_c3)3_ 452^3 = (a3 _ 68 - c2 + 2k)(aa _ js - c^ - 26c) = {a2-(6-c)2}{a2 (6+c)3j- = (a - 6+c)(rt+6 — c)(rt+/> +()(a — 6 - c). Hints and Solutions. Most of these examples can be factored by the method of Art. XVII ; but a few solutions by the method of next article may be given. 1. {lx + 6y){x-y); {7x+S){x-2) ; (-y -z){6y+S) lx-\-6y, 7a;4-B, 61/4-8, x—y, x — z, -y — z. Hence the factors are, &c. 8. Factoring the partial products as in Art. XVIII, and arrang- ing in two lines ; ■t CHAPTEU THREE. 50 ,3 4,'/- +13 8^2 + IM, a;"- y\ x^ - 1, - if' - 1. Hence the factors are tS^^-t-4//-4-13 "-"^^ ^^— //*"' — 1. Note. — In oases whore numerical terms appear, connidor these terms as involving z^, 5. (9.c+8//)(8aj-y); (9x-20){8x-l) ; (8// -20)(-»/~ 1), and arranging in two lines as before. Qx+Sy, 0a;-20, 8// -20 8x~ y, Boj — 1 , — y ~ 1 /. the factors are, «&c. ' 9. Omit terms involving z, Qx* -lOij^ ■^Ux'^y^ ^lOi/^ -15x^ Factoring the first three terms we have i^x^ — 2//^ 2a;2 + 5/y3 ; but the last two terms show that the third term of secoiid factor must bu — 5, hence, &c. Exercise xxx., page 75. 1. Quantity = x*+lx^ + */ - V = (-^^ + 'iV - V . which gives the factors a;3 + 1 ±.^^/5; 2x^+^±W^' 2. x^+ly^±y^i/5, as in last Ex. ; quantity = j'^{36x^ -\-Q0x^!j^ + 25!/'^ - ISy*) = j\{{6x^-{-5y^P-lSy^}, which gives the factors 3. Quantity =i(16.c4 ^40x-3+25 -13), which gives i(4c3 +5± j/13) ; writing k for aj-j-y, the quantity =3 3k*-^5k^z'-{-!i^, which factor as in second part of Ex. 2 above, and restore the value of k and factors are 4. (a;2 + i2/2)(a;2 + 6^7/2); (a;2 + yy/'-')(a;3 + 3y2). 6. (2.^2 ^4^y2)(2a;2 + 1^3); x{^a^h)2 + 5c ±i/13}. 6. Multiply by 4 X 3, quantity = V<^(3Gu;4 + 96.^2^2 .{.55^4) = V,(6a:2 + 5y2)(6;c2 4-112/2); )li ■' itfl ! 60 OH AFTER THREE, V 1:1 %t:" ■ »r til ■•■'!!. ! 1 •>- ■ ( I i '! Second part deiived from the first by putting ,v = 1, .-. factors are (B.r- 4-5)((;a;2 -f 1 J ). , 7. \{5x'-hlO±Si/10); (2^v'-^-f3±2p/2). 8. i^']oni second part of last Ex. we write at once {2{x f//)3 + (3±2i/2>2} ; from first part -^{10;*;a + (iO±8/10)?/2}{10ai« + (20-Oy^lO)//2}. 0. Quantity =l{dlx*-hl26x^ +36) = l{81x'^-\-VA6x^ + 4^-18) = i(9x«+7±i/13). In second put m for y-j-z, and afterwards restore the values ; factors are i{2x^+{6±VU)(y+zy}, 10. I{2x^-h6± a/6). Tn second quantity = l(49;c* + 28()a?3-H4()0-85), which gives |(7a;^4-20± a/85). U. Quantity- ^(]6a;4 + 72ir2v'^+58v/4) = Ki^'>«'^+72.c2//2 +81^4-23^/4), which gives 12. |{7(a-6)2+8r2-fcv/29}; ^!3^<2-|.52 ^g}, 13. i{3i*.'aH-(3 + ]/3)i/2[ ; in this write a + b for «, a- J for y .-. i{3(^ + 03+(3±v/3)(a-6)2f. 14. {7a^-i-[6±\/U)b^\; {5m^ +9n^){om^ +3n^). 15. {7(w+u)-'H-(6±|/14)(w -71)3} Exercise xxxi., page 77. 1. {x^±2xy-{-Sy»; (x^+xy-y^): x^±xy+y^). 2. [x^±2xy + %r-\ (4:c2±3a;//+//3); {Ijx"' ±xy^y'^), 8. (a^2-tv2.i- + l); (a;2±i/6a;.y + B^v'); ld=2t/ -4//^). 4. (x'-^d=3a;+l); (a;2± v/C.c+3) ; (^a;2zh2a:2/+,^3. I •vi 6. Expn. = ?/-^ + U;c2//^ + 1 a 1 X*- 1 25 4 a;4 = 7/2 + ya;2±«a;2l-^±^\xiiV'6'd 4:c4 + l-i2.«--^ + l)2-4r-=2a;24.l-i-2a;. 7. a;2"*4-8/y2'"4-4^:'«// aj 2 '" I '0 ,i~'i + 2i/-"'-±2x"'u r"! . 1 } '2' \u-±xus/^. 8. 2.^2-14:2^;; Expu. = - (|^;4 _ o.i,-2,y2 + ;3G^4 _^2^2) - _/ I ,.3 Expn.=u;'* + 2a2u;2//2 4.a4^y4 _2a-'a.'2//2 =a;3 +a3//3±rta;///2 9. w/a;2 -ny^ ±xii v> ; expn. =u;4"'4.2 . 2'"-'i/2"'-2.22"'-J2/2"* 10. Expn. = (4a;» -3)2 -a;2 = 4a;2 -3±a;; 2^2 -2±2a;|,/2 ; Expn. = +(9.^;4- 12x2^/3 ^^^4)=:: - (3.^2 - - 2i/2 +^;^ ;(;J^2 _ 2,y2 _ r^yy 11. (2.< 3 f Ixy - 3//2) ; a;2 ±2.^ + 5). 12. a4+64^(rt+/;)4 = (a2 4-62,3._,,2/,2_|.(,^,4./,)i_,^253 _(fl2 _^„/, + /;2)(,.2 _ab + h^) + {a^-\-%ab^b^){a^+ah-\'b'') c (,(2 4.a6-|-&2 ),2r*« 4-2.^/> + 2/^2^ If 6 = 1 we get the second part 2(a2-|-rv+l)2. 13. {(x-^y)^ ^^(x-]-y)z^z^}{{x + yY -'6{_x-^y)z-\-z'''\. 14. Expn. = {a 4 /O* +7'-2(./ +Aj2 ^ ^^.4 ._ 4,5^4 15. {4a2 + 5a(6-r') + 2(5-c)2}{4^/»-o«(o-c) + 2(&-c)2} 16. {2(a+'>)2±3(rt^-/>2)-3(«-/>)2}» 17 18. 19. 20. 4(a2 + 5ab-'lb^){b^-\-^ab - 2««). From 13th, {(a;2 +y2 -a://)2 -|-,Si>2+//2 _a;_yUa:+,y) +ra-+j/)2 } From 14th, {d'^ -\-ah + h'^)->r\{a-h)-^ ±'^{a- h)-i^'r^\, 4a,2±2a+l) ; (.c2±7.c + 4). (a;2zt9a;yH-9/y2) ; (l±32+5z2). 1 V{ I, il- PI "ft (32 CHAPTER TimEE, = (0^2 _4,._^.2)(2.c-'-4x + 6) = 4(8x'2-2a;+ 1)^x3 ~2x + 3). Exercise xxxii., page 7H. ± 2f.«--'4-3)(.r2+.f-:}). 3. ix^' -lri)ix^ + -3x~-^) :=(x'^ +4){x-i-i){x-l). 4. (x^ -4){3x^+.r.+ i'2) = {x+2)(x-2){3x-'^ -i-x+m. 5. (u;2-3)(5^'+4,/;+lo). 6. (a;2 +0)(10.c2 4.5x-- 60). 9. (5u;2-8)(7.1x2 .-(;.-c-12). Expression = l^(25./;4 — 04 1 - C)x{^)X ■ 8V 10. 7(9^*--lG)-18^<3^3-4) = (3.c3-4)(21.^'9 -13.r-28). 11. 10(81a;4-i) + ^a;^^.^2 + i).:(18.c^-f 1) {:^x'^ + 'l^.c+(^). 12 2(121.^4 -l)-3.c(lla;2 + l,' = (na;3 4-l)(22a:S-3a;-2). 13. (ia;3-|)(^.c-J + i^+^). 14. S{;x^ -2u'^){l{)x^ - 4:X!/ + '10f/-). 15. (2.c2 -5//2)(12.c-' - 6x1/ + 30/ =* i. 10. (u;3-16^2)f2^2 + ^^_,y + 32//-) 17. (.e3_.n)(ii^2+i()^4.Y>). 18. 10(a;2 + 2)(4.c-+3.c-8). 10. {x^ -6^M(13a;--12.c//-f-7S//a). 20. [x^ + 4 // 3 ) (3a;3 + 3.c.y - 1 2y/ -' ) . 2i. [x'^-dy^){5x''-rix>/ + l5y-). 22. 2(a;2 - 2//-')(2.c-^ - 7xy+2y'^ ). 28. {x''+y^){x'+SQxij-l7f^}, 24. (a:a~6y2)(2a;^_icz/+12^^) CHAPTEn THBEE. 63 28). Exercise xxxiii., page 79. 1. (a-2+27)2-6^^r2+27)-27.v'2 = (a;2 + 27)2-6a;(^2+27) + 9a;2-36u;3 = (a;2 +3a;+27)(^2 _9^_,.27). 2. a;4+8a;3-ya;2+2x'(^2^4) = {(a;2-f4)2 + 2u;(a;2 4-4)4-a;2} -6x^=.v^-hx{l± s/3) + 4. 8. {a:3+l + i(l±i/5)^}. 5. {2x3_^.2-3a;zt-c/(2H)} 7. (4x2_2)(4u;2-n.r-2). 9. (x3+7a;-2)(; 0. {.c^-^\o.r-rj)(x-' -x-Ty). 8. (a;2 + 8x4-4)(a;2-3a;+4). X' X -_ '^ ). 10 (x^-i- 5xfj + 3// 2 ) (a;2 - xij 4- 3//2) (a'24-10a;-l)(iC^4-2.«-l). 12. (.^2 ^7xj/ + i/''){x^ - 3a://+y2). 13. (2x2-o//2)-^+2a;/y(x'^ -5?/2)+x2y2_4Ga;2^2 = {2x^ -{-xy - 5?/2 db.^//i/46.) 14. {x^-{-lx>j-y-){x'' -xy-y^). U. (a;''+2//2)2+0.i-?/(a;2+2//2)+9a;2i/2_3a;2^2 = (x2f2//2+3a;?/±a;//i/3). 10. (3a;2 + 10a:.v- 'ly^)(^x^ - ixy - 2v^). 17 j\{l21xA+4:S^x^y^+^84y^-^h)xff{nx^ + 22i/'') ^4:6^\x-^y^\ =^\{llx^+^l'2,y-^-^Dxy±-i^xy , 11}. Exercise xxxiv., page 78. 1. {y-z){x^-tj). "A. ax{hy-\-c) + {by — c)(-'y-\-c) = [hy-\-c)[ax-\-hy -c\ 8. {z^-Jra){x-\-n){x-a). 4. {2x-a){x-'lb). 5. (^+8a)(ic+2/.), 6. (x-62j(a;_a)(^ + a). 7. («-6)(x-|-/\x-a)(x« + aa;+a2). ^13 I! it I liM >' H t' .1 64 CHAPTER THREE. 8. {2x + Sn){ix + 5h). 9. {a+hx){a-hx-\-cx^). 10. {a + bx){a- ox) i- cx^{a + l'X) = {a - bx){a + bx+cx^)' 11. Group the terms coutainiug a, also those contaiciug b. .'. {i(x — d)\J>x'^-{-vx-j'). 12. {p.c-q){x- ~X-l). 13. If written a^ +ab-\-''lac-1ir- ^-^hc -^c^ it becomes (r<-6-6')(tf4-2/>-f ac). 14. x^-^iy^-\-x+ax^-\-ax+a = {x + a){x'^+x + l)* 15. ( /;/.c — 7i) ( ^j.f - -f (/.« — r ) . IG, ^x3 -/;^- _(,j5-|-k') -a(a;2 - /^a; - «a; + 6c) = {x — a\x~b){x — c) 17. .^:='-(/> + ''u;"+/^(v;+rtu:2-nr(/;4-r),c4-re6(; Group terms as in last examj.de. 18. .-. {x+a){x-\-b)ix-cy ID. a2 (.x'3 _ aa;-// - ./;// + (Hj - +zix^ — ox^ij —xy-\-ay^) =:{a--k-z)[x'^ -ax^y-xii-^m/^) = {a'-\-z){x-ay){x^—y), 20. ahx{ax-^bij}-\-cdyi Tx~\-by i — ej'z[ax-{-by) • (t//^.C + cdy — efz ) (dX -f 'ry ) . 21. ax{i(X^ - bx-{'f) + r{(ix^ — hx-\-c) = la.v.-\-c)[nx^ — bx+c). 22. mx{x''^ - ^^ ) — "'.'/(.<•'- - //^ y + '"^i-*^ — .cjc ¥ a) -\-m{'tx — bcx-\-a) — {m.x-\-n){cix—hcx-\-a). 25. c^^ft/'b'-^ —b-xy -a-yz4-xirz) —xz[a"b^ — b^xy - a^ yz-\-xy-z) - {c^ —xz)(b^ —ya){a^ — xy). 56. Arrange in three groups, terms in m-^, terms in a, and remaining ones. x^ {x^ - (n-n^)} -m^x^ {x^ -n-7i^} •^a(x^ -71-71^) f {x^ — m^x^ — a){x 2_^_„a ')• CHAPTER THREE. 65 liug b. les c-h){x-c) rz). "X-\-cl). in a, and 27. {l+x-x^)-r,x{l-fx-x''i)-{-hx'[l+x-x-) -cx^{l+x-x-} = (l+x- x''){l -ax + bx-^ -rx^). 28. Group terms in d, .-. ax{a'^x^ — ahxy +acx!/ — bnj^) — dy[a^x^ - abx>/-\-(iexi/ - hcy^) = {ax - dt/){ax — bi/){ax + cy). 29. nix{ni'^px- —ni>x-{-ni''^nx -n^) -\-q{m-px'' — niix-\-tii"nx— n^) = {mx + q){px + n){in^x — n). n^ y^i'p^x^ + //^x- -hq-xy -\- )(r,+6). 9. (x2-2/2)(a;-'+2a;?/+2/2) = {;c-//)(x' + v/)3. 10. {x-y-hl){x'-^+xy+y^). 11. (/>-2.c)(2 + &a;). 12. a;3 - 1 + 3(u;3 - 1) = (aJ - 1 )(^' +4.x + ^) = {x- l){x+2)^. 13. {p-q)(p-^-2qn r-:(a-l)(rt'-^+2a + 2). 15. 3^364 - 2ab^ - 1 = 3fl264 - 8^62 +.,/ 2 _ 1 = dab^{ah'-i -l)+{^,b-^ -l)^{ab^ -l){'dah'^-irl). 16. //(//3-l)-20/-l) = (//-l)iv/3+//-2) = (//-l)3(,v + 2). 17. 2^/3_rt2/^_^//2^.2/,3^2(rt3-j-/>3^_a/>^a+^) = (a4-6)(2(r-'-3«/> + 2/'>--'). 18. 63'"-i+62"»-l = (6«— l)(63m_|_2/,m_|_2,. 19. 7/3" _2?/2"2'»^-2//"2!2" + 23" ^ (y»4.2H)(7/3«_3yn2,n_,_g3»). I .1- §:; ^ 66 CHAPTER THREE. 20. a{a^~b^)-Qb^{a-b) = {a~b){a^-{-ab--Sb^). 21. ft'X«'"-^'0 - 2r"(a"'-c") = K-c")(a'«-2c»). 22. (^.c-6H;«3_^a;-/>). 28. 35.>;'-"' + b')rt2a;''-2L/3a;"-9rt* = 56"(7a;'*4-3a2)-3a* 24. rt3/,2_(/,^;_^;a)2 = {ab-\-bc - m)(rt6— 66'+c'a). 25. (?»2 - /;2 ) (ft _ ,/t) = (,y?, - 6 ) (»/ + />)('« " ^«) • 26. (i-3a3)(l-9a3) = (i-3a2)(l-3a)(l + 3rt}, 27. (x-yy^-[x~!/)^z + {x-u)z-z^ ^{x—y)^{x-!/-z)+z{x-i/-y) = {x-y--z)(x^-2xii-\-!j^-\-z) 28. 6?/i(4m2 + /i2)_7H^4,;,,3 4.,i3) = (6;ri-7w)(4w2 + «3). 29. a;"'(a;" + /rj+<(.^" +//'") = (■'•'" 4-?/")(^"+r )• 30. xi-{-'lxh/+x^y'^-{a-'x^-[-'2axy^ +y^) = (x^+x!/)^-{nx+y^)^ = {x^-\-xy + ax-by'^}{x'^+xy-ax-y^). Exercise xxxvi., page 84. 1. (x^^y^){x^-\-y^) = {x-y){x+y){x:^ +xy-\-y^){x^ -xy+y^) ; {x-l){x^+X'{-l)\ (.'c + '2)(u.-2-2.c+4); (2(^-3.cj(4rt3 + Gf^x4-9.c2); {2-['ax){i-2ax + a^x^). (3a-4)(9a2 + 12a4-lG) ; [a^ -hi){a^-\.b^) ; {x^-^y){x^+2x^y-{-ix'y^+8x^-y^ + 16y^). 8. (a-&). 4. {x+4.y). x^-y^ {x-y){x+y)(x^- +y^){x^-\-ii^ ) 6. Expn. = = ., ^ x-y x—y / CHAPTKll THREE. 67 ij Ji 1 G. {(3i/2-2;K5)-far2-2v^)}{(3.v--2;^=)2 + ('8v2-2a^3) (3i;2-2//2) + (8. iz/-)n (,,4 _4/,2)(,,4+4/,2)^ (,,2 _ 2/^)^,^,0 _^2/>)(./,4-|-4/)2). 7. (a; - ?/)(^2 +.C// 4- v^ ) ~.^{x + //) (x - //) +y{x - //) 8. 6(.r"5-aa)+nfa;(a^2 „^^J^a^[x-a) = {x-a){b{x- -\- ax + ^/ - ) + ax{x -f ^; ) -j- « 3 1 = {x-a){{a-\-h)x--{-ax{n+h)+a^{,tJ^b)\ = (.1; - a)(a;2-}-«.^ 4- a2)(a-{-A). 10. (a;3 •2/3)(a:4+.cy+.v4) + 2..v/ua4.^...^'>+y4) = {x^ -\-xu-^y^){x^ -a;//+y3)(.r2 -L2.f// - _v^). 11. (m2 -/).)-f-26rO{(«^-'>^-)^-2/>r(a3-^c) + 4i-'c2} = 12. (a;-«)3+63=:(^-ri + /;){(a;-r*)3_(a;_r/)/)-f^;^}. 13. (x+2//)(:c2 _2.c?/ + 4//'-^) + 4a:/y(;c- -2:c,y-f.47/-^) = (a;3—2a^/y + 4//-')f>c + 2^+4.7/). U. 8.^3+27//3 0.i-//(2a;+3//) = (2a;+3//)(4.c- -6.^^+9^-' -6a;_., = (2a;+3?/)(2a;-8//)2. 15. l-2.«+4u;3(l-2a;) = (l-2.r)(l+4a;-'). 16. The exiDressiou a G-],Gc6 (a3^h^c.3)(a^J{-h^c3] a -be (I — be z=z{a^+abc-[-b^c-){a + bc){a''~abc-\-0"c^). I 4 ■At M ■m ^ ^ 68 CHAPTER THREE. l! t|!? M vfi ! U n 4, h 1^ . ■ ^ 'I:;. • ;* ^■i':r . 1 Mwv Exercise xxxvii., page 89. I. Putting x-^y = the expression vanishes. Put x=l, // = !, 2= 1» .•. 2-i = 8/n .•. '/' = 3. .-. Expression = 3(.t; + '//)(?/ + 2;) (2 -hic). i2. hc{h-c)-^ca(c-a) + (tl){a-b)=z{a-l))(b-c){r/-c). . 8. Proceea as in 1 above, 3 (^^^ _6-)(6- -o2)(c-3 -a^). 'i. (^• + Z/)(:/+2)(z + .c). 5. 3(rt4-/>)(6 + c)(r+a). 6. See Hand-book, P::x. 5. {a + h+o){a - h){b-c){c-a), 7. (a + &)(/;4-'-)(^." + ^0- 8. Put c-/j2^0, .-. a^(c~fr-)+h^(a-h*)-{^h^b-a^) +ab^{ab^ - 1) =aA3 _67+fe7 -«26«+ft-/>a _rt/>3 = . . c — //- is a factor, &c., result is (a^ —b){b^ — c)(c^ —a). 9. {a-\-b){b-^<-)[c + a). 10. Expu. is of three dimensions and vanishes when a — 5 = 0, .-. {a—b)ib-c){r — ii). II. {x^+y-{ir+^^){z^+x^), 12. a — h is a factor, .*. b — c, c — a are factors. So that {a — h){b - c){c — a) is a factor : the remaining factor is of tiro dimen- sions and si/miiietrical in a, b, c, and must .*. be of the form m{a^ +b^-\'C~)-{-n{ab-\-bc-{-ca)y where vi, n are numerical, and may be positive, or negative, or zero ; to determine them put c-0, .'. {a — b)'>-\-h'^ -a'' = - ab{a — h) {m{a'^ -\-h^ -\-nab} multiplied b/ some numerical quantity, ■}), suppose, (independent of the letters) to be afterwards determined. Hence -5a6(a-/>){rt3+rt/,4-/,2 -2ab\= - pab[a - b){m{a^ -\-b^)-\-nab) whence a^ -\-b'- —ab = m{a^-\-b^)-\-iiab, .-. m = l,n= ^1 : .*. the quad, factor required is a'^-\'b^-\-c^ — ab—bc—ca» CHAPTEIi THREE. 69 6 = 0, 18. Put a4-6+c = 0, i.e., a-\-h=-c, &c., ;. ~nhc-ahc-ahc -i-(a + /^)3-fc3_3rt/>((,-H/;)= -£ (/;c' — 6'3 + 3^f6f + c3 = o. As in last Example, the other factor is a symmetric quadratic in a, 6, c, and .-. ol the form m{a'^-irh^+c^)^n[ab-^-bc+<:(i). Put c = 0, .-. «i(r<4-/>} + (^^^+^3^ = (ct+/> + 0){m(rta+^8_|.02^ + 7!(rt6-hO+0)} multiplied by coefficient which is independent oiw, n, and to be afterwards determmed: hence ah-\-{a^ -ab-^l/'^) = vi{a^-\-h")-\'nab, .-. Mt= 1, ?^ = 0, and required factor isa^^-/*- +c2. 14. {c-b^){a-c^){b-ai). 15. (x''i -y^){y-^ -z2){u;^ -::^). 16. (a;-|-y + 2)(a;-?/4-3)(^-z4-.<:j(2+^-a'). r/. {a-b){b-c){a-c). 18. By formula [8] expn. = {2(^*4-6+0)} » = 8(a+6 + c)a ; Or substituting a-\-b-\-c = we get the same. 19. Substituting a'^ for b the expression vanishes .-. a^ -/> is a factor. 20. (x+y)'' -x'f -y^ = 7x!/{a'-{-y){x*+2x^l/+'dx'y-'i '^2xy^+y^) = 7xy{x+y){x^-\-.ry-}-y'^r^. 21. Substitute a;^ = 5^- - 6, 5x'-^ -6x- 45;c + 54 + 26..; - 24. Substituting again 25.c - 30 - 6.^ - 45a;+ 54 + 26.c - 24 = 0, x" — 5x-\-6 is a factor. 22. Substitute a^zb-c .-. {b-cy\b+c)-b^+c'^{'lb~c)'Y bc{b—c) = {b2-c'){b-c)-b3+c'\2b~c)+bc{b-c) = 0., .'. <* — 64-c is a factor. 23. Substituting a»=z- 3b 962 + 12a64-96S- 9/^2_i2rt644.3/,«=0, .-. a^-\-Sb is a factor. 24. {a-b){b—c){a-c)(a'^ + b-^+c^-i-ab-{-bc+ca). Note. — The quadratic factoi found as iu 13 above. Exercise xxxviii., page 96. 1. (a-2)(a3_7rt + 2). 2. {x-2,}{x-S){x--4:). 8. (a;-3)(a;-2)3, 4. (.i--2)2(ip+4). 5. {x + l){x^A-2x-\-S). 6. (u,-2 + 2x'+3)(a;3 + 2a; + 3). i'< ■ 1 '■'• :|i sr^r ;• i:. jijjij 70 CHAPTER THREE. i 7. (3r-f2)i'a5-l)9. 9. {m-H){m"~2nm-2v^). 10. None R. ix^'^2x'hH)(x^-.2x + 3). 11. {m-n){ni -2n)' 13. -(m_7t,2(,„s_ •W)? 12. (A + H6')(/>2- 2^^4-18^2), + ??). M. (a + 2M(^-2/>)(a»-7rtA4-468), 16 i;«+2*(:?3+Jla;+l). 15. (a;-r/i(u;-3)3. 17. (^-l)(-l)(;>4-'l). 19. (a-l)2(a + 2)(r7 + 3). 20. (rr"- l}(rr"-2)(a="'-.S). 21. <,2^.[h'i^j„h, 22. (a--6)2(a2-|-2^t/>4-263). 23. (p-2)(;>5J.- 2/5+2). 24. (a:'*-l)(x'-^" + 5a;" + 5). 25. (?/--2)(?/3- 87/^ + 2^+4). 20. None. 27. (r/-/>)(./3 + 2a/) + 3/>3). 28. (a» + i)(2rf2'»_3,v-+2» 20. {x~2){x-'6)[x-ij)[x-l). 30. (a;-?/)(a;--2//)(a;-3//i-' Exercise xxxix., page 100. 1. 2. 3. 4. 6. 6. 7. 2(«-l)(a;2-9a;+10); (a;-2y)-i a; ^2/). (4.i,'+3//)(3a;2-a;//+//2) ; (.t-- l)(4.c- 2)(2.»- + 3). (a;---5a)(3a;2+rt2) ; (2.t' + 3//)(x-' + 3.tf// -T/a). (6-f-^)(6-4c)(2ft2--/>r-+r3); (5rt+-i/>)(3rt^+7a6-363)- (2;;+^)(2/;4-37)(i^2+^8). (10a;-9//)(15.c4-lC//)(a;^ - rrxu+Sy^). {2x- 3y){2x+'6!j){ox + 4//)(3.r - 5?y). (i5.c-22)^:^:i-3a;'// + 3a:^ai- 12^/3). It .4 CHAPTEii iV. 1. Dividend .-. quotient 2. Dividend divisor ; .*. quotient 8. Dividend: .-. quotient 4. Dividend .*. quotient 6. Dividends .*. quotient = 6. Dividends .*. quotients 7. Dividend r- .'. quotieut - 8. Dividend = .*. quotient = 9. Dividends: .•• quotient = 10. Dividend = • •. quotients 11. Dividends .*. quotient = 3i3l«PCise xl., page 108. = (l-a;)-|-a;3(l-a;) = (a;a-l)3. = (l+2a;-3a;a)(l-2aj + 8a;3), = l-2.« + 3a;2. =^(a-x){a-\-x)^. xi-'^-j-l/^+z^-^-xy-^yz—zx, {a + h){Sa + b){2a'^ ~3ab-i-U^), ix^y)i2x+3!/), (7^2 -3a6+263)(8rt2 _ ab-^b^), 7a3-3a64-262. !i" -si ,.|*l •!l 72 CBAPIli^B FOUR. i.( I m I Ml . 12 Dh:(lon(1 = (2-»«4-7rf4-a)(a-7), .*. quotieii' :=ft-l. n. Dividends -(«-/>)(/> -r)(c--rO(« + 6+'-). ,-, .iuotieut= -{o - />)(6-r)((-a), or (^ 15. i)ivideud = (.^;2 4.2->)a_(,/U4.i)a, .*. quotient = .<;'"' +2 ■-^4-//'^ + 1. 16. Dividend = ^jj^^— ax'+ />)i.c -c), .-. quotient = J', X'- —ax-\-b) 17. Dividend = a;'-' +//-. 18. Dividends --:p^ = •- - ^^^^^ .*. quotient = (a;— //)(.*;- 4-//'^ )• 19. Dividend = („a _ ;,a )a _ (^3 ^. i)2^ .*. quotient = a^ -/>-4-c- 4-1. 20. Dividend = i;'r^ -/r"5-c»j((« -264-3c), see Exercise XXXIII. .•. quotient = (»^ -62—c'^. 21. Dividend = 6(fj2-.tr3)+a;(„2_^a), = (x+h)[a — x){a+x}j .*. quotient = 2_^.2(,e+/;^3^ divisor = {ab - ca — bc)[ab - ca 4- ic), .'. quotient = ab — ra — be. 24. Dividends (.f-f/y—l)(a;24-/ya4.1-a:?/4-a;+2/), . • . quotien t = uj - 4-i/' ■^ 4- 1 - .<;// 4- a; + v/ . CHAPTER FOUR. 73 25. Divii\m(\ = (x^—2){.r.^ + l) = {x'^--2){x+l){x''-x-\~l), .'. quotients (a;-'* —2 )(^+ 1). 2(5. Dividcad = (^'-^ -5^ - 7 )(aa 4--'»'' + 3), .-. quotient = //'* -}-5^t-j-8. 27. Dividends {(2.c—?/)a3 - {x-^!f)"), .'. .r2 4-/;2 = 2a6, or a^+b^-2ab = (a — h)'=0. .-. a = 6. .'.the other quantity is same as given quantity. 19. Since x-\-a is a measure, and also x—a, ■—a^+pa^ -f/a+r = 0, (1). and (t^ -\-j)a^ +qa-[-r = 0, Adding, 2/;a2-|-2r = 0, .*. a' ~ - Substituting in (1) ar P P — 1 — qa+r=:0, ar^2)(]a, r=spq. 20. Since 0-2 —2rt.i-+rt2 is a factor of both expressions, there- fore if 2ax — (i^ be substituted for x^ both expressions will vanish Therefore, 'Zux^ - a^x-\- qx-\- r = 0, 4a2x — 2a* — a^x+qx+r = 0, CHAPTER FOUR. fB? also, 4a^x-^n^ -a"j:+nix-{-n = 0. .*. {m — q)x+n ~r = for all values of x, .'. m — q =Uandn — r = U, .-. m = q aud n = r, .'. q^n^ =7n^r^. Note that the expressions ave proved iden- tical. 21. Let x+b bo the other factor of x^ + »?;c+n. Then {x-\-a){x+b) = x^-^mx-\-n x^ + {a-{-b)x + ab = x'-^ +inx-{-n, .*. a-\-b = m; oy b = [m~a) (1). Also, L.C.M of x'^+px-i-q, and x^ +mx-f-ii {x^-{-lJX+q){x + a){x-\-b) ~ x-\-a ^(x^-hpx-{-q)(x + b)=x*-^h,r^-\-px'' + (hp + q)x-{-hq...{2). Also, since x-^-a is a factor ot x'-^ +px-jrq, .'. -~a^—iM(-i-q = Ot .'. q = a^-\-pa hp -\-q — mp — pa-}- a ^ ■\-pa ~a^ -\- mp bq =a[m — a){a^-\-p), ,-. (1) becomes x^-\-{m — a)x'^ -\-j>x^ + {'i^-\-mp))x'\-a[ni-a)(a^ -\-]>) 22. Since x+a is a factor of x^ + qx-\-l and also of x^+px^ +7.C + 1 .-. ./2 -qa-{-l = (1) and— a^+fa^ - qa + 1 = .-. a' ~ { /> — ija^ =0 .-. a = [p-l). Substituting in (1), (!-/>)' -^(l--/>)4-l=0 OY[p-iy'q-{p-i)-{-i=0, 23. From 21, if we let {x-\-b) be the other factor of x^ + mx-\-n. The L.C.M. in this question will be {x^ +px- +'/)(x-+6) = «* + (6 A-p}x^ + Opx^ -i-qx+bq. Also as in 21. b = m — a. ilil '111 Ail 1; I 1 11 Mi-! I ■'K. 76 CHAPTER FOUR. ( I ! i ' i ' ' But since a; + a is a factor of x^ +px- -\-q .'. — ti^ -\-JJa~-\-(]f = 0' Substituting those values for h aud 7. x^ + {l>-\-p)x''^ f hpx- +f]x -\- hi} becomes a;'^ + (»/ ~(i-\-f))x^ +p{m~-a)x^ +a^{a—p)x -{- ti''^ [in — (t){a — p). 24. Since aj --» is a factor of both x^-{-px+l undx^+px"^ -hqx-hl .-. a'^+pa + l-O (1) aud a3-t-j^a--f-(/a-j-l=0. Multiplying (1) by a, a^+/>/^^ 4-c/ = 0. ,-. a{q-l) + l =0, .'. tH)(6 - c)((- - «)(r/ + //)(/; 4-f)(c'+rt). But a — b, b — Cf c — a are fa.ctors of the other expression ; also {a-\-b) is not a factor. .•. also (64-c) aud (c+a) are "^ot. .♦. H. C. F. is {a-b)[l) — c){c — a), 26. a, i3, y, each contain 6 as a factor. .J-, ^, and — ., have 6'^ (V- 52 no common factor, .*. _L ^ ^''' or (J 3 cxfjy 53 a/Sy or _^ tJ3 a/Jy and a6 a bS 6'' f^ 21 or —^—, have no common factor, .-. L.C.M.= L.C.M. of numerators abcS afiy aijy 27. X- ■\'ax-\-b=^{x-\-c)[x+a~c) ) :. L.C.M.= xr-\-a'x + b' = {x+c){x-\-a'-c)]{x-\-c){x + a — c){x-\-af-c). 28. H. C. F. is l-a;. /. L. C. M. = (l+//+3+j/2)(l-//-;^4-//0)(l_a;) = Ul-^.'/2)+(//+^)}{(l+//^)-(^4-^)}(l-^) CHAPTER FOUB, 77 Now substitute for y. xz=7/^+z^- i/^z-, aud the L. C. M. becomes (1-y'^' -z^ +y-^z^){l -!/^ -z^ +y-z') = I 3 \3 29. a;8 + 2a;« +'Sx^ -2x^ + 1^ (.i;^ +0^2 ^i):i _ .^-^ If a;* + «» + 2^ + 1 be a factor of Gx^ ^x"^ + nx^ -Ix^ -2, we see at once that 6x^ must be the other term ; then to get x'' the second term must bo x^ : also there is no x^', .-. the tliird term must be —Gx^ ; also x''^-hl=x^ and —6x^ x2x= —12^;^. to f?et - Ix^ the next term must be ix and the last term —2. ,■, x^ hx^ -Gx^-\-4:X-2 other f;ictor of second expression. H. C. F. = x-t+.'--^ 2x-f 1. 1. Exercise xlii., page 111. x^-lx+Q {x~l){x~(j) x-1 2 x^-2x--Sx-0() ~ {x''\'4:x+lij){x-(j) x"+^x+l(i' Sx!/^-rdxi/-\-Ux a^(3j/zJ)(y_z.2) __ a^(%-7) 2/(72/ -310/^2) - 2/(7^ -3j' , (0-" - (?iK+a2 ) (.r3 ^ax + a^ ) 7V/3-17//3+6//" 3. X- - ax-\-a ^ _ _^±^-^'^^ _ (ag + 4) (a;-3) _ _jH-4 {x + rt) (a;— a) ' 0^3 -"5^";! +7^^.33 '- {x -iyz^x-'d) ~ {x-l)^' a;3_ 3.^4.2 _ (.i--l)(a;-l)(.f+2^ _ ia;-l) (a;+2 ) . a:4 + 2^^ ^9 a;*-4u.-*r^a-'-^-0 2+/>.r (a; — 1 ) ( a; - + oa;+ o ) ic- + 5a;+ 5 (a;-+8)~--la;-_(x--+2a; + 3)(a;2— 2^-f3) ar2"(:«-2)2"~9 ~ (o.-^ ~2:c~f3x^;;^^::ii^ a;3 + 2a;+3 a;- - 2^-3' 2+/;a? 1 "• 26 + (62 -4a;)- 2/'a:-' ~ (l')-2.^•)(2-f/^a;) ~ 6 - 2a; " .'c3 4-2a;-4-2a.' a;(x-' + 2a; + 2) a;24-2a; + 2 «■'' i- 4a; a;{(a;2 + 2)3 _4a;'} " ^x'^ - 2.r + 2j(^-- - 2x- + 2) 1 1 ill ■ y\ :'i^ -j(; \\ • "f* i 78 CHAPTER FOUR. 1; I'll I'ii i 5a»-f 10«4.,.-f Txr Sa^ia-j-x)^ (c^x + ^a 9// 3 ,.-' I -> 'l((.(-''^+. x{a'^-\-j;^+'lJ X' >r x-y + // X i(CX 't-2ro;3('2u2^j4 --- c)[c — «) The numeratoi" vanishe=! when a — 6 = 0. .-. a — 6 is a factor ; n,Ur> /) — (' and c ~ « are factors, and the numerical coeflicient is — 1. .-. numerator = —(a— 6)(6 — 6*)(c — a). ^(^a-h){h-c){e-a) __ _l abc(a-~b){b ~c){c — a) ~ "" ~J)^' (a-^-b + rY'i (a^b + c)^ 9. a^{b-c)~\-b'^ [c - a) + c a ]^a - b) —{a — bj{b - a){c—a ) {a + b -]- r) rt-f /; + c CHAPTER FOUK. 70 10. Let —x — b — c, —i/ = e — a and coDsequently x + y = a—b. Substitute these values in the results of Ex. -1 reierred to, then 2(^- ^xi/-{->r) x{x-hi/)+ir+x^ + [x+i/yy 6xi/{x-^?j) 6x1 i{x -f //) ^xii{;x+y) {a-bY+[b-cy-+[c-a'^) 6(a — b){b—c){c~(i\ 11. (x-^yY' -x''-y^ = 5xy(x + // ) (x - + .r// + ,v 2 , ; {xi-yY -X' - y' =7xy{x^-\-'dx'^y + r)x'^y^ + r:x^l/^-^^xy* + y'^) = "Ixy { {x^ + y'' } + Sxyix^ 4-//^ ) + ^^-i'- //'-' - x -f //) } = 7xy{x-{-y)ix'^+'2xh/-{-3x^~y-^ + L'^;/y ^ +//^) = 7xy{x + // )(.(•-" 4.^y/-^-_v--i )3, (r +.'/)■'- a;* - ?/ ^ _ y.f// (u* + // 1 (x - f .*■ v + /y - ) 12. This may be inferred from Ex. 11, in same manner as Ex. 10, from Ex. 4. Exercise xliii., page 113. l-^{l-^(l-..)[ l-Ui-ii^-x)} Q-3-^{l-x) 4:—x nominator by 6 .-. o-:2 + (l_:^;)*= ^^x' a-\-h a~~b + -z-n Multiplying numerator and denominator by ^2 4/>2 1. multii)lyiug both numerator and de- a-b + rt + 6 (i + b n'-b — a-b a-^b 2^/./; {a + bY-{a-bY ») ^•+// x-y ^ ^ [x + y x-y) '2.r ( 2.r_) iC'3 2a: 2.*; = »; ' ,1 '■X: I: ■■' ^ i. Pi '■! 80 CHAPTER FOUR. 1 1_ 1-a i+ a r (i-^^)ir^ a _ 2a a (I 8. 1 + rt a «(H-a) 1 + 1 , o . o^ = 1 I i I I •' *» 2nd result = a?. ""2<72~ ~ a'+h2 _ 2r/3(a^+/72 ) _ 2a2 ^ 63 aa+/>3 2r/=^ (f/- -6'-^)3 _2r^_ ^+77^ 26 -(.^-+^") 13 aS 2]^ /> + l ^ rt+6 rt — /> c-7i "^ c + d a(r-3 rt(rt 3 -I- 6 3 _ a » 4. /, 3 ) _ ^, (« 3 4. 534^^!_^3 ^ (a + 6Krt^ + 63) (rt+6)(a3"-f63) 2(.,24-a6 + 6g) -2(/6(((2-/;a (cHr6X"34-6s O 4 4.^,9^2 _Li4 a6(a-6)'3~ i' 1 : . f* "% 82 CHAPTER FOUR. lU W I ■ I I . ! \ I M 1 10 -h 1 6 + r - a a 2 -aS "26c {h + r-\-a){h-i~r:\-a) 'Ahr 64-C + 6Tc~~ a [b + c+a){h'^r-~a) {a-i-b-\'C = a 26c 2/- >o 11. The first fraction = ■■ 1 -x 1+. 1 — .<; -»- 1 + 1 l-a l-rc lH-« The second fraction = - (x — a X -f- a x — a + 12. "L j_ iJuJL x-\-a ' x — a\ 4.(1-^ x^__ 1-^ x-\-a\ 3 ■^-'^ {{x ~aY - (a; + 0)^ \ ' ,■3 I. - i + L i + 'l ,4» Cancel numerator of first fraction into numerator of second, &c. Then 1 ~ x-{-i/ x-y 18. The fraction = \ C/ + 6.' \a^b) 26 v3 + 6/ 26 \3 / a + 6 _ \^ / 26 V \a-6 / la -6/ a - /A* CHAPTER FOUR. 83 14. The dividend = x^ — y'' X — '/ xty x+v ! 1 H- - x—y x\y — y x-\-y x-^ry X - y 15. The dividend == 1+. 1-^x-j-x''^ + l-x .1 .' o {l-[-x+x-\i-x + x''^) The divisor = 1+.;; j(i-u;|-a:3) ,*. the quotient = 2 2.3 16 = {l+x+x^){l-X'[-x'') ■ (l+.t;+.i;-)(l -.(.•+;/ 2^ a h \ lb x^ + n[a—b) nib — a) A 4/> ) l-~h 18. =1-1- l^lVia-hx) ya{l-\-c-^\- c) ^{a + hx)- \/[a — hx) \, a{l+c—l~{-c) Exercise xliv., page 117. 1, [x — a) -7- 5. 2. a + h, 3. -i^- X ^=^''^ = lQ>a^x^{a^-x^)\ • a*— a;* /> a 6 a* —a;* 4. = a + 5. = _2- rt-|-6 ' ft — /> 7 a — b + 8 16.C-4 = 0. c-2 "^^ x + 2 ^^ a;3-4 6. ^. 1 I 2a3(a3-a;3) + 2a3(a-+a;2) = 1 ^ (a^-x^). 1 1 1' 4?! 84 CHAPTER FOUB. M I • Gx 8.f-l f 0a;3 - 4^5 + JB + 4x2-1 x{2x-l) ^ x{ix^-l) ^ x,HJx^~l) 4a;'- +2 y. a;(lGa;'^-l) -4 x+2 X 1 + 9 X -6 2(;«4-3) ~ 2(.c-f2)uX- + 3) _ , to this add {x-\-'l){X'\''6y aud result = — u; + 4 2(.f+2)(^^3) — . , to this add ^-^, and results 1 -J- (.f+l)(a; + 2)(:c+3). 2(x + l) 10. First aud second t'nicLio'is= !(,<;- +//-') h- (a;^ — 2/^)» to this add third fraction aud result = IG.t--//- -r- (.j;4 - y/4), to this add last traction, result=--4(;c't + 4a;-//'^+y'*) -^ (^* -^'^j- 1 2. Combine tirsfc and last fractions (of dividend), and take result with second, and that rcsn't with third. .*. IGa'a; -r (a® — a;^). Similarly, divisor -8a^a;- -r- a^—x^y .'. 2/ -f- x. 18, = 1 + 14. 1^(0-6). 16. = 10:.- 7 236 -77a: isaL-c-s)' 15. 15a(8a - x) -f- (ya4-2a;)(a+3.u). 1 .. 17. 2. IB. /(2/"-a;"). (2;c-5)(a;-l) (2a;- 7)(x*-4) 19. First and third combined = {a - b)-''+{)2"4-2. 20? Combine first and second, result= 1 -r- (.t;--fa'-')(cc-+62). Combine this with third, result = 0. 21. Combine first three, result = 2:^;: -, this with last, result = 4CC2 -J- (^.12 .1). 22. -{a^+b^\a^-ah-^h^)-^{a^-h^yaij^ab-^h^). CHAPTER FOUn. 85 Exercis'* xlv., page 121, x{x - 2// ) 3 - //(// — 2^ ) ' 1. = -^ 1^+^)" 2. = rt(a+2/>)3-//(6+2a)3 ("-6) 3 = x-//. (See Exercise IX., Ex. 10). =:^ + ft. 8. L. C. M. of deuominators is {a — h)(b-c)c — a). num. is a^ -b'^, second is 6- -c^, tliird is .-. First result = 0. — ij - 4. = - A + c — c4- n — a +6 = 0. {a - b)[b - c){c. — a) 6. L. C. M. of denominators is {a-\-h){h-\-r){c-^a) \ first i\v.^\- fractions give numerator — (a — />)(/; -f-cjlc-j-'i) + aual - anal : of which a — b is found to bo a factor, .-. b — c, and c— a are factors, and it becomes — (a — /')(/; — c)(c — (/), .*. result = 0. 6. L. C. M. of denominators is {a-\-h){}> — c){c+a){x-\-a){x-^i) (ic + cj, results [{a-^h\[c-{-a)x'^ •\-^l{ab+hc-\-ca)ax — 'la^bc)-r- {a+1}) •\-n){x+a){x-\-h)ix-ir '•). 7. =^{x-y){ii-z){z-x)-^[^x- y){y -z\{z-x) = l. -.a^{h-c)-b^{c-a)-r^{i -h) 8. = = a-f-6-*-c. {a — b){b — c){c — a) 9. = { - bc{h - c) - m(c - a) - rt6(a - 6) } -^ (a - i) (ft - c) (r - o) = 1 . 10. a;3_j,3. 11.0. 12. =—{a-b){b-c){c-a)-r^ {a^b){b-{^c)ir-\-a). 13. Sum of u inner ators = —a^{b — c){x — b){x — c) — a,nn\~VLirAl. which vanishes for a — b, .'. a — b, b — c, c — a are factors, and the other factor is of the form mx^-'rii. See Ex. 2 in Hand-book. ?i = 0, m — l, .'. result is x^ -r-ix — a^x — bYx-c), 14. Numerator = (u; — ?/!(^2/~2)(2 — .<-), .-. result is 1. 15. Numerators { — (a+6; + (6-c)-|-(c-+-«)}=* — 0, b{x-\-a—h)+ax i(5. ab-^{b -a){x — b) lil I . , • U F^ IMAGE EVALUATBON TEST TARGET (MT-3) V. ^ ^K^ /. 4^ ^f^ 1.0 1.1 121 mm U^ m .,. 1^ ^ LS. 12.0 y^ III u 11.6 ^-i VI ^>. ">>• Photographic Sciences Corporation 23 WIST MAIN STRUT WiBSTER.N.Y. 14S80 (716) 873-4503 ^^ ''f CHAPTKR FOUR. Exercise xlvi., page 127. 1. a'* a -h 1 a T c c2 - cd+d^ cd-id^' a' 2. ,.,— ,.. — r/,2 — r^ n a — c n -{■ c 6'-* d'^ l/^-d'^ b h-d b+d '• 62 " \^^^/ xV+Tll ■■ bi~J>-\b-dl ~\b-\-dJ a' U. 1 o — 1 — a^ + c a 12 d A2-f.,/2 T=T = ^ «' ;K 3 2x^ -x^y-\-y'' y/ 2 «2y-H«i/2+2i//' a;' +-+2 y^ y 27 9 22 23* 2 nia — nc—pe mb — »ti — ^>/ w6 /JLl < b —nd-pf mb a wi6 — w(i — ^/ CHAPTER FOUft. (a—mr-^nr) * = J 6 (-7- I— '{b^md+nfi [ -^ ' md r, ~7 4- b — )nil-{-nJ' ■in' J 7. [i-xj [l+jH-x^l = J^jl-^x+x^\ il-x + :t^\ a [l-x-j-x^l \l-i-x^x^j or -^^ b_ a l-x3 ■ 8. ^^1"^.^^ :^ ^^T^) _ iL 2V{a^_x) _ tf + l 2V{a-x) ~ a-i' a+x (rt + l)2 "2 h — a b+a [See (6), p. 122]. a— ii* 2^ 2a 9. wx-^a+b nx+a+c (a- 1)3 (rt + l)2-(a-l)2 - 4a_ (a+l)3 + (a-l)3 - 2(a-^+T) 2a2 nx-b-d " h4-c4-d X — {iix+a-i' c) — (/ix — b — d) a = 1 mx-\-a + b^nx-^ra+c, {m—n)x = c — b. .-. x - n—m 10. ay-^rOx ~ bz+cx ~ ct/-^az ~ a.i--fr-by+cz {u-b) + {b~c) + (c-a)-{-{a-^b-^c) {ay +bx)-\r{t z+cx) + ( cy + az) -j- [ ax + by + cz) a + 6 + ( 1 {a-\-b-j-c){x-i-y-^c) x-^y^z i 'Ki H> 88 CHAPTER FOUR. 4'!!: I 1 .'I m\ 11. Let each of given ratios = m ; then («+6)+ yi£ + '^ = r«{(a-^)+(6-c)+(c-a)} = 0. Clearing of fractions, Sa + 95+ 5c = 0. 12. V^(^) + l/(^'-^) ^ J_ ■ V/'(a) — |/(rt-a;} a 18. Each ratio = l-g 1 + a a l-«\ • l-f-a/ • diffen;nee of numerators difference of denominators ^.2 _ ,y2_j_2.(x — y) x-t/ = a;+.V+2. 14. From the given ratios the value of xy is found to be ?/2+2t/ + l a:+v4-4 j;+?/+2 2 - 1 16. 3.4.^4.4-6 a;-h//+4 2 25:r2-_l_6 __ (5a;4-4H''>«- 4) 10a; + 8 "" 2(5a; + 4) 3(a;2-4) 8(a;+2)(a;-2) -3 5a; — 4 2 3ra;+2) XI/ — 3 2(a;-2) 5a; -4 2(a;-2) . ~ 2 ?(^-j:?) ... 5^.-4 = 3^+6. I- s.. X — - 2a;=10; a: = 5. •" ^ a;+2 16. 46c 2/ = ;, + C' 26 ii 7 2c _ 3 y_^2b 6 + B) = 2. 17. Let each of given ratios =w. Then — -.— =vi{a^^h^) 4 5 = m(62— c») C2 + rt2 — g— =m(c8-a«) a2 + 6a 6a + c8 c2+«2 Multiply through by 60 ; then 15(rt2 + 62) + 12(/;2+c3)4.io(<-2+rt2) = = 25r/«-f27ft8 4-22c2. a^ />2 r2 rt2 4-A2^c2 18. a;2 - 2/2 2/^ ~ ^-^ 2(2 — a-?/ ar^-f-Z/^+a^ —xij — yz - zx a' a^x h^y c^z ° x^ — yz x^ — xyz y^—xijz z^ — xi/z a^x^-h^y-\-c^z a^x-\-h^y-\-c^z x^'-\-y^+z^ — dxyz (x+y-{-z^x^-^y"-^z^ — xy—y^ -zr) a2-|-ftHrc2 a^x-hf>^y-{-r^z x^ +y2 4-52 ^xy — yz-zx~ {x-\-y+z){x^ -\-y'^+z-—xy—yz - zx) ... a2+ft2+C== - ^-^ x-\-y-\-z Cearing effractions a^x+b^y -\- c^z = {a^ -\-b^ + c^)[x -\-y -{-z). 19. Let each ratio = m, then x = w{a •i-b)—c) .*. {a—h)x = m[{a^ —h'^)~{a — b)c} [c — a)z = m{c^—a^ — (c — a)h] ' u i S it' J I 11 V M. ■ Ml;'- < t ll 1^, ;■: t 90 CHAPTER FOUK. .|-©+-S)-/=©|.^|(|V':H«;+^)r=-. 62+(i2+/2 again, tt 21. Let each of given ratios = w. Then bx+aij = m{a-b) &11A bx-hay-^rff-ybz-^-az+cx ^vi(a — b-\-l) — c-\-c —a) = Oy .'. bx-\-a}/-\-cy-^/iz+az-\-cx-^{((x + hi/+<^z) = ax+by'\;'C8f .'. x{'i-\-h-\-c)-h!j(a-\-h+c}-]-z{a-{-b + c)f or {a-{-h + c){x-\-y+z)=: ax 4- by + cz, x^ — 5x^a — 5,ca^ —a^ 22. — i- (ic — (/)3 —2xa{x — a) {x-\-a)^ + 2xa{x+a) x^+x'^'a+xa'-^a^ 2xa{x — a) sum of numerators. 2xa{;x+a) sum of denominators. (ar-a)3 Ix—aS^ x — a ^ Ix-aV^ ~~ {x-\-a)^ \x-^aJ ~ x+a ' ' yx-j-al ~ — ±1, .*. ;c = 0, or a = 0. x — a x-^-a Note. — If the sign before 5xa (in numerator of given fraction) be — , the result is 5. 23. Inverting each fraction, 6(a-f6) 5(6+^^ 10(c-\-a) a -b c — a Let each of these ratios = w ; then C)(a+b)+5(b+c)+10(c+a) CHAPTER FOUR, 24 «*_+2a-.v2+?/3z2 _ 1-7/3 Dividing both by x'^ ■- y^, 1-2-' _ 1_ Clearing of fractions and transposing^, l-x^-y^-z^-2xyz = ^, .: u^'^^,;2 ^^2 _^2xi/z=:l, 25. Let each ratio = m ; then a-\-l)-^ <• = m{x—]i) -^m{y ~ Z) -\-m{z- x) =^ iH\^x — y-\-y -z^z-x) = (i 26. ac Td a . a-irb ^/{ac)-\- \\bd) nr, la-{-}nc-^ne lb-{-md^n/ W l^\ + ,„, II.) + ,. '■(7) lb a H- in lb-\-tnd + nf d {1\ 4- nf /^) !L.^lh^mdJrnf) 91 If » y (Ib + nid-^n/) 2a + 26-c 2b-^^c-a ~ 2c + 2rt-6' then 2<^ + 2^-^ _ 2M-2r-a _ 2cH-2a-6 X ~ y " z _ 2(2a + 26-c) + 2 (26+2^-a)-(2r + 2rt-6) ^ 2;f;H-2//-a 2(2^> + 2c-a ) + 2(2r + 2rt-^ )-(^a+26-c) ~^ 2v + 2«-aj ~~ 2(2c + 2a-6)+2(2a + 26-c')-(26 + 2c~a) .2a-f2a;-// a T ■\f ' hi iV- I* I M ■"» w t I \; . > < ( 1 '"'' 1'' ! 'Ill k t . if ! . 1 -iJkL 92 9ft CHAPTER POUB. dc 9a 2a;+2?/-3 2?/+2«-a? 2z + 2a;- « 2x+2y-z 2fj-\-2z-x z^-2x—7j a 28. Jl = a ItF a — & but 1/ 1/ c2"+t/ jn €)+^k:4-:) ca"_|.^2. ^ 3" i/ r .„(<''"+'<'") = 1 .2" (t.2n_J.(/2») a — & 7^ \ n i = V 29. Let each ratio = r: then X X ■- = r(2/+«) and — {y-r)^r{y^ -z^). y Similarly -r (z—x) - r[z^ ^x^) and — {x- y) X y ^r{x^-y^) .'. -_(y_2)4.-^(2 ^)^. (a._y) a = r(a;2 -22 -f-gs -a;2+aj3 -2/3) = 0. 30. Let each ratio = r ; then •a a -j^ = r(ny-rm) .-. -^ (?-«) = r(ny-m)(Z- a:) my m {m-y)=:r{Jz-nx){.a-y) and (n - 2) = r(jnx ~ iy)(»i — «) niXPTKK FOIUl. 08 •• u ('-•^^+.„('"-//H ,(^' .^) z rtlso, .^-'y- =(f/.r- -f/-), If' n it a{t/' - (t) z^ -//+.•' 2 Substitute value of x- + 11- +2- — .<;// — x:: - zif, then {x-irii+z) a- +/A' -fr -')=.,•-• .,•+ /A' y-{-r-V\ &c. •^w 88. — „- = ^Z' V vi^ .r- J2" * ;t ;/-' //» ,♦;• //' /.^ 22 m2 aj' w^ + r:^ + -:r = in-' n- t" rt- h- c^ 8w3 8(,„2^_„2_,.,.2) av -^- i-ij'+z- 84 a 8w ..S» a 3'i _ ..SI 63' but a 3" 63' ,8» ,I3> f/An .,/3' lUfl n' ((^ _ ^w ^_ ^1 /," _ r/» +_/•' ^rj'» — c" +< " \ 3 63' ^" +/ (( 3» 63' 6"^y" 'If '. ti 1^ i 1l .i iK ' il fe-:f 94 rUAPTFK FOUR. ,1 ■ "' I ! 1 > t i i 'Mr '. i (rt* _ r* -f <;*• ) 3 a"n'*e'* - («" - c" + *'») 8 Heijce the equality required. a.ftg -figrtg + ■..(-l)^-»tf„_iq. also siuce = a:u. Similarly ^=,^^,^^,^^ = ,^^,^,^,^^- rt| rf, \/((l^r/3)+^^,|/'((/3^^^)-^- Transposing and dividing by a — c A B C _. _L _1_ 1+^ + 2 + TX72 = 0- l-fa3 ^ 1 + /.2 Secondly, dividing both sides of (1) and (2) by a and c respect- ively, we get B{ a-h) ^ C{c - a) ^^^ B{h-c) ^ A {c-a) CHAPTER FOUR, 95 _ c-ctj C .-i ) /J6 a — c) ac a — c Transposing and dividing by " " we get Aa Bb Cc l4-a» "^ 1 + 63 + 1^.,. = 0. Dividing both numerator and denomiuator by a, h and c, we get A B + a + a b-^ + X' xh S7. .Q • „u — a a' r />3 + /.3 1 = 0. zl X y z xh a- A; '^ L ~ a' ~ xh' - + ^ + -r Tl xh / 9« again, ^ a 2 x^'h^ 62 63 /.2 A2 ^-' - ^3 -'^ • 64 xVi'-i a4 22/2 /« 2\ S U-/i fl3 63 c2 3a4 ^^^ ••• It -^ t -^t) =^b + F + f) Exercise xlvii., page 134. 1 x^-\-ah-(a-\-h)x-c^=x^-\-2px+p^, .'. 2|; = (a+6), p^ = ah—c^, .-. 4/)a = (rt + 6)2, .-. >-f6)3=4(a6-6'aj, or (a-6}2 -f-4c2 = 0. 2. 4(2a;2n) = 64a;» ; .-. w = 8. i( ; It <(•.: ■t '' i t>^ IT I III" 06 CHXPTKR FOUB. H. FiXtrftcfinR squars root (See Ex. 1 in Hanrl Book), we find romaiuder to be -M.f+HO, which miist = 0; .-. x=U). = 4a26y-f(«2-/>2)2 = (a-'-f/;a)J, .-. sq. root = aa+/>a. 5. 4x^ -x) -ix^ + iix"^ Ix^ - 9.x ) ix^-mx+n 4x2-2.cfl (2— m)x'+(w-l) .'. (2 -m)jj -f («—!) = for all values of 4» t .'. 2 — »// = or ///. = 2 and 71— l = Oorw=l. 6. Given expression = a;4 _ 2.t;a 4.4 -4j;3 =a;4 -6.r3+4 = (a;2 -2)2-2a;», which is a perfect square if 2tx^ be added ; also x* — 6x^ + 4 is a square if 5 be added. 7. (x* +mx-h())(x^ -^nx-^S) = .v*-^x^ - If^x^ ~ ix+iQ = x^ + {m-\-H)x^ -hitnn-^Uyx^ -{-{8m+6n)x + iS, .-. m-{-n = lt and 8w/-f-7^*= — 4, .*.?//= —5, n = 6. 8. Extracting the square root as in example 5 above, the remainder is found to be /80-c2\ rx 1+ -6— T + i- f HO- c'- i 64~ in whieh the coefficient of a? = 0, and the the last term = ; these conditions give c= ±12. The roots of the resulting expressions are 2x^ - '6x+l, and ^x^ + a«; - 1. CHAPTER POUR. 9T ii. Exprecsionsa -(„8^/^8)2(,.8^,/8)8^ ... squarerootis (^/. 24- />3)(c'a +i^h^n^ - 4p iucxi-Hhc-\-ir^ 12. Expression = (x^ -^nix-\-p) [r^^-^nx-j-q] — x^-^{m-^n)x^ + (p-^(j+nm)x'^ + {np-\-Nty)x+pq ; equating coeffi- cients, .*. m-\-n =r. —4 (1) p-\-q+mn= — 1 (2) np-\-mq - 16 (3). If ;> and 7 are rational integers, their vaiaes must be, since pq= — 12, 1 and 12, or 2 and 6, or 3 and 4, one positive and the other negative. Now, n 3 and - 4 be substiiiuted for p and q respectively in (2) and (3), the resulting values of m and n, obtained from (3) and (1) are —4, and respectively, and these values satisfy (2), .*. factors are (.7;3 — 4j;+3)(a;2— 4). By similar reasoning other values of p, q, w, n, may be found, x giving {x^-Sx+2){x^ -x — 6) and also («2+a;-2)(a;2-5u;+6). 13. Let x^ -{-pax-\-a^ be the other factor. Then {x'^+pax+a^){x^-^max-\-a^) = x* + {m+p)ax^ + (nip-{-2)a^x^-\-{m-\-p)a^x-^a* .*. »w-f />= —1, and mp-^2 = l or mp= — 1. il !• P.'» w ■I :: pi 'ijN 98 CHAPTER FOUB. AiWp = — 4 m^ — ''Imp-ifp^ = 5 m — y— |/5, .'. m ± V^(5--_l) 2 14. Let it be the square of | x'^ -ric + ^ \. \ 2 a / Then / a;^ + ^ a; -f-l f = x^-+ax^-\-bx^ ■\-cx+d, \ a ^ 4 / a3 2c « = c- .*. — + — - = 6, and -- = rf- a 4 a- .;. 8c = a(46-a2), .-. 64^ = (4A -«3)3. 15. {(aJV'(«)+//i/(^0+'5V(c)}2 = aa;2 -i-6//2 +rz^ -f 2a;?/i/(a6)+2//^|/(6c) H-Sga; v^(ac) .-. 2,/(«6) = ./, 2|/(6c) = ^, 2i/(ac)=/, 4a6 = t/2, 4k = g^ 4ac=/3, 16. Let x-\-p be tiie other taotor, Then {x+p){x^+'Iilx+d^)^x^ -h{'id+p)x'' + {d^+2pd)x+pd^ .'. 2d+p:=a,d^\-2pd = b. pd^=c pa = d{a-2d) =^b-d^)=~ 17. Lei {x^a- ^d)^ = ax^ -Ix^+cx-d. ax^-Sx^f{a?d) + 3xf^{ad-')~d = ax^-bx^+cx-d, .-. 3^(a2rf) = /,, 3if(2^2) = c .-. 27a-d =63, 27ati2=c3 a 6» a c** CHAPTER FOUB, 00 IS. The remainder on dividing is (SoS^dah^ nc)x-\- rf-3a36~2a4, .-. Sa^ +6ah + f^c = (1) and 2rt4-|-3rt2/; =(j, ^2). From (1) (a3-fi)2 = />2_ft. . f^om (2) d = 2a^(a'-ii-2l>) -~o"-h^ -2ac-a^b, .\ c^ ■-bcl = c^-{-2ahr-^n^h^ = {ah j-c)- = (ab-a^ -2nh)2 =a^a^-+h)2, .-. iih^ -ac){c2 -hd) = 4{ac-b^){bd-r^) = 4:(r-{a^-\-h)2{n^ +hy-i=Ar-{a-'+h,^. ad = S(iSb~^2a^ irom (1); -bc = a^h-j.±ih2 horn (2), .'. {ad-bc)^ = {2a^+ia^b+2(d/^)^=4:a^{a^+2^ or 4;>3 _|-27(/ = 0. 20. Let x--\-max + a^ be the other factor, then x^-\-(ix^+a^x^ -\-a^x+a^ = {x^ Jr^utx-{-n^{x^ +wax-\-a^) — x^-\-a{in-{-n)x^ + a - (2 + wn)x'^ ^a^{m +n)x+(( ■* . .*. m-\-n = l, 2 4-mn = l ; from these equations m — 7i=± v 5^ .-. 2w = l±^5, .-. n^-n- 1=0. 2L 'Letx^-\-ax^+bx^+cx-\-d={.v+m)^(x^n)' = .*. 2(m+n) = a, m^7i^-\-4:mn - b, 2imi{m-^n) = c, mH^ = d; .'. {4:b-a^)^ = {i{m+n)^+8mn-i{m + uy\ = 64w2w2 = 64rf. Also (46-a2)a = 16wm(m + w) = 8r. 22. Let X* -f />a;3 + qx^ + rx-{-s =lx^+^ x + y 's f = II f Ml' 411 4-1t .4 .1- 100 CHAriEL FOUR. V 9 q = -.- -|-2|A, y = i>^ = hi/f y --- !>-is. i :' i ■): }\ Ji u '»♦■ ."i 28. Dividing by ax^-^-'lbx+r, the remainder is found to be (--^>^"-© 2c = 0, or itc = l>^. (1) and d — — = 0, ovud — hc (^2). (t From (1) ^/ = i/(^/c') /. ^c')+r = (a;]/a4- \^cX' (l)x(2) .-. h^c^a^cd .'. i3 = a2(/and6 = (/M* « 2f/2 Also — — =z a« /.2r2 ad^=r-^ 1 8 = a' d .'. ax^-\-Sbx^+nrx + d = iu-.^-^Said^.>-'^+'Snhllt'A-d = {a^x+d^) 25. Let ./•+ — be the other factor, then x^ +p.v- -{-(fx-\-r:= {x-^2r^{x + l.) = x^+{i^ 4- ^) .«2 + (/-f 4) x-j-r. p = A + -—', v = /-+4 .-. 4yy=lG-fr = 7 + 12, or4(/9-3)-:r7. ' *- 20. If divisible by iix'-\-2px + g, it is divisible by x^ - ^px+ hf. Let x-j-m be the other factor ; then (.6'-f-w)(a:2 + i|/y.r-|-i 3 x-\- tn~x-\- 3 CHAPTER FOUR. 101 Exercise xlviii., page 137. 1. If an' = hhi=:cc' then Vfi\\{a-b')a'=z{b-a'W {b-c)b''={ci-b')csi.nd {c' -a')r = {ft-c)a' ... (a-b')(h-c)(c'-a') = {b-a')[c'-h'){a-c). Nos. 2 to 7 may be proved in like manner, or thus : 2. Interchange b and b' in 1 and it becomes- U ua' = b 'h = cc' then will (a-b){b'-c){c'-a') = {b'-a'){c'-b){a-c) which is 2 with the members transposed and the factors in differ- ent order. S. Interchange a and a' in 1 and it becomes — If a 'a = bb' = cc' then will {a'-b')(b-c){c'-ii) = (b-a)){c'-b'){a'-c) ft mere variation of 3. 4. Interchange c and c' in 1 and it becomes — If aa' = W = cV; then will (a-b'){b-c'){c-a')=^{b-a'){c-b'){a-r') which differs from 4 only in the order of the factors of the right- hand member. 5. Divide 4 by 2, member by member. {a-]/){bj-c'){c-a') _ {a-c'){l>-n'){c—h') lb-c')(c-a}{a'-b') " {c-b'){b-a){a/--c') Rejecting factors common to both numerator and denominator, {a-b'){c-a') _ {a-c'){b-a') {c^ajia'^b') " {b'-^a){a''-c') from which 5 may be immediately obtained by a transferrence of factors. 0. Divide 4 by 8, member by member. ii ; ! f>i '■i«t' • 4 ■». ' ! 1 :. H M' 102 CHAPTER FOUR. 7. Dlflde 4 by 1> memDer by member. 6 and 7 may also be obtained from 5 by operating with the substitutions {abc\a'b'G') anf. {'ibc\a'b'c')^ respectively. 8. From 1 by actro,! multiplication (ac> 4-a'b' - aa'-b,'c'}{b -c) = {ab -^a'c-aa' -bc)(c' - b') :. a{bc>-cc')+a'{bh' - b'c) -baa'-\-b'cc'-^eaa'-c'bb' = a(bc'-hb')^a'{cc'-b'()-bcc'+b'aa'^chb'-c'aa' :. {a+a'){bb'-cc')-\-{b+b'){cc'-aa')+{c-^c'){aa'-bb') = 0. Hence the equation in 8 is merely another form of that in 1. The equations in 2, 3, and 4 may be formed from that in 1 by interchanging 6 and 6', a and a', and c and c' respectively, hence variations of them may be formed from the equation in 8 (proved to be a variation of that in 1) by the same system of interchanges. But these interchanges have no effect on the equation in 8, i.e., they leave it unchanged, hence the equations in 2, 3 and 4 are merely different forms of that in 8 ; and consequently the equa- tions in 1, 2, 8, 4 and 8 are all merely different lorms of one and the same equation. Also 5, 6, and 7 have been shown to be formed from 1, 2, 3 and 4, and consequently they may all be formed from 8. (The example and exercise of Art. XXXVIII. are important geometrical theorems — See Chasles' Traite' de Geometrie supetieure). ; '4 CHAPTEK V. Exercise xlix., page 138. 1. 5, dha, -3. 2. -4^, -a, 2, 10. 3. a+6, c-a, b-c, 3. 4. -2, 6, -6, 12. 5. - 14, a-Sb, 2a -86, 66 -3a. 6. 7, 4, a, 6. c 5 7. -1-. -, 0, 1. 8 a 9. 6 — a, a+6. 10. -, 11. 26, a. a+i' a-b a3 + 6a 12 a+6, -1 , a+b a—c 14. a + 6, a2-[-a6 + 62. 16. 18. a-t-6 b a » 12' ac' b 20. 10, 12, 4, |. 6c 22. 9/^, ab, a 24. a a — h (a-f6V LO 3 a2-6'2) 10 26. TI~. Tv2 > 2, Q (a+6) — ac Zo« 1^, T ' 30. 12, 1. 1 33. — m 13. ^, 6+c. a — 6 15. a2-a6 + 62, 1. 'e+6 2 3 17 e-b 15 14 a262 6c 21. 1000, I, h 62 6 23. — ' c(a+6), — (a+6). 25. -1, -1. 6 ac 27. «6, --> -r-' 12. 29. 9, 2. 31. 3, 1. 34. 1„ 2a -1 2a'+2' a6 + 6c+ca aa+6c + c2 82. 2a+2'^- 85. lit <&i?.| m l t» I .! 1 %:W f. ' "i!# (IT,; 104 CHAPTER FIVB. 86. cr.2_f.i3^^» 37. {a-\-h-\-c\ 39. 1. 40. 1. 41. 1. 43. IG^. 44. 6. 45. 6. 46. 47. -^. 48. 0. 49. _ 25 T5« 38. 8(l-]{3(8a;-2)-2}-2)-3 = (3{3(8.6'-2)-2[-8 = 0, .-. 3(3a;-2)-S = 0, a- = l. 42. 15. 4fi npqra -{-pqib + *c + iff mnpqr 50. 1. Exercise 1., page 142. 1. 2, 3. 2. i, i. 3. ±2, U. 4. 1, 1^. 6. ±^ =b/'^ + ^), «. 6. 4, 5; 2, 2^. 7. -3 or 2; 4, -3; 2^, -1^. 8. 1 ; f or f ; i or 3. 9. _|. or 3 ; ^ or 6 ; I or -f . 10. - 1, 2, -i, 1. 11. 0, -6,86. 12. «, ±rti/-l. -I-a/6 -1+V5 18. 1, 14. ±a. 15. ±6c, -{b+c). 2 2 16. Factors are {x~~a){x — a — 2h), .♦. fl; = rt + 26. 17. 6 or ±a. 18. a;4-2rt6 is a factor, .*. x=-2ab. The other factor is 3a;2 — 2«6a;- 2a- b^, which gives «= irt6(ldzi/7). 19. a— 6 is a factor, .*. also x—a, b — x are factors; one linear factor remains which must be symmetrical in a, 6, x, and is .*. ;«-f-«+6; .-. x = af b, — (a + 6). 20. Transpose 1 to left-hand side, which then vanishes if X -a = 0, or x — b = ; .". x = a, or b. 21. Left hand = fc!L)('^+l)^ = x-a, which = a;*-a8 .-. a; = a, or 1 — a. 22. =(a;4-«)(a + />)far + 7)-^ + />) = 0, &c. 'I'd. .'. ah{a — b)+bx{h - x) -{-axix - a) — {x- a)(h -x) = Oy .-. x-(i=0, h-x = 0, .'.x = b{l~b)^{l-b+a). rHAPTKR FIVE. 106 '^ 24. (a;-5V.r4-0)t«-7)=.c3-6.'---37.C'^'ilO. 25. (x - n){x - 4n)(x - i)a){x+ia) = x*~U(x^-Vi)a^x-+VAa^x--iH,i^. 26. a;iaj-l)(.i; + 2)(.c-4) = 0. 27. (u;--2u;-l)(.c=^-2a;-2)^-.c4_4;^3^.,.j_^0x+2:-O. Exercise li., page 146. ANSWERS. 1. 4. 2. -V. 8. -107. 4. 8. 5. 3r/. 6. -^Vo- 7. 5fg, 17. 8. 22, 46^. 9. 7, 3. 10. 10, 10, 11. 11. 0orll;33. 12.3956-8971. 13. |(15±,/190). 14.3. 15.3. 16.4. 17.22 4-16. 18 1^. 19.3}. 20.4. 21. ±8. 22. 11. 23. 2 and -l±V-ii. 24. 2^. 25. 0. 26. 3a. 27. f. 2^. |^. 29. 3. 80. 10. 31. 0, 1, or (-5± v'-23)-^8. 82 102». 33. (-11 ±1/4681)4-20. 34. 2, ^, or |. 35. -4. 30. 0, or ±i/{a^+b-). Hints and Solutions. 12.c-f2 1. Combine first and last fractions .*. 13.C - 16 25 18 , &c. 9a;+15 '13 „ 2. As in last Example k _ o" — "i - » &c. 3. Completing divisions .-. X 72 70 = ..4-2' *«• 11 2^-9 33 21 &c. 6. Multiply through by 6, and complete the divisions, Qa x — a 12n x + a = 0, I l«'"i HI ■■m^ 1 -I i m M-" 1 A " . 1 f » , 1 : 1 ' ' ». 1 lOG CHAPTER FIVE. I, [i !1 1 1 • ', 'i 1 ( 1 0. .-. 29 24 6a;+5 " 6a;~2 -^.•. 84=-58(8a;-l), by subtraction. ,,17 16 8 , 15 ^ 4 , a- — 17 a;- 18 2 2a;-i-5 a; - 2 12 13. .-. 5a?- 6 280^^ ' a;-3 289 30 &c. 3a;- 2 6 a;8-5a;+6 a;2-5a;+0 ■" a; -5 2a;4-5 5x 7a;+5 a;2— 6a;H-6 x^ - (,x 48 1 . 24 36 14. — - -L — — a;+l ^ a;4-3 ^• + 1 g , &c. 3 a;+¥ = -^1- = h &c. 15. . = — . (By the usual divisions). •• 5a;-4 7a;- 10 ^ ^ ' \- L 16. a;3-9a;+14 a;3-7a;+6 , .*. &o. 17. a;+2+ -— -}-a;-|-2+-J_ = 2(a;+2).f -- , &c. a;— 5 a; -9 a; -8 18. <- 835 + 12 • 8a; + 12 16a;2-|-4€a; + 35 16a;2 4.48a;+32 , .*. &c. CHAPTEK PIVB. 107 19. 7 + 1 2x-4: '^ 2x-9 ~ 2x-H "^ 2x - ■f or, 4a;- 18 4ic-19 4x^-'I{)X + 36 ~ 4a;8-2Gx + 40 4.^-13 = 0, &c. * ■* 4 "'"14 23a;- "^ i ~ 21 ~ 42 "*" iV 4 ' 14 23a; -0 . 2a;-|-4f £va;-0 • • r r'— I = 7; &C. 21. .-. L_ 4. 1 + 7, - ,',»» or a:^-6 "^ a;3-12 " ^a _ 8 "^ a;a - 10 __2a;3-18 _ 2a;3-18 . .^^..lo-o An a;4-18a;2 + 80 = ;^-r8~^+8' ' * '^'^ ^®-^' *''• oo . « 01 W oa; 2 52 K9. ^ 1 Ji ~ -^ on + . ^ - 5fa; 2i 50 39 "^ L9 ~ 52 = — — 4- —1, .'. &0, 3 26 52 ' 52' 23. Multiply through by 18. 6(1 - 2a;) 9(1 +a;) " a;2-a;+l "•" a;2 + l "^ a'+l ~ x^ + 1 . -6(2a;-l)(a;+l) + 3(a;3-a;+l) 9a;+7 x^ + 1 a;2 + l' •• "-^ -■^^* 24. Completing divisions as usual, -2 1 •*• 2^"^ "^ a;-l -5 a;-4 — 5 + ^ 2 2u;-o , &c. •*• (2a;-7)(a;-l) ~ (a;-4)(2a;-3) a-b ^ a-h (a-h^) 25. .-. 1 - or 2+ 2^(n-x) x~b (a-b)^ I. 1 I. _ _ x — a {x — a){x — b) a-\-x (a-b)^ 2(a -A') {x-a){x—b) {x -a){x — b) a-^x) 2(a+a;) = 2(a-ij;). &c. n ^ II i^v. h% IN r R S«V' \f' I i 1U8 26. CUAPILK I'iVJi:. 1111 27. ••• i;.},^^_o "♦■ 13.i«-lli " 18U'-8 "*■ 13iiB~l() 3J 28. Sum of first two fractions is ., , , , „ .•. u; is a factor, 1 _ 1 _ 16 2('a;-i)a ~ 2(u;^+lj ~ {x-l){x^~+i) X , &c. 2y. |- + 2 - 2i + — = 3 2 -, &c. , , . - , ... 0(3a;-l)(8.+l) 80. Multiply by l and factor, .. x- ir^x__Y)T^^Q\ = ^^' Sic''- 1 ,„ ^ ,. . . 6r) _ , -19+a;; perform divisions .*. "Tq "**' *^' u;+3 81. Perform divisions, .-. 8 a; -1-3 1 5a;+l 2(a;-l) 2(a;-fl) " (a;-!)* x-\-2 5a;+l a;+2 5a; + l :2» 01" - V = : -T .*. a: — 1 = 0. (a;-l): a;-l x-1 2x 20 23 32- T - T-^^-i--^ y+^'0r^^33 ia;-3 23_ 10 21 2 2. 19 1 83. .-. 84. - or 18a; - 22 1 5x 13 -2a; 1 16 8a;-l " 4a; -7 52a; -23 = Y + T = 2 + 5a; 4 50a;- 19 , &c. 12a;2-25a;+7 50a; -19 (3a;-l)(4a;-7) ~ (3a:- 1) (4a; -7) 4a;-7 = 0, &c, •. 8a;- 1=0, CHAPTFB FIVF. 109 5 5 , 4.r+lP) « _6 2a; 4a;+6 4*3 + 82a; + 68 [To .*. 4a;+16 = 0, &o. 2.r 86. Sum of first and last fractions = ^..j n,-¥h^)' ^"'^ "^ 2.C ) f 1 1 second and third = ^2 _(«_/,)2' •'• 2^' ]x'^'"r(;i-f '/>)-^ +.,-' - (rt-7>)^ ) =0, .-. a; = 0, or2a;'^ = («+/>)''^+(a-6)3-:2(ua+62j, &c. Exercise lit., page 150. .2a; 1-a ..v . 2a; w + 1 ^ ^ 2 1+a 2a wi-1 (c) .-. 2ax m-\-n 26 ~ w — n 2, (a) (a -6). (6) 0. (c) a a — x b-x ~ a—b ~ .'. a; = 0. 4. (a) .-. (a;-1) 3.6; !!;^;A. rt+6 „ . i_^ =0, .'. .r = l. (6) - 1. (c) _ — = = —, • ' ^ - ^' ^ b—x b-\-x 5. .*. 2a; +8 2.r+3 1-, .-.3;= -I, or -1. 6. /. 2a;3"-7x'+3 a;3-9a;+2 2^a; _ (6-6')2 + (6 + «^)3 ^^^^^ {c-h){h'^ +c^) 2(6 -c) ~ {b-cY-{b+cY' '^cibc 7. Add and subtract, .-. l'^}r'^'^\ = ''-"^^ ; divide each side 2A/(a;-i/) a;-2/ by V(^+y) ... ^y ^1. ^{x-y) " x-y i 'I < I .til" Hi h •.p no CHAPTER FIVE. - f ■ F H .'. x= 14. .-. 0^ = 27, a; = 4i. 8. ('* nx^ + l n nx .-. a;=l-^H(w-l). 25. (rt). Each fraction = difference of numerators divided by difference of denominators = 5, .-. a; = 4. V**)- ^2x + 9 #'2u;+15 6 ^' a y^x-^2a _ \/x+n 26. (a). Each fraction = -^ = \/x-{-h ~ i/x 1^3^. ^-^^2 .-.. = 8. ... X = a2fc8-4-(a-6)3. (6). 1 = 2 27. X 4a /« « + ^ a (1 + a) < ■i I I ' ,'i m r. m. I ■•y FW^ ■ > ■ ! 1 *, -1 ' I* 1 M i! f! ■i- 1 i ' V % ■>' \ Jl- ■1' ; ♦ i 112 CHAPTER FIVE. usual 28. Divide numerator and denominator by ^/(c/jr + l), then as ]/{flx) + l) 6 + 1 29. As in 28, 1 -?- v/{l - N/(l-aj)} = (l+a)-J-(l-«), &c. 80. Square both sides, then a^-i-{x^ + ^a:c) = 46-H(/!) - 1)2, &c. 31. Proceeding by addition and subtraction we find x-\-l x-l = &c (a; + l)«-^(.c-l)« = (a« + 5a4 + 20(<3-f.2)-r-(5rt*-a«), or Exercise liii., page 153. 1. We have the identity a;+4— (u;-3) = 7, divide this, member by member, by given equation, .•. x = 8. 2. Transpose and square ic=0, which satisfies the equation 3. x = '>}. 4. i/x{Vi}i~-\/n) = m--n, .♦. x={Vm-hi/n)^. 5. \/bx — y^x= — V{ab-\-bx) sq^imve, ^., x = nb-ir(l — 2y^b). 6. x = 7. x- ^^_2. 8. x = 18962 12393* 9. l/a 10. (i-ha;)'"^ — (62 + a;-) = 26a; identically, divide this, member by member, as in 1 above, and add, 2(6 fa;) = 'ihx 2bc^ ^3 +c -'-^-202 -2b' 1 1 . ft" = -i-. 12. 2a; — (2x — 27«) = 27tt identically, then proceed as in example 1 above .r = 18a. 13. Cube by formula [VI] 1 -x, + l-{-x-\-S if (l-a;2) x ^/3 (by substituting cube root of 3 for its value) = 3, .*. x^ =|y. 14. Cube and substitute as in last, .*. 3f/7^{d—x^) = l, .'. •C- It 3 V 2T* 4 15. Proceed as in last two questions, .'.«=+ — v^ — 11. 16. a: = ±l/{«'-^^2^T 17. a: = 0. 18. x = ^j,a. CHAPTER FIVE. 113 ;«t;»- 19. Proceed as? in example 3, page 152, (c—a—b)^ = '2>7abc. 20. .-. xV[a^-hx^) = i7i-l)(t2-x^,Rndix'^2n-l) = {n-l)^a\ &c. 21. 16a;j/ = (n-4;c-?/)2. 22. 1 + x— {l-\-x-^-[/ (l—x)} =1/ {1—x) identically; divide member by member, and add given equation, .-. 2|/(lH-a;) = ^/(l-a;)-l, &c.,a; = 0, -|i. 23. i/^x is a factor, .•. x = 0; dividing, .•. v^(|/a;+l) — 1/ ( \ ;c-l) = «-5-|/(v'u; + l), and y'x-\-l — (i/x—l) = '2, identically, 2y{Vx+l) .-. V{^'i' + l)-\- V'iv'x-l)^ a :.2V(Vx-\-l) = ^V()/x+l) a + 777T^)---(2«-%'(V-+l) = a '• 2 y{Vx- a^^i/{yx+l)y &c., x= {20^'^ ~ ^j ' 21. l4-a;-f a;2 — (l- a; 4-aJ-) = -^', identically, divide, &c., 2 l-m2 2 v/( 1 +i» + a;3) = — H- wx ; ic = — \/ , „ " 2 25. a2-ic2-a;2(a2_i) = (,2^1_^2j^ identically divide, &c., /. 2v (a2-a;'-^J-a;/(a3-l) = li I ...a;= '*-r-- {a« + 2±v/(a3+l)}. \/bx-\-c 26. Eeduce first fraction, .*. ^/ix•+c= — o, 27. 2ic2_f.5-(2a;2-5) = 10 identically, then as in former examples. 2y(2a;2 4-5) = 2>/15, x=±5. 28. 3.r3 + 10-(3a;2-10) = 20 identically, .-. 2y(3a;2 + 10) = Vyl7-f/3, &c. 29. 3a;2 4. 9 = (3a;2- 9) = 18 identically, &c., x=±5. 30. 3a-3i+a;'»—(2a-26+.i;-)=a-6 identically, &c. u;=±%/(36-2a). «4l'i'C i ■*y '\ m ll i ! -if 1 , f ' ^ i 1* j, V ' , r ^ * jv ■" \l' ft i ■■' .r ! u, >i- ^1 1 i I 1 t ; '■'■■ r 114 CHAPTER FIVE. 81. 4a»-852-2a;2-(3rt3_363-aj2) = a*-a;2 identically, &o. 32. Cube and substitute as in example 21 above, {2i/-\-2z - 2a;)3 + 216a;//2 = 0. 33. Eqnation is >/(a+a;) + N/(a-a;) = 2.r-hi/{a+y(a»+a;«)} ; we have identically a+o;— (a— ic)=2a;, divide, &c.,« = fav'G. 34. Eomove the factor ^a;, and reduce the fraction, .*. aj = 0. A.lso, ;u + 2rt — (a3— 2a) = 4a identically, divide as in former cases and clear of fractious, .'. 2(x'+2a( = — (a;+2a)+w«; x = a n i 2n-i 36. x = a^+2a ■ 36. (2rt.+a;)2 + ^>3- {(2a -a;)3 + 6'-i}=8rfa; identically, .'. 2x/{(2a + ;i;)2-|-63)} =2« + 4;c, a;=±>/(3a3-h63)-r-v/3. Exercise liv., page 157. 1. Eight-hand member is (x^ -((x-\-a^){x—b) ; divide by com- mon factor and clear of fractions, x^-{'ax-{-a^ =x^ —6*, .*. a; = &c. 2. Right-hand member=(a;— ^)(a;3-|-2aa;-|-2a^) .-. a;2 + 2a'-*-2ry- \-J - 3 '- = a; a; 3- - a (, a (" a \a — bj \ i (rt— 6)2 ) .*. x — ah-^{h — a). 7. Denominator of left-hand member = {x^ — 3flfa;— a^) [^jc^^Zax-a^) ; invert both members, .-. x^ — '6ax-a^ is a factor givinjj; x=\a{'Qa±_y/Vd)\ result of division is x^ — {^a-^l)x = a -, &c. 8. Equation may be arrangerl, multiplying by 2, thus : ( —2. \-x~a = 0, where x — a is a factor, &c. \x-^nj x-^a 9. Eight hand member = aj3 -{a-{-h->tc))x'^-\-{ah-^hc^ca)x-abc .'. x=i{a-\-h-\-c). 10. Transpose negative term in right-hand member and com- 2bc + 2ca + 2a<'-{-a^'\-b2i-c2 _ {a + b + c) - ~ 2 bine the fractions .*. "Aabix abcx = 1, x^l-^abc. 11. Multiply through by abc, then a+b-\-c-{a^+h^ -\-c-)x {2ab-\-2ac-^2ca)xy .'. ((i-\-h+c)^x = a-\-b+c. (t 12. Transpose and combine terms involving — p- ; x = {a-h){ac-2b)-^{a-{-b)ac. 13. a;3 + (6+c)3 + 36(64-('V<.-— 63=0, .-. x-\-h+c-~h is a factor. 14. As in last example f^x-f^a - ^6 is a factor. .'.x=--{fa^-fbY. 15. ll(u;4-16)-fl0x'(a;2-4) = 0, .-. x2-4 = 0, &c. 16. 7- a (a-b) J X- ( ^■' ) a-6| ~(a + 6)2r «-6 ) a — -•f-sil 116 CHAPTER FIVE. ^ Wj! 17. The equation can be put in tlie form a-h I \x (i + b \ \ + rxj 1 — ^^1 „ 2r'.'c3 1 + cx a — h or X- 1+cxi \ i + cx at I is a factor, &c. = i(2a;+rt)(2a;3-2«2_5«a;) .-. 2a;3 -2rt2-6aa; = 0, or 2a;2_2r/2+5^/a:=i(ic+a)3, x = -^a. 1 \ 7;i-3 14 a -h h 19 • x--l\ u;~4 + or x~lOj ~2{a;-4)(a;-10)"-' (a;-4)(a--10) = 7:c3-=-2(a--4)(a;-10), .-. a;2^4. 20. Proceeding as in last example, 8(2:c-10) ar4 . .. . .^ « . V TV/ F\T n\ = ^ tt; r7^ or a;4 = 16, «* — 16 = 0, &c. 21. Add, term by term, the identity used m example 8, page 155, then ^x H- 2a; a-\-c {a~b){c-a) ^ {b-e)[c-a) {a-b){b -c){c-a) .', 2x[b — c-{-a — b) = &G.,x = i{ 1' 22. As in example 4, page 166, left-hand member as 8{x - a){a - b){b — x) = x^ —a^ , a; — a is a factor t Also X = {dab -3b^- a)H- (1 + 3a - 3//). 23. Left-hand member is ^^ — ; \- ' which = 2a .. x = a, (a; — «)^ 24. First member vanishes fora;+rt=xO, a-i-b=0, and aj— 6=0, and becomes 3{x + a}{a-\-b){x—b) = &,c. 25. Left-hand member vanishes for 6 = 0, x-a = 0, and a — 6 = ; numerical coefficient is found to be 6, .*. 66(a; — c!)(a- 6) = (a -6)0-2, a; = (c2-f 6a6)-r-66. CHAPTER FIVE. 117 26. First member vanishes for a— /> = 0, a+b = 0, x -a = 0, and numerical coefficient = 6, .*. 6{a-b){a+b){x-a)z={a'i-h2)c, .-. a; = jl(r-f-6a). 27. Clear of denominators, x{x^ + a^ — {x^ — a^) = a*, .•. x = ha. 28. First number vanishes if x = 0, a = 0, or 6 = 0, .•. dbx is a factor ; one linear factor remains which must be symmetrical in a, b, X, and is .•. x+a-^b ; numerical factor is found to be lii, .-. Vlabx{x+a-\-b)^12ab{x^4'{a-thy^} .-. x = a + b. a b e 29. Arrange thus «s _ be + 62 -ca -IM c*2 — ab ] + , . ,- ,--" +anal.H anal. = ab-\-bc-{-cal ' j be a^ -i-ab-^ca aj ] n — I x/ 1 , "."", \ +anal. -f anal. x{a-{-b-j-c) ab-\-bc-^ca a + + ::t-~: where the a^ —. be b^ - ea c'* — a6j ' quantity within the { } is common to both members of the equa- tion; strike this out, and x = {ab-{-b(;-j- ca)-T-(a-\ b + c). 80. First member vanishes if 6 — a 2 =0. and by symmetry, if a; — 6'-* = 0, or a — a;2 =0, and the numerical factor is found to bei' -1, .-. -(6-a3)(a;-63)(a-ij2)=(rt_ ;c3)(6-a2)(6+/3j, .-. a-a;2=:0, &c. 31. l+ic+oj^isa factor, and .*. ic=— ^ + i\/-3; also from l^x-\-x^ ax-\-l the quotient we get Y-s^^ = ^:ri' 14-^2 add and subtract X = ax, &c. 32. Transpose, square and perform division in left-hand mem- ber then a — b x^'b 1 ia-b\^ a-b\^ + a- a — b c-b a or 4(aj+c)8' x-\-b 4(a: + c) x-^c , &c. divide by tv - 6 . J 1 x-^-b x-\-c m- ■»fi srcr #. Kf,, ':'U ■ • '■ I , ' 1 - 1 . 118 CHAPTER FIVE. 83. Complete the divisions, square and transpose, 2 a—h c-h (2a; + 6-hc)3'°^ x-¥h a — h 1x -h 6 + c x+h 2a; + h-\-c .'.x={c^-ab)-h{a-\.h-2c). 34. Factor, then |/{(a;+12)(a; + l6)} - >v/{(a;+12)(a;+14)} = V^+12 •'• (^ + 12)v/(^+15)-(aJ+12)(^+14) = \/(a;+15), .-. (a;-|-ll)>/(a;+l5; = (a;+12)|/(a;+ 14), square then a;2 -f 29a;+201 = 0, a; = ^ ( - 29 ± a/ST). 35. Put a;+a=m, and |/(a;- +2rta;+2A2) = n, then (m+ri)3 + (m. - n)^ = Ihn^ or 2m(m3 +391^) = 14m*, .•. n^ = 2?m2 restore the values of ??i and n, and 2a;-4-4aa;+2a2 = ;<;2-^2au3+262, or (»+ «)^ = 262-aS&c. ■ . 36. Left-hand memher is of the form {x + yY + {x — y)^ .-. 2(aj+«)2 +2(ar3 - 2rta;+262) = ^2 _53 4. 2rt(a-6), 87. Puta;+a = w, anda; — 6 = w, .•, a-\rh^m^nj m* %if}—n which gives wi + n = 0, or 2a;+ = w?, and \/{x^-\-a'^ — h^)=in^ then the equa- tion becomes [m-\-n)^ -\-[m—n)^ = %m^ which gives m = 0, .-. j;+a — 6 = 0: also m^ ■{■'dn^ = -^m^ n^ = lu'^ , i.e., {x+a — by^- x^-\-a^ --b^, which gives x = b, 42. Factor denominators and transpose, then L _ 1 __!___ _ _ A _ ) =3 _ 1 _ 1 L_ 1 . ^^.fl^-j-t (aj-^a — 6 03— a — 6J x— a-\- b[x-\-a — h x-\-a-{-h 2rt -J^h {x-\-a-\-b){x + a-b){x-a-h)~ {x — a-\-b){x-\-a- b){x+aVb) b a a- +62 •• x — (i + b a-\-h-x ". a-{-b (39 39 ) or 89 41 41^) _1_ , J__ 1 , 1_ 8 + ^ + 2"^ ^ "^;« + 3j ■•a + l+a; + 4-u; + 2"^.i;-f 3' . . 2a; +5 is a factor, &c. 44. Divide and proceed as in last, then f 61 ^ 61 ) 51 16+ — T- 9 --—7 ( X— 1 a; — 4 151 ^1 27- — :.-7 -— . «— 2 a;-3 or 51 X7+-863- 61x20 + x—1 «— 4 (a;— 2 x — 51x61x x-df 3 -1 2 , , . or -i; — ^ — ,-7 = -^- v---—^= o=l by subtraction. a;3 - 5a; + 4 a;^ — 5a; + 6 2 -^ a; a_5ar+4=-l, a;=i(5±^/3). I T^J '1.. if ' '■ ' V m i -^ 1 ! ! 1l ' 1 fijV ! !■■ 1 ^ i. i '■ ,■ ■ ■•■ : +c) = a3+62-|-c2, &c. 13. a;=(a+6+c)-r(a2+63^c3). 14. ;c(a6 + 6c+6'a) = a2+6«-j-c«). + — ^ab + bc-^-ca CHAPTER FIVE. 121 15. 4^|«'+6'^ + (a+6)c-| x= ^^_j-^ja3 + />2 + (a + 6)c} x = n+ b a —h IG. or rt6 f ah ) ( rt6 > aft a+6 rt6 1 2 17. — = -9-'-« = ^^- ^^' 12a; " 12 •'•*'- ^* 19. Equation reduces to — ~ — , .-. x-4. X 4 20. Transpose last fraction to left-hand member and combine the three fractions. 119 .-. 6 = 7 + -, .'. x= —140. x+2r 21. (10+3 + 7)-^2(a;+3) = i, .-. a; = 17. 22. Transpose second fraction of left-hand member to right- hand, and take the three fractions together. ... ^^+^ - i5 = 1, .-.^=10. 8a;- 16 "" 15 23. 24. ax-1 1^ X ajj+l a^dx-a^e^ ax-1 1 . _ _ — , . . x = ci. aa; + 1 a hdx-he^ -c'^x 25. a; = 3i; a;=0. ab — X f • • X •" e^(a^-h) 27. a + b a^d-\-c^-l>d 26. a; = 3^-|. ; as second equation stands it is a quadratic ; right- hand member should be c^—x^, then x = {ab-c^) -r- (a+b). 28. a;=— 6; x=:a. 29. a; = 0; a; = 0. 30. x = i{a+b-c). 81. X ab a + b when a; — cf=0. 32. x-d. Equation becomes an identity I' mi. iHXJ tf(t m W" 122 CHAPTER FIVE. -1 ! ) ,; 1 i ! ( ! 'H ' * I i. 83. {a-h)(a-^h)9.r. = nb{a-\-h), x = ah^ {a^ ^b^). 84. --- = -^ ^ , .-. by division = (Ck+IH) - (./•2H-JU-4-2) .-. x= -3J. MH. Apply formula li. x = — 3 afi. As ill last Ex., x= ~3i|. 87. Equation reduces to 50 = 54, .•. there is no finite value of a;. 88. Apply formula B, x' = 10. 89. Apply formula B, x = ahc -f- [ab 4-6^-1- ca) 40. Apply formula By x = {ab -\- hc-^ca — ad — bd — cd) 4- 41. Apply formula B to obtain product of second term, .-. x^-2x^a-\-xa^-x^+2x-a- {{u-h){a-c)-\-{a-r){h + c)+{b + r){a-b)}a^+(a-b){a-c){b •{"-,) = (a2 +l,r){h + ('), or (/>2 +c'2 4- hc~ah—ae)x = = (6 + c){a(/>+6-)}='K6 + r)2, .-. uj = &c. 42. Reduciug as in last example, the coefficients of the third and second powers of a; cancel, and. we have (ab-{-bc-\-cd -ad-b^- c^)x = bc{d - a) + (a - b){b-c){c - rf), &0, 43. As in last example, x{ab-{-bc-ac-b^) = bc^-h^c-ac^-hb^d-abd+acd. 44. x^~2x^{a-{-b-\-c) + ix{((b-^bC'^ca)—8abc-x^ + 2x2{ni-b+c)-x{{a+/>}{b-^c) + {h-\.c){c+a) + {c-ha){a+b)}-^ {'(-^b){h-\-c){c -j-n) := {a + b + c){ft^ -i-h^ -}-c'')-d)(6+c)(c-|-a) = {a-^b+c){ab + bc-^ca) — abcy the equation becomes -x{a^-{-b^-{-c^—ab—bc—ca) = {a+b-{-c){c^'{-b^-\'C^-ab-bc—ca), .'. x= -(a+i + c). CHAPTKR FIVE. 123 45. Reduce as iu last example, then x{a- -\-b^-\-c^—ab - hc-ca) = {uA-b-^c){a^-\-h^-\-c^ -ab-bc-cn), ,'. x==a-^b^-tt. U'\-b-\-c 1 1 u-t-^> + t' 40. ahcx-^-{nh}-hc.-\-ra)-ir — - — "" 77 + la ~ ;/• ' .'. x = {ab-\'hc-\-rn)-7-iihc. 47. {'la-\-b^c)x-\-u[b-^r) = {'1<1 ■^h-Jrc)x+tl[b-\-('), .'. 2{a-d)x = {d-a){h-\-c), x= - }.[h-\-c). 48. ax(2b — c- ab) - b{c + ab) = <(x( - 'lac-ab - c) +af(rt/> + ac), .-. ax{^lb-^%.ic)-{ab-^c){ac->rb), x=^[iibrc)-^'2,a, 49. By division and caucelliag, 6 22 _ 31a;-J^64_ '^rj + ~x~'^ ~ ;c--i27+82' where the denominators of both members are equal ; .-. 27*-128 = aix-104, ^ = 9. fiO ^ _2__ ^ _ 5 1 T ~ a;"+l "" 2 2a;- 2 " 6 30^-3 0u;2-6 9.c-fl lO.c-1 6(a;3-l) ' 6(.c2_i) 51. Complete the divisions , and dj = 2. 18 + 22a;+6 2a;-9 ^ x-\-'6 ~ 23;3-8a;-27^ where denominators of both members are equal ; .-. 25a;-15 = 22a;+6, a; = 7. 52. 45a;3-49a;+26 6; -4a;= -16, a; = 4. (3aj-2)(3a5-l) 53. Taking together fractions of like denominators we have 6 4 + 0, 8a; -7 2a;- 5 ^ 9a;- 25 ... 27a;8-10a;+30=:0; a; = 7V(5± v'785). m tX' 1 i?.' .1 , ■ r ; i ,! ! ) 1 1 ii: •►«■ *• it 124 CTTAPTER FTVl. 54. {a) Divide bj^ «, .*. ir = 0; quotient is i^ ^ _J_\. 4.^-!±^; divide 1+a; 1-a; x^-l \Vx "^ 1 8 6 ('>) a:— rt — = — -, .-. ar = 4. - a; / x^ - I x—h h—a , a;(n — m ■\-a — h)= nm — m&. x—m x~n 71— m If the sign between the quantities is + (as first given in Hand- Book) the equation is a quadratic, and x = \{a-\-l)-^}n-'rn±:y/{{a + h-\-m-{-u)^ — 4(a«+6»i)}. 55. {a) u;=jV- i'^) Completing divisions, a c be c a ah . _j_ -_ 1. __ I a[(ix-b) a c e a c c{cx - h) a{fix — b) c{cx — b) 5G. Multiply terms of first fraction by 2x, those of second by 3a;, and those of 8rd by 6aj, then Sx-Ji Qx+2 9.r-4a; ., or lOa? 5x 4 + Oa; (3a;+2)(2a;+3) 4+Ga; .-. x = 0, and (3a;+2)(2a;-l) =0, or a;= -t, and h 67. By division 11 12 x-1 2ic 12 132 2a;-ll x-4 - 5»-28'°^ (a;-7)(a;-4)^ J ^ = ^ — 7.- = 7, .*. 11a;- 66 = 10a; - 56, a; =10. 6a; — 28 a;^— 6a; x—6 a ap cq a c = — + 58. (a) By division f — 7 \-\ h~? .^— . .*. (apn^ -{•cqm^)x = apnq ■ - cmpq, .*. a; = &c. Before dividing, both members may be multiplied by mn to avoid fractional quotients. a ap-\-mb (6) — + c cq-^nd m{wx—p) n n{nx—q) a m n CHAPTER FIVE. 126 ^_ . . _ .... a-\- h a —r 69. (a) By division , + •*. {a^h-\-a—c — ^lu)j'= —oc-\-ab-{-ac — ah^ .'. x = a})-i-(b -c). (h) — ^-- + , X — il ft-\-b c + h x—c ' x—c (a+b) {-^- ~] + (^+^') [-^ - V^) = 0, or 7 w X + } 1 <, > = U, or {x—a){x — c) {x — b){x-c} x-a x—c = 0, or {a^+b^-c^+ab'-bc-ac)x = 60. c{rt2 + (6_c)rt-6c}+a(/>a-c2), .-. &c ax b hx ax ax —b ax -j-b ax-]-h ax—b b{ax+h - ax^ 4 bx) (ax^ - 2b)b — &0. a^x^-h'^ a'^x^-b^ n-i Tk a- • • « ap—mh c cq — nd 61. By division, f- —f r H + -/— — - -f 7)1 m{mx—p} n r{nx — q) {bn'{-dm)x - (bq-^-iip) a c h — w n {mx—p){nx — q) .'. {apn^ -cqm^)x — mpcq-^apnq, .-.Ac. 62. Take together the fractions whose numerators are alike, thus, m\ l+wJ [x—a x—c) [ x—o x-a [J" L^l = 0, - '"("-^^ , x—c x—b } P + + n{b — a) {x — a){x — c) {x — a){x—b) p{c 6) ^^Q Clear and transpose : ^ ix-c){x-b) {Is 'U> •A X -rl^; Iti lit :?'•:;"' yH 'I 1 .■5 "■ .; '" . ... j I ' f , 1 '»fti }l 126 CHAPTEP F'-'-K. .*. .'jr { ^«(^/ — c) -\-n{h - a) +p{c — b)} T=bm{a^c)-\-cn{b — a)-\-ap{c-b), .*. x=kc. o8 (1) By subtracting uumerators and denominators each traction = '*""-, .-. by (5), page 122, '*"-/,.- = -,-, u-\-b ax — Ab rt+o .'.x=^[a^+b^)^(ib. (2) Multiply terms of first fraction by aaj, and those of soconci , x — a ((x — 1 f^ («) x+a (tx+1 4u;-2' X = 81 13* 04. (1) Proceed as in last Ex., each fraction = — , nx—p .'. x={ap — cm)-T- [an — bin). (2) As in last Ex. each fraction = ax — d .'. A- d{n — q) — q(h — d) mx — q a{n—q) — m(b~-d) 65. ^1) Multiply terms of first and third fraction ly four. 1 — 4a; 4a; . ^ 14.4a; -^ * l+4a; ' ^ (2) Transpose and add, .*. ^aj-i- (|_a;)=:4 ; x~\. QQ. By transposing, 21 / — - _ — = ^ ^ ° U-98 .^;+44/ 1 71 la; -94 a;— 52 ,_ 21x142 _ 71x42_ ^^ (7-98)(u; + 44) ~ (^a;- 94) (a; -52). .*. denominators are equal a;- 94 _ x-\-i^ x^^ - a; -IT ^2 4G -1, .-.a; = 100. 1 07. (1). 9 8 6a; -78 X - -12 a;- 7 X- or 11'"' (a;-6)(a;-12) Qx 78 (a;-7)(a;-ll) 0a;-78 = 0, a; =18. CHAPTER FIVE. 127 I (2). As in 66, take together the fractions where numerators are 824 ^^"^ ___ alike, then j^^if^^ZJE) - '(jc^si){x+Sl) x-Sl x-15 _ 66 _^ — 1 V>h' — ^> •*• •*' — 111. •• x-61 a;+81 ~ 182""^' 6 18 4 4:»r-44_ 68. (1) --Zq- - ^rio = '^^Zf - ~^~d ' ^^ (x- (J){x-T6) 4t — 44 - . 4r-44 — a-=ll 8 4x-28 4a; -28 oi = (2). ic — 6 aj — 2 J- — 5 .-. 4a;-28 = 0, x = 7. x-S' 11 1 \ /I t \ 69. {m-n)[~~~--^] = {a-h^^_,^^^^--_^j^ {m — n) {a -b) (a-h) (m —n) ^ x-v _ .r - h _ b-n (x — a){x—bj "^ [x - m) {-.~n) ' ' x-a~ x-in~ m—a (])y subtraction), .'. x-{cv-{-b — m-n), 70. Transpose a-\-b + r + f/ a + c 4- ft + r« a;-6 "^ a;-(f(4-/>H-c + rf) — 6 a;-c ' x-(a-\-h+c+fJ) c Combine the two fractions of first member; also those of second, x[a-\-b+c-\-d)-J^a + h){a^b + c+(J) - h(a-{ -h)-b{c-^cI ) {x^h) {x- {a. +6 + e -fd) - b} x(a + b+c +d:)-{a+(^) {n + h-}-c-{-d) - c{a ^ c) -^ c{b + d) ^ (a;-c){a?-(a + 6+^'Hr^)-4 Divivide hy a-\-b-]rc + d x-a — 2b x—a-2c {x—b){x—{a'\-b-^c + d)-b} {x- c){x—{a+b-\-c+d)-c} 2{b-c) {b-c){x-{a+b+c+d)+b{x~b)-c{x-c) {2b -c) by subtraction of numerators = (b-c){x-{a-^b+c-\^d)-\-x{b-c))-[h-\-c)b-c) 1*. ■ r ' ■ it',, ill ^il i ' ■ I ' ii ■ ' ;jr' )'' If I -l h',, ''I 128 CHAPTER FIVE. X — {a-\-c-\-i'-\-d) ■\-x —b—c 2 2x-a-^h-2c'-d' x — a — 2b by cancelling 6 - c, •• {x-h){x-a--'^h-c~d) (2x-a-2b-2c-d) M.u\ti\)\ylug hy x — a~2h -c—d, 2{x-a-2b) _ 2x-2a-^b-2c-2d 2{c-d) 2{x-b}~ ~ x-a-2b 2c-\-2d 2x -2a -4b -2c -2d _,„ ., 6 oT 6 — J = r o"~T J substitution, 2x—a — 2b — 2c — d a + 2c+a * x — b x—b x — a — 2b a-ir2c-\-d a-\-2c-\-d invert terms a + 2c+rf 2c + 2d •• by subtracting terms a— A = x—b a—d {a + b){a+2G+d) x = a — d a^ -\-2ac-^ad-\-ab + 2ftc-+ bd+ab - bd a — d a^^2ac+ad+2bc+2ab a — d 71. The first number vanishes if x — a = 0, and since it is sym- metrical with respect to x, «, 6, it will also vanish if a— 6 = 0, or b — x = 0. Since the expression is of throe dimensions, x — n,x—b and b—x are the literal factors of it. V^ii x = 0, a = l, 6 = 2, and we get the numerical factor —6. .*. the - ri becomes 6{x-a){a-b){b-x)^{a-b)cK X - {x—a)[b — x) : x^-x{a+b) + a^b ^j{a-by c2 (a+by 4 + \ X- ~x{a-{-b)-\-ab = ~ — , 6 ■" 1 4 6j ~ ' c2 . f a-t-6 1 1- - -2- 6 ^4 6 « + 6 ±v (a-6)i 6 CHAPTER FIVE. 120 72. Both members vanish if .r = 0, .-. this is one solution. Since first member is symmetrical with respect to the letters x, a, h, .'. it will also vanish if rt = Oor 6 = 0, .-. ubx is a factor. One factor of two dimensions remains, which must similarly involve x, «, h. It must therefore have the form ni{x^ -\- a" + h'^ ) + n{xa-\- 180 CHAPTER FIVE. 1 1 1 abcx a^ib + c)'^ u^r + a)'^c^{a -fT) "- ab-\-bc^a ab-^hG-\-('(i a be = 1 X - 75. Transpose and arrange thus : I x-'2a \ I X -2/> \ / x-2c ^ \ ^ (h-^^t, - i) + (,+.-3, - 1) + (,^+6:r, - 1 ) = x-a — b — c x — (i\~b — c x — a — b — c b-{-c — a c-\-a — b a-\-b~c :. x — a-b — c = 0,x = a-{-b-\;-c, 76. Transpose and arrange as in last example, x—2u I X ^\ -1+&C. = 3 , , , - 1 ' 77. a — x a' be a-\-b+c +&C.+&C =0, or ab-\-bc-i-ca-x{n-\-b-\-c) — 7-s — y-T: — rr','~\ — + anal + anal = 0, /. x={ab-{-bc-{-ca)-T-[a+b-^c). 78. Combine the fractions in pairs as they stand. -2ax + iab(b + c) 2ax~iab{b-c) 79. lb+c + a)[b + c-a) (a_{6-c})(rt+{6-c}) -x+2h(b+c) x-2b{b-c) Abe ^^ ^^.^. -(64.c)2_,,¥- = ^2_(6_c)2 =4ir. = 1 ^y addition, .-. a; = 262-2k + rt2— 62+26c-c2=62^^2_c8. a;+6 -c + 1 + x-\-a — c a—c ^ b —c +1= :7TT+l+:7T-: + l, or x+b x-\-a x-\-a-\-b—c x-\-a-\-b — f. x-\-a + b - (. xi-a+b — c • • — ' ■ j ■ ■ "™7 m^m • — --■ — .. ^ .. » a;+6 — c ic + a-c a; + 6 x + a 1 *s CHAPTER FI\n5. 131 81. Let x^-llx = z, a3-0a = w, and /)2+13 = w, then the equation lednces to (2+iB)(z + :-30)=~(7/?+12)(;/? + 40)-(n-22)(n+20)-f (y/t + 22)(m+3U) + (n-10)(7i-f 8), or 4Hz+510 = O'4O, .•.2 = 0, i.i'.,x''^ — llx= ), .*. x = 0; or 11. Exercise Ivi., page 172. 1. il = 0, or ^ = 0. 2. ^ = 0, or B = 0, or = 0. 3. a; = 0, or a -6 = 0. 4:. x = 0, or i/ = 0. 5. In the first case either x — 01/ = 0, or x-4:fj +3-0 ; m the second c^se both conditions holrl. 6. .j; = 0, or x — (t. 7. « = 0, or u-= -h. 9. a; = 0, or x = o. 11. a; = 0, or j;^ =a=*, .-. «;=d::a. 6\ 8. x = a, or x = c -^b. 10. JB = 0, or x = rb) = 0, a; = M«+^^)' ^^ h{b-a). 44. (2a-6-a;)2 = (a + fe-2a;)'-^-0(«-/>)2 = (4a-26-2a;)(4?>-2a-2a;) = 4(2a-ft-a;)(26-a-a;). .-. (2a-6-a:)(3a;+6a-9/>) = 0, and a; = 2a-^*, or 36-2^/. 2a+2c-a; 3a-6+3c — 2a; 2M-:b^ ~ 2a + 2c — a? 2(a+ c-fe- a;) _ { a-\-c—b-x) 26^+^ " 2a+2c' — a;) ' ... (a4-c-6-a;)(4tt+46'-3a;-26) = 0, and 4rt + 4c- 26 a; = rt+c — fc, or a;= q 45. 46. "Ja — b — ^x Sa—Shj-x da-5b+x ~ 7a-b-Sx , .'.by subtraction of terms, /L{a+b-x) ^ zl('*±^Z^), {a-^b-x){na-lb-5x)^0, ba-5b+x la-b-Sx X = a+i, or na-7b Ih- :J' I I ' I. I til 31 iH f0 i 1 - '^ ' ■ /■-: i ! •' ^|: 1 : , • * 4 ■ ^ i^n m \ '< *t? ' ' ■, # :, n-' m \ > ■■ '"■ i 'i i r:; % '{ (•i«. ;; j| \ • ■"»' 4^ ■' ' .' : % \ \ ■ M ^ - ' . i ■ $ ■I- 134 CHAPTER FIVE. 47. — _ .'- , .'. as in last Ex., 5a-{-Hh-Sx Sn-^b-x 5a + 36 "8a; Sa + 2h-'2x , .'. x = 4a -\-h, or u-\-b. Sn-^b-x 48. c(h-'c)x^-b(a-b-{-b--c}x+a{a-b)=^i), It {(b- c)x — {a — b)\{cx — a\ =0, a — b a X = , or — . b— c 49. Equation reduces to {h-c){a-b)x^-{a^+b^+c9-ah-hc^ca)x+(a-c)(a-b)=0. 50. Equation reduces to (use formula B) 2t;«4-18u;3 1-84) = 96, oy x^ -Ux''-\-^G = 0, or (a;2-9)(.c3-4) = 0, .-.a; =±3, or ±2. 51. Proceeding as in the last Ex., the equation reduces to 2(a;4—40.^;2 + 135)4-18 = 0, or «* -40a--3 + 144 = 0, .-. (a;2-36) ;x2-4) = 0, and .= ±6, or +2. 52. ^-^■^i 3 =0,.-.(p:-3) (1-3^=0. .*. a; = 3, or o' _1 a-\-b rt+ b 1 - . n 53. x — / + """ — ~ I =0, .*. a — X a — h rt+6 X = a or X ~ ia+b\ bj a — b CHAPTER FIVE. 135 1 ^^ a+a; _ a-\-x 55. , , - -2+ , ^ 'I = 0, ... |«±^ _2| 1-2^+-) = a;=6 — 2^/, or a — 26. 0, and 56. Let 2 = ^ -— , the equation becomes « — - 4- — - x-h 2 z (a-6)=:0. I i = 0, or(2-|)/l-i-| =0, .-. 2 = f, tfud |^ = 1. T> . 1 t A 2a 4- 36 , 3a-f26 Restore values of z, and x = or — - — 6 •» 7 v« *;: 't. M "■ 1 If 67. ^.rJ^ » ^ _ a-.r 6 4- a; n T+x m. _ = 0, .*. as in I'ormor cases, n x = {mb+na)^ {m+n), or {ma-nh) -^ (m + n). 58. — + — = — , m + 2w a; rt a2+a;2 %rx m in lm-{'2n\ a'\-x ; m+2n\ or = J X- . rt_a. \ \ni~2n! (a — x) 2 m — 2n ;. x = i/{{fn-]-2ii)-V{m-2n)} -f- |/{(m +2n)4- \/(w.-2/0[. a;2 4.rt3 2aa; aj3+aaj4-«^^ 59. -;•-' — .— ;; = C a; + a _ \/{c-{- 1) a; — a -;jy and x = a |/(/, + l)_;/(.-l)j' a;2 4-a2 /(2c-l)}. t . 1]^ « 1 -iii mr, r ■ .J 1 i ■ 1 ' •t 1 ' ! 1 hi hi h 186 and a; = i(a+26). (n-xY'i + ix-hy CHAPTER FIVE. 9 n-b = 1 •• a-\-b-~2x " 8 1 m gg '" 7^_-^y '/J_^v — = ^- , which is exactly the form in last example, , —^ = ^•-- - a r 2a;=(rt + /))i/(wi-f-2n)-(a-6)(m-2n) 64. a + l>\2 [n-\-h\^ x-\-a a-\-b b x-\-bl ~ [ii-bj "x-{-b ~ a — » » X •— 2/> or 2a 65. As the equation stands it reduces to a quadratic; but change sign between the squares in numerator of left-hand mem. ber, and between the squares in denominator of right-hand member, and it can be solved as 64. 66. or, {a-x)^- \-ix -b)9 x{a - x){x - b) a-b __ _8_ fl+6-2.i? - Y 84 . [a-x^-x -h)^ "— • • • 16 r {a — x — x+b)^ 4 64 or — 8a-f56 86 -5a X = — ? — , or 8 8 g^ ^a^+a{a-x)-\-(a-\-x)^ __ c-fl or. 4a2 4.a;3 ao; 4ag4-(^j;4-a?- _ c-fl ^a'^ — ax-^x^ ~ c-l' (2r7+.c)^ (2a^a;)2 ~ c-4' "* 2a -a; ~ -/(c— 4) a; = 2a. {|/(c-f4)-v/(c-4)} ^ {p/(c-f4) + i/(c-4)}. c+4 2rT4-a; \/(c-\-4) or ^ ' CHAPTER FIV1?. 187 68. Let?* = 5 -a;, v = x-2 (See example 12, page 170), u+v = H, ?fc'*4-7J'* = 17 81 = 17 + 80(5 -a;)(..,'-2)-2{(5-a:)(a;-2)} 3, {(5-a;)(^_2)}2- 18(5 -a;)(ic- 2)4-81 = 49 (5-.T)(a;-2) = lG or 2, .-. a; = 4, or 8. 60. (1). u = x, v = a — x, u+v = ay u^-\-v^=r. Proceed as in last example a d we fiud a52(rt_aj)3_2,<3(,t-x>-fa4 = 2 — •*• x{a-x) = a3 ± yj — ^ — = m suppose, &c. (2). Substitute in the values of x found in the last example, rt = 4, and c = 82, we linda; = 3, or 1. 70. Substituting as in last examples, we find (a—b)* = (a-b)*-\-4:{a~b)^{a-x)ix-h)-2(a-x)^{x-b)^ or (a-x){x-h){{a-x)(x-h)~2{a~b)^} =0, .'. x^=a a,ud X2=b ; and a\so x^ —{a-^b)x-\-['2a^ -ab-{-2b^) = {). 71. Jjet u=:a — x, u-^-v^a — b ; v — x-b, u'^-^v'^ =c; ov,{a-b)'^=c + ^{a-b)^{a-x){x—b)-5{a~b){a-xy{x-b)^. het (a—x){x — b) = tj and = r. 5{a-b) 2 Then t^-'{a-b)^t-r = 0, t = But (x — a){x — b)= -t, i.e., x^ -{a + b)x-\-ab-j't = 0, a; = a+b±i/{a + b)^ -^ab-^t) 2 , ) ■ li *€. Hi 188 CHAPTER FIVE. . I * 'I ( m,' m V ii 72. (1) Let u = x, «4-»' = '^ v = a—x, it*-fi'* =^/*, and pro- ceed as in last exjiuij)le, then a** = a'^ + r)(t''^(a - x)x - .'>//( = ± ab, .'. a; = 0, or a+6, or i{(rt4 b)±\ y/ {a - h)^ - \(th} , 74. Let a — x = m, b-\-x = n^ and .•. w/+w = «+/> ; the equation reduces to inn {w^ -^ti^ -\~mn{ni-\-n)} ={m-\-n)c, or ;/m{(w+?i)3 — 2//m(m+n)} = (*« -f- n)t', .*. ?//n{(//i+//j^ — 2/;/?t} =(>', or (m-«-n)2mw — 2w3n' =c, i.e., {a-x)^h+x)^-U(t + bj ^ .x){b-hx)=~ -1, ... (a_i^)(/,+a:)=:i(a4.6)3 ± VU''+^)* -84 =r suppose .". x^ — {a - b)x-\-ab = r, &c. 75. Letrt — a; = m— «, x — h = m-\-Zy and .'. a— 6 = 2m; the equa- tion becomes Wi4+6M'-'22-f24 41 =^ X 4»/i2or 5z^ - Um'^z^ = 36m*, .-. (522 - 9m2)(22 _4^,i2)^o, .*. «2 = w8, aud 2= ±m Eud 2=+2wi, .*. a— a;=+m, or Sm,= — i(rt — 6), or |(a-6), .'. x = ^{Qa — b), or ^(36 — a). 76. As in last example let a — x = m-z, x—h — m-\-z, and .*, a-b = 2m ; then substitueing these values the equation becomes m^-}-10)nH^-{-5wz* 211 m'^+(im^z^+z'^~ ~ 97 ^ ^^' °^ clearing and transposing G324_1562w222-325m4=0, i.«., (63«2 + 13?Ai2)(a2_25m2) = 0, .-. 2^ = 25^3, and 2= ±5w. Aud a—x-m — z= —4m, or 6m, .•. a; = 3a— 26, or 86-2rt. CTIM' "•••'• 'i-n'll!. 130 77. ATiiIce same snbHtitutions aH ju last two exnraplcR, and a (liiadnitic in z'^ is found, .-. z is known = < suppoHf, 78. From same subHtitutions as last, z is found =/, snppoHe, 70. Ar in example 70, a quadratic in «2 jy found, .-. « = ^ suppose, &c. Make same substitutions as in t'xamplo ITt, and hlr-hand mem- ber becomes (^•*+0m'-'z"^4.,„i)--(2.' -,^3), a quadratic ina^ tlu^n proceed as before. 81. By same substitutions, loft-liand monibcr booomos wliich giv s a quadratic in z^. 82. As in former examples, we liave left-hand member (z^-h(hn.-z''-\-m^)-^4z-'i, and right-hand member is known ; a quadratic in z^. 83. As in example 70, (5mz^ H- 1 (h,i H'^ + ,n •"' )^ Iz^ = &c., a quadratic in z^. Note.— For full solutions of these and following equations see Appendix, "if^ CHAPTEB VI. •i '■ ! I 'Ml.'. V 1. X- 4. X- 6. x- 9. X- 11. Exercise Ivii., page 182. 7, 7/ = 9. 2. a; = 2, 2/ = l. 8. x^S, y = l, 9, ?/ = 5. 6. u;=— 10^, y = 5i. -2, y=:^. 7.x=-l,y = l. 8. a;= -2, 2/=--3. 3 -4 -4 3 -1 +24 -18 2 5 -25 12. * -i A 8 _ 1 _ JJ¥_ _ 7 5i" 13. « 15. X 17. = 10, = 18, 5 1-7 8 I) V ?/ = 12. 2/= 13. -4 -2-2 200 1 6 . 2 I 9 iB=i2, j=a. i * »/ 14. a;=12, y=15. 16. a; =3, y = -2. 1 5 7-9 1-7 11 6-8 31-6 2-2 -4-2 ) -29-4 7 II 0) -17 39 5 -37-8 CHAPTKR SIX. 18. a; = 7, 7/ =-3. ' 19. .^• = 7, y^i 20. 1 1 1 1 1 -^ 1 -1 -\ ;^ 1 +^ -i — 'I -1 ~l 11 II 1 1 X y .' r='I, y = S. 21. 8 8 -3 8 16 -4 - 12 -4 16 -32 -45 120 12 -12 -132 1 - 44 -33 " i i II 11 1 1 X y x^?> y = 4. 22. a;=4 ?/=A- 23. .^—-3 //=}. 25. .5 7 3 - u v* 10 7" U - 31 V' 460 7 -'A _ fiO T f -54 — 1 •"' 50 7 44 7 1 447 10 140 7 iV 10 II II 141 24. ./— 12 /y = 15. St ■^ x = X \J5 V 2/ = A * ■ r :,'■ I" J Hi" ■1 1 r Ml i 1'"' ■ H 142 CHAPTER SIX, 26. a; = 8, ?y= 9. 27. a;=3, ?/=l. 28. a: = 7, y = 9. 29. ic = ll, >y = 7. 30. a: =17, 2/ = 13. 31. .« = 5, //= -4. 32. ic=-^-^, y/=-|^. 33. aj=13, // = 10. 31. a; = lj, 2/ = 3iV 35. x=ll,y = 6. 36. a; = 7, v/ = 5. 37. 20 8 -15 -12 5 20 20 8 — 210 -120 -300 GO 40 400 -120 -210 -360 2 3 II » . y 38. x = 5, ?/ = 3. 39. In this question the second equation is not a new one, but an equation that can be found from the other, hence x and y can not be found. 2 3 40. 1 4- - - = 1 + x-\-l 2/4-5 2i + 2 = 21 + 3 // + 1 ' ■ 2.C-3 3a;-2y-7=0, 4.7;-3//- 9 = 0, x = S, y = \. 41. 1 - JL = 1 _ _?_, 1 4- _^ a; -3 ?/4-7 x+'l = 1 + .-2' ^x-y-l^ = i), ic-3//+8 = 0, x = l, ?/ = 5. 42. a;=:0, y = 0, 43. ^.x^ly, x = 'd, y = 0, 44. x = 0. y = 0, a; =13, ?/-ff-. 45. Adding three equations, x+y-j-z = 4:2. Subtracting each, x= 17, // = 2(), 2 = 5. 46. ..= ^2^, y=-iU, z = m. 47. ^=11, 2/ = 7. z = d, 48. Multiplying second = ;i. by 2 and adding loa; = 315, x = 21, // = 22, s = 23. CHAPTER SIX. 143 49. From first equation 2a: = Sy ; adding second and third equa- tion, ia;-iy=_a,a;=-,i5,2/=-o,«=- 8. 60 Multiplying first and second, and subtracting second equa- tion trom the result, 3?/ - 2f « = ; ?/ = ^s. Substituting in third equation, z, 5 ; y, 4 ; x, 3. 61. Adding three equations x + y+z = 87. Subtracting each equation, x, 12; y, 15; z, TO. 62. Adding three equations and afterward multiphing second equation by 3. i. ./ o « 3a; -f 6?/ + 142 ^47 3x-+0.v+lL>2=:45 S3, a;, 4 ; y,U\ «, 2 = 1, a;-C, i/ = 3. a 7« 54. Multiplying second equation by 7 and subtracting from first equation, 20//= 100 ; 7/, 6 ; .r, 3 ; «, 7. 56. Adding all three equations, l()x~ 110 ; a;, 11 , y, 13 • « 17 66. Adding three equations, and then multiplying second equa- tion by 3, 8a;+6?/ + 102 = 43 8a;+6//H- 92 = 42 2 = 1, 2/ = 3, ic = 5. 67. X, i); y,7; z, 8. 68. Adding three equations, and then multiplying third equa- tion by 4, 4aT+47j-\.4z = \){) 8a:-f4,// + 42=128 4a; = 29, a; = 7i, y = Si, 2-9;^. 59, Adding three equations, 21U'-f//-{-2) = 135 From (3) 7a;-f 21//-f 2X2= 91 14a; = 44; a;, 3j; y, 2| ; 2, 1|. H ^ Ht >»t f 5 IS i ■ ■ I' , tf I' '' , - ' f ,1 I r.. I ^1 ; - I ' r •ir-' < ! i M ! i 14t CH APT Kit SIX 60. Adding 1 and 3, multiplied by 2, and again adding second and third, we have two equations in x and z. X, '2i\ • y, 3-4 ; z, 4 5. /.-I o*» . o/\ rrn nn 8 80 I (I !» ft I 1) 4 . ()1. X, 30; ?/, 20; 2 70. 62. x, g^ ; y, ^-.^ ; 2, ,,9 ; 63. ;c, 30 ; ?/, 12 ; 2, 70. 64. a;, 6 ; y, 12 ; 2, 20. 65. :f, 5 ; //, 2 ; 2, 0. 66. a;, 1 ; 2^, 1 ; zr, 1 67. .♦•, 11 ; ?/, 9 ; 2, 7. 68. a;, 5 ; y, 3 ; 2, 1. 69. Multiplying the first and third equations by 3, and adding to these results the second equation, — = 7, x = 2, ?/ = 3, z = l. X 70. Changing sign of third equation, and adding all three, 20 ii\^i -~ 4. 2 = 5. ?/ = 4, x = S. z 71. 1 X + 1 = 5 . 1 ■ y + 1 z = 6, 1 X + 1 z = 7, . . 1 X + 1 y + 1 2 = 9, adding 1 + 1 3 = 6, J_ = 3; a;=i, 2/ = i, « = i. 72. a;, 5; t/, 4 ; 2, 3. X 2a;+7_2;v + l a; + 3 _32 + l j/ + 3 ^ 2+2^ 73. ^_f2"- '7 ' a;- 2 ~32-l' y'4-1 ~ 2 + I 3 _J^ 5 2 _2^ 1_ •"• ^2 ~ y v-i ~ 32 - 1' 2/+ 1 ~ 2-f 1 * X- 37/4-2 = 2a;-162 + l = y- 22 -1 = a;=+7, v/=-|-3, 2=1. CHAPTER SIX. 145 lino; second 74. .r, 2; ^. 8; «, 1. 76. Ec^uation reduce to *- + — - = q X y 8 6 - — H ^-2 21^ 2/ 5 = -8 y= -1 adding?/, 3; a:, l;a, 5. 7^. 0!, ; ;/, 1 ; 2, 2. 77. Equations reduce to « — %+B0 = O, 5a;-lly-z = 0, 21j;+ 31^-1-410 = 135; from which // = 300 h- 349, a;= 1755 ^098. 78. First two equations become 2^//4-6,^:«-4//2 = 0, Ox//4-3;r^- 3^5 = 0, .-. 6xz-^z = (), or5a; = 3^; so3//-2 = 0, or J5 = 3y. Substitute these values of x and 2, in third equation = «' 5; ?/. F? 2?' 1- 79. Eliminate «, between second and third and between second and fourth, and add results ; IO2 = 10, z = 1 ; a;, 5 ; v/, 4 ; n, 3. 80. Eliminate u ; 5.«+2,y - 3z = 22, ^ + 4// - 3z = 11, 4a;-2i/ = ll, 4.^+28*/ = 110 ; x' = 4?, // = 3^V 2 = 2t2<„ t* = |^. 81. M, 21; a;, 31; v/, 41 ; z, 51. 82. w, 8^ ; u;, 7; 2/, 4^ ; z, 4. 83. M, 30 ; a;, 20 ; y, 10 ; z, 0. 84. u, i ; a;, 11 -^ 24 ; ^, i ; z, 1 ^ 24. 85. X, 270-M17;v/, -52^ 117 ; z, 15 ^ 117; i*, -126-117 80. U = x=y = z = 'lli). Exercise Iviii., page 189. 1. y = (a'o-aci) . {a'b-ah'). 2. y = h{cn—dm)^{ad-bc). 3. y = h{d -~c){d-a) ^ d{b -c){b—a). z = c{d~ a){d - b)~d{c—a){c - b). 4. y = cz-\-dii-{-ew-'fax, z = du-{-ew}-ax-\-by, u = ew-\-ax+by+cz, iv = ax+bi/-^rz-\- dii. ks. ii i; I, ■'' 5. 1 n„ I 1 1 : i^^- „; 1 ■ 1 ' 1 i 1 •' i ' i f : I * • i 'i t ■ II 'J' r I 1 1 ! !nm ^ r . 'J i i i 1 1 146 CHAPTER SIX. 5. (1) (2) (8) X VI n X + = a n + — ~ h y=lx{h-n-\-a) (l)-(2)4-(B), — = a-h + n 1- - r= c z=ipiG-a-{-l)) III p 2« m X = lin(a- b+r), y, z, by symmetry. 6. (1) x-\-ay-^bz = m (2) hx+y + az = n (8) ax-\-hy + z=p ft(3) - 1 (rt/; - l):r+(^2 -«)?/ = ^V^- r? (3) - 2 (a 3 _ /^).,;'4. ( ^,7, -l)y = ap - n m x y = {p{a^ -h)-iii{ab-\)+n{h^ r-(i)} 7. ( 1) rt? + ay = 1, 1 - («)2, x-nhz = l- {,|3+/,3_3,,^_|_1}^ mn. (2) // + /'^ = (8) z-^ru = m. n. -\- aJ){S), a;-f- ahcu = /. — am-\-abn. — a/>r(4), .i; — abcdw — I — am-\-abn — aicjo. (4) u + dw=Pf -\-abcil{5). (5) M;+''a; = r. x-\-cihc9? -f- nbcdr) -H (1 ■\-ahcdr)^ y = {jn — bn-\-bcp—bcdr-\-bcdeI) -i- (1 + abode) . 8. iC — by — cz = 1 // — cz — ax = 0. 2 z-a.r-{-by = 3 (1-2), (l+rt)x-(l + % = 0, (l+rOx-=(l + %=.(l+r)3, Dividing the terms of (1) by these equals, Y+a 1+^^1+c 1 ^ rt 1 + a ' 1+Z» l+c CHAPTKR SIX. 147 ©• T — ay — nz-O ^ 1 y—hz—bx = i) ^> z~cx — r;/=0 ;^ &(3)f2, {i~bc)i,-{h^bc)x = 0, 6'(2) + a, {l-br)z-.{r.^hrlr:^.(), X y 2 '• l-bc~b{l+c)~'c(l + h) •*• ^ividinsr (1) by these equals, l-bGz=:ab{l+c)-hac{i-{.h), .-. 1 = ^/ /> + />c + c't/ + 2r/,6c'. 10. 1-2, :.x-y = b!/-ax, x(l+(f.) ^ /,{l-^b), .'. x{l-{-a)=y{l-{.h) = z{l-\-r)-=n{l-\-,l) =irn+r), dividing the terms of first equation hy these equals, 1 _ _J^ _c d f> . 1 ^ . __L. . ^ . ^' ^ •• '- i+« + 1 + 6 + 1:^7 + TT^ + T^e~' Since -:r-] — = 1 — ^r— — 1+a 1+a Exercise lix., page 192. 1. x={nc- hd) -^ {na - bm) ; y = (wc - ad) -^- (w/; - 7?a). 2. a; = (wc -f 6^/) -r {an-irbm) ; // = (wr - ad) -r- (im -|- an). 8. a; = c(n - b) -f- (an - mh) '; y = r(//,: -a)^ (bm - an). 4. x = {b-c)a-^{b-a); y = h{a-c)^{a~b). X = ah-^ {a-{-b), y = ah-^\[a-\-h). 7. x = ac-^{a-\-b), y = bc-^[a+b). a b b a a^—b^ Q j -y„ — _j ^^ = am~bn, X y x y X X = [a'^ -h^)^{am -bn),y = {b^ - a^)^{bm - an) ■4. 11 r '.• I ' . :' .•I h 148 9. CHAPTER SIX. (n -^r).v — (4 — r)j/ = %ih 1 — {(( —h)x-\-{a + b)ii — 2(1 c, adding (b-\-c)x+{b-\-c)!/ = 2a{h-^c), ;»•+// = 2a, (a — c)x-\- {a - c)i/ = 2(i{a — c) adding to 1, 2ax = 2a[(i'\-b — c), x = tf-^-b — Ct y = b-\-c~a, z = c-\- a 10. X — G y-c a (1) .-. tZ^ = x-y a~h y But {x — y) = a — b, l^ ^ <* a — b a — b y —c a x-T-y^ a y=sb-j-c, x = a-^c. ■y + rn _^ d y-\-7i y = b{cn — dm) -r- (ad— be), x = a(cn — dm) -r- (bd — ac). 12. x + y a+b-{-c 2/-fl ~ a-b+c ••■••••• B «••••• I y -1 a — b—c x\-l a-^h — G (1) (2) by adding « + // &c. x + 1 a-\-b—c numerator to denominator, divide (1) by tliis and multiply result by 2, -{b + c) //-I ^ _ «/ + l 2(rt — c)(a — ^ + 6')' 18. X — a -\- r y —a-^l) x + c y+h ^ b x — {a + b—c) ~ c ~ y— [<(■ - 6 — (!) a-\-b X —{a-{-b — c) a+c ~ y — (a — b-\-c) ' From 1, ic— (a + 6 — 6')= — {i/ — (« — ^4-t")}' (1) (2)- CHAPTER SIX. ••• •'• ~{a + h-c) = and // ~{a~b -\-c) = U, i.e., x = a'\-b-.c, y = a-b-\- •. . , « -f- c w -f /> "■ ,.+«-i+;,+3 - 1 = (i). x — h y~c From 1, ^-^-^tf + y^JlZ^Zl _ ^ From 2, ^~Z^;±1 - /^t^^JI^izi _ n a-j-b ^ a-b - "» . / 1 , \ I ^-^^ a-c\ ^ ^ ^ \ a+b a + bj -"• .*. a; = a+6 — c and ^ —c+a — b, 15. -'"-. + J' =1, _5_ + -J^ -1 a -- {vi ~a){n - a) +y ~ (m - b){n - b) = 0. :.x={)H-a){n-(i)-T{h-a), {m-b){n -h) y^ — ^-6" — 16' ir + // + c; = u ., 1^ '^+c)a; + (rt + .)y-f-(r^-f./>)2 = j 2, (a+&)l_2 (a-r)^-4-(/>_t.)?/ = (a-^)(a-c)a;=l, ic= tt-Tw r (rt — Oj(a — c) y {h~c){b-a)' ^ {c--a){c~h\' 149 1 '«■-* ! I Iv ![ 1: P4f"\ = tijj,i kl ''^ 160 CHAPTER SIX. 17- a;+2/+2 = / •(1) •(2) j_a ^ <-4 ^ l-e (8) (4) 2 — c(l), {a — c)x-\-{b-(')y = 7n-k "T (Z-c)(/-a) ^ {l-c){l-b) ~ (i-c) a;(c— a) y{b—c) I — a + Z-6 = c. .(5) a;= {(m — ftc)(Z-rt)} -r- {(c — a)(i + m^+*i»" + c, « — /^ />Z4-w*9+wr -f a. CHAPTKK SIX. 151 x — a y—h z- c .'. ?(.c — rt) -!-;«(// -A) +z(n — r)= 1 -{la-\-mh-\-nr), comparing with 18, jM { 1 - {la -f vih 4- nc) ) p{\ — {hi-\-mh-\- nc) \ J/= 20. ax-hj^n^-h'^ (1) ax—cz = n^—c^ (2) ax-\'hy-}-:z = m^ (^) (2)4.(8) 2ax-\-bi/ = m^+a^-r-'i (4) (4)4-1 8rta; = »i2^.2aa-/>3-c^ „j2_|.2r/2_i!,a_c'2 m2 + 262-r2-a2 «= 8^ ' 2/= 86" ' ' 21. y = a—b + c, z and x by symmetiv. 22. x+y-\-z = (1) ax+b]i -{-cz = ah-\-cb-\-ca ( '2 ) {h-c)x + {c-a)u-\-{a-h)z^(i (il) 2-c(l) {a-c)x+{h~c)]j = ah-\-hc-\-ca, 3_(,<_fc) (1), (2&-a-c)a; + (6+c-2% = 0, » (6+(?-2a)(2/;-«-c) ^ ^ (/;-{-e_2a)(2ft-a-p) . x^ («&+^^ + ^^)("^Z^)(t+c-2a)4-(6-c)(2&-a-c) = (6+g-2a) (26 -a-g) 7/= (a5- + ^c + ca)^^^_^^^^^^_2,,^_,_^^~^.^(26-a-c)* }^OTE.— First equation should be x +!/ + +nnq,) z = {mqr) -=- {anp-i-bmp +C7« ' 7 symmetry. 26. x = )r{b-\-c — a), y and 2 by symmetry. 27. From second set of equations, a h c Substitutmg, — — — . X = r, dr ma-^-nb+pc-^-qd 111 T + 7 + 7 = «+^+^- (2)» rHAPTKU SIX. ica Divide (1) hy xi/z, ~^ 4- — z 1 1 X tl n / 1 1 I 2 — -h — + - o 1 a + 6 -t- c 1 1 1 1 // = 2(/>-a), 1 1 — + ~ = <«, — = 2(/> + 6'-rt), a; = I -r (A+('~a.) — 4- — = 26, — + — =2fr, a z y X X 2 = <«, — y X .'. (rt — ft)(a;4-r') — ^'(// + 6)H-6(3 4-a) = 0, aud a{x-\-c)-\-{c-a){y-\-b)-c{z-{-a)==0, .-. {(a-6)6'+a/>}(a;+r) + {(c-a)6-rt4(?/+6) = 0, .-. (ac-6c + a6){(aj+c)-(?/+/^)} =0, .•. a;+c = //H-i = (by symmetry)^ + a, .•. since x4-^/+« = 2(a + 6 + c), a; = rt + 6, y = c+a, z = b-\-c. 30. Adding, 2(a.t'+6?/+cz) = 3, «a;H-6//+c2= :}. Subtracting first equation, cz=i, 2=l-i-2c, a; = 1-^2-7, ?/ = lH-2&. 81. Multiplying (1) by n, (2) by m, (8) by /, Iny •{■niTix = n^ , mnx-\-lntz = m'^ lnu + /ny — l^. 2 Adding, mnx+lny-{-lmz — Subtracting first equation, Iwz = l^^,„2._n^ 2 2 = /2 4„,2_^i3 aj = /// -'-fn2-^3 2iw 82. Adding, x-{-y+z = 2?Am a + i-f-^' "~2 y "^2«i( Subtracting second equation, x = '^"'^ — . -r t 1:. r • |i T J ' ff ! ' >, ill-' 1 ■ ' 1 f , 1 r 'Mj. '■I!' 154 CHAPTER SIX* mn In 83. Adding, X'\-y+z = -r- + — + — • (■ m n lyt Ltft Subtracting first equation, 2x = — -i_ — , •ni n I III X = I. - m^ -\-n^ 'linn y = m 2nl 34. Adding, — -l — 4- — = (a+o-l-c) X y z Subtracting first equation, — = 6+5 -a, X ^ 1 ^ J b-\-c-a c+a — b 85. Adding, — 4- — + — X y z 2 2 2 — + -r- + a 2 2 2 Subtracting first equation, — - JL ^ — , X b c X = be U ca b+ c " c-j-a 36. x = b-l-c — a; y, z, by symmetry. 37. X, a] y, b', z, c. X V Z 38. Adding ^ + c "^ c + n "^ fl-f-/^ ■"^* Subtracting first = n, = n-b, a-\-b z, a^-h^; X, h^-c^ ; y, c^-a^. 39. x = \{a-\-'lh-('-\-'dd)\ y, z, v, by symmetry 40. u = ^f{4:a-{-b-\-'dc-2(l+5f), x, &c., by symmetry. Exercise Ix., page 202. 1. Add t^e equations, .-. 2a{x-\-y) = 4:a^, subtract, .% 2b{x — y) = 4:b^; x = a-{-b, y=.a~b. h 2. x+ij = ((, x — y= ^, x:=^{a^-{-b)-i-a, y=^{a^-b)-i-a. CHAPTER FIVE. 155 3. Factoring in second equation (:»+v)(2^-37/) = n3, .- from first, x-^-y^n^m-, a; = (3n^+^^')-^5yM, y ^ ^n^ ~,n^)^6m, 4. Dividing first = n through by aS _ ^,2 ^ ,., ^' ^ 1 a-\.!, ,/_5 ^j_^ add to second equation, .-. 1x^{a-{.h) = 2a~{a2 _ ^aj . « 6 X, a-b' 5. From 1st equation, {a:^ --h^)xM ^A-h\,^a^h->r\ ' add second equation, .-. (^2 -^a _i)^^(^3_ ,, .,_i^ ^ ^^_^^' . 6. Substitute in first equation the value of x from second. ... {(a+6~c)3-(a-6+c)3}y = 4a(6-.j(a-6H-c') or, 4rt(6-c)?/ = 4a(Z?_c)(rt-6+(;), 7. From first equation, x^{a-\.h~c)ij ~ {a~h^c). Substitute this value of x in second equation and we have .'. y = a-b-\-G, x = aj-b-c. 8. First equation gives (a + b)x-(a-h)y^2,h; second gives y, (a^-ab-\-b'^)^(a~b). 9. From the first, 0.- 2/ = (" + l)-(r^-l), and from second ^.+y = {b + l)^(h-l); x={ab~l)^{^a-l){b-l); y,(a-b)^{a-l){b~l). 10 From first, (.;+l)^2/ = ^,, or x = ny-l; and from second af^(r/4-l) = l-^/>, .-. {ay-l)iy-\.l) = l^f,; x = (l^a)-^(nb-l). y, (l + h) -- (ah-l). 11. From first, x^(l ->/}== (a-\-l)-^{a^l) ; from second 3?' It 4 : - rl ( 166 CHAPTER SIX. < ! I fi: ., .1 ft. ''''^' H 4- 12. X, a(a-{-h)', y, h(n — h). 13. x = a{b[a + b) -c{a-<-)\ -^{a^ - ^'c) ; y = {a — b)'\-c{ii + c) } -i-(<' ~ — ^c). 14. a;= — (rt-}-^^)» y = 0'b. 15. Adding the equations and subtracting each separately from sum, a; = i(6 + c), 2/=U6'4-a), z=^(rt+6). 16. Add all the equations, .-. l'd(^x + y + z) = {ti-\-h-\-c)\ 17. Adding the three equations and subtracting each — n, 8e|>«- rately, x = l -i- {b-\-c); y, 1 -i- (f+a) ; z, I -i- [a-^-b). 18. See question 29, Exercise 59. 19. y = {c-\-a)(c-a), z= {a-\-b){n- b), x = {b+c){b — c), 20. Subtracting first equation from third, z = a^—b^; X, b^—c^] y, c' ■ a* 21. Proceeding as in example 7, page 194, assume X y ^-^-:^^ = {t^-\-Bt^+ct-{-D)-i-t{t-l){t-2);h}it{rom the equations it is seen that the left-hand member of this equn ion vanishes font = rt, or 6, ore, .-. t^ -^ Bt'^ '\-ct-\- 1) = {t — a){t - b){t. — c)] multiply equation by t and put i = in result, .-. x= Uibc ; simi- larly multiply by t-1 andput i— 1 = 0, .-. i/ = (l -a)(l — 6)(1 — c) ; Again, multiply by i — 2 and put this = 0, .-. 2 = (2 - a}{2 — h){2 — c). 22. Equations are xy = a{jc-ry), &c, divide the first by axy, the second by byz, the third by czxj .*. _1 2_ 1 a x y \ 1_ J_ 1 _ 1 J_ b y z c z X subtract the second of these from the first and add result to third. 1112 ^ , .*. H"T-+ — =7- .'. x=^'Aabc-i-{ab-^bc — ca), y and z by sym- metry. CHAPTER SIX. ^^ X-1 y — 1 2-1 157 aj — 1 V — 1 ~-l iC — 1 // — 1 2—1 24. + ■ — -f , a X mx ny fz qrt = 0, .*. x = y = z=^l. a = 7.n = :,t= »; = .,,/- = ("w+n//-f /-z+?'/)-H y«a 7i/> 7K- (/ (ma-\-nh-\-pc-\-qd) =r~-(uia-\-. .). j^ = ar -f- [}na-^nb-\-pc + (//), y=hr~-{ ), z = &c. 25. Substitute in last equation the values of ll derived from the otliers, .-. u o Id rl^ 63 6-2) [a — ll w, .-. w = 0, or 26. x^{h-\-c-a)-^{a'\-h-^c\\ y = {h-a — c)x-^{n — h — v). 27. x — \{a — h+7n — n); z, ^i{a- h - m+n). 28. ic=(4ffH-2c-^— 8/>) -:- 40, //, z, n by symmetry. 29. As in Ex. 3, page 193, the polynome i'^ + uf^+zt--^yt-\-x vanishes for t = a, t = b, t = c. and t = tl^ and .-. = {t-a)(t-b)(f-r){t->i) = ti-^n+h-\-c + -l)[=(«/^-l)-^(^/-6), .-. •i^=(rt6 — l)-7-(a — 6)— a^, substitute in (l); x=(2nb+a+b-\-r) -^2{a — h), where r^ =4a(/>- +/->-}-l) + (3rt- 6)i^36-a). 2. From(l), (ajS+cc+D^i^B^ -2aj-|-l)/>^ = (v/3-|.v+l) {2i2yy2^2/>-^?/ + 263 + (/y-l)3^, .-. (a;+ 1)3 -:- (»;_ 1)2 = {462(//2+y/ + l)-0/-l)2}^3(y_l)a;from(l),(a;+l)^- (.«-l)2=a2(?/-}-l)2-i.(,/_l)2, .-. this right-hand member = {4/>2^y2^^ + l)_(yy_l)2^^3(,^_ip^^yljej.g^_l cannot bezeroifjcistinite, .-. 3a2(_v + l)2=462(^y2+^y_j.l)._.(^^y ._1)2^ &c., x=.ar-\'l)-^{ar- 1), where /-a =(62 _ l)-v-3(a2 -62). 3. Multiply the two equations together, member by member, .-. (H-;'-)2 -4- (1 -a;)-' = (l+a)( 1+6) -^ (1 ~a)(l-6). T,aking sq. root, and then sum and difference of numerator and denominator, .•.«;={^(l + a)(l + 6)- v/(l-a)(l-6)}-- {^{l^a}[l-^b)-^ v'(l-a)(l-6)}; ,y={^(l+a)(l-6)-|/(l-a)(l + 6)}4- Ul-hu)(l-6)4-v/(l-''Hl4-6}}. CHAPTER SIX. * i^rt 159 4. Applying (6). page 122, we have from first equation xy-^i-x-y "" {x-i)(frT) = ^• Similarly from second equation, ('^^^^ )(//-!) _ b^ ^ (aj-l)(// + l) - ^■ Multiply these results together and take square root, .-. (x+l)~(x~l)=.ab-r-cx0. And again applying (6), page 122, ^ = (.A- ,/,) ^ r,,j,, ... dividin:, tlie results we get ?/ = (./?+6a) -^ (./i_/,^). 5. Proceeding exactly as in last example we have {\/{a-{-h + e){a+b--c)- y j_ ^ NoTE.-Rationahzing the denominator of tliis the value of . appears m the form i2ab-4A) -^(.. ^.^^ - ..) ^^ , !, "' ' , " product of the /..^r factors involved. l^^^^ = the 2(x4-/y)(l-f-^^) - 2/« ~ 'm^' ^ow proceed as in ex- amples4and5above,.-.^/(l+,,,^^,^^,)p_^,l^ ^ ^^ Similarly the second equation gives {(•'• + l)(^-l)}2-^{(a;-l)(yy+])|3. Multiplying these results, /. /~ P = i (iL±^'^' ''^ +_") ) :. = {ti±Z!0(^±_*!UiHiLl_!^-^0'} + actly as in last example. ^loceea ex- 10, 11, 12. See examples 14, 13, 12, pa^e 197, Hand-Book. ■«l* '" if ||4';>l- ■/ I ' ' : A ' I r I." ! 1. 1,1 ^ »fl.) .it- *, 160 CHAPTER SIX. 13, 14. See example 15, pa^ife 198, and example 11, page 196. 15. 0^ = ./, y = h, (2)--(l), ,^3V//4-l == 63+T+l' "^ ^^^^^ substitute the value ofi/ from 1. IG. a-^b = [x--\-!/^)-i-X!/, .-. (r/+6)-v-(a-6) = (a;2+a!?/ + ?/2)-j- a-{-h a 4-/) a + b (a;-?/)2=: ^-^ (rt-26), .-. 2x- = &c. 17. x^-{-xi/ -}-//- =/>, and by division, x^—xy + y^ = ~, .-. x'^+y^ = ^ {b-i- fj, ici/ = i(6 - ~y /. {x+y)^^i ( 6 + -yj + /> - -^-- = i [36- -^] , and 18. {x^ —y^){x^ —xy-\-y^)-=a, xy{x^ —y") — h, .'. {x^ — xy-\-y-)~-xy = (i~-b, .'. {x-^ y)^-^(x — y)^ = (a-\-m-^{a~b), .'. {x-\-y)~{x-y)= V [a+Sh)^ y^[a-b)~m., sui^pose, tind ;<; from this and substitute in second equation, getting a quadratic in y^. 19. Add the equations, .". 2xy = {a + b)x^ — {ci — fe)?/^. Puta;=:t'^, .-. 2vy'^ = {a-{-b)v''iy'^ -(a-b)y^, or 2v = suppose. Substitute in first equation. 20. Subtract second from first, .'. x^ ~y^ = {a — h)x'^+(a — b)y^ -{•{a- b)xy = {a — b){x^+xy + 7j'^) .*. x — y = a — b; similarly, add the two equations, and we i^nd x-{-y = a-hb; .•. x = a, y-b. CIJAPTKR SIX. , page 196. in which and lOi and . &c. titute in imilarly, a, y = h. 21. A(ld(l),tnd(2). ic(:c^^^>r~)=^2.ir->j)2. Vi^^~<-)^Vc;x-^y={ ^y^a - .) + ,/,} ^ j ^ ■ ^^ _ ^j __ .^. , = m suppose, ... .f = .,^, which substitute in ;2), &c. 22. Subtract second from hi-st, and x'-^ - /y- = |(-^+--//+^^)i..4-,). If ..+,, + ,. ^0, then from (1) and (2) . = , = ., or =U-l±y -3).; if ..M-., + ,^ .snot zero, dwKle the above result by It, ... ,^,= _i(^+^^, or (.-,; .^ (, + ,, -;) !J - '"// sup- 2(/ pose; theny3_,,3^ ^^^K^^+^i^ _^^(.,^^^^, ^ ^3 23. From(l) (^•+.i--+//+//2) -^ U.-yV4-.y4-l) = r.. i- 1) -- ^"-^^ ^^)- From(2) (.-//K.•+//-l)-^ (. + .,-,, -f^) =.(/>_l)_^(/,^.l) :..(^^^ (3)X(4) ...,:.+ , + 1),: ,,.^^_:i^ = (« + l)(6-l)~-(.,-l)(/. + i), ... x-+.v^|(a + l)(/.-i)_u (a-l)(/>4-l)}-{(«-l)^/> + l)_(, + l)jy,_l,,_^.,,^^_^.^_^^^_^^^^ ... >/ = {ab~l) ~ (^,i-Ij)~.c\ substitute this value of ij in either equation. 24. .'. (xi-y)^ ^ (x~yr^ = (a + 2-~(a-2), or o' + y^ v/(a+2) - yia-2), ... u;-^^=/^/(,,4..2,+ ,/(., J2)} -^. {|/(rt + 2)-i,/(rt -2)} =;j suppose. Similarly, (l + .ry) ~- (l-.^v/) = ,/(/. + 2) -/(/,- 2), .•..^7-{/i/>+l^)-|/(6-2j;- { ]/{0-{-2) - V{h-2)}= m sup23ose ; xy x~ = x^ =:pm, X = zt y/'dmi). it' u» '/ It, I Ir .i* !'■-■ I ■( »i;i^! \ \\ 1 . f I:' I It' Mi I i 1-1 : ■ 1 ■ !1 t * .] ■' Li Mi ■,.t > *■>•■• 162 CHAPTER SIX. 25. Add the equations, a5// + .v:+2;,'-= .\('/ + 6+<^), .•. xi/=h{'i + h-r), i/z, z.'\ by symmetry ; multiply these, x^i/'z'^ = k{^i-i-b~c){h + i'-n){i; -i-a — h), from which x^jz is found ; divide this by x>/ = & • . !/z = &e., g.f = (fee. 20. liee example 1, page 1H7. 27. .-. - //2 -=• - ^3 = V suppose, then x= V'('^-'')» '/= l/i'''"" '')» ^= V{c — i'), sabsti*>uting these values i n xij + //^ + zx = V we get l/{a-v){f>-v)-{-s'[h-v)(r~v)-\-i/{c-v){a-v)=v, :. {^/{a-v){h -r)-{-^/{h-v}{r - r)}^ = {,:-i/{c - >;)[a -v)}'*\ or 2|,/(c-i')('^ — i') = ^'"-"/>-6'' + '-^'^". ••• 4r'fc-4(rH-r/)f4-l'"- = (ca-a6~6r)3-|-4/;((Vf-«/>-6c)*'H--l/>2,r2^ a quadratic from which V may be found = ±ni suppose, .*. a-x^ = zt'n, &c. 28. {z-\-r-!/){x-\-y-z) = a, {y ■\-z-x){x-^y - z) = b, {y-{-z -x){z-\-x-!/) = c', multiply together and extract root, .*. [x^i, - z){!/ \-z- x)i;:+x - y) = |/rt/>c, .-. x + y-z= y{ahc)-^c, ,l-\.z -X = y'{abo)~a, z+x -y= \/{ahc)^c. Add .^ast three equa- / 1 1 . 1 \ tions, .'. x + y-^z=i/{abc) (^ + ^-+ ^j ' .-. x=h ^\ahii) // + ]j-\ , !/ and 2 by symmetry. 29. x'^-\-y'^=az\ from (2) and (8), x^ ^^ y^ = i{h-\-cYz^ -\- ^{h-cYz^, .-. ,|2!3{(6+c)3+(6-c)2}=r/2, and 2 = 0, or 2«-^- (/;2+c2); if 2 = 0, then a; = // = 0; if z = %i-r-{b^-\-c'^) then a; = i-(6+t') X 2rt.^ (/;2+c2) = rt(/>+tO-^(/>24-^''), andy/ = rt(6-c) -f- (/;2+c3). 30. From (1) and (2), l-f-a;^ = («+/>) ^2^, and 1 -f-?/2 = (a-fe) -^2^ .-. l-v-:*'+l-f-//={i/(^'->-6)+ |/^a-6)}-^2=l-7-f, .-. 2 = cW{ 1,: ' !■ '• h W' i 'I: ■,,,:t CHAPTEE Vn. Papf.r I., page 201. 1. Add coefficients of x, {(t-\'h+c)x, and by symmetry, the Rame coefficients for ?/ and z, .', {n + h + (']{x-\~!i-{-z). 2. (1) (a- + 2//)3(.ir-2//)3 = (^3_4^3)S, &c. (2) 2{u-ib'^-i-h-i:^ + r^(r-)-a*-b^—c*. 8. (1) a;4-(U'3 + 18a:2-12^ + 4. (2) Factors of dividend are :c- + 3-|-:c~^ and ^2 _3^jc"';. n M)tient = &c. H^-n u2 — 2« , ^2 — 3n , p ■i-x +x -f &c. (3) a:' 4. (1) ^x-'^-^.x (2) rt c a 5. Cube by formula [6] , and for the sum of the cubes suD^ti- tute its value (3) from the given equation. .-. 2 + 3if(l-4a;-)x3 = 27, .-. 1 - 4a;-' = 15625 ^ 729, .-. a;=dil4|/'(-19)--27. 4 4 12 (2) .-. + ^ ^ x+2 ^ X--2 x-S :. 4-T-(.r2-4) = 3-j-(a;-3), .-. Sx^-4x = 0, a; = 0, orl^ 6. a; = number of oxen, 320 -7- a^ = cost of each, 820-f-(a;+l) = supposed cost; difference of these costs = 4, .'. aj2+4.c-320 = 0, or (a;+20)(a;-lC)) = 0, /. ic= -20, or 16. The negative result indicates r. Add result to third. .-. 2.4L' = 74, aud .!/>' = 87, C'.l-r,2, Ii( =Vo. Paper Tl., pagr 2()K. 1. {{a-^h {-c)-v,)^ = {a-\-h)^. 2. Factor first quantity, {it-\-}>)'^' -\-<''^ —^ii^>{a-\-h-\-c)^ {a-^h-}-c){(i'^-\-h- -\-r'- —ah — l/c- (II): secona quantity = ( -fr-)- = {a-^l)-\-c){a ~h — c), :. {a + l)-{-(i)-^(n- b — r). 6. (1) Given function of a; divided by x — a, i^ivrs remainder (found by substituting « for x), \ wbicli, by the question, =0, ;. the given function is exactly divisible by x~a. (2) Put a--i), the expression becomes (6 + r)/vr- (b-\-rj[jc=:0, .*. rt is a factor, and by symmetry b and c are frtctor^, .. expression = wa6t; ; put a = 6 = ^ = 1 and // is found = 1. 1 1 'i- , ( ln ; the interest of %b iov the time «— w, must equal the diseoimt of ^a for the time m -x, or h[x ~ n)'0o - a{m — x)'Oo-i- {l-i-{in - x).0o} i a quadratic ior Je- termmg x. Paper III, page 209. 1. (1) (a;4-?/ fzV^-?/ -2)x ^'"^'^ '^ = {x-z—y){x-z+y) = (^-;s)2-v/3 = -117. (2) rta(2-:t;)-+(a;-?/)a/;-f-{//-z)/>-. 2. r/+/''"+f*.'*^^. which may be found by factoring, as in Art. XXIIL 8 — '■ix-\-l.i''- — liJx = square root. 8. (1) See Algebra, page 145. Equation reduces to form 1 1 T^ + .^Q' •'• ^^'-^i- x-\'l x + 4: x+2 x-t3 (2) a; = H, // = !), 2 = 12. 4. Let a; = number of dozen bought, ;. 45.r = price paid, 5. Let a; = number of bushels in lower priced, .-. 75— a; = do. in higher. • • 40{x-+:;) = 45a;, .'•. x=^\%^ (""ozen = 160 oranges. 78 75 = $1.05 lower priced per bushel. — = $1.20 higher 75 ^ (< (C .-. 105aj=(75-a;)120, .•'. a; = 40;35. )n rerlnces to •■•9 + 2^=17, ben rent of $i for J time m-x, Iratic lor Je- , as in Art. to form !e paid, 75 — a; = do. ingGs. CTr\PTRR SEVF.N. |Qy 7. Equation reduces to (40.*:- 101)(^'-.8) = 02(0,^ -7)/j._.2) whence X = 5. or i. Si.ico the given ,.qimtion has e<|nal roots, ax'^ - 3Gu; + 81 is a coiupl. tu square, /.f., i„ x 81 = (30)^, .-. « = 4. y. See Algebra, pa;^'o 121. 0. Substitute ~^(h+c)iora, --{e^n) hi h, mu\ (w-f-Mforr. in tlie given expression, and it bee nucs .'. a+b-hc IS a divisor, &c. See Algebra, page .i:{, prob. 12. Papi;u IV., i)age 210. 1. {(o + h)2-r^}{(a+cy^-h-^}^ )^a^: (/. + ^m -'[=(., + /. 4.,.)-... 2. T e tirst and second fractions requning to be inverted onr, and thm' times respectively, will finally stand inverted, while the third, requiring two inversions, will stand unchanged. .• result = a b. 3. (1) {2x~dij)(2x + S!,),(^lx~S>/){2x -%j), (2.y-3//)(3;.-2v). .-. L.C. M. = (2.«-8//)(2.f+8v)(2u;-2//)(3^- -2^;. (2; 1st. ={lJrx)[l+vx)\ 2nd. = (H.y.,.)^2...+3v/;.3), .-. G.C.M. is l4-;s*. 4. (1) Ji/3- 1,1/6. (2) Extract the square root and the remainder is 17.^--' //^ -^u-^y/S, which must = 0, .-. r = 17. 6. Left-hand member reduces to 4, .-. m = 4; m = some quan- tity involving X would make the given expression an equation. 6. v/(H-.t')-v/>j ^ 2-fa;-l-.c v'(l4-^V+ v^i'^") l-hx -X = 1. 7. (1) Eeducestol : (u.4-2)-l -- (..-2) + 1 --(,;- 1), whence « = 0, or 4. '1.:.. ■J. 111 '.. '. y hk I. f lit ■ ' 1 1 ■ t| i I ,1 • t ■<■ : I IGS CHAPTER SEVEN. (2) Clearing second of fractions and combining result with first, .-. 7[13i/-5x)= -ddxi-lO'2'dij, or Sy-z, whence a; = 8. 8. ic = distance by carriage, // = distance by train ; thenar -^ lO + z/ ^ 8G + (22i-x'— //) -^ 4 = If,; and x^4: + y/-3G4-(22^~x-//)^ 10 = 301 -5- 120; whence a;=7o, // = 12. 1). See Paper II., prob. 7. 10. (1) Divide by x - 3, remainder is 6ij^ +9?/+ 18, which must equal ; .-. y =^ [( — 3ib V - 39. (2) {a ^b)= -l,ab = l, .\ a^ + ab-j-b^ ={a-hb)- -ab = 0, .'. a3 ~b'^ =^ (a-b){a^ +ab-\- b^) = {a - b) xO = 0. Paper V., page 212. 1 1. Sx^-^ YoF' 2. Substitute — 2/ fora; and the expression van- ishes; or expression is divisible by (/i^b-ewce of quantities, *.t',, by (*ic-.v) — (;c - i//) = h{x-{-y). 3. (1) Expression {{a-\-by'i -{a-by^}^ = 16a-b^ ; (2) Expression = (^/+^ 1- c){a + b-c)[c — {a-b)}{c+{a — bf] = {(«+6)2 — 6'^}]c2 - (rt-6)^}, expand and substituto a-'i+b^iorc'^. (« + 6)' nb^ - X - X a^ — ab-\-b^ b lab " a3-f-63 ^ 4:a{a + b) 5. Let it" = rate of " Rothesay " per hour in miles, "2/= " "City Toronto" 35 35 12 42 = 5^ and -; — ^ -h - =6 J. X y !/ From these equations a; = 15 and y = 12. 6. (1) Multiply second equation by 2 and subtract, « a .'.x= -g- and j,.= -^- CHAPTEPv SEVEN". 1G9 iiig result witJi wlicuce A* = 8, and 0; '8, which must 0. 'Pression van- titles, i.e., by id subsfcitutu (2) Add 2R to each side and ti-Hiispose. Patx^ + 5..-^28^.^ ••-'^-5.^2.4, .-.az^Sor-S, substitute iu given equation, and .<; = 4 or - ,u, ^ a(x~ un{x~. n). (2) Let r = each root, .-. 2r= --- vahie of x in the second equation. .*. / = 2^, wiiich is the 9. ■tn.'^ n' X' y X^ ni-i ?/2 ,^2 „2 ,„2 VI ■> .•' I 9 » or 6* ^/-- y- Paper VI., page 213. 1. Numerator of first fraction is {a{x^ -}i^) + 2h,r>/^^ = ^/2(^2...,^2)2+4^i^^^^^2_y2)4.4/,2^2^^o. ^^,^^^.^^01. of second (by symmetry) ---.6-^0/3 -x^^}2+Uaf/x(!r^ -.x^-)+4ah,^x^ ... their Snmis{a-^-i-b^~){(x2-y'2y2_^4^^-,y2i^^,,2.^f,2^(^2^,^-,p^ of which the latter factor is cancelled by the denonnnator. 2. (1) By common division or by factoring, a^-^h^+c^-i.ah-{~ar,-bc. (2) This is the same form as (1), and has for one factor, n +x-+.c^) - {1-x+x^) - 2,4 = 0. Or by formula [0] the first three terms ate seen to be the cube of {l+x-\-x^)-{l-x + x^') = {2x)3^&Q. 170 CHAPTEH SEVEN. ■A r j1 ' 3 (1) x'^-hi/^ + ley. {'2') {7x+6rj-9)(x-y+i). See Ai't^. KX. aiid XVII. 4. (1) -20. (2) 0. See Ex. 1, page 44. ±:(l — b) v^c/. So secoad term of numerator is fonnd to l)o + (1 — />)]/<^ .*. i/a caiicols from both terms of resultiug fraction, and result is h, or 1-^6, 6. (1) l^a-\'a^{l^a) = 2-l^[l+a)yl. (2) (a2 + />2)-^a/>>2, ^•.^., if a^ _|_/,-' >2.//j, or (a- 6)2 >0, which is tile case, since the square is positive. 1 1 7. (1) From second equation, aj^ + i/^ = -^^^i/^> •*• from tlie first equation, x^y'^ = 6 ; whence from first equation x = 4 or 9, and from symmetry, y = 9 or 4. (2) Adding {x-\-j/-z) = 6, and (1)- (2) gives -2x+y-{-z = '^, thence x' = l, v/ = 2, 2 = 3. (8) Take together the first and fourth factors, and also the second and third, .-. (.c^ +7.^4-6)(.f2+7a;4-12), or (aj2+7,, + o)2 4.G(.t;2+7.,;+6)^lG, .-. (0^24. 7.,, 4. 6) = -3 + 5 = 2, or -8; .-. :»= ^( --7±|/33), or ^( -7±:i/ -7). 8. Let x-1, X, x-}-l be the numbers, then (x-l)^+x'^ + (x'4-l)3 = 10? .r j^ + 1), ic is seen to be a divisor of first and third cubes and .•. of the equation, .*. x = 0; result is 7aj2 — 38a; = 24, which gives ^ = 0, or — i, .-. the required numbers are- 1, 0, 1 ; 5,0,7; or -V, -h ' T 9. (1) Irrational and impossible roots enter in pairs in equa- tions whose coetftcients are real and rational, .'. the requireu equation is a:(.6-- v/ -3)(a;-l- i/-3)(a?- l + ]/'2) x (x— 1— v 2)x /(x) = 0, where /(x*) = contains the other roots. CHAPTER SEVEN. 171 See Alts. fonnd to lie of resultiiii' e. from tlie x = 4: or 9. Qd also the x + 6) = 7±|/-7). k and third -380^ = 24, e-1, 0, 1; 's m eqiia- e requireti l-v:^)x (2) y(pq) is a root substituting tliis valne for x in the equa- tion we have pq -h P]/{j>q )-}-(}= 0, or (J^|J ^ 1)2 =^ j/^q^ or 10. Let .K = nnmher of mile:^ per hour of train from .4, nnd // = number of miles of train from IJ. Then the whole distance = 1^ (x+y) and l^(u;+?/)-^.r = first train's time = second train's time + r;2:V minutes = 1{, (a: 4- //) -f- // + !-; or by division, 1^+ U\ ^~j =--l.V + li I — ) +i, or 12//-4-u;= i-i.^-:-i/T7, multiply by Paper VII., page 214. 2. First two terms = because e is a factor of euch, numerical value •01. See 3. The quantity vanishes, when x-—a. or —h, ka. Algebra page 41. 4. (1) Second quantity = if3 + l +2.^2 - 2:= r.r+ ' )(^2_|.^_1) . the tit'st quantity does not vanish for .c-f-l = i) ; and X'-^u — l is found to be the H.C.F. (2) First expression when factored = (.r-f-y) ['.'•- !n{(ix + iij)} ; Second " *♦ " =yx-~ y)^^\x->rrM'C-l'U)\■ .'. a;2--?/2=:H. C. F. X- i) Vlx GO . ^ 5. (1) First two terms reduces to lo, < -^ ' ^ i -.\-tx~ — J) 1./. — 9 .'. whole expression = X +5 X - 4 __ 86./;2 4 1 S,,; -f (2) Bring first two into one fraction and do similarly with second two, after cancelling we have { (:c - a ) [x + 6( -f {x -i-a){x-ln} ' 1 Jtll I IHi'.: m I : ;. ■<"■: - ' ■ij ■•! ,« 172 CHAPTER SEVEN. 0. (1) x = 9. (2) Equation reduces to (-^+1) +5:.-4- ((-^-1) + 7.^1ol = ^2^- ^^-^ ■ (3) Subtract second equation from first, .*. X- —y^ =ax-\-hy —bx — au, .\ {x+y){x — y)=n(x—y)-hh(?/ — x). Divide through by ic—y, .*. a;+?/ = a — />, and since x—y is a common factor, .-. x = y, .-. &c. It is seen from symmotiy that X = //. Papkr YllL, iiage 215. 2. (1) Product = (.e^ -y^r^ir-y) = x'-x'^y-2x\//^+2x\f/^^- ^y^—y'^' (2) The quantity must vanish for .r+p = (), and for x-\-q — ^, and.'. =.''''^ + (/'' + '?)«c+M' •'• ^'=p-^^h f^ = p(i- 3. (1) Dividend = (o;-^ + ?/3 )2 =(.:c4-^)2(^a;2 -xy+y'^y^ which divi- ded by {x + y)^, gives [x^ -xy+y^)^. (2) Dividend=(rt4-|-/;4+«o/,2)(,^4_|_^,4_a2^3 ; divisor = a* + ?>4-j-f,2^2 •. quotient = &c. 4. Let wbe the number, dien n^ + {i{n^ — l)} - =n^-\-i{n'^ = 1)2 = i(4?i2-fyrt-2;/2 4-l) = i(n4 4.2?i2 4.l)=|i(n2 4.i)|'j, 5. (1) Arrange in descending powers of x; or, by inspection. ax-^ -{-bx+ ex -I 3 6 7 .12 (2) ia;^_la;^ + l;^;^^ 6. (1) Let u;4-)y? be the C. M. then m^ —am + h = 0, and w2 —a'ni- /> = (), (ct-}-'^*')"' — ^-^' ^^' 2//^ — ((t+^i^ ) = ^> w \(a-\-a'); o?- the C. M. nnst mecsnre the sum of the quautites which is 2x'^-\-{a-\- = (8a-f-6j)M, a4-26^ = (-a+462)m, wlience, &c. 8. Transpose and square, the equation reduces to 16(a; + l)(a? + 6) = 9a.'2+00.c+225, andx = 3,or -43 --7. Api^arenily neither value satisfies the equation; but any quan- tity; has two square roots, and if the negative root of \^{x + l) {.f-f-6) be taken, the valiuj 3 satishes the equation. Papeu IX., page 217. # 1. Qimntity = (n^-\-n~}{2' ni-\- a -'Ian- n) + an(2n + 1 - 2rt-- 1) = (^a'^^n'-i){u ~ ,i) - 2an{a - n) = (a - n){<( 2 -f-7i3 — 2an) = {a -n)'^, 8. Let quotients be x-k-r and x + s, respectively, so that x^ + ax^ + h~ yx^ 4- mx+n){x~\-r) = x^ + (in + r)x"+{n+mr)x-j-nr, and x^+pxi-rj^ {x'^ + >iKC+n}{x+s\ = x'^ + (//< + .s>;- + {n-\-iihs)x-{-ns. Equating coefficients, (1) a = in + r, (4) /,7 + 6' = 0, (2) n + mr-^0. (5) n + m.^=p. (3) b = nr. {u, 71.^ = q. (3)-(0), b-q = 7t(r--s) =.na, by (1) - (1) ; /. ^b-q)^ = n^a'\ Also, (l)X(G), bq =. nhs :=: n^ [from (2) and (4)], .-. [b-qf^ =a^bcj. 4. ,. •'■st quantity =:(){ax"t/ '■\-a^'x bxy'-b'^ir'), ,'. c>xi/i~OisE.G.F, lil:- r ;,, 4: '•' !■ ■, I ' t m ■ .1 i ■ I- f. '1 ' I m k i' Iff ' M' ■i' (^ ''\ 1 . 1 ' r ' * 1 t ; . - 1^ if! ' * 1 • * :. ^.i: 174 CUAPTI,1{ SEVIiN. 5. Let rt = left land uigit of A, and 6 the right ; let r? = lefi hand digit of B, a d d the riglit ; .". 10r< f /> = A, and lUc+^Z = 11 . a — c = x, li)a + h — l()c — (l=//, (■-\-il — (i--h = z', .•.da-dc = ij+:: = dx. J, 77; />, (JO. 6. Let each ratio = .(; ; .*. a = l\r., b = rx, c — djo ; .*. abc = bcdx^, and u; = |j^'-— . Also -}- r 4- (/}^, .*.&(}. 7. (1) Equation, on division, &c., becomes a a X X x — a x-{-a J- , &Q. x=zt Vab. h — X l)-\- X (2) /y= s [u-(j)^y[ — 7 fa^ = 0; also when y; = 1 , .-. !■+ i>-{-q-\-((^ =0, from these two q = and l + /'4-'<^ =0. Second, the quantity must vanish when x-a, .-. r/4-|-.)a--f 7rt-j-r«- =0 and also wliena;= -a, :. a-* +p(i^ — qa-\-a'^ =^0, and these two give q = (), and l+p+a^ = 0, the same as before. Or substitute in given ex])ressiou the values of p, q, from former, and result is plainly divisible by x'^ —a'^. 3. 3(first quantity) - second = 3x^ - lOx'^ - 8 = (dx^ + 2)(:r2 _ 4), of which first factor is not a CM. x^ — 4: = [x-^2){x — '2) ; tlio quantities vanish when a; - 2 = 0, but not tor a; + 2 = 0, .-. x — ^A is H. C. F. let c = lefi '. abc = bcdx'^. Vab. 'i~b). of 6 sheep, ach remain- -q fa2=0; g-0 and len x~a, .-. -qa-{-a- — 0, as before, 'om former, -2)0r2_d), .*; - 12) ; the CHAPTF.P. SEVEX. 175 4. Let :. = distance, the,! j^-4.-| + -|- = ~. + -^ +1, ,. = 24. 6. (1) Square root of first term = 5.;^ of k.t 4./^ r.vV. j-roduct is40;c^a2. b„t expression contain.^ 4.\),c-a\ .-. Sl.,;-,.^ is square of third term, .-. 5x'^~3ax-\' ia^ Sec example 1, page G2 of Iland-Book. (2) Expression = ~- + 2 + '4- (-- ^ -'^) ,/2 + I- ^ y 1 '. y X X )/ \ ■-' ^ — + ---_. y/1 1 ^ _., square root ^kc. // * / 7. (1) ^+14- 5^...4-.^+l +7,li,j, .•.7«:-10=:;^.r -1:^ = 3 (2) (4a;-2)2 = (5u;_3)2~(a.-l)^ = 2(2^-2i(4u;-2), .-. 4u;-2 = 0, x=},. 8 Lot 2a?, Qx, bo number sides respectively, then 4x'--l = number right angles in fii-st. 6^- 4 -number in second; .-. (4a;-4)-f-2ic, and (Ca;-4)-^3^ are the magnitudes of one angle m each, taking one right augio as unit, .-. (4x-4j — 2.t; : (G^-4) -^3.1-, :: 3:4, and 2x- = 4, 8x- = 6. Paper XI., page 219. 1-2. 2. Let a; = time for A, ?/ = time for B ; then 1 -f-.P = part done by A in 1 day, and 1 ^?/ = part done by B. Also 4 -^3.r and 3 —- 2/y represent parts per day on second supposition, 1 - + X If o 1 7,4 .) — = — , and — -|_ 24 dx %j 50 144 , or 1 ^ . .. .' 'All 16 8. (1) 3;d factr)r=(2/+3//V; 4th = (2/-3//)^ .-. product = {2x +3// )3 X (2.« - 3// y^ ■= {l/ ~ 0/y';3 ^ ,tc. entire 176 CHAPTER SEVEN. ii-' ■ »' Et , [ M: . ' I i;;i!^ -i 4. (1) a;8+3 + 9^--. (2) By xlorncr's Division, or by factor- iug, X' ~ [a + h)x - c. 5. =^'2"-i(a;-'«'--"+3_ 1), which vanishes for a; = 1, and .r= — I. since 2/*i— 2»-f- 2 = 2(_ >» — >? + !) is necessarily even, 6. Add the fractions ; numerators ^^ (,,2_/H;)(/,4.c.)-j-(/,3_.c«)(r + fl)-f-(r2-rt6)(a + ft), wliich vanishes for (/ = (), and .-. for />--0, and c = 0. Also it vanishes for a +i = 0, .*. for 6-fr = 0, and r-fa = 0, i.t'., it van- ishes for six differe t I'actors, .*. it vanishes identically. 2 3 7. (1) Equation reduces to ^^:^+ ^j^^^ ^:^:^^^^ or -a:-^(aj+l )(.(• + 2)^ -^H-(^+8)(^+-l), .-. « = 0, or -2^ 21 ( X I 100 ( y ] (2) --.u^+^s)-, = -~.|(;,2_y.)._[ or G;0a.4 4.o3.-'//2 = 400ic-?/'^-10()//3, or (7x'--25//2)(9^2._ 10^8)^0, .-. 9uj- - 16//- = 0, and a;= zh;^//, substitute in first equation, //:^± 3, or ±1/ — Paper Xll., page 220. 2. {a+h){a-h) = a'~h^\ {a-b){a-\-b){a^ -rh^) = a^ -b^, &c., i.e., the product of three factors = «'*64 ; oifour~a^—b^ oi jice, ^a^&^fji6 ; .-. of ;? + l, a'^-6". 3. Apply Horner's method, 2 I 1 ^1 I +2_ -f2 + 4 + 4 +8... + 8 + '1 +-1 +2 or l-\-x + 2r^-\-ix'^-+-Sx'^-\- In this series note that (1) the index of a:' in any term is one less than the number of the term ; (2) the coefficient of any term is that power of 2 whose index is two less than the number of the term. (3) the remainder after any .erm = next following term. .-. (/•-i-l)th term = 2'"~V, and the required remainder = 2'U;''^i. OHAPTEU SEVEN. 177 >r by factor- and 07 = — I . = 0. Also it i.e., it van- Ly. 3 or ±i/-0 *-64, &c., -6» oi Jive, 3 note that number of lower of 2 irm. 4. aj and S-a7 = tho parts, then x- - {'d -.r)^ = {x-^H - x)x {x-(S-x)}='S{x-(S-x}\. 5. (Difference of the quantities) --:- fir - 1 -;*;,/• r- Tm--. which divides ihe second quantity, and also the hrst, ^'iviiiL,' iiuoiieut, 1 nx-u\ .'. L.c.M. = ^i-3a;--u--5)(i-2.«-iat' + 3dt-J-24.r4). 6. Expression is x^ i-2mx^ -{-xV in-\-)lii -\- f>)-{-.rr2wn-\-(i)-}-)i- ; the first two terms of the root must he .r- +ui.r and the last, teim n, .-. the expression must be {x^ -{-'Inix-^ii)" = x'^4-\liii.i:-^ -^ x^{hi^-\-2n) + 2,uinx + n'-^, equating coGthcieiits wo have /y/-+2/<- m-\-'2n-\-p, .-. >n'^ —m=:p, B>\ii\.'2i)ui-'l)iin-\-i], .-. (/ = (), and // is independent of the others and may have any i-elafion ti) them. 7. (1) Reduce the fractions and add, .-. 2+ U-'-' /(l -^••'), x:^ = [a-'2)^{a-^l). (2) Eoduce first fraction, .-. \/ax--b = {yax~l>)~-n-~c, .*. i/x= [l>[n- Ij-cn}—- v^'tin-l). 1 12 2 3 3 (3) Equations reduce to 1 = — n = — -] -1, .-. X y z x z y /I 1 \ S , ^ / 1 1 3 =-r and 3 — \ — 8. y « y l,.-.// = 2J; x=l^,z= -12. ax by "(?/4-2) alz z "~ g ~ z ~ (a - l);s " {(a-- fj6 — «j2; ~ ah (^r:iK6-i)-I* 9. .x'=tens, // the units, then I0.c + // = the number, .-. a: + // - U), 2Ux'-f-2// — 1 = 10^ + X, whence x = , y = l. Paper XIII., page 221. 1. The factors of the nuiaorator are a- -6-, .-. the exjDression^: {{a^ -b^){b^ -c-^){c'^ --'-f')} -^ {('(-}-h),)=«: -''a"n. .-. a,+rt2+a3 + rt^= "'-Z"2'*n, 8. artiu;2+/>ai^--fmj=0 (8) ar/j»;2+,,Aj,,;4.f^(.j ^r () (4;, (3)-(4) .-. {a,h^ah,U=:ar,-a,r, ,. .,= "'')~:^, (5)-(6) (rtP, -a^c).r:i = {h,€ bc,)xov{ac^ ~a^o)x = {h,c~h';] ar, -rtjc *' ,ic^—a^r ~ a^b — ab^' 9. Let ic = left-hand digit, // = right-hand digit, ;. (10x-f-//)+(10.v-f.6-)=:121, .'. x+,i = n (8)* (10a;4-//)-(10//-}-.c) = 9, .-. .C-//-1 (\) From (3) and (4) x = (), {/~o, .-. number required h^ 65. ^^^ > >. ^>r^^. IMAGE EVALUATION TEST TARGET (MT-3) .<^t^ 1.0 I.I 1^12,8 125 ^ U£ 12.0 IL25 i 1.4 i 6" VQ V) / Photographic Sciences Corporation ■ •^ \ «> v> 4 33 WEST MAIN STREET WEBSTER, N.Y. I4SS0 (716) 873-4503 \ •si,^ / '^""l^ ;" .^% ■ 180 CHAPTER SEVEN. til jir, ■ A 1.^'/ I*' iU . 1. ' ,■ ll Paper XIV., page 222. 1. The expression vanishes when a = 0, .*. rt is a factor, and bv symmetry, b and c are factors ; /. the expression = nalx- where // iu numerical. Let a = 6 = c = 1, and then ?7 = 12, .*. the expression = 12a6c. 8. ^2+^^24.22= _2(.T//-f?/2++2.tV hence a;'22/5!-a;=//'2'+2/'=^z.^-.'/23'.»:'-h2'2a;//-z2a./2/' (i) = a:2a:/(2'+2/')-:r^-'-(3-}-y/) + //2^'(.»'+2!')-2/2/'^(2+«^) = x^x'{-x')-m'H-x)'\-7f',l'{~yi)-inj"^{-y)+z^z\-z') -2r'22(-2')=a;2a;'2-a;2rc'2 4-&c. = 0. -¥{z^^-z'^){xy-x'y') = xyz{x ■¥ y -\- z) - x'y'z'{x' + y' -\- z ' ) -\-{x"'yz-xh/'z'+y'^;:x-y^zx'+z'^xy-z^x'y') = 0. 4. Let ic+dist. and y = Q's rate, then -^.'5—4 = lx~-(y — 1) .-. i/ = 6, also■^x•-^3 = ^a;-^.v4-•|■f, .-. ix = ^\x+^^, dj = 8mls. 6. (1) x{7nn — n^) = am^—amn,x = n (2) Add twice the second equation to the first', then (a?2+i/2)2 + (a;3 + //-)- 182 = 0,.-. U'^ -f //2 _ I3)(ar2 4-j/2 + 14) = 0. ;. aj3+t/2 = 13 ora;2+^2=:_i4 (gj ;. a:2't/2= 49 -13 or 49 + 14, u:^=+G or ±8 \/ 7 (4) taking x^-^-y'' - 13 and a^= ±6, (a;+//)3 = 25 or 1, .*. « + j/= ±5 or ±1. {x — yY = 1 or 55, x~ y = ztl or ±5, &c. CHAPTER SEVEN. 13] C. Let X and y be the numbers, theu hix ~ ii) = .',; (x i- >/) - ? j. 7(^-/7) = i^'//= 6^, .-. ^a; = |27/, .-. since a; cannot =U, // = 2, 8-2-1" =^= "5 •. j;=-i(). Paper XV., pa^'e 222. 2. From formula [8] tbe expression = 0-{-(( -a)3 = i. 3. c3(c2-6(/)4-^/2(/;2-«t')+m-(rt'/- '''•). 4. Let X represent tbe time after noon, then mx-ihr -f-2, .-. x --= 2H-(m~?i). Ify/^.-n is negative, x is uo<,'ativo, wbicb shows that they were together before noon. If /// n = 0, x is infinite, which shows that they are never together. 5. (1) {x-^a){x~'2a), x'X^x+u), -x^x-u\ .-. the L CM. is x^{x + a){x — 't)(x-2a). (2) Quantities are {x-a)(x-ha)(x—y), and (x-\- ->--{-, I ) -^ {n~r^-^-c-{-d), :. by symmetry the denominators of the others at. *be same, and the numerators also can bo written down at once. .'. xesnlt is {a -\-h-\-c-{- Sd) -^ (i(-^h-{-c + U). (2) {x^y+z)-^{x-y'hz); 3a-^{a-{.b). 7. Multiply sum of first and second by h, and arrange result, ; .-. ab{x+y-z)-^bc{x — y~z) = 2h-{c — a)-^ah(a-^h)- 6c(6-Hr). and by symmetry two similar results may be written down, ; the three results, .-. 2aHb-c)-\-'Ib^{c-a) + 2c'-{a- b, =0. 8. I0x + y= P. lOy-hx=Q, . l^-hQ=ll[x-^y), and i'-{4- 'i){x~y), .'. \l[x+y} X^[x - y) =^i){x- ^) X ll{x-*- y), an identits . w i » , K'l' ,ip : l»r . I /t :>■ *ii i, !■' . I. ♦ ' I t, ■ J ) ! ' . '■ ; ' ■ 182 CHAPTER SEVEN. Paper XVI., page 224. 1. (flr2 + />2)(.r2 4-?/^ )(«*-' - /.2 ),;a;3 -?/2) = (a^ - b*){x* - y*) ; {(a-^>)-+-(6-r)4-(c-«)}2=0, .-. ( ""ah (2) 8./,2 2^A-10r(c'-86-' + 2r^' + o/,^. 8. (1) 11" .c-hrt is a factor of x^ i-j>.r-j-q, thenx^+px+q = {., + a) L + 't] ^ hence p = a-{.±, ... ± =._„, :c+7 = (...4.,o(.-<;+;p - «). Similarly, x^-{-ju'-[.ri' = {x+a){x+j/ ^a). L. C. M.= product aivided by H. C. F., = (^+rt)(.c+'OU'-W>-")(-^-+/j'-ff.) x~f QQ If pendent of x. 4. By Art XXXVIL, bx+cy+dz ^ (a + M-^+c/V; + aual. + anal. x-\.,, + z b + c^d-a ^a'■]~h + c^\■~d) '" ^ ~ 2 ' 6. Substituting 3 for x, the two sides of the equation become equal. 6. .a ^^ + 3 (Wl^ = (.+ l|^ Sectio.II.,,]. (») Difference = — + ^^ ^ ^,^ ^ 3, ,„„, ,^ .„^^, .c c Hence (a; -f — j =?>i, and (^- - Jl) "* = «, -t) :c .'. {c + — = ^w, and ic - J_ = a n ■ i ' At '3 1^- vf. V I;. ;.;.. U ! Mr ■f-^ 4 184 CHAPTER SEVEN. ^ .» 2 2 2^ = ^m + i^'w ; and — = ^ni - f'n ; .'. m^ - « ^ = 4. 7. If 1 1 a ^ b J_ _ 1 _ c ~ n + b — c' £ 1 r (I -^h ~C Now, let a = - />, and expiessjou becomes zeio ; 1 1 Then — + -r a = 0. Hence a-{-h is a factor, Similarly a - (.' " And h - V. Hence, expression = m[ii 4- />)(« — '•)(6 — c) = o ; .'. one, at least, of factors = o; .'. a, 6, c*, cannot be all difrerent. 8. Let a; = number of gallons drawn from first cask. Then />+c — x= " " " second cask. Hence - — x -\ :- (fe + c — a;) = 6. .*. X - --- — — {in-\-u)^ mq—pn J . , mc — hn , , X ando+c — w= (/^+'/)- mq—pn Paper XVII I., page 226. 1. [1) (l-a;)(l + a;)(H-a;2)(l+.«*)(H-.c«) = l-a;i«. (2) Theory of Divisor, page 39. 2. Since a(ft-c) 2 -c(/> + <')2=t> c ~ \b-cf V^ b-c Va- V c ^/a+ yv _ b _1 _ 1 b v/rt - V c c «/ b. Let a; = As age, and 3 = B's age. CHAPTER SEVFV. -,q^ loo Also, father's age = 2,;+5,. n„,I mothe.-'s ,„, ^ -^ _ ^^^^ •■• "iotl'ti's ago is ,•.• of iather'a. 4. The difference between tlie c,nantities is « - 6. for ;.3 ,, „„t j„,^,y^, ,,^. ^ _ be -or («+6,= will not cliviclo „■.(„ + /„ „„r ,,=,„+;„ Heni? V*/, J qnanht.es have a common measure it m, Jt bt J-« - ;. tl by trial, ,« found to be a factor. ' 5. Identity holds if (n + /y-,„.i +(,+„ _ ,,,3 ^ 6. (1) Book work. (2) Multiply out and extract the root in the ordinary way ■ tnevaultisa'-'-rc//-u'2-//3. 7. 1 + , — u — o a «. (1) x=9a. (2) Square and simplify, and or(2^-2_i)^2.f2-;-i) = o ''* (2^;- — 1 =0, or »• = a- -H,T = 1. 1_ \^2 9. Let :«=:oue parf, then 21 -.; = the'otl'ier • Ilf % i I: ;1i'' • '.. ' si '1. ■ I: I [ 186 CHAPTER SEVFN. Paper XIX., page 227. 2. From the conditions j'i^-{-i/z-\-zx = (1), and x^-{-y--^ t'' = (2). Also by formula [8] and (2), (u;'"»4-2/--f-2^)^=^;"4-//"+3"-.%,'2v322=0 (8). U'+//+ ')•'' =-^'^ +//*'-f-2^ -y.<7/2 = <)...(4) now divide (3) by (4). 3. (1) (5.c-l)(4.i-+lj, and (5a;2-lj(16.c2+3^-t-3), .-. 5a;-' -1 = H. C. F. (2) {x-\-i/y=7x!,{x+f/)U'-+X!/+y^)^ (x^-!/^)^ = {x-y)^{x'i-\-Xf/+'t/'')^, .". the latter factor is the H. 0. F. 4. First eliminate xy, and find the value of x+y, hence (x-^y)^ is known ; second, eliminate x + //, and find the value of 4a;//, which subtract from that of [x+y]'^ and (x~y)^ is found. Or, ab- {a-i-h){x-\- y) -{--ixy = (1), ed-{c+d){x-^'y}-{-4xy = (2), and let {x-y^^z (3). (2)-(4) .-. {a-\-b-c-d){x^y)z=ah-cd (4) (!)-»- (3) .-. ah-(a + b){x-^y)+{x+y)^ =z, or {a-(x + y)}{b-{x+y)]=^ (6), But {a^-h-c-d){a-{x-\-y)} = a^ — ac - ad-\-cd= {a- (')(a — d) (6). Am\ia + b~c-d){h-{x+y)} = {b-c){b-d) (7). (5), (6). (7) ■.(a-^b-c-dy^z = {n-\.b-c-d)^a-{x-{-y)}x {b-{x-^y)}--=(a-r){n-d){b~c){b-d) (8). (3), (8) .-. {.v-y)^ = {>t-c){a-d)ib-c){b-d)-r-{a~{-b-c-d)K 5. Add (1) and (2) and z is found at once = bx+ayj which substitute in (2), and wo get . y^(l-a^)=x^{l-b^) = {hy symmetry) z^ {I- c'-i) 6. Apply (6) page 122. a; = (a+2)2H-(4a2 + 4a). ^w CHAPTER ^'FVKV. 7. Extract square root, remainder i« wliencc^- = (7/,-' _,,-)_.. 12,,. 8. Let r = number women tiipn ,.j_i . = number children .-. 4.4")n ' a 7J'"""'"' '"'="' ^ + 1* Paim;ii XX., paf;e flK 1. (l+,„U4-(l-„).v. (a)Therem.i„deri,„„= „ „^, +»«-/,«+,■, which mns. vanish for all vah.Jsof ,r"-""'+'^''^ . . p^{mn + r) ^n; equate these two values of . arid ' theH. C. F. ^ ^^•^--'^'^•+/>). .-. x-»-l i« 4.(1) =(:.+.v-.10/+,_^.)^,^,,._^^)_^^^._^^^^ ^^ 6. Equation reduces to 2 ~ (8.,; - <)) ._ i _^ / 1^. " o> _ 6. a"{cb>-bc ') + 6''(./. ' - a ',)+c\a 'b ~ ab ') = 0. 8. Let :« = number of first kind, .-. c-.. = that of second. X c—x JL a ^ b =^ h whence x = <({b~c) -^ (h -a), c-x^b{c-a)^{b~a). 0. Let ;. miles per hour = rate at starting, .- 4+ • :,.,h^« required to walk distance, an//4-,'2)2 4.(rn/ - M- = (r?= + /'-')(r2 4-//- )4-''-2'4-2r^(//). Also Now add, .'. ttc. 0, p. M. RyEx. r.,p.ll. 2(./2+A^-a;2)84-('',}}- -M'lh + u^o^y-i = {{<, 4-h)2 -{a, -6i)- X {a-b-{-ai+bi){(i-b-(iy --/'i); .-. 2{(a3+62-.c2)'-' + ^,,'^.|-//; -a;-^)-'-2(.»6 + rf,/>,)2| Bimilai-lyibr 2[(r/-+-|-2c + (/.j}"'" (6i-t/).f-a(nf + 2/^+2r+ti)-26c :. ^t- + 2c4-c/)-2k_U, unless the denominators are equal, i,e„ 6(rt4'6+26'+(/)v-6'(«-t-2^+c-i- tij, which reduces to 6 = c, or a-i-64-c + t^ = ^. ADDENDA. 189 80, p. 165. m/ ^ _ ix—m x~q I "^ n 7 «(^-fe-~| -I- .(.-.) ^-__^ ^_^^ .-d)p- . _L.) = .'. a; = {or, "'(rt~/>)fw-r/)-f-ra:->/?j(a;-y)4.&c. = 0}. Pp. 175, 170, 177, Exercise LVI. 75-87. Lr-t a~x = /.-+i/, x~h = k-.y, a = Ic+m and.-. ~b = k--iu, from wLicli it follows that 2k=.a-h, 2m=a-\-b aud y^i(a-^h)-.cr=.m~x. Also let j bo dcliued by the equation 7* --f 1 =0. 77. (_y:.'/)** + (>^^~Z/)'» _ (^■+_m)* + (k-w)^ .'. y3-w3=o, a-d .-. ;y,+m = 0, or?/,-m = 0. (By J) or Ic'^'-{-(}/i-^nL--^)a^ - /,.2_|_,,i2 , aud .'. JHok^+m^)/r- •■• .'/3=i^^'''. 1/4= -jki; in which r-' = (5/j3+m2)-H(/.-2-}->,?2). 78. .-. ('^■ + //')'* + ^^'l:://)* __ (/.•4.,;;)4^ (/._„, )4 A;4 + GA3//i2 + ;/,4 - A3 4-3A-/^/'3 ^4 + G/.3,/,2 4:„i4 - /.^J+y/.va-J- .-. ?/3-w = 0, and .•. v/j +?m = 0, 7/3 -m = 0, or .l!±i!^+l^_ _ _1_ or v^ + ^^(3/.^ + m2) _ ♦^ ^ /.2 . o,r^ "• k^ + iik - wt3 4. /;, 4 /,. 2 J. '^,,1 2 /v3+3?w2 f 190 i I f' ti f 79. 01 80. 81. 82. APDVNDA. (/l-h?/)'-»-(^-?/)* _ (^ + m)*-Kfe-m)* (A- + /7) 3 4- (A- - //) 3 ■" (A ^- /// ) 3 4- (A; - m) • A» + lU/.-3m3+5/.w,4 "• P-ttto» 6(/^-m4)-hlO/.-2(/y3-m») _ %'»-n^») 2/« -m^ =0, aud .*. ^, -f-m = 0, y^ -w = 0, _5(//a + ;//») 4- lOA-a 8__ (A+y)4 + (A;-//)* _ (A;4-m) ^ + (/g-m)* 7/3-A-3 " m»-k» AM-OA-^//a + 7/4 y^-yfc-^ A:'*4-6^-m.'-'+/M4' o o n V=^ 4-^34-0^2 1 (A;'3 - ^2)2 - ^^,i _ ,^^a^ " - A; 84-3//2 _ A;4-2A;2y/2 4.y4 8(^-' ///2 _ k^ m4 _ 2A-2(//2 - m2) A;2+8m2 ^3 — m2 = 0, or X;4 -'llc'^ni'i -j. ,n4 3 2/24->//2_2A:9^ 4^2 A;*4-6A;2//2 4-//4 k^ 4- 6A;2^2 4. „iT - ,7^ ?/ m' y3_w = 0, or ^4 7/4 _ „, 4 4.6A;2 (y2 _ w,2 ) ^ y 8 _ ^2 • "T44-0Aj3wi3 4-wr4"" ^24.^12 ^.6A;2 _^ 1 6/^'wi2 4.' w74 -f m' i.e.f m2y2— A:* = 0, yj4-w = 0, y2-"^ = ^> wy3 4-A;2=0, my^-k^=0. 68. ADDENDA. 191 // 3 A*4-10p"//r-'-f"6w^ ~ m3 or J(.'^l±'«')+1^)'?^'' 1 84. ... i^±y)'H^~!f)' ...3 .-. 6y* + 2^y-7/.-4 = 0, .-. (//^->{:'.>)(.^,2 4-7^3)^0, ••• (y+^)(y-^)(5.v+;/cvaoj(5y-.yX^,30).-o. 2y ~ ^2 - ,/3 .-. ^4-y4=c, ory*-{k*~.c) = 0, .-. («V + 5/t2)2-{25-6.sa+.s4}/t4 = o. J m ^e vj!, I m S c ,. i ' II 1 192 ADDENDA. 87. (k+y)S^(k^y)i .-. (3^4-c)7/4-2/t2(7c_3c)//2_;^4(^._c) = 0. Lets3 = 8/.+c, .-. si!/^~2k^{k-Sc)s^y^ -k^{k-c)s" =0. Let /•2 = (A--3c)2-L(/_c)(3^ + g), ,'. sy±k \/{k-3c±r) = 0. 88. Expand and collect, 3a;2(a:2 4-2a:4-l)— 8 = 0. .•- 9;c3u:4-])2-24=.0, .-. 3a:(x'+l)--j/24 = 0, (By B.) or3a;(.c-hl)4- V 24-0, .-. Ga;+3±:i/(9 + 12v/24) = 0. (ByD.) 89-102. Work with a new variable w such that M;.'c = a;2_|.i, Having determined Vie value of w, that of a; may be determined thus x^ + i w. IX + 1 so XJ+-1 x-1 = +r r+1 w-\-2 w-tl = *.2 = r^ say ^1 = r-r _ r-1 ^ r+1 89. (a;^-f-l)2-2a;2 ~2x{x--i + l) ' tt,4_ 4,4.24.4 ^ ,,2 ~"4a'2 ~ W a T IV 2 a 2w ti'4 + 4ir2-|-4 = i («) 4i(;' ^ -!_+21^ = A say, 62 68 "^ m;24.2 2^y'~ ±6- 6" (^) .-. (a)-f(^), **' = « + s .'. ^ — _ ^".Jl — jzif =r r2, (as above). w—2~ a-'lb-^s ^ ' 90. .-. _(^^^ + 2)«- a i(j-+2ti' a {w-'2){a--l) ^ T ^^ w^ - dw+2 ^ T w- +2w a 5iv - 2 "" a — b ) w-2 a {w-l}^ ~ b . W'2.. (;„,_!_ 9 6-4a ^ - s^ say, 92 93. W-1 1-1-.S •"•("' + 2) ^ b ^ ' ^ a~b 94. (^+^i! ^ ± %v b = .. M'2 = b— a « 95. .' x-l ?r+2 <+4 _ rt "i2^ - J = sS T tt- - 2 .'•+1 a-Qb = ± s or + ys = s*. say x-i Let < = ?r - 2. t2.f-lGi + 64 16a4-^> t- 96. H-8_ «-M _ l + .s ir + 1 II' — 1 M'2-l a = -1- .-. w^ = b 7r+2 i7-2* 4«-6 = s2 say, m;-2 w-2~M,-2-4~ 6 ••'^■- ,,l6 -(a-6)2 ^^•>'' ±« «- + 2 2(^?— 6)±s 97. .*. w = i(;2-4 a — 6 a w 2 — ••u-2 ~ 2{h-ii)±s 4a - 36 -- .- — ... - . — ■ ^ a -b 194 ADDENDA. '.i I' if »;> Nil ..'. v» 98. M>8-4 w A. . / w»+4 \ » +4\» 6a + 16a3 a' = -7^ say a ?(; = ~2a~' 99. (u;4-2)(ti;-l) a tv^+ io -2 (w;-2)(m' + 1) ~ T~' ^ M'*-*'^ m; - 2 "* w2-2 rt-H6 /M;2-h2\2 (a+/>)2-t-8(a-6) « w a — b w ■ (a -6)^ ~(a-6)a ^*y» ti;; rtH-6±:s 2(rt-6)' 100. PTa;2--a;+l " (x^ + 1)^ -x{x^ + l)-x^ X 4_.^3 w^ + u;— 1 w^ —w— 1 a MJ2-1 u; m;2+1\2 (a+6)2+4(rt, - 6)^ r-^^)= w = 2(«-6)* 101. (i(; + 2)2 m;2-2 = =ts, (M;+2)g w+2 its oa a+ 46 = .«2 ss-* 8ay, i« -2 + s 103. (fl-a;)(a;-6)u; = (a-a;)2 + (a;-6)2 2 (rt-a;)4+(a;-6)*}-9(a-a;)(a;-6){(a-ar)2 + (a;-6)2} + 14(a-a;)2(a;-6)8=:0, .*, 2(u;3 -2)-9m;+14 = 0, or 2wa_. 910+10 = 0. or (w-2)(2t(;-5) = 0, w = 2, 2a'3 = 6, .-. 2{a-x){x-b)='a-'X)'^+{x-h)' ADDENDA. 195 and.-. {(a-x)-(x-h)}^=.0, .-. x,:.i{a+b), and.-. ma-x)-{x~b)}{(a-x)~2ix-b)\ =0, .•.x^ = ^{2a+b), x^ = k[a-{-2h). 104. ^{{a-x)*-2{a-x)^[x-by^-{.{x-b)*\- 9{a-x)^{x~b)^ = 0, •'* mia-x)^ -{x~by-} ~S(a-x)(x-b)} X [2{(a-a:)2-(a;-6)2}+3(a-a:j(,;_6)] -0, .-. ma-x)-\'{x-b)}{{a-x)~-2{x-h)} X {^{a-x)-{x-b)}{{a-x)+2{x-b)}^0, ,\x,^2a-b, x^ = ^{a + 2h), 2:3 = |(^2 1 + />}, ^^-26 -a. 106. a;*-12a;3-l-49x2-78x + 40 = 0, .-. («3-6aj)2 + 13(a;2-6a:) + 40 = 0, (x^ ~Qx-\-b){x'-Qx+S)^{}, .-. (^-l)(a;-5)(ic-2j(,t-4) = 0, /. a; = 1 or 2, oi' 4, or 5. 106. (^•*-l)-6u;(a;2_l) + 7(u;2-l; = 0, ... (a;.-l)(a,.4-l)(a:-2)(a;-4) = 0, .-. x=-\-l, -1; 2, or4. 107. (a;3-5a:)2 + 10(a;2_5a;) + 24 3=0, .-. {x^-5x^4:){x''-5x + Q) = Q, .\ {x-\){x~i){x-2){x~^)=:0, .-. r=l, 2, 3, or 4. 108. 2(4a;2 - 3a;) 2 - 7(4.^3 _ 3^) ^ 5 ^ q^ .-. (8a;2_6a;-5)(4a;2_3a;-l) = 0, .-. (2a;+l)(4a;-5)(4.r+l)(a:-l) = 0, .'.x=-\, -\, 1 or |. 109. (a:3 + l)-6(a;3-l) + 5(.^ + l) = 0, .-. (a?+l){(a;2-;c+])-6(a;-l) + 5}=0, .-. (a;+l)(a;2-7a;+12; = 0, or (x4-lj(a;-8)(.T-4) = 0, .'. a; = —1, 3, or 4. 106 ADDENDA. = 0, hetx^-4:ax = a7/^ .*. 10 8 9 = 0, .-. 7y2_ll7/-120 = 0, or (y-5)(ly-j-2i) = 0, .'. x^-4ax = 5n^, or 7x^ -28ax-\-'na^ =^0, .*. {x-5a){x-\-a) = 0, .♦. Xi =5a, ajg == — «. (Two other values.) 111. Let 25?/ = x2--35. 14(20 + 55) 5(5-M0) 4(25- 10) __ + 2o?/-1100 ^ 25^-200 207/4-250 .7^ = 0. 70 77 4- 15 1/-44 ^ 7/-80 2/ ^ = 0. 10 2,2_2y_lG3 = 0, or(2/-hl2)(^-14)=:0, aj2-35a^+ 303 = 0, or a;^ - 35aj - 350 = 0, (a;--15)(a:~20) = 0, a;= 15 or 20. Two other values. !■, 112. 5a X In x — a + i« X- 2a X — 3a da X- Aa + 5^ X— oa >. 't •> 'i Let a^y — x^~ 5ax, .'. 4?/ = 25, .-. 25 1/ 27 + 2 2/+4 " // + = 0. 4a;3-20«a- + 25«3=o, (2a;-5a)2=0, .-. x = 2^ a. ' 1 113. :4-2 + + X -1 a;-2 "^ a; -3 + « 5 Let ^ = a?'* — 3a; and collecting pah-s of terms as in preceding solutions. 14 U -10 G + h 2/ y/T2 = 0, 52^2 _ 14^ _ 24 = 0, .-. (2/-4)(5y-f-G) = 0, «3-8a;~4 = 0, or 5.^3 -15x-f- = 0, (a;-4)(a;-Hl) = 0, .-. a; = 4or -1. ADDENDA. ^ay lit. X..ua..y = :r^ -Gx, and collecting in pair.s an.l roclucin;.^ (the term _i^ must be written J^ Z'^^ \ the equation becomes .1. _ J!l_ . _r^ 1 .-. 41;/^ +7nv - 2520 = or ( ./ - 7)( i 1 v + IJGOj = 0, .-. a;3 - rxr-7 - or 41.^3 - 24f;.r + ;}G0 = 0, .-. (x-7)(x-hl):^0, .-. ar=:7or -1. 115. .-. i/(^^-«^-63) + i/,,. A^ .2,.., _^/,,,_^,^^,, ... 464a;4-4/>-'.f3(rt2//J^/^2,.2 ^-,.-\/-'+:^/,4,^ (,f2/;2^i2c2+^2„2_^2A4)2^ 4Kc'i-4/;}i^x-~i:) a- m^x v/(rt*-4^) "^ V('wAk2-4~) m4a;4=rt4 .-. mx= ±a 01 ±ja. 117. Work as in Ex. 115. Let 2.S = a +6 + f and 2.y '2 = rt2 4. /^e ^^3 ^ 2.s-(s-rt)(.s--i){.s -c) Square and reduce X = |/{(S"-^ -a3){.,/3_^2)/^./2_,.i.^)^ TV ^JLi, t Hi- 4^- ■ ■* ^! s^ V I- '!•' !(• t.ff^: I'f i 198 ADDENDA. ^(a«-}-/>a -r2)^ and the equation becomes of which (see Ex. 115 above) the solution is x=±i (b^c^ ^ C|^i a, + ^^1 118. .-. V{a-x)-,/{b-x) = c, .-. 2ci/(b-x)=a-b-c^, 2ab + 2ac8 + 26^2 - a^ - />2 _. c4 iC = 4f2 iiq T^(^-«^) _ *»-a ; ^ a-a; _ (a— a;)* (]^.r— 6) ~ a;-/> '* x — b ~ {x— b)^* .'. {a-x){x~b){{a-x)^-{x-b)^}=0, .'. {a-x){x-b){a-h){a+b-'2x) = 0, .'.x = a or b or ^{a-^b). 120 (w4-v)^=tt"' + y'^+5/ty(«t;+v)3-5i<;2y2(4y^_^). Let m; = V(«+sc) and v=&/{a-x), .*. it;H-a;= V(2:»), .-. 2a = 2«H-5V(rt2_^2).5/(3^,3)_5y(^t,2_a;S)2y(2/i), .-. 4a2(a2_.i.2) = (rt2_^ij^2^ ,., («2_a;2)(3a2 4.a;2j = 0, «= dia or ±ja-y/S. 121. = w*»4-^' (a'2+v2ji w^+2iv-v^ +v^ = w*+ivv{w^ 4-1'^) + 1 wv (m;-v)2=0, w s= or i; = 0, or if = v. Cube these. o— a; = Oora;-fc = 0, ora— a; = x = a or 6 or ^{a-\-b). X ADDENDA. 199 2L P m, ' Ol^li;)t(/y4-^i+i^+e.U^^^.-|-;H,';,aDd bu+cv+aw = ab-hhc + ca v = a, cu + av+bw==a'^^b^^+c2 u^i, &c. 16. p. 206, 2/^+y+i 6«+/.^.i •• (7^v/+/;+ir6^'+/>ri-(3)- Assume ;.-l=(._l), ,,^ ... ,- i = (,_i),, ,. ^^^^ v-l = 0, and.-. ;c = a, y/ = 6; or ^^-H = ^1±^J « + />+! *^4-l 62 + 6+1 ••• *^ = T-^/6 •• = {a'-b)-i-{l-ab), y = {b2-a)~(l-ab). Erkata. Example iii., question 28, *« Cube of the sum.*' (8) or X >; \ ' ■.! <; ^ • m ■i I n v; \{ ,1*1(16 I, ■I, I ! I •i 1 ■ .!• y«lOKS I'OU TEACHERS & STUDENTS, ' Bt DR. McLELLAN. "Ill our opinion the hest Collection of Problems on the American Coiitiiieiit."— JVo^ojiai Teachers' Monthly, N Y. mmmm papers in uirnMPiia By J. A. McLellan, M. A., Lh.D., Inspector High Schools. Ont., and Thomas Kihkland, M.A., iScieuce Master, Normal School, Toruuto. Fourth i^oinplelc Edition, Price, $1.00. Examination Papers in Aritltmetic-Part 1 By J. A. 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