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Fraser, y^ssistant Master in Glad- stone Ave. School, Toronto. Price, 15' cents. BRITISH HISTORY NOTES, for Srd. 4th and 5th Classes. By G. E. Henderson and C. G. Fraser. Price, iS centi. GEOGRAPHY NOTES, fop 3Fd, 4th, and 6th Classes. By G. E. Henderson, and G. A. Fraser, Hawkesville, Ont. Priat, 15 cents. EXERCISES IN ARITHMETIC FOR FIFTH CLASSES. By G. E. Hendeison and E. W. Bruce, M.A. Price, 15 cents. Teachers' edition, containing^ answers, 20 cents. 8XBRCISSS IN ARITHMETIC FOR FOURTH CLASSES. By G. E. Henderson, and W. E. Groves, Principal Church Street Model School, Toronto. Price, 15 cents. Teachers' Edition, con- taining Answers, 20 cents. EXERCISES IN ARITHMETIC FOR THIRD CLASSES. By G. E. Henderson, and E. W. Bruce, M.A., Principal Huron Street School, Toronto. Price, 15 cents. Teachers' Edition, cob* taining Answers, 20 cents. EXERCISES IN ARITHMETIC FOR SECOND CLASSES. By G. E. Henderson and E. W. Bruce, M.A. Price, xa rent< Teachers' Edition, containing Answers, 15 cents. EXERCISES IN ARITHMETIC FOR FIRST BOOK TEACHERS. By G. E. Henderson, and Miss R. Church, Miss A. Harding, Teachers in Church Street School, Toronto. Price, ao cents. (This book is devoted to the teaching of Notation, Addition and Subtraction.) KOTES ON PHYSIOLOGY AND TEMPERANCE. By G. £. Henderson and C G. Fraser. Price, la cents* Pp %l.i x-^ttr. "school HELPS" SERIES. :■'^■ ■•;, •- -f EXERCISES IN ri) ALGEBRA For Fifth Book, Public School Leaving and Primary Classes. ., • '> . .• >" ,; G. E. HENDERSON/ "; ;,: Editor of "The Canadian Teacher" and " 7^he Ettt ranee,'' AND '^'^- \r V E. W. BRUCE, B.A., Principal of Huron Street Fnhlic School, Toronto. reatiC IS. ardirigi ents. ion apii Priee, 15 cents; Teachers' Edition, with Answers, 20 cents. THE EDUCATIONAL PUBLISHING COMPANY, 11 RICHMOND STREET WEST, TORONTO, 1899. / Entered according to Act of the Parliament of Canada, in the year one thousand eight hundred and ninety-nine, by Thb Educational Publishing Comi'Any, at the Department of Agriculture. yf PREFACE. The authors, who have prepared these exercises in Algebra at the request of many teachers, most respect- fully request a consideration of the following points in connection with them : I. No Answers. In the Pupils' edition no answers are provided. These are in the Teachers' edition. II. Saving in Time. The time of the teachers is too valuable to be taken up in the dictation of problems to a class, when for a mere trifle each pupil may be pro- vided with exercises. III. Writing. The possession of these exercises by the pupils will tend to preserve their hand-writing by doing away with copying questions from dictation. IV. Problems Grouped. The subjects are ar- ranged in what appears to be the most natural order ; and, in accordance with the recommendation of teachers, the examples for exercises are very numerous. V. Book of Exercises. This book is not in any sense designed to displace either the teacher or the authorized text-book. It merely furnishes ready to the teacher's hand many bright, crisp, new exercises with which to enforce his teaching. The Authors. Toronto, December, 1898. It 1 EXERCISES IN ALGEBRA For Fifth, Public School Leaving and Primary Classes. Coefficient, Exponent, and Root, in Simple and Compound Expressions. EXERCISE I. If a = 5, b = 3, c = 2, x = 4, y=i, z=o, find the values of A. (i) 7a. (2) 13b. (6)152. (7)5ab. (11)23x2. (i2)37yz. (i6)bcx. (i7)3xyz. (3) 17c. (4) 19X. (5) 25y. (8) 8bc. (9) 9CX. (10) i6xy. (13) iixa. (14) I4ax. (I5)abc. (18) 4axy. (19) 7abx. (20) I3bcx. B. (6) y3. (11) i8z3. (i6)c*y6. (i) — ac. ^ ' 2 (2)x2 (7) Sa^ (I2)b^ (17) Cb. (2) -— abc. (3) y^. (8) 7c^ (I3)a2b3 (18) a'^ . (4) i\ (9) 9x^. (i4)b«c* (19) 5bc . C. (3)^b-. (4)[c (5) 5a«. (io)i5y3. (I5)x2a3. (20) I2ax'' 10' (6)l|y2z.(7)7b. (8)63 (9) 19^ (io)T:^6a2b^ 6 EXERCISES IN ALGEKRA. (ll)^ (12) b* (13) abc 64- 27 19 19 o o /.ox 3 (14)^(15)^. 21' 23 5 (16) ^^abx (i7)-oX2a2 (18) |-xa . (19) |-ba (20) D. jic_b^ ex (I) V(5a). (4) V(i2xb). (2) s/(27b). (5) V(i5ab). (3) V(2CX). (6) \/(i6xy). (7) V(2a2c). (8)y(^-) (9) // 2b-c3 \ V \5a«xV (14) V(2abc2) 3 // I5a2b2 (12) x/(3b2cx). (13) \/(5xyz) 36bc EXERCISE II. 25abc\ 6bcx / If a = 3, b = 2, c=-o, x = 8, y = 6, z = 5, find the values of : ' A. 1. I2a+i3b4-i4c4-i5x+i6y+i:z. 2. 9z-8y + 7x-6c + 5b-4a. 3. ab + bc-cx + xy -yz. 4. I4c-3a-5b + 2ix+i7y 32. 5. abc + bcx + cxy + xyz. 6. 5az + 6by - 8cx + 4bz - 1 3byc. 7. si-^+h^ + c^+x^+y^+z^ 8. x3+y3 + z*-a3-b3-c3 9. a2+b2 + c2+2ab + 2ac4-2bc. 10. x2 4-2xy + y2-2xz-2yz4-z2. 11. a^ + 3a'-^xH-3ax=^+x3. 12. y3-3y2b + 3yb2-b3. COKFFICIENT, ETC. 13- 14. 15- I. 2. 3. 4. 6. 7. 8. 9- 10. a*+4a3b + 6a2b2.|.4ab3 4-b*. 81x24. i44xy + 64y2 -492-2. i6x2 4-4oxy + 25y^-9a^-24ab-i6b3. b a V c a X z y 4a 9b 8b 3a — +4- — — — 5b 4x xy 7y 12a 3f_7ab 3xy 5x~ xy "^iab B. xb ig^tib 3b + 2y a4-b4-c X 4z ~ x4-y4-z a+_c x + b x4-y y4-z a-c x-b x-y'^y-z xy4-2xz4-3yz 4ax 4- 3ac ab4-2ac4-3bc xy-xz4-yz- a-h'^ + i _i+a^c^ 4x4-y- + y'^zg a2 4- b2 c2 4-a=^ '^' y^ a^ + b2 b2 + c2 c24-ai x I2a3- b^ 3a2~ '^ 2bx + 2x2 a4-b24-c3 x b^ 2CX a4-z 2bc ay yz az + :»■■ I' BXERCISe III. Ifx = 8, y = 6, z=i, a = 9, b = 4, find the values of I. 5 1,7 5 2- o^ ax 27 32 b2 8 6x abz' T lO EXKRCISKS IN ALGKHRA. zab-^ X 6. V(yab)-3y2 + ^- 7- i^^- \/(9b)->7(uO I2ax=^ V \ b-y=^ / V 3a 9- \/ifz)+\/ity^+\/{TI^y) • V Vb2/"^V \2o8xy2/+\/ V38a^b3 / ADDITION. EXERCISE IV. Find the sum of : A. I 2 3 4 5 6 7 8 9 lO 2a + 3b + 4c ; 5a + 6b + 7c ; Sa + gb+ioc. i2x+[iy+ioz; 9x + 8y + 7z; 2x4-5y+3z. a + 3b-4c; -3a + 3b + 3c ; 2a-3b + c. 4a + 3b-2c;-a + 3b + 2c; 2a-b + 3c. -4x + 3y + 2z ; x-2y-rz ; x + y-3z. , -x + y + 4z ; 3x-2y + 2z; 2x-3y-z. -2a + 3b-8c; a-b + c ; 4a + 3b-5c. - I4a-i8b+i9c ; I3a+i5b + 8c ; a + 6b4-8c. 26a-i6b-3c; i3a-iob + 4c :a + i9b-3c. - 17a- iib4-6c ; ioa + 6b + 7c ; 6a-|-5b \.,-. T 1 1. 12. 13- 14- 15. I. 2. 3- 4- 5- 6. 7. 8. 9- 10. AlJDITION. 'j 4ax + 8by-3cz ; ax + ^by-cz ; -3ax-2by4-4C2. 2op + q-r; 3P- 2oq + r ; p + q- 2or. 6m -- 1 311 4- 5P ; 8in + n - 9p ; m + 2n 4- 5p. V sab + 6cy - 3 ; - Sab - 3cy + 1 1 ; - 7ab + 1 2cy + 7. i3xy-5z + 8 ; -7xy + 5z-i8 ; -6xy+i32+io. B. - 4ab + 6bc - /ca ; sab - 3bc 4- 7ca ; - 2ab - 2bc 4- 4ca. I4ab-28bc-i7ca; i3ab-t-45l3c-3La ; I2ab-i8bc + loca pq-qr-rp; -pq-qr + rp; pq-qr + rp. x + y + z ; 3x-4y + z ; 2x + 3y-2z. 3a + 24b + 1 7c ; i6a - 23b - 9c ; a - 3b 4- c. 4xy-9yz4-zx; 23xy- I4yz4-zx ; I2xy- I3yz-zx. 47x-y4-63z; -27x-i-5y- 13Z ; -22x-47y- 19Z. 23a-i7bc-2d ; Sa-i6bc-i3cl; -a4-9bc-d. ax-4b/ + 3cz; I3ax-9by4-7cz ; - Sax + 7by- 14CZ. 2aH-3b4-c; 4b4-Sc-6d; 17a- 1304- I9d. I. 2. 3- 4. 5. 6. 7. 8. 9- 10. ri. 12. J3- EXERCISE V. Ad^ together the following expressions : A. 3ab - 2bc - 3ca ; 2ab 4- 3ca 4- 6abc ; ~ Sab 4- 2bc - sabc. 3x2-sxy4-6y2 ; 4x2+6xy-3y^ ; 4y2 4.6x'-2 -7xy. 3a--7ab4-Sb- ;.4ab-sa--4-3b=^ ; b^ -4a-+9ab. x--xy + y'-i; Sx"- 4-4xy- SY^ J 6x24-xy + 5y2. x2 + xy + y2; a3-a-4-a Sx'' -3x"^ + 2x i2x-*^ + Sx-8 a'^-ab-bc 5a"5-3c3 4-2d 5y^ 4-yz — y2 • x''^4-xz4-z^, r; 2a»-4a2-6a4-7; S'l''^ -4a" -a + 9. k'^ ; x*4-7x^-8x^ ; Qx"* - i?x-*- isx^. ; 3x^-4x2-6; rDx^-4ix-i9. , ab4-b3-ca; ca4-bc + c'^ b3 - 2a-^ + 2d=* ; 40^ - 2a3 - ^d'^ Sa- - 3^* 4- 2d'^ ; b-^ - 2a-^ + 2d=* ; 40^ - 2a3 7x3-8x + s ; 2x'^4-x-2 ; x2- 7x8 -3x4-9 x2 4-yS4-2xy ; 2Z--3y2_4yz; 2X2-2Z2_ 4X-* + 5x--y 4- 5xy2 ; - 3x2y _ 7xy2 _ 2x-J ; 7xy2 - 8 lO EXERCISES IN ALGEBRA. I! 14. a3-3a2b + 8abc;a2b-ioabc + c3 ; b3 4-2a2b-abc. 15. 20x-i + 20x-y-3xy^ + i4y^ ; - I7x=*+ iSx^y- lOxy^ -5y«; I9x3 + l8x2y+l5xy2-5y'M " I2X» -I3x2y-23xy2 + i8y=*. B. • I i I 3,53, -a--b+-c;-a+-b-^c;-a-b-c. 7 II 9 5 , 5 7 13, II I Iv._i-V. -,x«-3xy+-y^; --x-^+4xy--r, 16 5 X2 f I -xy .y2. |l a b c b c a^ ^ _ _^ . ^ 4- 7" 3 "^4 ' '^~i'*"4 '234' 5 8, I , 5. -2a + -Jc;-b-3c; --a-2b. 6. |--[xy+^y- 4x^ + ^xy-y- -^x^-xy + yy2. 7. -^x«+4ax=^-|-a%;xa-f ax'-'+'^a^x; - -^x*^ +— a^x. 4 8. -a3-2a2b-^b3;-^-a2b -7ab2 + 2b«'^; -|-a3 + ab2 +-b-^ 2 I. 2. 6. 7. 8. 9- 10. II. 12. I. 2. 3- 4- 5- 6. 7. 8. b - abc. - ioxy2 -I2X* -c. II 1 2 -X2 x2 _xy 2 „_^. I « ADDITION. II 9- 17 15 23 II 8m--^n + p; T^m + g- n- -p;- m + n 10. a3-4b3-2abc; b3 + 2c3-— abc; 4 23' ia3 -^V-9_abc. 12 5 Subtract : SUBTRACTION. EXERCISE VI. ■ A. I. 2. 3- 4- 5- 6. 7. 8. 9- 10. II. 12. I. 2. 3. 4. 5. 6. 7. 8. 2x + 3y + 5z from7x + 8y + 9z. '; 4aH-7b + 9c from I2a+I5b+ lie. 2a - 3b - 4c from 5a + 7b + 6c. 4x - 5y - 6z from 6x - 8y - 92. 3a - 4b + c from 4a - 3b - 4c. 2x-7y + zfrom 9x+ioy-i6z. I4a-29b + 8c from ioa + 4b + 5c. -9X-i2y+i3zfrom x + y-z. -4a4-3b-4c from2a-b + c. -i3x-i4y4-i5zfrom -9x+i3y~i5z. 3ab + 4bc - 6cd from 5ab - 2bc + cd. ab-cd + ac-bdfrom -ab + cd-ac + bd. B. 2ab 4- 4cd 4- Sac - 7bd from 2ab + 5cd - 3ac - bd. xy-yz4-zxfrom -xy4-yz-zx. 9p-i4q + 3r from 5q-3p + 2r. - . 8a - 3b + 7C from 5c 4- 2a -5b. 8-c4-b-a from a 4- b 4- c 4- 3. 2x-2y-3zfrom x4-y. 2a 4- 4c from a - b - c. 5ab - 1 7xy 4- 1 8 from 9ab 4- 3xy - 23. 12 EXERCISES IN ALGEBRA. 9. - 5a+ iixy- 19 from 2a + 8-2xy. 10. - f X + |y - ^z from |x - ^y + f z. 11. |x + fy + 2from^x-^y-H. 12. |a-|b-|cfrom -a-5b. From EXERCISE VII. A. 1. 4xy-7yz4-9zx take ~2xy + 3yz- I2ZX. 2. x^ + 12x2 + 5x- 2 take 3x3 -2x''^ _7x-5. 3. - 5x2y2 _ J 5xy3 + 8y* take 4x'^y2 + 7xy'* - I7y*. 4. - io + 4ab -6a2b2 take 5-3ab + 2a2b2. 5. a^bc + b^ca + c^ab take 4a2bc- 5b2ca + 6c2ab. 6. -sa2b + 8ab2-20take -3a2b-8ab'- + i7. 7. 4x'^y-3xy2 + 2i take 5x^7 -3x72 -20. 8. -a2-b2-c2-d2 takea2-c2 + d2-b2. 9. 4x"*y - 2x2y 2 4- xy3 take 3xy3 + 4x2y2 _ jx^y. 10. a"*-3a2b + 3ab''^ — b'*' take -a3 + 3a2b-3ab2 + b^. 11. iSx^- i6x2 + i4x- 12 take 10- 12x4-14x2- i6x*^. 12. - 3a2 + 4ab - sb^ take - 7a2 + 3b2 - 2c2. B. I I. 2. 3- 4. 5- 6. 7- 8. 9- 10. II. 12. a3 -f b3 - 3abc take b'^ - c^ - 2abc. 5x* - 7x3 ._ 2x2 ^ ^ take x* + 6 + 2x - 4x3. - 3a3 - 5b3 + c3 + 5abc take a^ -i-b^ + c^ - 3abc. X* - X - I + x2 take 1+ x - x^ - x* - x'*. 7ab2 + i6a2b + b3 take a3+b3 + 7a2b-8ab2. x-t _ 2x3— 10x4-5 take 3x*-2x2 + 7x — 9. Ja4-b take a-^b. |x2 - ,^x - U take -fx2+|x-T^. §a2 - la - 1 5 take - §a2 - a - f • ix2-|x4-^takeHix-Jx2. |x2 - |xy2 - y2 take ^x2y - |y2 |xy2. ^a^ - 2ax2 - |ax3 take ^ax"- 4- Ja2 — |ax3, . SUBTRACTION. EXERCISE VIII. 13 1. Add 4X + 77+132 five times in succession to x-4iy-72z. 2. To the sum of 3a -4b -17c and 4c -5b -a add the sum of a- 5b- 19c and a + c+i5b. 3. From 3x3 - 2x2 - 5X+ 7 take the sum of 8x3 _ ^^2 + 7x - 2 and 8x"^ -gx— 19. 4. Subtract S + a-ga^ + yaS from the sum of 8 -2a - I3a2 and 6a- iga^ -27a3. 5. Find the sum of r9a- 27b -36c and -28a + 27b -39c, and subtract the result from 2a -3b -5c. 6. Take x2-3y2 from 5xy + 7y2, and add the re- mamder to the sum of 5x2-9xy + 3y2 and-8xy- iiy2. 7. Add together 5x'^ + 7x2y-9xy2 + i8y3 'and -2x3 -5xy2-7x2y-y3 :uid diminish the result by -x3-x2y - xy2 — y3. 8. Take i4a2-i4a+3 from unity, and add 5 + 13a - 9a2 to the difference. 9. What expression must be subtracted from 19x2 - ^x +4y ~ 7 to leave x2 - y - 9 ? 10. What expression must be added to 5ab-iiac + 1 2bc to produce ab + 5bc - 6ac ? II. To what expression must 8x2 -9X + 5 be added to produce zero ? 12. Subtract 5X3+4X2-5X-9 five times in succession from x3+ 13X- 18. 13. From 5x2+6xy-5y2-i2xz-3yz-8z2 take 2x2 ~ 3y ^ + 4XZ - 5z2 + 6yz - 7xy. 14. From a5-4a3b2-8a2b3-i7ab*-i2b5 take in succession a« - 2a*b - 3a3b2 ; 2a*b - 4a3b2 - 6a2b3 ; 3a3b2 -6a2b3 -9ab* ; and 4a2b3 -8ab* - I2b6. 15. By how much does 2x - 3 exceed 5x - 17 ? 16. Subtract the sum of i5l-9mH-3n-p and 4m -5n +P + 1 from 13 I- iim-9n. 14 EXERCISES IN ALGEBRA. MULTIPLICATION. ' EXERCISE IX. Multiply together: 1. 5x3 and 4x2. 2. 4a* and 5a^. 3xy and 7xy. 4ab and ab. 3x and 8y. 17a and 2b. Sa'^ and a. 7x and 9x*. Sa^b and loab^. 3. 4. 5- 6. 7- 8. 9. 10. II. 12. 13- 14. 15- Qxy'-* and 8x^. iia'^b^ and iia^b^. 4a^b3 and 8a«. x3y3 and 6a2x*. 7a*b'^c3 and sa^b^c. xyz and abc. A. 16. ga^x and Bex. 17. 3a*bx and 7b3x*. 18. 5x^y2 and ddL^-m^, 19. 2a2b and a^b'^. 20. 7x^y^ and 4x''^yz''*. 21. 4a^bx and I2a"'^b2c. 22. iQx^yz^ and 4x2y*z3. 23. 7abc and 14 xyz. 24. a^b^c^ and 7a^bc. 25. ay^z and bxyz^. 26. Qm^n^p* and mn^p^. 27. 8c*x-^ and 7a^c*. 28. acx and cxy. 29. sa'^b^c^ and bcx^. 30. I4a'^ and \(i?L^'^h^'vci^ . B. 1. ab + bc and a^b. 2. x=^y + xy^ and 5x3y*. 3. 8x4-3y ^nd 7x"^. 4. 5ab-9bx and 3a2b3x*. 5. 4a2-3b and 5ab. 6. 8x'^-9xy and 7x*'^. 7. ax^ and — 3ax. a2 4. ^2 _ ^2 and abc. 8. — 3abx and - 9abx. 9. -ab2c2dand--3a2bc2d^ 10. xyz and -Sx'^y^z*. 11. —be and a'^bc^. 12. — 7x^yand — 9xy3z*. 13. 6m2n^p and mnp2. 14. -a^bc and -9. 15 16. ab 17. 5a 18. 19. 20, 21. be — ca and a^b' 9b2-iic2and — 3ac. 9x2y + 8xy2 -7y3 and 14 x^. 3x3y — 5xy3 and -7x8y3. 8x'-yz + 3xy2z— I3xyz2 and —xyz. a^bc — b^ca- c^ab and -ab. I. 2. 3- 4- 5. 6. 7. 8. 9- 10. II. 12. } 13- 14. 15- 16. i' I mm MULTIPLICATION. m^. 22. 23- 24. 25. r3_ r2 r3,4 x^y^-y«z* + z*x2 and 5x2y2z2. 2xy^z3--3x2y3z_5x3yz2 and 4xy2z. 5x»-4x2 + 8x-9and -13. -5a3-7a2+6a-ii and loab. EXERCISE X. Find the product of: I. 2. 3- 4- 5. 6. 7. 8. -9. 10. A. 4x2-7x + 6and 2X-3. Sx3-6x2 + 7x-8and4x + 5. 4a*-3a3-2a2-a-8and3a-7. 2x3 + 4x2 + 8x+i6and3x-6. x3+x2+x-i andx-i. 15 7 - a + 5a2 - ga^ and 5 - 6a. 13 + x - 3x2 - 7x3 ^^^ 2 - 9x. i , ; 5x2-4xy4-7y2 and 3x--4y. " \' 7a3 - 9a=^b + ioab2 +6b3 and 7a - 4b. * i =- x3-2x2a+2xa2-a3 and x + a. 11. i6a2 + i2ab + 9b2and4a-3b. 12. 8x2-7x + 9and 5x2 + 3x-8. 13- a2-2ax + 4x2 and a2 + 2ax + 4x2. 14. ioa2--3ax + x2 and ioa2 + 3ax-x2. . 15. a3-3a2b-h3ab2-b3 and a2-2ab + 3b2; 16. x3-6ax2 + i8a2x--27a3andx3 + 6ax2 + i8a2x + 27a3 17. 3x^+4x2H-5xH-6and3x3-4x2-f.5x-6. B. 1. a + b + c and a + b-c. 2. 2a + 3b-4cand2a + 3b + 4c. 3 4. 5- 6. 7- 8. 9. 10. x2+xy + y2 and x2-xy4-y . 3x2-2xy + 4y2 and 3x2 + 2xy-4y2. x2 + 2xy + y 2 and x^ - 2xy + y 2 . ab + cd + ac + bd andab + cd-ac-bd. '^' a3 + 2a2 + 2a+i anda3-2a2 + 2a-i. a*-2a3b + 3a2b2-2ab3 + b*anda2 + 2ab+b2. x2+y2 + z2 -xy-xz-yz and x + y + z. xy + x + y2 + y+i and x + y-i. i6 KXEKCfSES IN ALGEHRA. II. 12. 13- 14. 15- 16. 27a^-36a'x4-48ax2-64x''^ and 3a + 4x. .12 x«y2+x"y*-x-^y" + y*^ and x-*4-y2. a* + 2a3 + 3a2 + 2a+i and a2-2a+ i. x-^ -ax^+a^x — a'^ and x4-a. x*-2x^y + 3x2y''-2xy^ + y* and x2 4-2xy + y2. x2 + ax + a2, x2-ax + a2 and x^-a^x^ + a*. DIVISION. if; « Divide : EXERCISE XI. A. 2. x by x^. by x^. 10 3. 3a* by a^. 4. 27a* by 3a*. 5. 35x^ by 7x3 6. Sia^ by -ga^. 8. x-^y3 by x^y. 9. a^x* by a^x. 10. 4a'^b2c^ by ab^c''. 11. i^a^b^cfi by -3a*b2c. 12. -2ox^y'M5y — 4xy''^. 13. -48aO by -8a^ 14. -35 a^o by -5a^. 15. 63a^b'^c® by ga'^b^c''. 16. 7a2bc by -7a-bc. 17. -5oy"^x*by -Syx^. 18. i8b2yx2 by -3xy. 19. ~24a*b''c^ by -3a3b*. 20. 2oia3b^x3 by -3. B. 1. 4x^+6x* + 8x2 4- lox by 2x. 2. 3a*- isa^- i8a^ by -3a''*. 3- 4. 5- 6. i5a3b-^-6a'^b2 + i2abby -3ab. 36a3b3c2 -48a'2b3c3 + 6oa3b'^c3 by 4abc2. a* — a.b + ac by -a. a^-a^b + a^b^-ab^bya. I7x2y. 7. 34x*y''^-5ix'^y2 4-68x2y by 8. 72xSy'' - 45x*y3 - i8x2y2 by 9 cy". 9. 8im^n''^-84m^n«4-27m-''n*p by 3m2n2, 10 i69a*b- Ii7a3b2-9ia-b by I3a'^". 11. 36ib^c* + 228b*c^- I33b3^s by I9b3c2. 12. - i44x* + io8x3y-96x2y2-f 6oxy3 by - I2x. I. 2. 3- 4- 5- 6! 7. 8. 9- 10. DIVISION. 17 Divide EXERCISE Xil. A. 1. x2+ i6x + 6o by x+ 10. 2. x^- 17X + 70 by x~7. 3. a^— iia + 30 by a — 6. 4. x2 -49X-I-600 by x-25. 5. 3a2 + ioa + 3 by a + 3. 6. 5a2 4-i6a + 3 by a + 3. 7. 4X2 + 23X+15 by 4X + 3. 8. 4a2+a-i4 by a + 2. 9. 6a2- 31a + 35 by 2a -7. 10. I5x2 + i7xa-4a2 by 3x + 4a. 11. 9a2+6ac-35c2 by 3a-5c. 12. 6ox2 -4xy-45y2 by 6xH-5y. 13. 96x'-^-4xy- I5y2 by I2X- 5y. 14. 9a'^+3a2 + a- I by 3a - i, 15. 7x3-24x2 + 58x-2i by 7x -3. 16. x'^ + 2x2 4-2x4-i by x+i. 17. x3 4-4x2y + 3xy2 + i2v^ by x + 4y. 18. a* + 4a3b + 6a2b2+4ab3 + b* bya + b. 19. a^-Sa^b+ioa'^b^-ioa^b^ + Sab*- b^ by a-b. 20. x* — 4x3 _^ 5x2 _ ^x + I by x^ - 2x + I . ^' 1. X* - 5x"'4- iix^-r I2X + 6 by x^ -3x4- 3. 2. a* + a'^-9a2-- i6a-4 by a'24-4a-f 4. 3. x*H-iox-^4-35x^4-5ox + 24 by x2 + 5x + 4. 4. i8x* -45x3 + 82x2 -67x4-40 by 3a^-4x + 5. 5. I4a* + 45a3b + 78a2b2+45ab3 + i4b* by 2a2 + 5ab + 7b2. 6. X* -2X*- 4X3 -t- 19x2 -3IX+ j^ J3y x3 _7x+5. 7. x* + 8x3-4x2- 128X- 192 by x^- 16. 8. m^ +2m* + 4m3+9m2 -3im+ 15 by m'^ + 2111-3. 9. a3 + 3a2b + 3ab2 +b3- I by a + b - I. 10. b«+6b5c+i5b*c2 + 2ob3c3 + i5b2c*+6bc'''+c'' by b + c. i8 EXKRCISES IN ALGEBRA. i, II. 12. 13- 14. 15- 20. 21. 22. 23- 24. 25- 3a2 4-8ab + 4b2 + ioac + 8bc + 3c2 by a + 2b + 3c. x's-fyis by x + y. X«-y8 8ix*-i6y* by 3x + 2y. i6a*-8ib* by 2a -3b. x*-8iy* by x-3y. x^H-i by x+i. 16 b* by a--b. r; by x-y. 18 a* -1 6b* by a + 2b. ' 19 xfl-yfl by x^ — 2x2y4-2xy''^ — y". ai2+a«-2 by a* + a''^ + i. x3^.jx''5y4-3xy''+y3 + z'* by x + y + z. a^ -f-2ab + b=* -c'^ by a+b-c. a2_b*^-c''*+d2-2ad + 2bc by a-b + c-d. x^ + y^ + z"'' — 3xyz by x + y + z. i ! BRACKETS. EXERCISB XIII. Simplify by removing brackets ; a: 1. 3a-b-(2a-b). 2. a-(b-c) + a + (b-c)+b-(c + a). 3. a + b + (7a-b)-(2a-3b)-^5a+4b). 4. (a+x)-(b-x)-(a-b). 5. 1 -(i -a) + (i -a + a2)-(i -a+a2-a3). 6. (a-b + c)-(b + c-aH(c-a + b)-(a-c + b). 7. -{ a-(b-c) ^ + ^ b-(c-a) ^-^ c-(a-b) y. 8. a-[2a--{ 3b -(4c -2a) [>]. "3a- -(4a -(5a -2a) f 4b- -j 4a-(6a-4b) )- 9. 7a- 10. 6a- II. 2X 12. 3x I 2X- - ^ 2X - (2X - 2X - x) y]. 5y_^6z-(4x-3y) )■]. 13. -[5a -(lib -3a)] -[5b -(3a -6b)]. 14. a- 2b -[4a- 6b- ^ 3a-c + (5a-2b-3a- c-2b) )-]. 15. 8x- ^ i6y-[3x-(i2y-x)-8y] + x ^. 16. 3b--! 5a-[6a+2(ioa-b)] )-. 17. 2x-(3y-4z)--{ 2x-(3y + 4z) ^ - -{ 3y-:4z + 2x) ;•. 25. 26. 29. 30- BRACKETS. 19 -2b) H- -1-2X) ]-. 18. 3a2 - [6a2 - ^ 8b- - (9c*^ - 2a2) H- 19. -4(a + d) + 24 (b -. c)- 2[c + d + a-3<{ d + a - 4 (b + c)H- . 20. [a-5b- ^ a-(5c-2c-b-4b) + 2a-(a-2b + c )-]. Without removing ths brackets^ divide : 21. x® + (a + b + c)x2 + (bc + ca + ab)x + abc by x2+(a + b)x + ab. 22. X*- (5+a)x3 + (4 + 5a +b)x'' - (4a + 5b)x + 4b by x2-5x + 4. 23. x'^-(a -b)x2-(ab + 2b2)x + 2ab2 by x-b. 24. ax* — (ap-b)x^ + (aq-bp-c)x2H-(bq + cp)x — cq by ax*-^ 4- bx - c. 25. 2x3 + (2a + 3c)x2 4-(3ac-4b)x-6bc by 2x4-30. 26. x* 4- (a 4- b)x3 + (a2 4- ab + h'^)x^ + (a^ 4- ba)x + a^b^ x2+axH-b2. X* -(a4-c)xa+(b4-ac)x2-bcx by x-c. x«+px5-f (p-q + r)x* + (p"-* - pq) x» 4- (pq4-pr-qr) x2 4-p^qx4-pqr by x2 4-px4-r. apx*+x^(aq 4- bp)4-x2(ar4-bq4-pc) 4-x(qc4-br) + cr by ax2 4-bx4-c. x*4-2ax3- (n^- i) a^x^ + 2na^x-a* by x^- (n- i)ax4-a'^. by 27 28 29 30 Insertion of brackets. B. 1. Write the expression ax — bx + cx-ay4-by-cy ; first, with the x terms and the y terms bracketed re- spectively ; secondly, with the a terms, the b terms, and the c terms bracketed respectively. 2. In the expression ax^ — cx + 7-dx2 4-bx — c-dx^ 4-bx^-2x bracket together the powers of x so as to have the sign + before each bracket. 3. In the expression — a^x — 7a4-a^y + 3 — 2x-ab bracket together the powers of a so as to have the sign — before each bracket. In the following expressions bracket the powers of x so that the signs before all the brackets shall be positive : 20 EXERCISES IN ALGEBRA. 1 4. 6. 7. ax* + bx* + 7 + 3bx - 8x3 - 3x* - 9x. 4bx2 _8_^x + ab+ 6ax8 4- ex - 2x2 _ \)^q^ 4-8x3 + 6ax2 -3cx4- loax^ + Qx- I2x2. 3CX* - 4abc + 5dx - 4bx* - 2a2x^ + 3x^ In the following expressions bracket the powers of x so that the signs before all the brackets shall be negative : 8. ax2+4x -2a2x*-3bx3-bx*-3x''^. 9. 7x^ + Sx-^-3c''^x-abx^+9ax-abcx^. 10. ax2+a-x^-5x2-cx3-bx2. 11. 6b'^x'*^-2bx-3ax*-cx'^-5c-x-7x*. 1 2. 5ax^ - 7bx + 7CX''' - 6bx'^ 4- 3ax'' + 2ax + 4cx-^ . Express, by brackets, taking the terms (a) two, (b) three, together : 13. 2a-3b-4c4-5cl-4e + 3f. 14. -b-.5c + 6d-3e + 4f+g. 15. -3x + 4y-2z + 3a + 2b -c. 16. 4c-2d + 3e4-2x-y- 52. 17. -2m + 3n + 4a-6b-5x4-7y. 18. 3p4-2q •4r-5m + 3n-2a. J31 |. il SIMPLE EQUATIONS. EXERCISE XIV. Solve the following: 1. 7x + 5 = 5x+ii 2. I2x + 7 = 8x + i5 3. i6x-ii=7x + 7o. 4. 24x-49=i9x- 14. 5. 3x+23 = 78-2x 6. 5x-7=3x + 7 7. i2x-9 = 8x— I. 8. i24x+i9=ii2x + 43. 9. i8-2x = 27-5x. 10. I2X- i45 = 7x- 125. A. 11. 26-8x = 8o- 14X. 12. 266-6x = 2x- 186. 13- 39-3x=iSx-9- 14. 75 -25X = 30-20X. 15. 6x- 22 =14x4- 12. 16. 24+ i2x = 36-48. 17. i6x = 38- 12 + 3X. 18. 5x-35 + 63 = 9x. 19. 2x4-3=16-2x4-3. 20 63(5-x) = 72(x-s). 20. 21, 22. 23. 24. 25. I SlMPLHi KQUATIONS. 21 -s of X so gative : (a) two, 4x. 186. 9- 2.0X. ■12. ■48. ■3x- 5x. X4-3- -5). = 0. 4) = :0. 21. 27x-9(5x-6) + 9o = o. 22. 4x-(6x-35)=5x-(3x-7). 23. 24x-5(i8x + 6)+i2(7-8x)+i566: 24. 2x-i-2(3x-2) + 3(4x-3)-4(5x- 25. i3x-2i(x-3)=io-2i(3-x). B. 1. 6(i69-x)-2(78 + x)=58x. 2. 1 4x- 78 -20x4-30=200 -66x4- 52. 3. i63-i5(2x-5)=i57-2i(x + 3). 4. (x-|-i2)^x-8) = (x-6)(x+i). 5. (x4-7)(x-3) = (x-5)(x-i5)- 6. (2x-7)(x+5) = (9-2x)(4-x) + 229. 7. (x+5)(x-9) + ix+io)(x-8) = (2x + 3)(x-7)-ii3. 8. 5x-(3x-7)-^ 4-2x-(6x-3) l>==io. 9. i4x-(sx-9)- -^ 4-3x-(2x-3) )^=30. 10. 25x-i9-[3-^ 4X-5 H = 3x-(6x-5). 11. (5-3x)(7-2x) = (ii--6x)(3-x). 12. (3x^-2x4-i)(2x-i) = 6x-*-7x-+9x-6. 13. 7x--i(3x-2)-(5x + 3) H27-2[3x-(x4-2)]. 14- 5(5x-4)-4Ux-3) + 3(3x-2)-2(2x-i) = 2. 15. i.2x-.o5 = .o7x + .3x+i6.55. 16. 2.ix+5.25-4-4x + 5-5 = 3o8x-5.39- 17' •375x-i-875 = -i2x+i.i85. 18. .72x-i.o8 + .45x + 4-o5 = 2.8x-28. 19. (x+i)(x + 2)(x + 3)-(x-l-4)(x-3)(x+5) = 84. 20. x=^-|-9x'''+4(7x-i) = (x + 6)(x + 2)(x-M). 21. 7x--{(3x-2)-(5x + 3) l>=2:-2[3x-(x + 2)]. 22. (5x2-7x-|-4)(2x-9)-iox3-59x2 + 37x+ioo. 23. (x + 3)(3x-2)-(x+5)(3x-7) = 3x+i. 24. (x-|-i)(x + 3)(x+5) = (x + 7)(x + 9)(x-7)- 25. .6x + .75x-.i6 = x-.583x + 5. EXERCISE XV. SYMBOLICAL EXPRESSION. A. I. Ifx denotes a certain number, express :— double the number ; three times the number ; 5a times the num- i \ 22 EXERCISES IN ALGEBRA. ber ; the square of the number ; the cube of the number '. the square root of the number ; the cube root of the number ; the sum of a times the number and b times the number ; the difference between c times the number and d times the number ; a number that is 8 less than 9 times the number ; a number that is 13 greater than twice the number. 2. If a and b denote two numbers, of which a is the greater, express : — the sum of two numbers ; the differ- ence between two numbers ; the sum of the squares of two numbers ; the difference between the squares of two numbers ; the square of the sum of two numbers ; the square of the difference of two numbers; six times the product of two numbers ; \he sum of the squares of two numbers divided by the difference of the numbers ; the sum of two numbers divided by the differ- ence of the squares of the numbers ; the sum of the cubes of two numbers divided by the sum of the numbers ; the square root of the sum of the squares of two numbers. 3. If n be a whole number, express : - an even number ; an odd number ; four consecutive numbers of which n is the least ; four consecutive numbers of which n is the greatest ; five consecutive numbers of which n the middle one ; five consecutive even numbers of whicn 2n is the middle one ; five consecutive odd numbers of which 2n-|- i is the middle one. 4. If a and b are two digits, express :— the product of the two digits in as many ways as you can ; the two numbers of which these are the digits ; the product of these two numbers ; the sum of these two numbers. 5. If X, y and z are three digits, express :— the pro- duct of these three digits in as many ways as you can ; the six numbers of three digits each that can be formed by them. 6. Write algebraically all the numbers that can be expressed by the three digits 1, m, n. SYMBOLICAL EXPRESSION. n 7. A man makes a journey of x miles. He travels a miles by coach, b by train, and finibhes the journey by boat. How far does he travel by boat ? 8. If a man was x years old y years ago, how old will he be in z years ? 9. How old is a man who x years ago was n times as old as his child then ^ged a years ? 10. A man has a journey of m miles ; he travels a hours at x miles per hour, and b hours at y miles per hour ; how long will it take him to finish the journey at z miles an hour ? 11. Two places are x miles apart ; two persons start at the same time, one from each place, and travel towards each other at the rate cf a and b miles, respectively, per hour. How long will it be before they meet, and how far will each travel ? 1. A rectangular solid is a feet long, b feet wide and c feet thick. How many cubic feet are in it ? How many square feet on all the faces ? How many feet in the sum of the lengths of all the edges ? 2. A room is x yards long and y feet wide ; how many square inches are there in the area of the floor ? 3. What will it cost to plaster the walls and ceiling of a room a feet long, b feet wide and c feet high at x cents per square yard ? 4. A man works p hours a day for m days, and q hours a day for n days. He receives x cents per hour ; how many dollars does he receive for this ? 5. A grocer mixes a poujids of tea worth x cents a pounds worth y cents a pound, and c cents a pound. What is a pound of the pound, with b pounds worth z mixture worth ? 6. In a class of n boys, x work work at classics, and the rest are idle of workers over idlers ? If a and b denote two quantities, braically : — * at mathematics, y ; what is the excess express alge- 24 EXERCISES IN ALGEBRA. 7. That the square of the sum of two quantities is equal to the sum of their squares increased by twice their product. 8. That the square of the difference between two quantities is equal to the sum of their squares diminished by twice their product. 9. That the product of the sum and difference of two quantities is equal to the difference of their squares. 10. That the cube of the sum of two quantities is equal to the sum of their cubes increased by three times their product into their sum. 11. That the cube of the difference of two quantities is equal to the difference of their cubes diminished by three times thei" product into their difference. 12. That the difference between the squares of the sum and difference of two quant''es is equal to four times their product. 13. That the sum of the jquares of the sum and difference of two quantities is equal to twice the sum of their squares. 14. That the sum of the cubes of two quantities divided by their sum is equal to the sum of their squares diminished by their pro(|Mct. 15. That the difference between the cubes of two quantities divided by their difference is equal to the sum of their squares increased by their product. 16. That the square of the difference between the squares of two quantities, increased by the square of twice their product, is equal to the square of the sum of their squares. EXERCISE XVI. PROBLEMS LEADING TO SIMPLE EQUATIONS. A. I. Find two numbers whose sum is 38, and whose difference is 6. i \ PROBLEMS LEADING TO SIMPLE EQUATIONS. 2$ 2. The sum of two numbers is 83 and their difference is 27 ; find the numbers. 3. The difference of two numbers is 1 3 and their sum is 95 ; find them. 4. One number exceeds another by 27, and their sum is 289 ; find them. 5. Find a number such that, if 18 be added to it, twice the sum will be 114. • 6. Divide 60 into two such parts so that the double of one part may be three times as great as the other part. 7. At an election where 975 votes were given the successful candidate had a majority of 63 ; how many voted for each ? 8. The difference between two numbers is 26. When the larger is increased by 15 and the smaller by 12 their sum is 159 ; find the numbers. 9. There are two numbers whose difference is 61. If the smaller is increased by 20 and the other decreased by 16 the sum is 295. What are the numbers ? 10. To the double of a certain number I pdd 96 and obtain as a result 854. What is the number ? .il-^ II. A father is 30 years old, and his son is 2 years old ; in hosv many years will the father be eight times as old as the [>on ? 12. The sum of $216 was raised by A, B and C together ; B contributed $12 more than A, and C $27 more than B ; how much did C contribute? 13. Divide the number 63 into two parts such that .three times one part may be equal to four times the other. 14. Find three consecutive numbers whose sum \shallbe255. XI 5. Divide $2,426 among A, B and C so that A may have $23 more than B, and B $54 more than C. '" 16, A is 36 years older than B, and A's age is as much above 15 as B's is below 55 ; find their ages. 17. Twenty-nine times a certain number is as much above 99 as 189 is above 7 times the number; find the n umber. 26 EXERCISES IN ALGEBRA. 18. After 34 gal. had been drawn out of one of two equal casks, and 80 gal. out of the other, there remained just three times as much in one cask as in the other ; What did each cask contain when full ? 19. Divide $1,520 among A, B and C so that A may- have $100 less than E, and B $270 less than C. 20. A number consists of two digits whose sum is 1 1. The tens digit is 3 more than the other. Find the number. B. 1. A person buys 4 horses ; for the second he gives $48 more than for the first, for the third $24 more than for the second, and for the fourth $8 more than for the third. The sum paid for all was $920 j how much did the fourth cost ? 2. Divide 60 into two parts such that the difference between the greater and 64 may be equal to twice the difference between the less and 38. 3. A gentleman left $1,100 to be divided among four servants. A, B, C, D ; of whom B was to have twice as much as A, C as much as A and B together, and D as much as . and B together ; how much had B and D ? 4. A starts on a journey 20 minutes before B ; A^ walks at the rate of 4 miles an hour, and B at the rate of 4^ miles an hour ; at what distance along the road will B overtake A ? f^ . 5. The length of a room exceeds its breadth by 3 feet ; if the length had been increased by 3 feet and the breadth diminished by 2 feet, the area would not have been altered ; find the dimensions. 1r 6. The length of a room exceeds its breadth by 8 feet ; if each had been increased by 2 feet the area would have been ir • reased by 60 square feet ; find the original dimensions. 7. From a certain number 3 is taken, and the remain- der is divided by 4 ; the quotient is then increased by 9 and divided by 6 and the result is 3 ; find the number. PROBLEMS LEADING TO SLMPLE EQUATIONS. If 8. A man buys a horse and harness for $io8 and the horse cost five times as much as the harness. Find the cost of the horse. 9. Divide 96 into four such parts that the first increased by 3, the second diminished by 3, the third multiplied by 3, and the fourth divided by 3, shall all give the same result. ^ 10. A has $240 and B has $96, and each loses a certain sum. Then A has five times as much as B. What is the sum lost by each ? II. The sum of two digits is 9. Four times one of the numbers they form is equal to seven times the other number. Find the digits. y 12. A workman was employed for 40 days on con- dition that for every day he worked he should receive $1.37, and for every day he was idle he should forfeit 50 cents. At the end of the time he received $37.97 ; find the number of days he worked. .13. A certain number of men and twice as many women were employed on a work for a week. Each man received $1.45 and each woman 83 cents a day; their total wages being $317.22. How many were employed altogether ? 14. Find three consecutive numbers such that, if they be divided by 10, 17 and 26 respectively, the sum of the quotients will be 10. 15. A man sold a horse for ^35 and one half as much as he gave for it, and gained thereby ten guineas ; what did he pay for the horse ? FACTORS, MULTIPLES AND FRACTIONS. EXERCISE XVII. Find the Highest Common Factor of: 1. i5x*5, i8x2. 2. 2ia^, 28 a*. 2n3 6. 45m**n*, oom^n 7. 2oab2c, 36a"*bc*. 28 EXERCISES IN ALGEBRA. ,s 3- 4- 5- II. 12. 13- 14. 15- 16. 17. 18. 19. 20. 21. 22. 23- 24. a*b, a2b3. ,3xr2 x**y^z, xyz 2„8 24x3y. 8. 9. 10. I7p'^q^r, 5ip3qr*. abc, 2ab2c^. 3X-ZS I4x^y 7a2b*c«, I4a«b*c2 35a2b3x3y*> 49a^b2x*y3. 63ax2, 49ay2, sdaz*^. 34ab2c, sia^bc, 68abc2. 8a-x, 6abxy, loacx^y^. a-*x-y2, b'^xy^, c'^x-^y. 24a3b2c3, 48a'-^b3c\ 64a3b8c2, icxDxy^z, 75x2yz, 5oxy. a^pxy, b^qxy, c-^rxy. I5a*b5c2, 25a5b*c% 6oa3b''c^ 3oa2c'M5, 35a^c2b2, 42a'*cb'', 22p2m*n-, 33p*mn, 44p^m. 36x*y5z^, 48x*'y5z^, 6ox^y*z'^. I5xy, i6yz, ly xyz. x'^yz*. I^XERCISE XVIII. Reduce to lowest terms: I. 5- 6. 8a2 I2a3 8x« 36x2- loa^b^ 24a^b- * 4axy 3abc* i8xSy'»z2 45x^y2z* 7a5b7c^ 2 1 a" b"c^' 9. 10. II. 12. I2a*bc^ Ha^b'V^ io5m^n2p 42m2n3p'-2- I2a*b2x 13. Ti: i8a2b*y i6a^b«c 2oa'*b*d' 9bcyz I2abxy2* 14. 12a'' i6ab* 17. 18. 19. 20. 21. 22. I2mn-p I5m-np- abc^ a^b-c • 3x^yz'^ jxy^z- • mn*pq m^n^p-'- 39a2b*c5 52a'* b^c"' 38k2m*p» 57k^m^p*' FACTORS, MULTIPLES AND FRACTIONS. 29 7. 8. 5iay2z 6x^y"^z^ 8x^y*^z 8vf 15- 16. 2x-yz Sa'-^b lob'-^c' 46x*y^z* ^' 69x^y*z^ abxy ^"^^ x*^y=*cd • EXERCISE XIX. Simplify the following : 3x 7y 1. —X—. 4y 9x ■^a 2b 2. T><^' 4b oa 4x2 3y 3- -^x — • ^ gy^ 2x 8a^D^ ijxy2 ^' 45x''^y 4a3b'^* 9x2y2z 2oa'^b^c iSa^b^c I2xy-z " 3ab cMS 5- 6. 7. 8. 9. 10. II. 12. - 2cd ab' I2a2bc 36ab' xyz 48b2c'^ 3 8ab« 5a"\\ 3yz isa^bc * ya^b^ i8x2c . y . gax'-^y I5ac^ i5x"yz-^ . 5x^yz' i6m'^n 39m^n^ 42p'''k^ loxyz-^ a^bc 8mn2 I3mn* 35kV 5yz 3a^x 13- 14. 15- 16. 17- 18. 19. be ab 2 b2 X — . ac b-c a^b -^-x^r-x y'z z'^x c'^a x-'y 7a^b* 2oac 4c^d' 5c2d-^^42bd''3a*b3' 25 pm y ok'^m^ 25n3q X .-. o X ,, 4P''m m^x^ I4k=^n 75n''^q' 5im2p 7y32 2ix^y lyp^n pyz 27b^ 15c X abc 81C 4od3 ' I4d^" 4c-^ 5d» 8ax2 20. i6a2b2c2 b2 X — . I5d* 3c i6bdx 49cy2 X 21, 7by * I4aby 64dx'^ 3" 3 i5mpx* P^q-y ^ 36p^q lox^" 8imn ' 27n2x''y=' a^ xy ap * b^ ab ■ bx' 6c3x_^28a2c2^ _7af_ ^^' 5iuy'»'^i5b2xy=^^4xV* i8x2yz I5abc looa^bc ^' 25a i6xyz * I28x'*y'^z'* ^ 1 I 30 EXERCISES I^* ALGEBRA. EXERQSB XX. Jftnd the Least Common Multiple of: 1. 4a'b, 6ab2. 2. iSa^b^c, I2ab2c3. 3. Sb'-'x^ya, I2a2x»y2. 4. 6a**x, 4a''^x''^. 5. 3x5y, i2xy-^ 6. 4a=ibac, Sa'-^b^c*. 7. abc, 2a'^. 8. I2xy, Sab 9. Sa3b*c5, 3a*b3c2. 10. 4x^yz, 6x'-"y'*. 11. sya-^x'-^y, 76b''^x^. 12. 27m2n'^p, Sim^nq. 13. 8a2, 4ab, 6bc2. » 14. a^bc, b^ca, c^ab. 15. 4x3y*, 3X'^y2, 2xy"\ 16. 7x*y'*, 2x^y^, 3xy^. 17. 3oa''*c^b,35a'*c'^b2, 42b3ca2. 18. 8a^b2, lob^c'-*, i2abc. 19. 44a*b'*, 33b2c'^, 22c^ab*. 20. 35axy, 4obxz, 45cyz. 21. 5mnp, 4npq, 3mpq. 22. 39a2b, 52b"'^c, 65c=^a. 23. 34X, 5iy, 68z. 24. 76ax2, gsxy'*, 114 bz'*. EXERCISE XXI. Simplify the following: I. X X in- II. X 2 X X I ^7- 21. ^-l- 2. a a 12. a 4^" a a 22. 2a 3b' 3- 5 7 13- X X X 9 + 6- 23- 7a 5b I2X 24y 4- 2X 4X 14. 2X 3 ~ 3x , 5x 4^6- 24. a - -^y. cx 5. a b i3'*"39* 15- 8x 9 5x 7x 12"^ 6" 25. 2x2 - ^; a 6. 2a 3a 9 H' 16. 15X 16 3x X 8 12 . 26. 2x 3y 2i'^ z^ a'^c** 7. 6x 3x 17- 23X 2/ 5x 2X ~ 78 ~ 9" * 27. 7b 2ic' mm FACTORS, MULTIPLES AND FRACTIONS. 31 8. 5111 5 m 12 36 ■ 18. 5a 3b 7c 8912 28. 2X -4C. 3y 9- 7y 5y 8 5- 19. a+b 1 a-b 1 • 14 21 29. m n 3X 2X P X 0. II ab 5ab 12 8 • 20. x-y x+y 18 27 • 30. a b 411 511 c '6li- b3ca2. ab*. 3y_ L^C*" ON WRITING PRODUCTS AND FACTORS. EXERCISE XXII. IVrife the product of : 1. x4-7 and x+ 10. 2. x4-8 and x + 9. 3. x+5 and x+ 12. 4. a + 9 and a + 8. 5. a+ii and a + 3. 6. a + 2 and a + 1 7 7. m+ I and m4-2, 8. m+ 12 and m4-4. 9. m+ 1 1 and nH- 10. 10. C+13 and c+i. 11. (x + 2y)(x + 3y). 12. (a 4- 4b) (a + 7b). 13- (a4-9x)(a+i2x). 14. (b + 9c) (b+ioc). 15. (c4-i6y)(c + y). 16. (x + z) (x+ioz). 17. (2x + 7)(2x + 8). 18. (2a+5)(2a + 4)- 19. (2X+Il)(2X + 7). 20. (2a + 3) (2a +2). Write the expressions which multiplied give : 21. X^'^- 22. x2 + 23- X2 + 24. x'-^ + 2S. a'-*4- 26. a*4- 27. a2 + 28. y2 + 29. y'^ + 30- y'^ + iox4- 16. 12x4-27. 13X + 42. 13x4-40. 12a 4- 27. iia + 28. 17a 4- 60. i6y4-28. i5y4-36. 31. x^ 32. x2 33. x'^ 34. ^'' 35. a2 36. a2 37. X2 38. a2 39. a'^ 40. X'^ 4-8xy4-7y''. 4-i2xy4-35y-. 4- loxy-f 2iy*. 4-ioab4-24b2. 4-ioab4-9b2. 4- I5ab-i-44b'-. 4-14XZ4-48Z-. 4-7ac4-ioc'^. 4-7ax4- I2x'^. 4-i7xy + 72y'^. 5' 32 I 2 3 4 5 6 7 8 9 lo I 2 3 4 5 6 7 8 9 lo EXERCISES IN ALGEBRA. EXERCISE XXIII. Write the product of: (x-4)(x-8). (x-4)(x-9). (a-io)(a-4). (a -8) (a -7). (c-5)(c-9). (c-2)(c-5). (2X-7)(2X-5). (2X-9)(2X-3). (3a -5) (3a- 4)- (3a-8)(3a-2). 11. (x-7y)(x-6y). 12. (x-6y) (x-8y). 13- (x-9y)(x-6y). 14. (a -2b) (a -8b). 15. (a- lob) (a -2b). 16. (a-2c) (a- 15c). 17. (2a -3b) (2a -5b). 18. (2x-4y) (2x-8y). 19. (3a -5c) (3a -8c). 20. (3m - 6n) (3m - 911). 21. 22. 23. 24. 25. 26. Wtite the expressions ivhich multiplied give : m^ -6mn + 5n". x2- 15X + 56. x'-^ - 17X + 72. x2-i8x + 8o. a^ - iia+i8. a^ - i2a + 27. a^ - 16a 4- 63. 27. 28. 29. 30- 31- 32. p-'-9pq + 8q2. c^ -4cx + 3x*^. x^ - ioxy + 2iy2. x^ - I2xy + 27y^. a2-i5ab + 36b2. EXERCISE XXIV. Write the product of: (x + 9)(x-7). II. (x- (x + 8)(x-5). 12. (x- (x + 7)(x-2). 13. (x- (a + 6)(a-3). ' 14. (a- (a+9)(a-4). 15. (a- (a + 8)(a-3). 16. (b- (m + 6)(m-2). 17. (b- (n+io) (n-3). 18. (c- (p+9)(p--6). 19- (c- (b+12) (b-7). 20. (x- i2)(x4o). ii)(x + 6). io)(x + 8). 12) (a +5). ii)(a4-7). I2)(b + 9). io)(b + 2). 9z)(c + 3z). iod)(c + d). i7y)(x+sy). I. 2. 3. 4. 5- 6. 7. 8. 9. 10. 31. 32. 33. 34. 35- 36. ON WRITING PRODUCTS AND FACTORS. 3> Write the expressions which multiplied give : 21. X2 + 9X-36. 22. x'*+7x-44. 23. x2 + 5xy-5oy'^ 24. a2 + 7a-i8. 25. a''' + 5a-24. 26. a2 + 3ac-28G2. 27. c'^4-5c-6. 28. c2 + iicd-26d^.^ 29. m2 + iimn-42n^ 30. n^' + Sny— losy'-^. 3t. ^*- 32. X2- 33^ x2- 34. ^^ - 35. a2- 36. y2 - 37. y^^- 38. c2- 39. z' - 40. b2- 5k -84. 3X-88. 4xy-6oy2. loa — 39. i8ab-4ob2. 24y-25. 2iyz — 72z2. i3cy-48y*. 38Z-80. i6bc-i7c''. EXERCISE XXV. Write the square of: I. x + y. 2. a + b; 3- m + n. 4- a + x. 5- a + c. 6. x + z. 7. b + 8. 8. c + io. 9. d + 12. 10. 2x+3y 11. a4-3b. 21, 3xy-9. 12. x+5y. 22. 9mn-3c. 13. 5ab + c. 23. 4a + 6ab. 14. p-q. 24. I2x-iiy. 15. x-ab. 25. 7X+IOZ. 16. xy-z. 26. 13c- I. 17. ab-c. 27. ICX39. 18. ax -2b. 28. 1012. 19. pq — r. 29. 688. 20. 7a -4b. 30. 989. Write what each is the square of: 31. c2+2cd + d2. 37. 32. x2+4xy + 4y^- 38. 33- 8ia2 + i8ab+b2. 39- 34- 9x2 4-i2xy4-4y^. . 40. 35- i+2a + a2. 41. 36. 25p2q2+6opqr + 36r2. 42. x2- ioxy4-25y'. 4a-- I2ab+9b2. 25p2-3opq4-9q''^- 36a2-84ac + 49c2. 49b2-i4b+i. I6x2y2-96xy+i44- 'K f •34 M EXKRCISES IN ALCKHRA. EXERCISE XXVI. Add a term to each of the following and state what each is the square of: b2 + 2bc. I. 2. 3- 4- 5- 6. 7. 8. 9- lo. c''+6c. p2+2pq. p2+iopq. x2 -f.2X. a-b'-^+'6ab. y— 8y. X2- I2X. a^- I4ab. X- - 20XZ. 11. 4a2 + i2ab. 12. 9p''^ + ^4pq. 13. 16X- + 24X. 14. y2 + 38y. 15. 36a2 + i2a. 16. 25a2 4-2oab. 17. 4a2+4ac. 18. 8ix2-i8xy. 19. 9a2 - i2ab. 20. 40'-^- I2cd. 21. 25x^— 3oxy 22. I -2a. 23. a2-4ax. 24. 4a'' x 2 _ 28acx. 25. 9a'-m2 4.3oamxy. 26. i6c2-8c. 27. 4x2y''^ + i2xy. 28. a'-'x'^ - 2ax. 29. b"y'^+4by. 30. i69x2z-- 52XZ. EXERCI5E XXVII. Write the product of: 1. (x + y) (x-y). 11. 2. (a+b) (a-b). 12. 3. (a + x)(a-x). -13. 4. (x + z) (x-z). 14, 5 (b + c)(b-c). 15. 6. (x+8)(x-8). 16. 7. (a+12) (a- 12). 17. 8. (2x4-3y)(2x-3y). 18. 9. (2m-f 4n)(2m-4n). 19. 10. (3a 4- 5b) (3a - 5b). 20. 21. (x+y) + zand(x + y)-z. 22. a + b + c and a-fb-c. 23. 1 + m + n and 1 + m - n. ?4. i2a + 3b + 4c and 2a 4- 3b - 4c. 25. 5x + 2y + 3z and 5x-2y4-3z. 26. 3a4-4brc and 3a-4b + c. 27. x'^ + x^- 1 and x^ -x-f I, 28. a^x^ + ax+i and a^x^ - ax + 29. x2 + xy + y2 and x^ — xy + y2. 30. a-^ 4- ab + b'-^ and a" - ab + b^ . (i4-x)(i-x). (i+9a)(i-9a). (2 + 7c)(2-7c). (ab + 3c)(ab-3c). (i2p4-ioq) (i2p- loq). (7ab + 9x) (7ab-9x). (i2x4-6by) (i2x-6by). (9xy+i)(9xy-i). (3ab + 7cd) (3ab - 7cd). (a2 + b2)(a2-b2). I. 9- 10. Ji. 12. ■ ^ 'J- 14. 15- I. 2. 3- 4. ON WRITING PRODUCTS AND FACTORS. 35 EXERCISE XXVIII. tate what - 3oxy. I. ^ax. :2 - 28acx. + 3oamxy. -8c. '^ 4- 1 2xy. — 2ax. +4by- 2z- - 52XZ. 3c). ^ 2p- loq). b-9x). 2x-6by). ab - 7cd). -b2). Write the factors of: A. I. 2. 3- 4- 5- 6. 7- 8. 9- lO. II. 12. x2-9. x'-^ - lOO. y--8i. 3^^-144. 9-a2ba. 64 -c*. I2i-y2. 225-x2y2 x2-i6a2. b2-2Sc2.^ 36x^-252' j-9a2. a 13. 49p2-36q 14. 4m2 - 1. 15. 81- look'* 16. 25x2-4. 17. a2-9b^ 18. I6x2-y2. 19. a*b2-49. 20. x2-8iy2z2. 21. a2b2-4cM^ 22. 25a2-64. 23. p2q2-i44. 24. > ' 25. 26. 27. 28. 29. 30. 31. 32. 33. 34- 35- 36. x3-i6b*. looa^- I. I2ixa~8iz2. p2q2-49a*. 64X* - -> 25y». I2ix*-y*. ibx" -9y**. a2b2c2_x2ya, 25x10 - i6c*. I - ioox*y<'z^. m 2 _ ni2n»-225p''. 1. (a-b)2-ca. 2. (x + y)2-z2. 3. (a-b)2-c2. 4. (x-y)2-z2. 5. (a + 2b)2-9c2. 6. (x-2v)2-i6z2. 7. (x + 5a)2-25b2. 8. (3x-4a)2-36c2. 9. (a-3x)2-y2. 10. (9x + 3c)2-4. 11. 9a2-(3b + 4c)*. 12. c2-(5a-2b)2. .3. x2-(y + z)2. 14. i-(3x-4z)2. 15. (a + b)2-(c + d)2. 1. (a2 4-2ab+b2)-c2. 2. (x2-2xy + y2)-z2. 3. a2-2ab + b2-x2. 4. a2-(b2-2bc + c2). •9y^ B. 16. (a + n)2-(b-m)2. 17. (3a + 2x)a-(2b + 3y)2. 18. (a-2b)2-(3c-4d)2. 19. 4x + a)2-(b + 4y)2. -JO. I -(7a 5x)2, 21. (a-hb + c)2-i. 22. (x-y + 2)2-289. 23. (x-5y)2-8iz2. 24. (x + y)2-(x-y)2. 25. (x+3y)'-4y=*. 26. (2ix + 3y)2-(i8x-4y)2. 27. (9x + 3)'-(6x-8)2. 28. (7a+i)2-(2a-i)2. 29. 25a2-(4a+i)2. 30. (2x-3y+4z)2-(6y-5z)2, C. 8. 9. 10. II. x2 + 2ax + a2-y2. y2-c2 + 2cx-x2. x2 - 4xy + 4y 2 - 9x2y 2. a2 - ioab + 25b2- I. 4i| \m 36 5- 6. 7. 15- 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. EXERCISES IN ALGEBRA. l-m^-2mn-n*. 12. i6x^ -a^ -6ab-9b*''. x* + 2xy + y''*-9x'''y2. 13. 4m''' -p2 + 2pq-q=^. a2-2ab + b--4a*-'b!^. 14. p2+4pq + 4q'*-r^ X'' — V*^ "t" 2XZ -f- Z'^. (aM-2ab + b'2)-*(c2-2cd + d'^;. a^ --2ab4-b2-c'^-2ccl-d-. (2a4-3b)'^-(3b + 4c)''. x'-* - 2xy + y ^ - b- + 4bc - 4c*. a* +6ax + 9x- - y2 - 2by - b*. x2 - 2x + I - m- - 4mn - 411^. a2 + b2 + 2ab - c- - d=^ - 2cd. a^ - b2 + c'^ - d-' - 2ac 4- 2b'd. a^ 4- x^ — y '-^ - z"'^ + 2ax — 2yz. a^ -Sab+iGb'-^- i + ioc-2 5c2. x2+6abx + 9a-b--y2 + 8cdy-i6cM2. x + y2 — z'^ — w- — 2xy— 2ZW. 4a'-^b2 + i2abcd4-9c-d^- i6x2y2. 25a2b2 -36c2 + ioac4r- 8id^ 4x2 - c^ - d* +9a2 - 1 2ax - 2cd. EXERCISE XXIX. J*i'nd the value of^ I. (675)'' -(32 .5)'*'. 2- (73i)-~(03i)^. (650)2 -(350)2. (1613)2 -(387)2. (1922)2 -(578)2. 6. (4632)2 -(368)2. 7. (5 1 87)^ -(987)-^. 8. (iooi)2- (999)2. 9. (8176)2 -(8171)2. 10. (9876)2.(9864)2. EXERCISE XXX. Write the square of: I. a + b + c. 7. x + 2y + z. 13. 5p-4q + 9r. 2. x + y + z. 8. 3a-4b + 2c. 14. x2 4-y2+z2. 3- m+n + p. 9- H-2x-3y. 15- a2-b2-c2. 4. a + b - c. 10. 5-8y + 6z. 16. xy + yz-l-zx. 5. x-y + z. 1 1. 4-5x-6x2. 17- 5X--9X + 3. 6. a-b-c. 12. 3X- + 7X-8. 18. I - ax — by. I. 2. 3- 4. 5- 6. I. 2. 3- ON WRITINd l>ROI)UCTS AND KACTOUS. 37 19. Find the value of (x + y 4-z)2 +(x + y - z)- +(y + z - x)- + (z+x-y)-. 20. Find the value of (a + b + c) - + (a + b - c)'-* -f- (b + c - a) - + (c + a-b)-. 21. Without multiplication find the continued product of (x + y + z) (x + y-z) (y + z-x) (z + x-y). 22. Find the product of (a + b + c) (a + b-c) (b + c-a) (c-|-a-i3\ 23. Find the value of x^+y^ + z^ -xy-yz-zx, when x = a + b, y-=b + c, z = c + a. 24. Find the value of x-^-y^+z^+xy + yz + zx, when x = a + b + c, y=:a-b + c, z = a + b-c. 25. Of what expression is 25X'' -6oxy + 36y2 -70XZ + 84yz + 49z=^ the square ? EXERCISE XVXI. Write the cube of: A. I. x4-y. 6. X--3- II. 3x + y. 16. 5x2-4y-i 2. a + b. 7. a4-4- 12. 4x- -5y. 17. 3a-- 2b- 3. x + a. 8. a-5- 13. ab + c. 18. 5X-5+I. 4. x-a. 9- x + 2y. 14. 2xy -3z . 19. 7a'^- I. 5. X+2. 10. a - 2b. 15. 6a + bc. 20. ax-y^. I. a + (b + c). 7. 2x + 3y + 4z. IS- x2+y2 + z2. 2. (x + y) + z. 8. 4a- - 3b + 2C. M- a^-b^-c'*. 3- (a + b)~c. 9. 5x- -4y-7z. IS- 2x2-5y2 + 4z2. 4- x + y-z. 10. 6a- - 5b - 4c. 16. ax-by + cz. 5- x-y + z. II. 3p + 4q-8r. 17- a + b + c + d. 6. a-b + c. 12. 3a- -2b + 5x. 18. 2x + 3y + 4z + 5w EXERaSB XXXII. Write the product of: (x + a)(x + b)(x4-c). (x + a) (x-b) (x + c). (x+a)(x-b)(x-c). 9. (2X + 3)(2X + 5) (2X + 7). 10. (2x - 4) ('ix - 6) (2X - 8). 11. (3X+ i)(3x + 5) (3X + 7). 38 KXEiRClSfiS IM ALGEBRA. 4. (x+i)(x4-2)(x + 3). 12. (3x-8)(3x + 6)(3x-5). 5. (x + 4)(x-l-5)(x + <^^)- 13- (4a + 2)(4a + 3)(4a-9). 6 (a-3)(a-7)(a-2). 14. (x + 4)(x + 5)(x + 2) (x + 3). 7. (x + o)(x+i)(x + 7). 15- (a+5)(a+0 (a + 6)(a + 8). 8. (x-9)(x + 8)(x-5). 16. (x-9)(x + 8)(x-7)(x + 6). m ON WRITING QUOTIENTS AND FACTORS. EXERCISE XXXIII. Wnte the quotient of: A. 1. X3^.y3_. _^^.y^ 16. a'^-i25-7-a-5. 2. aS+b^H-a + b. 17- 216 — x'*-^6-x. 3. x^+a^-rx + a. 18. 64x^ - i25y'*-:-4x — 5y. 4. b3 + y3~b-;-y. 19. m''-i-n^-i-m4-n. 5- x3 + 1-^X1- I. 20. a5 4-b''-f-a + b. 6. a^ + i-ra+i. 21. x7 ^y7_^x + y. 7. i+c3^H-c. 22. a7_b"-r-a-b. 8. I+Z^-i^I+Z. 23. m^ -n'^'-T-m-n. 9. x3 + 8-f^x+2. 24. x''4-i-r-x + i. 10. a^'^^- 27-:- a 4- 3. 25. c'^-i-i-c- r. K. x3 + 8y3-i-x4-2y. 2 3. a^4-32-T-a + 2. 12. 27a3+b3-r3a4-b. 27. x6-243^x-3. 13- 8x"»4-27b3-T-2x4-3b. .^G. x®4-a^-r:i'-+a. 14. x3-y3-f.x — y. ^ "^ (a + b)3 + c-^-=-a4-b4c. 15. as-b^-f-a-b. 3c, B. 6. (x-y) — .^*^x-y-z. I. X* _y4-i-x-y. i6x*-8iy*-7-2x4-3y. 2. a«-b«-^a-b. 7. 625a* - 256b* -r 5a 4- 4b 3- m^ -n^-7-m-n. 8. Xfl -yfl-i-x + y. 4. x" - I-r-X- I. 9- a*- i6b*-T-a4-2b. 5. I -a^-M "... 10. (a4-b/-i-j-a-' b4-i. I. 2. 3- 4- 5- 16. 17- i«. 19. 20. 21. 22. 23- •1- ■ ON WRITING QUOTIENTS AND FACTORS. 39 EXERc!S)B XXXIV. Write the expressions which tnultiplied give ; I. x« + y». 9- a*-c*. 17- X-''y'5 + 5l2. 2. x3 — y"^. lO. a'^-i. 18. 343^i^-i- 3- a^-b'*. II. p'^-q". 19. I2 5x'^- loooy 4- a-* + i. 12. x" -729. 20. x' -y7. 5- i+a^. 13- a'ib'> + c3. 21. a«+b-\ 6. 8x«-y^ 14. 27X'' - 8y3. 22. 11^4-1. 7- a'^ + Sb^. 15. c'* -216. 23- i-c*. 8. 27x» + 8. 16. 125 +z-"^. 24. p'^q-' 4-271-'^. r'5 RESOLUTION INTO FACTORS. EXERCISE XXXV. Resolve into factors , I. 2. 3- 4. 5- 5x2 r5x. a'l-a-^. 8x- ■ ya^x 6x 4x .-2 I4ax^. 6. 7- 8. 9. 10. A. 6p'-^ + p. x"'* 4- xy. X5-X*V2. 16. I2a"'b -9ab2 17. I5x*y-6x2y3. 18. 8m3n2p4-2om2n"p. 19. x-^4-x2y-xy2. 20. 4x2y3-6x''?y'^4-4x-'^y2 21. 15X'' - ioa-x3-5a-^x^ 22. 38a''^x^4-76a'^x* + 57a*x*^. 30 23. 3a"b-3a2b2 4-3ab•■^ 31. II. 12. 13- 14. 15- i2ab-8. 344-5ix-y. Ka^ 225a2 - lOX'*^ 24a"-^ - 27ab''^. 25x^^y- lox'** 3x^y'^ — 9xy* i64-24x"'^. 24, 25 26. 27 28 29- 39P''^q^ + 26pq - 52p 2oax- -4oax4-45a. 6.\""*-9x*4-4x". 35m-x- - 7omx2 + io5x^. 7a- 14a- 4- 7a*. 22m'^ — 33mn4-44n". ax-' -abx2 4-acx. I4a''^b3c* 4- 7a'^b*c2 -2ia*b2c3. B. 1. a2 4-ab4-ac4-bc. 2. a^ -ac4-ab-bc. 3. x5^ 4-xy 4-xz4-yz. 11. x^ + x24-x4-i. 12. mx + ina-nx — na. 1 3. 1 5ax - lobx - 1 2ay 4- 8by. .^o exercisp:s in algebra. 4- 5- 6. 7- 8. 9. X- _xy — xz + yi'. a^b^ -i- abc + abd + cd. a^c^ - abc ~ acd + bd. a2 + 4a4-ab4-4b. nix + iny — nx — ny . 3ax - 2bx 4- 3ay - jby. lo. pr - qr 4- ps - qs. 14. 2x2- lox- xy4-5y. 15. 3ax2 4-2axy4-3bxy4-2by2. 16. axy — bcxy — az 4- bcz. 17. a'-^x2 + a2y2 + b2x2 4-bV- 18. y''-y2-f y-i. 19. x* 4- x3 4-3x4-3. 20. ax 4- bx — by - cy 4- o - .\y. EXERCI5E XXXVI. ^\ lit Resolve into factors- 1. x2 + 3x4-2. 2. a'- + 7a 4- 12. 3. >:2-f 20x4-96. 4. a2H-23a4- 102. 5. p2 + 3op4-2i6. 6. X'^y-4-3ixy4- 130. 7. 774-i8x4-x2. 8. x^-f 13x4-42. 9. x2 + 5xy4-6y2. 10. a2 4-9ab4-i4b2. 11. x2 + 49xy-f 6ooy2. 12. m2-|-43inn4-39on2. U- p2 4-35pq4-2i6q-^. 14. a4-5a'-^4-6a-^ 15. 3x2+45x4-168. 1. X2 -I- I IX -80. 2. x^-i-x- no. 3. a2 + i6a-26o. 4. a2x2 + i4ax-240. 5. x2 + 7xy-6oy2. 6. a2 4-ab-42b^ 7. m2+m- 1S6. 8. a2 + r2abx-4';b2x2. 9. x*4-8a2x2-jr6a*. 10. 3ax2 + 36axy- 255ay2. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. B. X2- 11x4-30. X2- 19x4-90. X2- 15x4-56. a2 -2ia4-iio. a2- i8a4-45. a2-2ia4-io4. m2 •22mn 4- 105112. x-y- - i7xy4-52. x-y2 — 23xy -I- 132. a--2oab-f-9ib2. a2-2iabc4-9ob2c2. iio-2ix4-x2. a2x2-24ax4-i43. 7x2-105x4-392. a2x2 - 2iax4-98. II. 12. 13- 14. 16. 17. 18. 19 20. x2 - 9x — 90 2 _ 2_ x"- I IX- 15; a*-32ab- losb'^. a2-2oab-96b2. m2 - iini-26. m2-mn-56n2. ;2_ 3xyz- ioz2. i65-4ftx~a2x2. 420- a -a*. 5a'-x - soax'^y — J95X f 3y2. ik y- y + 2by2. bcz. 2+bV- h ex. - ay. ;n2. 2. 2c2. 3- .2. 3 ,,2 rgSx'y RESOLUTION INTO FACTORS. EXERCISE XXXVII. 41 Express iu the form of a square: I. 2. 3- 4- 5- 6. 7. 15- i6. 17- i8. 19. 20. 21. 22. 23- 24. ^;. ^8. 2y. 30. x2 + i8x + 8i. V ; 8. a'^H- 263+169. , 9. m^ + 34111 + 289. : '\ 10. y2 + 2y+i. ; ■ II. z^+roz + ioo. ^ 12. x* + i4x2+49, , 13. x2 + i2xy + 36y2, 14. 4a2x''' -28acx + 49c _ o o . _ m2 + 22mn+ I2in2. x" + 24x'^ + i44. a2- 36a + 324. p2-32pq + 256q2. a2-3oab + 225b^. 4a"'^- I2ab+9b^. a'^-4ax + 4x''^. ga^m^ + 3oamxy + 25x-y2. tAo2v4 _ SoKa^3v2 A. K4^fi i6a2x* - 8ab-c»x2 + b*c' 1 6x2 y 2 _|. 24xyz + 922. i-2x3 + x^. 9a2- 1 2a + 4. a2-2abc + c2b2. 25-4on+i6n2 2v2v2 J. 4x2y* a*x2+4a2x2y ,^^ ^ 49X* +42x3yz2 +9y2z4. a*b*-a2b2+^. 3a2b2-i8ab + 27. "^ (a+b)2 + 2(a + b)c + c2. (x + y)-2(x + y)z + z2. (a-b)^ + io(a-b) + 25. (2m - 3n)2 ~ 8(2m - 311 )p + i6p^ EXERCISE XXXVIII. Resolve into factors: 5 '• X* + x2 + I. 19. 4a* + 7a2b2 + i6b*. i 2. a* + a2 + i. 20. x* + 4y*. 3. x* + 10x2+49. 21. a*+4b*c*. 4. X* + 2X2+8I. 22. x^+x*y*+y^. 5- a* +2532+625. 23. x^+x* + i. 6. m*+4m2 + i6. . 24. 4a*-ioia2b2 + 25b*. • « liil 4^ £xercisp:s in algebra. 7. a*-i8a2b2+b*. 25. 36x-i-97x--2y-+36y*. 8. x*+x2y2 + y*. 26. 144m* - 289m'^n2 + loon* 9. 4a*-8a2 + i. 27. 4c* + 1. 10. 8ic*+9c2d-+d*. 28. 9p*-iop2q2 fq4. II. . ' "y''*y2 + 9y*. 29. X*- 113x2 + 16. 12. 9x :''y2 + i. 30. 9x*-73x2y2-j-i6y*. 13. a*+.. . 2 + y4, 31. a* + 5a^c2+49c*^. 14. x*+3x'-^y2 + 4y*. 32. x* + 7x2y2+64y*. 15. a* + 2a2x2+9x*. 33. m*-7m'- + 8i. 16. a*+4a2x2 + i6x*. 34. x*-7x2 + i. 17. x*+x-4-25. 35. a*- I4a2 + i. 1 8. a* + 3a'^ + 36. p6. x'^-5x2y2+4y*. EXERCISE XXXIX. Resolve into factor s: I. x®-y«. 16. 8a-'^b'^ + i25c3. 2. a^-b". : 17. x'^y-^z'*~ I. 3. a"b"-i. 18. (a + b)-» + c3- 4. x^y'^+z-^. 19. (x-y)--^-z=^ 5. 8x3+a«. 20. (a-b)3-i. 6. x« 8a-*». 21. i-(x + y)^. 7. x^+y". 22. a*-(b-fc)*. 8. a«+b«. 23. a-^ + b^+a + b. 9. 27a'^-64b-^. 24. x-^ + y^ + 3xy (x + y). 10. p-^q"^ - 27r^ 25. x3-y"*-3xy (x-y). II. x*"' +32y°. 26. (c + d)3+(c-d;-*. 12. x'-*-y''. 27. 729a"b-ab'^. 13. m^o-bio. 28. a^x''-64a-y'^. 14. X3 2_^,32, . 29. a^ + 3a2b 4- 3ab2 4- b*^ - c'' 15. aia-Ibi'-. 30. x3 -3x2y + 3xy2 -y3 -z-' I( I i: EXERCISE XL. Resolve into factors : A. I. 3x24-5x4-2. 13- 6x2-31x4-35- 2. 2a24-5a4-2. 14. 6x2 -23x4-20. 3. 3x24-10x43- 15. 6x2-7x4-2. RESOLUTION Into Factors. 43 V. 6yA. - loon*. 4. 5- 4a" + 9a + 2. 3x'-^+8x + 5. 16. 8x-'-34xy + 2iy2. 17. 4a'-- i3'i + 9- I 6. 5X- + 14X + 8. 18. 7 - 10111 + 3111-. ■ 7- 2X-+9X+IO. 19. 4x- - 14X+ 12, ,r4 8. 3a'-* + ioa + 8. 20. 3x-'-23x+i4. y • 9. 7x2 + 16x4-4. 21. 56x2 - 67x7 + 20Z-. • 10. 4a2 + 23a+i5. 22. 24b2- i7bc + 3c-. • II. 3x2+41x4-26. 23. 24x2 - 5oxy + 2iy2. ' 12. 8m2 + 38111 + 35. 24. 561112 - 229mn + 2on2. B. I. 4x^ + 1 IX -3. II. 6x2 -7x- 3. 2. 3a2 + i3a-3o. 12. ^a2-a- 15. 3. 3x^ + i4x-5. 4. 21112+ 1 5m -8. 5. 3C2 + 7C-6. 6. 4p2 + p- 14. 7. i2a2 + i7a-7. 8. 5 + 32y-2iy2. 9. 12x2+ i3xy-35y2. 10. 39a2 + i^ab - 26b2. 13- 14- 15- 16. 17. 18. 19. 20. 3a- -19a- 14. I2b2-3ib-i5. 12x2 _ j._35. 2x2 - 5xy — 3y2. 24a2-29ab-4b2, 20 -9c -20c 2. 4-5n-6n2. 39x2 _2oxy- iiy2, EXERCISE XLI. Resolve into factors : 1 . 6x'* + 1 7xy + 1 2y 2 + 22xz + 3 lyz + 2oz2. 2. 6x2 + i9xy+ (oy2 + 13XZ+ i6yz + 6z2. 3. i2a2 + 3iab + 2ob2+46ac + 59bc+42c2. 4. i8a2+42ab + 2ob2+48ac + 38bc + i4c2. 5. i8x2 + 59xy + 35y2+66x + 72y + 36. 6. 6a2 + i9ab + i5b2 + 24a + 38b + 24. 7. 2x2 +7xy + 3y2 -7xz- iiyz + 6z2. 8. 28x2-43xy+ioy2 + 5xz4-i4yz- I2z2. 9. ioa2-9ab + 2b2 + i9ac-7bc- I5c2. 10. 35x2-26xy- i6y2 - i3x + 38y- 12. 11. 27a2-3ab- i4b2-87a + 85b-44. 12. x-*+y'^ + z3-3xyz. 14. 13. x^+y^ — z^ + 3xyz. 15. x3_y3 +z3 + 3xyz. X3 _y3_23 -^xyz. I 44 EXERCISES IN ALGEBRA. FACTORS. MULTIPLES AND FRACTIONS. 1 If m EXERCISE XLII. Jund the highest common factor of: A. I. 2. 4. 2_v2 x-^ + xy ; x^ — y (x + y)2 ; x^-y'-. 2a'' + 2ay ; a"* 4x2 _ gy J . 5x- - 9xy. a'-^y. 7- 8. 9- lO. x2-4y2 ; x2 + 2xy. a'^ - a^x a* ■- ax'^ 5.aab2_a2b;»;a*b3-a3b*. ii.xy ,12 a'-'bx + ab^x ; a^b c2-d2;c2d-cd2. y ; b3. 6 13- 14. 15- 16. 17. 18. 19. 20. 21. 15x2 + 8x4-1 ; i2x"'^ + x-i. 22. 3x2 -7x- 6; 2x2-7x4-3. 23. a2-ab-i2b2 ; a2 4-5ab4-6b2. 24. ax2 4-2a2x4-a® ; 3(ax + a2)2. x*y — xy. x^-l-yj J x'*4-X''y. x2 — 9x-ih2o; x2-x — 20. x2 4- 2x - 1 20 ; x2 - 2X - 80. X2 -9X-36; X2- 15x4-36. x2-9 ; X-- 18X + 45. x3 4-8y'^ ; x24-xy-2y2. a2-i7a-6o; a2 4- 23a 4- 60. x2 + 7x4-io; x^+6x4-8. (x-3y)2 ; x*-27xy^ a*-x* ; a2x4-x3. 1. x^ -6x2 -86x4- 35 ; x'^- 5x2 -99x4-40. 2. a'^4-2a2- i3a-f 10; a-^+a2- ioa + 8. 3. 2x^-5x24-11x47 ; 4x^- 11x24-25x4-7. 4. 4X^ - 3x2 _ 24X - 9 ; 8x'* - 2X2 - 53X - 39. 5. a3-a2-|-a4-3; a* + a'*-3a2-a4-2. 6. 7x^ -46x2-22x + 7 ; 49x"*4-49x2 4-5x-4. 7. x*-2x3 4-x2-8x4-8; Sx^ -24x24- i8x-2. 8. 4a"* -32324- 85a- 75 ; 332-15324-153+9. FACTORS, MULTIPLES AND FRACTIONS. 45 9. 36x""*-6ax2 + 36a2x-2ioa^ ; 6x^ + i4ax- — 94a^x + 9oa^. 10. I5X-'' - I4x2y + 24xy2 -7y3 ; 27x^4- 33x^y-20xy" + 2y^. 11. x^+x'^y'-^+y* ; x* + 2x^y + 3x2y2 + 2xy3 + y*. 12. x*+4x- + i6; x^Hhx*-2x^ + i7x2-iox+20o EXERCISE XLIII. Reduce to lowest terms: 3 4 5 6 7 8 9 10 ax a-x^" ax !o(x3-y'^) 5^^ + 5xy + 5y--i' 4a 2 _r2 4a"-^ + 4ac + c'^ 5x-^y+ lox^yS 3x-'y-+6xy=^ lob^ +2obc+ IOC 2 5b-» + 5b"^c x^- I x^ — 3x-^ + 3x— I x^ + a%"+a^ x*+ax'* -a'*x — a* abx+ bx2 adx 4- dx" * X"^ — 2xz2 X* -4x^z2+4z*° r2_ 5x A. X--4X-S 12. 14. 15- 16, 17. 18. 19. 20. 21. 3x'^ + 26x4-35 2x2 + 17x4-21' (a+b)2j-c2 (a4-b4-c)- * 20X* 4- x^ - I 25x*4-5x2-x~i' 4ag -- 9bg 4a2 4-6ab* ay4-y'^ abc 4- bey' 3a2 4-6a a'-* 4- 4a 4- 4' xy — xyz 3az — 3az''^* x^y4-2x2y4-4xy i-x2 i4-x4-y4-xy 27X4-X* I Sx- 6x24-2x8' w i6 EXERCISES IN ALGEBRA. II. 3 7 8 9 lO, II 12 14. IS- 16. 3mx + 5nx=* 6my + ionxy° 22. X®+IOx2+29X + 20 B. X^4-2x2-3X + 20 4x"'*+9x- 10* 3 _ x2 + 3x + 5 5x^ + iix- 15 x'y ■ xy*^ x^ + x-^y + xy-^-y* x^ + 4x'^ + i6 . x*-x3 + 8x-8 • 4. 5. 6. 6x3 -17x2 + 1 IX- 2 6x3-23x'' + i6x — 3* X'* — 2x- — x + 2 x'^-6x- + iix-6* m^ + 3m^ + 5m + 3 m^ + m^+m — 3 X* - Sax'^ 4- 33a''x^ - 763.^x + 77a^ x^ - 5ax* 4-2a'^x'* + 32a3x''' — 7oa*x4-49a'^ x* 4- 4x3 -5x4- 2 x^ - X* - 3X'* 4- Sx"-* 4- 3x - 3 * 3x*4-4x3-6y2- 12X-5 x^ - 2x* - 6x3 ^ ^ x^ 4. 1 3x + 6* 1 5x3 -38x2 -2x4- 2 1 3x3- 13X24.23X--21 a3-a2b-ab2- 2b3 a3 + 3a''^b 4- 3ab2 + 2b3 * a3 4-a2-ioa4-8 a'^4-2a^- 13a 4- 10 2 + 4x2 4-2x3+4x* 3 + 9x2 4-6x3+ 9x* 3x3 _. 27ax" f 78a-x - 72a3 6x3 ^ 2oax2 - 1 2a2x ~ 144a* 3x3 - 13x24- 23X- 21 15x3 -38X-- 2x4-21 ' factors; multiples and fractions. EXERCISE XLIV. Simplify: 47 I. 3- 7. 8. X- y X — ; X- x'^+xy X 2_y2 a2+4a X a2-3a 4a2-i2a 3a- + 12a ' i6x2-9y2 x--^-4 x-2 X • . 4x-3y X- — 7x X- + 2X x2+X-2 "" X- - 1 3x + 42 ^'-5x^ ,x2- IlX + 30 X- - 3x ''^^ x^-6x + 9 x2+2X X— 4 1^2^ -\i^ 3ax + 2x _£ — x*^ T— = 5a + b - 121 •-Jvl2-y1v2 9a ''X X+II 4x-^ '2 x"-9 4a2-i x + 3 2a + 1 ab + 3 * a^b'-^ + Sab' lOo II. 12, 13. 14. 15- 16. 17- x*'*-25 * x"^ + 5x * x2 _ 6x - 27 ^ x^-4x-4 5 x"-*- I2X-45 14X-15 x2+4K 2X- + 5X+2 X 2X-+9X + 4 X--4 — ^-^(x2 + 7x+i2). x + 4 X(x2- IIX + 30). 5 -6 x-s — y! , x-y (x + y)2 * x:f y* (a-b)2 . I a'f • (a-b)3 (a + b)^-cg ^ c^-(a-b)g a2_(b-c)2''^c2-(a + b)2* x'^ +2xy + y^'- z2 'LlZ.'ilf x"'^-2xy+ y'-^-z^ x + y-z* ■u 4S EXERCISES IN ALGEilUA. 18. TO. 20. 21. 22. 23- 24. 26. x2-(m-n)- (x-n)2-m2 \^-{n-m)'^ (x-m)^ n- x'' — X — 2 x''+2x-8^ x*-8x-9 x-20 x+i x2 + ex' ■ •■i _ x'-^ - 25 17X + 72 'x--9x4-8 x"^-i X-l -y3 xfy (x-y)2 4-xy x3-}.y3 (x-y)^ (x + y)'— xy* ( xf-a)^(x^b)g , (x + a)3(x-b)a (X + b)'^(x - c)» • (x + b) (x - c)--^ X«-y« x'^+y x + y x* + 2x2y2-|-y* "^ x'* - xy + y'-^ "^x^-y* 4x 2 _ 4x2- 16x4- 15 . 2X2- 17X f 21 2x2 + 3x1-1 * x2-6x-7 '^4x2-2ox + 2s" (a--b)a m^ + n'* ( a2-2ab + b2 a2-b2 7X+I2 X m + n 3_T m" mn — ) + n2/- X--X x^ -4X x2 + xH-l EXERCISE XLV. lu'nd the least common multiple of: x2-i ; x-^+x. a2+ab ; ab + b2. 2X2 4-X . 4x2y-y. 6x2 - 2X . gx2 — 3x. a + b ; a-^ + b^. I. 2. 3. 4- 5- II. 12. 13- 14. 15. x^-27; X2-15X-I-36. r2 _ 6. 4a2b-b ; 2a2 4-a. 7. x2 -4 ; x2 + 2x. 8. (x-l)2 ; x2- I. 9. x2+4x + 4; x2 + 5x + 6. 10. x2-6x + 8 J x2-5x + 4. I ix + 30 ; x^ 4- 2x - 35. x2 + 8x + 15 ; x2+9x + 20. x2— 9X-22; x2-I3x + 22. 2X24-3X+1 ; x2 — x-2. FACTORS, MULTIPLES AND FRACTIONS. 49 1 6. (x + 4)'' ; x3 + 64. 17. x"'*+x-2o ; x"-* - iox + 24 ; X- -x-30. 18. x- 4- 3x4- 2 ; 2x''^ + 3x4-i ; 2x=^4-5x + 2. 19. x''' + 5x + 6 ; 5x''' + iix + 2 ; Sx'-* + 16X-I-3. ^ 20. x2-4; 3X--X-14 ; 3x''«-i3x+i4. 21. i2x''' + 3x-42 ; I2x''^ + 30x+i2 ; IUA--20X-14. 22. 2x2 _ ^xy - 7y2 ; 4x2 - xy - 5y'^ ; Sx^ - 38xy + ^Sy^. 23. a-^ + 2a2b-ab2-2b=* ; a8-2a2b-ab2 + 2b». ' 24. x'* + 13x2 + 56x4-80; x» + i2x2+47x + 6o. EXERCISE XLVI. Find the value of; 4 5 II 12 13 14 A. 4X + 7 , 3X + 4 5 "^ 15 • X4-I , x + 3 x + 7 Y 1 , 2 5 .10 2x- 1 3X-2 4x- 3 6 2X + 5 X4-3 17 X 2x 8x2 x+3 x- 5 x-2 7. . 8. )7x 34X 5 IX a -2b a - 4b a - 8b 2a 4a 8a * ab b-c c-a 5X + 8 2X-3 X + 2 12 9 6 ^ ab be ac 6 ' 2X-14 x-9 x+3 30 !5 45 4x + 5 Zyi-^ 9 10. ~ ^+ ^ 5x 15x2' 3x-2y ^ 4x - 3z ,5 xy xz X a2-bc b2-ac c2-ab be ac ab 5a + 2b 4c -3b 5ac-2c2 3c ac-' a-c x-3 2x2-i8 8-x3 5x 20X' 15X 3 • si (i, ''\ 50 16. 17. 18. 19. 20. 21. 28. 29. 30. EXKRCISKS m AUiKURA. x-2y 3>1:: Ji 3x - 2a xy ay ax I I X + 2 X + 3' 3 + ' x+4 x+5 _4 3_ x-5 x-4' _3 L. X -6 x + 2' a b X - a X - b a b x + a x + b* 23. 24. 25 26. 27 x+y x-y x-y x+y" X X x+y x-y x-4 x-6 x-3 x-5" x+5 x-5 ~5~x + '5* X X I -x- I +X''^ 2a'^ 2a • a--b'-^ a + b* x'-^ + xy + y- X- — xy + y2 X + y a3 + b=» x-y a3-b3 a'-2-ab + b'-2 a^+ab + b'-** I I x(x-yy y(x + y) * B. I 2X + 2 _ v2 * X + y x-y x-^-y 2a 2 2 2. — + a + x a-x a- -X- a + X a-x a- - x^ a-x a+x a- + x^' x+i '^x + x+3 Factors, multiples and Fi< actions. 51 5- 6. 7- 8. 9- H 10. II. 12. 13- 14. 15- \ 16. 17. 18. 19. 20. - + X- - I X- I X+ I 2a_ jb 8b2 2a + 3b 2a -3b 4a''* -9b'''* 1 . 5 . 7x x-2 3X + 6 x'^-4" x^- 9x4- 20 x'-^-iix + 3o x-5 x-7 x!^-7x+i2 x2-5x + 6*. a+b b+c c + a (b - c) (c - a) (c - a) (a - b) ^(a - b) (b - c) 2 4-^r2 x^+y x- xy xy + y- x-+xy xy-2x' I X — y x-y ■ x^+xy + y- x^-y'* ' 10 12 15 X24-5X + 6 x2+9x+i4 X-+IOX4-2I a + c b + c (a-b) (x-a) ~ (a-b) (x-b)' .a b + + • (a-b) (a-c) (b--a) (b-c)^(c-a) (c-b)' a- -be _ b2 - ac c^-a b (a + b) (a + c)"^(b + a) (b + c)'^(c + a) (<^h)' a+b 2b b + c (a-b) (a - cT "^(b-a) (b-^'*"(c-a) (c-b)* I+X I -X 2 I+X + X2 "^ r^'^ + X^ ~ I+X'-i+X* * / a + b a-b \ _^ /a+b a-b \ \a-b''"a + b/ * V^T^'a + b/ _i ^L_ _^_i ^3_^ x-3a x + 3a x + a x-a* MM fej 52 EXERCISKS IN ALGEBRA. FRACTIONAL EQUATIONS- EXERCISE XLVII. 2. 3- 4. 5- Solve: X 2X— 10 3x — 6 X + I o + 23 x+ 19 X X — 4 X— 10 14 ~ 10 15- 8. -±UIZJ^',, 7 Z x-8 3 3-x ^7 21 3 • x + 5 X f-3_x+ I 6 4"~ 9 2X 7x II. — +5 = — +10. 3 12 10, 12. X + 2 X— I X — 2 10 14 4 * 6 3^-3 _ , .^+^ 8 8(x + 2) 5 14 + 6 I ox 13 2X 4x 13. - + 13= --+7. x + 9 4x 6x - 1 2 , 14. '^+- = +6. 27 5 7x 9x ^5- 8--4=ro-7- , ^Q-IIX 28x4-14 17-3X 16. — 4 -t / J 21 X X X 13 ^7- 4'-6 + 8=24- ,fi 9X.3-X 29X 18. -4-—-=-^ 43J. 5 8 19. 20. x4-5 x-2 x4-2 17-2X 4x4-2 5x =1. 3x FRACTIONAL EQUATIONS. S3 21. X X-2 X+23 lO + X 22. .3 4 x-9 3 5 ^ / • . X+15 8 -7-(55-x)-i-4 = -^-g-. . 23. ^(x-5)-f^(x-i3^)=i5-^-(i9-3-) , 24. 2x-i+^^-3^±^=7X:ii+5xl^. i. 3 5 5 3 25. .5X + .6X-. 8=^.75x4-. 25. 3x + 2 3x - 2 4x + 36 26, 27. x-3 x-f3 X--9' x-4 _ x-5 _ x-7 x-8 x— 5 x-6 x-8 x-9* ^8 ^3x-5 ,, 2X-4 25. X + = 12 29. 3 + X =H^-t) -h+[-{"-T)- 23 -X jf / ox 4 + x X- I 30. 7 =^ = -(x-8) +- — + . ^ 5 5 • 4 7 PROBLEMS LEADING TO FRAC- TIONAL EQUATIONS. £!XERCISE XLVill. A. ^ I. Find a number such that the sum of its sixth and ninth parts may be equal to 20. 2. What is the number .whose eighth, sixth and fourth parts together make up 26 ? 54 EXERCISES IN ALGEBRA. 3. What is the number of which the twelfth, twentieth and fortieth parts added together give as a result 76 ? 4. There is a number whose fifth part is less than its fourth part by 36 ; find it. 5. Find a number such that six-sevenths of it shall exceed four-fifths of it by 18. 6. Two consecutive numbers are such that one- fifth of the greater exceeds one-seventh of the less by 3 ; find the numbers. 7. Two numbers differ by 16, and one is eight-ninths of the other ; find them. 8. Find three consecutive numbers, such that if they .be divided by 14, 9 and 20 respectively, the sum of the quotients will be 23. 9. Four-fifths of A's money is equal to B's, and two- thirds of B's is equal to C's ; in all they have $595 ; what sum has each ? - 10. To a certain number I add its half, and the result is as much above 527 as the number itself is below 563. Find the number. 11. The width of a room is three-fourths of its length. If the width had been 3 feet more, and the length 3 feet less, the room would have been square ; find its dimensions. 12. Divide $1,3 59 between A and B, so that B's share may be seven-eighths of A's share. 13. Find a number such that one-half, one-third, and one-fourth of it added together shall exceed the number itself by 4)4. — 14. Divide the number 112 into two parts, such that if 2 1 be added to the less, the sum shall be less than one- third of the greater by the third part of unity. 15. What sum is that from which, if $46.20 be sub- tracted, one-half of the remainder shall exceed one-third of the remainder by $50? B. I. A certain sum consists of two digits, such that the right hand digit exceeds the left hand digit by 2 ; and, PROBLEMS LEADING TO FRACTIONAL EQUATIONS. 55 if the sum of the digits be increased by ^ of the number, the digits will be inverted ; required the number. . 2. Divide the number 360 into four parts, such that the first increased by 2, the second diminished by 2, the third divided by 2, and the fourth multiplied by 2, shall all equal the same quantity. 3. Find a number, s"ch that if 21 be taken from it, and the remainder divided by 8^, the quotient will be 5. 4. What number is that to which, if 11 be added, two and a-half times the sum shall be 85 ? 5. "What is the height of a house wall in which a window 6 feet high has under it ^, and ab jve it ^ of the whole height ? 6. A number consists of two digits, the first of which is grealer than the second by unity, and the sum of the digits is one-sixth of the number itself ; find it. 7. A, who walks at the rate of ' miles per hour, starts 18 minutes before B ; at what id n per hour must B walk to overtake A at the ninth mile-stone .'* 8. A, who travels 3X niiles per hour, start >- ^4 hours before B, who is going at 4^ miles an hour in the same direction. When will B overtake A? 9. How many minutes does it want to 4 o'clock, if three-quarters of an hour ago it was twice as many minutes past two o'clock ? 10. The sum of $1,650 is laid out in two investments, by one of which 15 per cent, is gained, and by the other 8 per cent, is lost, and the amount of the returns is $1,725. Find each investment. 11. A person goes from Hamilton to Toronto by boat at the rate of 13 miles an hour, remains an hour and a half in Toronto, and returns by rail at the rate of twenty- six miles an hour. He is gone altogether six hours ; find the distance from Hamilton to Toronto. 12. The sum of tw^o numbers is one-fourth of their product, and if 6 be divided by the first number and 3 by the second the sum of the quotients is r ; find the numbers. 56 EXEKCISKS IN ALGEBRA. SIMULTANEOUS EQUATIONS. EXERCISE XLIX. I. 3x + 7y = 27^ 5x + 2y=i6 2. 7x + 2y = 47. Sx-4y=i 3- 5x+ 4y=58 6x+i4y=i34 4- 5x + 8y=ioj^ 9x + 2y = 95 * 5- 6x + 35y=i77 8x-2]y = 33 I. X y -'^ + i = 8. 3 2 2. 2X x + -^- = .4. 3- |x-y=3 x-iy = 8. 4- I I —X y = 4 2 5^ I I — X + \' = 3 7 15' ^ A. 6. 2x4-7y = 52^ 3x-5y=i6' 4X+ 9y = 79. '' 7x-i7y = 4o 8 I5x-i3y = 78. ■ 7x- 4y = 55 9. i3x-2y=57 ^ 5x + 9y=-i88 10. ,72x+i4y=33o. 63X+ 7y = 273 B. 6. X y 3 -+- = !-. 7 5 7 x + y ~F x-y + 5 = 10. I + 7=9t 7. 8. 2x 5y 7 3 3x-,y=o. 2x->:^3=4. 5 3v= 9 X-2 SIMULTANEOUS EQUATIONS. 57 lo. y + 3 2X-- — - =7 + 4 8-x I 4y-— =24- I I - + - = a. X y = b. X y 3y -2x 5 2y+ I Problems: EXERCISE L. I ^1. The cost of 7 lbs. of tea and 5 lb-, of coffee is $5.15 ; the cost of 4 lbs. of tea and 9 lbs. of coffee is $5. ,0. What is the cost of i lb. of each ? 2. Six pounds of tea and eleven pounds of sugar cost $3. 54, and eleven pounds of tea and six pounds of sugar cost $5.64. Find the cost of tea and sugar per pound. 3. Five turkeys and four geese can be bought for $5.76, and seven turkeys and three geese can be bought for $6.66. What is the value of each fowl ? 4. Thirteen horses and eight cows can be bought for $1,166, and nine horses and twelve cows can be bought for $1,014. What is the value of each animal ? 5. If the numerator of a fraction be increased by 2 and the denominator by 4 it becomes equal to \^ ; and,, if the numerator and denominator are each diminished by 3, it becomes equal to f . Find the fraction. --f 6. Three times B's age exceeds A's age by 72 years, and one-half of As age is less than B's age by 17 years. Find their ages. 7. The sum and the difference of a number of two digits and of the number formed by reversing the digits ?ire 143 and 45 respectively. Find the number. 58 EXERCISES IN ALGEBRA. 8. The wages of nine men and eight boys amount to . $133.20 for the week ; if 4 men together receive $16.20 more than 6 boys, what is the wages of each man and boy for the week? 9. In 13 hours A walks i^j4 miles more than B does in 12 hours, and in 9 hours B walks 6^ miles more than A does in six hours. How many miles does each walk per hour? X 10. A farmer bought 100 acres of land for $4,122, V^art at $37.50 an acre and the remainder at $45.25 an acre. How many acres had he of each kind ? MISCELLANEOUS EXAMPLES. A. 1 . Simplify 2b — ^ b - (a + b) - [b - (b - a - b)j -f- 2a J^ . 2. Find the sum of a + b-3 (c-l-d), b + c-4(d + a\ and c + d-5(a + b). n _ 3. If x = 8, y = 7, z = 6, find the value of \/3x + 4y + 2z. 4. Find the square of 2 - 3\ + 4x''^ - 5x*^ 5. Subtract 2x2 - 3V'- -42- fiQj^^ the sum of 5x24. ^y^, 4y* - 5z'^, and 6z- - jx'^. 6. Solve ^ (2X-7)- -- (x-8) = i'^+2. 'o 3 ^ 30 ^ 7. Find the H.C.F. and L.C M. of x» + 3x--4 and x* + 2X-' --2X-f-4. F. Simplify I ) ^ -^ \x + a x-a/ X- +ax x-a/ x-4-a- ' 9. Show that the simi of 2a H 2b 4- 12c, 24a 4- 12b- 2c, and -I4a-2b4-2c is twelve times the sum of 25a4- 13b ~8c, - 13a- 13b- c, and - i ia4-b4- loc. 10. Find the factors of (a) loa- 4- 79a - 8, (b) 729a^ - b®. 5x4-3 , 4"<- Ji8 2\-f 1 1 . Solve —-- + ■ — r,— ^ = 3 - 17 if 5 MISCELLANEOUS EXAMPLES. 59 12. If a=i, h— -2, c = 3, d= -4, find the value of v/a2-4b + d2 - JsL + b'^'+c^ + d. 13. Find an expression which will divide both 4x'' + 7x=^ - 3x - 1 5 and Sx^ + 5X - 9 without remainder. 112 , 304 „ 1244 , 14. Divide 3 - 15X+ ^x2 + iox3 ^x*- 15 75 15 15. Simplify 16. Subtract 49a'^b"c2-35b3c« 9ia^bc-65a*b'''c''' x + 3 r._.„. ^ + 4 1 '4 • by I -x X' X2 +X- 12 from X--X 12 17. Find the H.C.F. of ax'M ab - a^ - bx^ and a-b fb'-^c - abc - ab^. 18. Find the factors of x2 4-3xy-4oy2 4-x-5y. 19. Resolve 4(a''^x» -8a^)-9 (b^x^-Sa'^b^) into four factors; also m^ -64. 20. Find the value of I I __i (a-b) (b-c)~(b-c)(a-c) (c-a) (b-a)' 21. Divide a*b2+b*c3+c*a2-a2b*-b2c*-c2a* by a-b + b'^c + c2a-ab'^-bc2-ca2. 22. Simplify + b2 + , (b-a) (c-a) '^(c-b)(a-b) '^(a-c)(b-c) • 23. Fmd the value of ~j^^+ -~-^^ . y. . 24. Find the remainder when x*-2x3 + x-7 is divided by x + 2; also find the value of this expression when x== — 2. 25. What number must be added to x^ + 2x2 in order that the expression may be divisible by x + 4 ? 26. If2s = a + b + c, show that ) (s-a)-+(s-b)2+(s-c)2H-$2=a2-(.b2+c2. ^^i—. 4 1 'It k^^' I? r ■ 60 EXERCISES IN ALGEBRA. 27. Find the continued product of x + 2, x^ + 2x + 4, x-2, x2-2x + 4. 28. Divide the product of x^ +8x4- 12 and x + 4 uy x'-^ + 6x + 8. 29. Divide the product of 6a2 + 23a +20 and 20ii'- -47a+2i by 8a"'*+6a-35. 30. Multiply (a'^+a+ i)x-a- i by (a- i)x-a- + a- i. 31. If x + y = 2a and x-y = 2b, find the value of X-^-2x2y2+y*. 32. Find the value of x* -2x''*y + 2xy3 — y* when x = a+b and y=a-b. 33. Resolve a^^ -b^® into five factors. 3X-2 6x-3 34. Solve = — -— • •^^ x+s 2X + 7 « 35. If x + y = 29 and X — y = 2i, find the value of 4xy. 36. If a4-b + c=io5, find the value of a(a4-b) + b(b + c) + c(c + a) + ab + ac+bc. 37. Find the H.C,F. of 2x'*-7x-- 15X + 27 and 2X--13X+18. / Q c- i-f (y±zi2x)2-(z + x-2y)2 38. Simphfy ^_^_,-^- 39. Solve (X- 3X-5I ' 2X-I _il + I / I X \ 2 3 4 ^ "3 40. Employ factors to find the result of dividing a* + b2c2-a2c2-a2b2bya2+ac-bc-ab 41. FindtheH.C.F. of7x*-iox3y + 3x-y'--4xy3 + 4yi and 8x*- I3x3y4-5x2y2 -3xy3-|-3y4. 42. What are the factors of x^ -88x4- 161 2 ? 43. Divide k-P-m(k2 4-m) l4-km'- by kl-m, and 43 ^^ ^2 I i a 4 4 removing the bracket, or bringing the fractions to a com mon denominator. 44. Simplify I a«-4a* + pa- ■} ■J o 2 a 4- 27 by a 4- 3 without \ n / N»r^4-n7 Nm n / T MISCELI.AI^EOUS EXAMPLES. 6l 2+2X + 4, i x + 4 Ijy and 20a'- a^+a- I. value of - y* when ; of 4xy. )f a(a4-b) + 27 and -)• f dividing 4xy3 + 4y^ > I — m, and -3 without s to a com- '--■)■ 45. Divide the product of a — b-fc, a + b-c, and b -f c - a by a^ ' b2 - c2 + 2bc. 46. Factor 6x» — 4x2y- 3xy 2 + 2y^. ■^X'** -f- 2x" — X -|- I 47. Breakup- — ^ into simple fractions. '^ xy ^ 48. If x = 3a-4b + 5c, y=5b-4a-3c, z = 4c-5a + 2b, find, the value of x + y + z. 49. If x = 7m + 8n + 9p, y=5m-3n-2p, z = m-n + p, find the value of x — y + z. 50. Ifx = a-f2b-3c, y = 3a-4b-l-7c, z = 5a + 9b-iic, find the value of 3X + 4y + 2z. 51. If x=i7a-3b-i2c,* y=i6b-4c~5a, z=i5c-3a - 2b, find the value of 5X - 7y - 9Z. 52. Use the formula (x + y) (x-y) = x2-y2 to find the value of (a + b + c) (a + b-c) (a-b + c) (c-a + b). J3. Simplify (i) 3(54- 2x) 3ax4-by+i5a b xy axy + 2a* X 2$(x'^-y2) ^ i2x(x-y) , 5 S^'^ 36(x-y)2 x'^ + xy " 3x"-^ ' 54. Divide X*-2x2y2+ y4_x2- 3y 2 _2y by X- -y2 -2y— r. 55. Find the factors of x^ -60X + 891 and a''^-4b2+a + 30b - 56. X— I X — 2_x-4 x-5 (Vx" 56. Solve *r ^ X 2 X— 3 X— 5 X -6 57. Find the H. C. F. of 35x8 + 47x2 + 13X+1 and 42x*^4ix3-9x2-9x- I. S«. Solve A(2x + 7)-^(2x-7)-if-J^(3x4-4).--:<^ 59. Express in wo^-ds : x" -y" is divisible by x — y always. ' 60. What number added to 4x^ + 34x* + 58x^ + 2ix''^ — 123 X — 41 will give a result divisible by 2x4- 13 ? 61. Find the value of lox*- 1109x3 - 109X-- 2 i2x — nil when x=iii. ^62. Find the H.C.F. of 4x8 -3x2 -24X-9 and Sx^ -^^ -2x2-53x-39. . r \4 ,-!^ t 62 EXERCISES IN ALGEBRA. ^63. Solve i^ + iin.^=ii^^^^ '^- ^ ^ x-io x-6 x-7 x-9 64. Find the factors of x-^ - 6x''^ -f i ix - 6. 65. If x + y = m, and x-y = n, express x^ + y^ in terms of m and n. 66. Find the value of x*-f x^y^+y* when x4-y=2a and x-y = 2b. 67. Find without j^ctual multipHcation the product of 2x* - 3x^ + 4x2 _ 3x + 2 and 2x* + 3x'* + 4^" + 3x + 2. 68. Simplify (a2+ab + b2)2-(a2 -ab f b-)'-^ + (a2 + ab -b2)2-(a'2-ab-b2)2. 69. Write down the continued product of (2x4-5) (2x - 7) (2x f 9) (2X - I I ). 70. Simplify sx'^ 4- 3xy - 2y2 -(3x2 - 5xy-7y2)-(x2 — 2xy + y2) and find the value of the result when x=2 and y = 3. 71. Find the value of (x-y)2 + (y~z)2+(z ■x)2 when •x=2, y==3, z=4. 72. Find the coefficient of x in the expansion of (x — 3) (2x-4)(3x+7)- / 73. Form the square of x^ + 2x2 -x + 2. / 74. Find the coefficient of x in the expansion of (x-a)(x-2b) (x-3c). 75. Simplify (a + b)8(a-b)2-(a2+b2)2. 76. Multiply 4a2-3ab + 7b2 by 3a2-2ab-9b2, and prove the result by division. 'j'j. Divide 8a^ — 64b^by 2a — 4b, and prove the result by multiplication. 78. Divide m^-2m3 + i by m2-2m-l-i. 79. If a=i, b = c==2, d = 3, find the value of ab^d -a2cd2+bcd-b'-(3ad-2b2+4c)-b^(4cd-3ac2). 80. Solve the equation : (x-3)(x-5) + (x-6)(x-7) = (2x--9)(x-4)-23. 81. Simplify x^ - [(x - y)2 - -^ (x-y-z)2_(z_x)2 J.]. 82. Find the H.C.F. of a3b2c, a2bc2, and abc* ; and also of 5x^yz2, i2xy*z, and 2ox2y2z». 83. Find + 3X+I. the H.C.F. of x3+x2-|-x4-i and x-^ + Sx^ MISCELLANEOUS EXAMPLES. 63 in terms x4-y = 2a iroduct of r (2X+5) 7y2)-(x'-' rhen x = 2 x)^ when of(x-3) ansion of 9b2, and the result J of ab^d :-4)-23. -x)2 H. ibc* ; and i x3 4-3x2 84. Simplify (x-y)2-z2 z2-(x + y)2 x2-(y-z)a x + y + z X (x + y)2_z^'7z-y) VrV-i - v'-i ^ y2 _ X — x''^-(y4-z)- x-y + z 85. Find the L. C. M. of 7(a-b), i4(a-~b'-'), and 2i(a'«-ba). 86. Smiplify 87. Simplify + ., x4 I i-x x- - I + r + I (x-2)(x-5) (x-3)(5-x) (3-x)(2-x) 88. Solve the equation : x-2 3 x+io X 3 ^4 5 4 89. When x= I, y = 3, 2=5, find the value of I2X*'- y + 2Z- x + y'-' + z^ 3X2 . • x + y2 5y3 90. Divide x* + iox3 + 35X- + 50X + 24 by x + 4. 91. Simplify 25a- I9b-[3b- ->]. 93. Multiply X* + 2ax'* + ssl^k^ + 2a.^\ + a* by x-~2ax + a^. 94. Divide i by i -x + x^ to four terms. 95. Add together a + 2x-y + 24b, 3a-4x-.2y-8ib, x4-y-2a4-55b ; and subtract the result from 3a 4- b + 3x + 2y. 96. Simplify -{ x(x + a) - a(x - a) )■ ■{ x(x - a) - a(a - x) }-. 97. Multiply 3a=*+ab-b- by a2-2ab-3b-, and divide the product by a + b. x-2 X + 23 lO + X 98. Solve X = -. ^ 3 4 5 99. If x= I, y= —2, z = 3, find the value of 3x2 — 2xy + 5y 2 + Sz^ + 2yz + 2XZ 4x2 + 2xy 4- 3y ^ + 2z2 + yz — xz - 100. Find the value of x^ - io2x5 + loox* + io2x3 — 99X- — 201X when x=ioi. t#i* (.A KXliKClSIOS IN ALGKURA. B. 1. A certain number consists of two digits whose differ- ence is 3, and, if the digits be inverted, the numljer so formed will be \ of the former. Y'xnd the original Find a number such that iff of it be subtracted from number. 2. 20, and ^^ of the remainder from J of the original num- ber, 1 2 times the second remainder shall be half the original number. 3. A fish was caught whose tail weighed 9 ll)s., his head weighed as much as his tail and half his body, and his body weighed as much as his head and tail. What did the fish weigh ? 4. A can do a piece of work in lo days ; but after he has been upon it 4 days B is sent to help him, and they finish it together in 2 days. In what time would B have done the whole ? 5. A cistern can be filled in half-an-hour by a pipe A, and emptied in 20 min. by another pipe B ; after A has been opened 20 min. B is opened, and in 12 min. A is closed, and B remains open for 5 min. more. There are now 13 gallons in the cistern. How much would it hold when full ? 6. Find the time between 2 and 3 o'clock at which the hour and minute hands of a watch are exactly opposite each other. 7. There are two bars of metal, the first containing 14 oz. of silver and 6 oz. of tin, the second containing 8 oz. of silver and 12 oz. of tin. How much must be taken from each to form a bar of 20 oz. containing equal weights of silver and tin ? 8. A starts at 4 a.m. from X to walk to Y, a distance of 50 miles. B starts from X at 5 a.m., and, passing A at the twentieth milestone, reaches Y at 5 p.m. When will A arrive ? 9. A man bought an equal number of two kinds of wine at 3 shillings and 4 shillings a bottle respectively. If he had spent his money equally he would have had EQUATION PROBLKMS. 65 two more bottles than he had. How many bottles did he buy? 10. A starts from a certain place and travels at the rate of 7 miles in 5 hours ; B starts from the same place 8 hours after A and travels in the same direction at the rate of 5 miles in 3 hours. How far will A have travelled when he is overtaken by B ? 11. A general on attempting to draw up his army in the form of a solid square finds that he has 60 men over, and that he would require 41 men more in his army in order to increase the side of the square by one man, How many men are in the army? 12. A and B made a joint stock of $4,000 by which they gained $1,280, of which A had for his share $256 more than B. What did B contribute to the stock ? 13. The tens di^it of a number is 2 less than the units digit ; and if the digits be inverted the new number is to the former as 7 to 4. Find the number. 14. A man can walk from P to Q and back again in a certain time at the rate of 4 miles an hour. If he walks at the rate of 3 miles an hour from P to Q, and at the rate of 5 miles an hour from Q to P, he requires 10 min- utes longer for the double journey. What is the distance from P to Q ? 15. The breadth of a room is two-thirds of its length. Had the breadth been 3 feet more and the length 3 feet less the room would have been square. Find the dimensions. 16. The sum of two numbers is one-fourth of their product, and if 6 be divided by the first number and 3 by the second the sum of the quotients is i. Find the numbers. 17. A man goes into business with a certain capital which he finds has doubled itself by the end of the year. He then withdraws $1,000 to pay expenses and the remaining capital doubles itself during the second year ; he t^en withdraws $1,000 as before, and so on for four ye^rs. He finds that he begins his fifth year with $5,000, How much had he to commence with ? 66 EXERCISES IN ALGEBRA. 1 8. The ingredients of a loaf of bread are rice, flour and water, and the loaf weighs 15 lbs. The weight of the rice increased by 5 lbs. is § of the weight of the flour, and the weight of the water is ^ of the weight of the flour and rice together. Find the weight of each. 19. A debtor is able to pay his creditors just 5s. in £i ; but if his assets had been 5 times as great, and his debts § of what they really were, he would have had a balance of ;^ 140. How much does he owe ? 20. A boy selling oranges sells half his stock and one more to A, half of what remains and two more to B, and three that still remain to C. How many had he at first ? 21. In a garrison of 2,744 irien there are two cavalry soldiers to twenty-five infantry, and half as many artillery as cavalry. Find the numbers of each. 22. Divide 150 into two parts such that, if one be divided by 23 and tlv^ other by 27, the sum of the two quotients may be 6. 23. The first digit of a certain number exceeds the second by 4, and when the number is divided by the sum of the digits the quotient is 7. Find it. 24. The length of a floor exceeds the breadth by 4 feet ; if each had been increased by a foot the area of the room would have been increased by 27 square feet. Find the din-ensions. 25. A person has travelled altogether 3,036 miles, of which he has gone 7 miles by water to 4 on foot, and 5 by water to 2 on horseback. How many did he travel each way ? 26. I wish to enclose a piece of ground with palisades, and find that if I set them a foot asunder I shall have too few by 1 50, whereas, if I set them a yard asunder, I shall have too many by 70. What is the circuit of the piece of ground ? 27. A man could reap a field by himself in 20 hours, but with his son's help for 6 hour«; he could do it in 16 hours. How long would the son be in reaping the field by himself? . . EQUATION PROBLEMS. 67 1 1 28. Divide 144 into four such parts that the first in- creased by 3, the second diminished by 3, the third multiplied by 3 and the fourth divided by 3 shall all give the same result. 29. The f.ides of a rectangle are 12 and 20 feet. What Is the breadth of* the border which must be added all round that the whole area may be 384 square feet ? 30. Find the price of eggs per dozen when two less in a shilling's worth raises the price one penny per dozen. 31. The difference between the squares of two con- secutive numbers is 1 503. Find the numbers, 32. What number is that, the double of which exceeds its half by 24 ? 33. A post is a fourth of its length in the mud, a third of its length in the water and 10 feet above the water. What is its length ? 34. A is twice as old as B ; 22 years ago he was three times as old. Required A's present age. 35. What sum of money is that from which, if 546.20 be subtracted, one-half of the remainder shall exceed one-third of the remainder by $50 ? 36. Divide 162 into three such parts that the first divided by 2, the second by 3, and the third by 4 shall give the same quotient. 37. Divide $2,481 among A, B and C so that B may have $72 more than A, and C $539 less than A and B together. 38. Divide $1,107 among A, B and C so that B may have half as much again as A, and C third as much again as B. 39. If 1 17 be added to a certain number the result is four times that ni.mber. Find the number. 40. Divide the number 132 into two parts such that five times one part may be equal to six times the other. 41. The sum of $745 was raised by A, B and C to- f ether ; B contributed three times as much as A less 30, and C half as much as A and B together less $20. How much did each contribute ? 68 EXERCISES IN ALGEBRA. I < 42. A gentleman left $750 to be divided among four servants, of whom B was to have twice as much as A, C as much as A and B together, and D as much as A and C together. How much had each ? 43. The sum of $18,259 was divided among four per- sons, so that the first and second together received $5801, the second and third together $8,023 ^nd the third and fourth together $1 2,4 58. Find the share of each. "{ 44. Find two consecutive numbers such that the fourth and the seventh of the fir.st taken together shall be equal to the fifth and the sixth of the second taken together. , 45. A herd cost $1,050, but, on 5 oxen being stolen, the ' rest average $10.50 a head more than at first. Find the number of oxen. 46. A person buys 8 lbs. of tea and 5 lbs. of sugar for $3.71, and at another time 6 lbs. of tea and 7 lbs. of sugar for $3.01. Find the price of tea and sugar per lb. 47. A farmer sold to one person 25 busl.-ls of wheat and 30 bushels of oats for $27.20, to another person 35 bushels of wheat and 20 bushels of oats for $31.70. Find the price of wheat and oats per bushel. 48. Two trains, 92 feet long and 84 feet long respec- tively, are moving with uniform velocities on parallel rails ; when they move in opposite directions they pass each other in 1 14 seconds, but when they move in the same direction the faster passes the other in 6 seconds. Find the rate of each train. I / 49. If one of two numbers be multiplied by 3 and the 'Other by 4, the sum of the products is 43, and if the former be multiphed by 7 and the latter by 3, the differ- ence between the results is 14. Find the numbers. 50. Five men and six women earn $1 16. 10 in 6 days, and six men and ten women earn $108 in 4 days. P'ind the daily earnings of a man and a woman. C. Find the product of: 1. 3x-"*4- 5x"'* + 7x + 9 and 4X + 6. 2. 4x-*+6x''*+8x4- 10 and 5X-H7. J 2; 3 3. 3 3 ^m MULTIPLICATION. 69 3. 5a"* + 6a'^ + 7a + 8 and 6a-8. 4. 6a» + 7a2+8a + 9and 7a-9. 5. 2X''* - 7x- + 3x - 6 and 8x + 2. 6. 6x^-5x2 f4x-7 and 9X-4. 7. 9x^ + 4x-+7x-|-6and3x2+4x + 5. 8. i2a^ - ioa'- + 8a-9 and 4a- -7a + 3. iia3 + 9a'-^-7a-5 and 5a-+6a-7. 3x-+7xy+i9y2 and 8x- + 5xy + 2y'^. Sa.'^- I2ab + 7b''^ and 3a- --4ab + 6b''*. Iix2-5xy-8y2 and »x- + 5xy+ i ly-. 4x* + 5x'^+6x^ + 7x + 8 and 9X-+10X+11, 9 10 1 1 12 13 14 15 2x*- 3x^+4x2 -5x4- 7 and 8x"-9x+io 7a*-6a3b + 5a-b--4ab=^ + 3b* and 2a" -4ab-f-6b*-. Find the continued product of: 16. 2X + 3, 3X + 4, and 4X + 5. 17. 5X-6, 6x-7, and 7X-8. 18. 3x + 4y, 5x + 6y, and 7x + 8y. 19. 4a- 5b, 6a -7b, and 8a -9b. 20. 6x2 + 7x + 8, 7x-f 8x + 9, and 8xM-9x+io. 21. a2-2ab + 3b-, 2a-~3abf4b-, and 3a''* -4ab4-5b-. 22. 9x2-8xy + 7y-, 6x- - 5xy + 4y-, and 3x--2xy + y-. iix'»- iox''' + 9x-8, 7x*'^-6x+5, and 4X-3. 3X-4-4X + 5, 2X + 3, 3x''-4x + 5, and 2X-3. 4a'- - 5a 4 7, 3a - 5, 4a- + 5a + 7, and 3a + 5. 23 24 25 Multiply together: 26. a-x, a4-x, a^+x^janda^+x"^. 27. x-i, x-3, x+i, and x + 3. 28. x24-x4- 1, x'-^ -x+ I, and x*-x2+ I. 29. The square of x^ - 3X 4- 2 and x^ 4- 6x 4- 1 . 30. The square of a4-b and the cube of a-b. 31. x''^-xy4-y-4-x + y f-i andx4y- I. 32. a'''4-b-4-c'-^4-bc4-ca -ab and a + b-c. 33. 1 i;x-* 4- i8ax - 14a- and 4x''^ ~ 2ax - a'-^. 34. The square of 2x- -3x4-4 and y.C^ - 5x4-6. 35. The square of 3a- -h 4a -9 ^'^^ the square of 9a''^-4a4-3. t.. ?d teXERCitSES IN ALGEBRA. D. I. 2 Find the quotient and remainder of: ( 2. (24X'* +31x2 +31x4- i9)-r-(3x + 2). 3. (3Jx3 4-87x^+94x+57)~(5x + 6). 4. (28a3 + 88a2+89a + 26) + (7a+i). 3' 3^ 3: 3^ 35 9. (24a2 - 43i3b + 26ab2 - 29b3)-i-(3a - 5b). 10. (84a^ - 59a2b - 53ab2 +36b3)-r(7a- 2b). 11. (6x* + i7x3 + 34x-+33x + 26)-r(2x2 + 3x + 4). 12. (i5x* + 3ix^ +7ix'-+6ix + 59)+(3x2 + 2x + 6). 13. (8x*-iox-* + 3ix--5ix + 4i)-i-(4x--7x-H3). i4- (35X* - 52x» + 23X'- + 41X - 8i)+(5x2 - 6x + 8). 15. (i8x*-39x-^- i3x2+43x + 7)-r(6x"-5x -7). 16. (6x* - 7X'*y - 2 ix'-'y'-^ H- 17x7'* + 1 5y*)-r (3x2 - 2xy- y^). 17. (i2x*4-4x^y -37x2y-+85xy3-39y*) -(4x2+8xy-7y=^). Find the quotient of: 21. (a8 + b3 + c3-3abc)-f(a + b + c). 22. (a^+b»-c34-3abc)+(a+b-c). 23. (a'*-b8+c^ + 3abc)-r(a-b + c). 24. (-a3+b3 + c3 + 3abc)+(-a+b + c). 25. (x"*+y^+8-6xy) — (x + y + 2). 26. (x'»+ya-27 + 9xy)-7-(x + y-3). 27. (x'*-y' + i +3xy) + (x-y+i). 28. (8x3+27y»+64z=*-72xyz)4-(2x + 3y + 47.). 29. (27x3 - y** + 82* + 1 8xyz) -4- (3X - y + 2z). 30. (i25x® + 27y*-2i6 + 27oxy)-: (5x + 3y-6). h ADDltlOM AND SUBTRACTION OF FRACTIONS. 7 I Divide the product of: 31. x'**-i2x+i6and x**- i2x- 16 by x- - 16. 32. x3-- 3x4-2 and X--2X+ I by x3-3x'^ + 3x- i. 33. x-4, 2x'-^ + 3, x2 + x-i,andx2-x-ibyx*-3x= + l 34. a3'l-x3 anda^ + ax + x- by a^ + a'-^x'-^+x*. 35. x2 + 2xa + a? and x* -4x«a + 6x2a- -4xa» + a* by x* — 2x^a+2xa^ — a*. Perform the additions and subtractions : 2. 3- 4- 5- 6. 7- 8. 10. 3x-y x-r3y 5x + 3y 7x + 9y" 23x-'+ i8x y+i7y^ i2x- + 3!xy + 2oy''^ X - m X - n , J X - n X - m (x - m) (x - n) X --X+I 2X'X-i)" 2x^^_0*. x'-i+x+i "^x^-hx'-^ + i x«+x* + i ' I I 2a 4a^ 8a^ _ 2x + 3 y 3x + 4y' (m - n)2 1 -a a i+a i+a' b + i+a-* i+a^ 2a -2b 2b — 2a /I I \ , /x + y x-y\ Vm + nA^ + y^-V-l^""^^ 15 72 2(X+1) lO(X-l) 5(2X + 3) 3 + 2a 2 -3a i6a-a2 2-a 2 + a a^-4* I y X 1 i . X + y x--y" x-+y- EXERCISES IN ALGEBRA. II. 12. 13- 14. 15. 16. 17. 18. 19. 20. 3x !ax X - a X 4- a x*'^ - a*-^ 3x-4y _ 2x - y - z I5^_4? _ x-4y 7 3 12 21 ' x + 8 x-l-7 x + 6 x2 + 5x + 6 x^ + 7x+i2 x"'*+9x + 2o* x+ii x4- 10 x4-9 • - + X--8X+15 X--1IX + 24 X-- 13x4-40 x-9 X- 12 x-4 X24-4X-2I x2 + 5x-2v~x'-* + i5x-t-56 a^+a^b a(a-b) 2ab x + y 2x x^v-x' y 2x x^y-x^ x + y x-y-y^' a- 2a8+bS-ab2 (a + b)(a- + b=^) aa_b2 a'- + b- I+2X I-2X . I -2X I +2x' I-2X I+2X • ^ I+2X I-2X a"-( b-c)- b' ^-(c-a)g c'^-(a-b)'> (a + c)--i-b2"^(~a + b)'-^-c-*'^(b + c)2-a«* ANSWERS. EXERCISE I.— (Page 5). A.-(i) 35. (2) 39. (3)34. (4)76. (5)25. (6)0. (7)75- (8)48. (9)72. (10)64. (i')O' (12)0. (13)220. (14)280. (15)30. (16) 24. (17) o. (18) 80. (19) 420. (20) 312. B.— (i) 25. (2) 16. (3) I. (4) o. (5) 125. (6) I. (7) 625. (8) 56. (9) 576. (10) 15. (u) o. (12) 8r. ([3) 675. (r4) 432. (15) 2000. (16) 16. (17) 8. (i8) 625. (19) 45. (20) 3840. C— (i) q. (2) 40. (3) 35. (4) 24. (5) 72. (6) o. (7)343- (8)7776. (9)19- (10)360. (11) I. (12)3. (>3)^. (14) ¥• (15)0. (16)5. (.7)380. (18)384. (19) 202 J. (20; 253|. D--(0 5- (2) 9- (3) 4. (4) 12. (5) 15. (6) 8. (7) 10. (8) i^. (9) ^. (10) X. du 15. (12) 6. (13) o. (14; 10. (15) I (16) 4 (17) 2|. (18) ij. EXERCISE II.-(Page 6). A.-(i) 363. (2) 51. (3) 24. (4) 236. (5) 240. (6) 187. (7) 138. (8) 818. (9; 25. (10) 81. (11) 1331. (12) 64. (13) 625. (14) 13175. (15) 3555. B.-(i)2M' (2)inVo' f3)i3. (4)2fe. (5)2oi^. (6) 33l?. (75 29T?fV (8) 4,^V (9) iSlM- (10) 4- EXERCISE III. (Page 7). (r) 3J. (2) 10. (3) 5.1. (4) Sh (5) 37. (6) 6. (7) 4h (8) 5^ (9) 2-/,. (10) 2^. EXERCISE IV. (Page 8). A.-(i) I5a + i8b + 2ic. (2) 23x + 24y + 2oz. (3) 3b. (4) 5a + 5b + 3c. (5) -2x + 2y. (6) 4x-4y + 5z. (7) 3a4-5b- 12c. (8) 3b + 350. ^9) 4oa-7b-2c. (10) -a+iic. (11) 2ax + 9by. (12) 24p- i8q-2or. (13) 15111- ion +9p. (14) - loab + I5cy4-i5. (15) 132. 73 Ml 74 fe5^KRCtSES m Alx;KliRA. B. — (i) 3ab + bc + 4ca. (2) 39ab-bc- loca. (3) pq -3qr + rp. (4) 6x. (s) 20a -2b + 90. (6) 39xy-36yz -zx. (7) -2x-43y + 3iz. (8) 27a-24bc- i6d. (9) 9ax-7by-4cz. (10) i9a+7b-7c+ 13d. EXERCISE V.-(Page 9). A.-(i) abc. (2) 13x2 -6xy + 7y'*. (3) -6a'^ + 6ab + 9b2. (4) iox2+4xy + y'^. (5) xy + yz + xz. (6) 8a'^ -932 -6a +15. (7)i5x*-8x3 -21x2. (8) i5x3 + i6x2-36x-33. (9) a« + b'» -f-c'». (io)a-^+b3 + c3 + d3. (ii)2x3+x2-iox+i2. (12) 3x-i + 2xy -3xz-2y--4yz. {1 ^) 2x^ - 6x^y + S^V^ • ('4) ^^ + b^+c^-3abc. (15) iox^ + 38x2y-2ixy- + 22y3. B.-(i)ia-^2b-J^c. (2)Y(fa4-Mb-Wc. (3) ^^ -H^y-W- U) T^^a+^b + Ac- (5) -^a+iib -ic. (6)iax«-|4xy + ^y'-^. (7) -^x^-fax^ + fa'-'x. (8) -a3-^a=^b + iab-i + b^ (9) ^i^ + is^ + irP- (»o) -2a'»-ib-' + /^c»-Habc. EXERCISE VI.-(Page 11). A.-(i) 5x + sy4-4z. (2) 8a-+-8b + 2c. (3) 3a+iob+ioc. (4) 2x-3y-3z. (5)a + b-5c. (6) 7x+i7y-i7z. (7) -4a + 33'3-3c. (8) iox+i3y-i4z. (9) 6a -4b +50. (10) 4x + 27y -30Z. (11) 2ab — 6bc + 7cd. (12) — 2ab + 2cd-2ac + 2bd. B. — (i) cd-8ac4-6bd. (2) -2xy + 2yz-2zx. (3) -I2p + i9q-r. (4) -6a-2b-2c. (5) 2a+2c-5. (6) -x + 3y + 3z- (7) -a-b-sc. (8) 4ab + 2oxy-4r. (9) 7a-i3xy + 27. (10) 2x-^y+fz. (n) -"M^-riy -2}.}. (12) -fa-J^*b + Jc. EXERCISE VII.— (Page 12). A. 4-21ZX. (2) -2x3+ I4x'- + 12x4-3. (j) + 25y*. (4) -i54-7ab-8aab2. (5) -5c^ab. (6) -2a2b + i6ab2-37. (8) -2a2-2d2. (9) 9x3y-6x2y2-2xy3. (10) 2a3 -6a2b+6ab2-2b3. (11) 34x»-30x2+26x-22. (12) 4a"'*+4ab-8b2-t-2c2. B— (i) a3-abc4-c'*. (2) 4x*-3x'»-3x^-2x + 3. — (i) 6xy-ioyz -9x*''y'-* — 22xy* -3a2bc + 6b-ca (7) -x^+4i. (3) (5) -4a3-6b3+8abc (4) x'' + 2x*4-x3 + x2 -2X-2. a»+9a2b4-i5ab3. (6) -2x*-3x3 + 2x2- 17X + 14. _a3 ANsNvb:rs. ^i (7) -Ja + ^b. (8) |x2-ex-^. (9) fa*-=-iH-i4}. -^a'*-2^ax'-^ + |Jax3. EXERCISE VIII.- (Page 13). (i) 2ix-6y-7z. (2) 4a+b-3ic. (3) -5X8-5X--3X-I-28. (4) 3 + 3a -23a'*-34a*. (5) iia-3b + 7oc. (6) 4x''* - I2xy + 2y=*. (7) 4x3+x'^y-i3xy2+i8y3. (8) -23a'-* +2 7a 4- 3. (9) i8x2-3x+5y + 2. (10) -4ab + 5ac-7bc. (11) -8x*^ + 9X-5. (12) -24x^-20x'^ + 38x4-27. (i3)3x2+i3xy -2y" - i6xz-9yz-3z2. (14)0. (15) -3x4-14. (16) -3l-6m-7n. EXERCISE IX.— (Page 14). A.-(i) 20x«. (2) 2oa9. (3) 2ix'-^y^. (4) 4a2b^ (5) 24XV. (6) 34ab. (7) 5a«. (8)63x7. (9) 3oa'MD3. (10) 72x*y2. (11) i2ia"b^. (12) 32a7b3. (13) 6a'^x7y3. (14) 35a^b^^c*. (15) abcxyz. (16) 72a2cx2. (17) 2ia*b*x3. (18) 3oa^x^y2. (19) 2a"b**. (20) 28x7y^z^. (21) 48a^b-*cx. (22) 76x5ySz^. (23) 98abcxy^. (24) 7ai2b7cfl. (25) abxy^z'^. (26) 9m'*n^p". (27) 56a^c'*x"'. (28) ac^x-y. (29) 5a2b3c«x-i. (30) 224a'"**bi6m-. B.-(i) a8b2+a2b2c. (2) sx^yS + sx^y". (3) 5<^x3 4-2ix2y. (4) i5a^b*x*-27a'^b*x^. (5) 2oa3b- isab'-^. (6) 56x*-63x3y. (7) -3a2x3. (8) 27a=^b2x'2. (9) 3a8b3c*d«. (10) -8x3y*z«. (11) -a-b-^c*. (12) 63x*y*z*. (13) 6m3n*p^. (14) 9a2bc. (15) a"MDC + ab»c-abc3. (16) a3b*c*-a'^b*cS-a3b3c«. (17) -I5a^c + 27ab2c-f33ic3. (18). I26x*y+ Ii2x*y2 -98x3y3. (i9)2ix«y* + 35x*v«. (20) Sx^y^z^ -3x'*y*z2 + I3x«y-iz3. (21) a3b2c4-a2b3c4-a2b«c'^. (22) Sx^y^z^ - sx-y^z*" + sx^y'-^z^. (23) 8x2y*z* - I2x3y*z* - 20x*y3z^. (24) -65x8 4-52x2 -104X+ 117. (25) -5oa*b-7oa3b 4-6oa2b- iioab. EXERCISE X.-(Page 15). A.-(i) 8x»-26x24-33x -18. (2) 2ox*4-x"*-2x^4-3x-4o. (3) 12a* -37a* 4-i5a3 4-iia2-i7a4-56. (4) 6x*-96. (5)x*-2x4-i. (^>) 35-47a4-3ia2-75a3 4-c4a*. (7) 26-115X-15X'* 4-i3x" + 63x*. (8) I5x'^-32x2y4-37xy2-28y3. 76 KKKRCISKS IN ALfiKBRA. x'^a (9) 49a* -9ia^b4- io6a'-b- +-2Hb^ - 24b*. (10) x* - + xa''^-a*. (i i) 64a'' -27b^. (12) 40X'* - i ix-* -4ox''' + 83X-72. (13) a*+4a2x- }- i6x*. (14) looa* -9a''*x- + 6ax-"^ -X*. (15) a^ - Sa-'bH- I2a''%"'* - i6a-b=* + i lab* (16) x''-729a''. (17) 9x" + I4x'^- 23x^2 -36. (i) a2 + 2ab + b2-c2. -3b«. B.- - i6c^ - i6y*. -b^cl2. + z^-3xyz. (10) C12) xi^ + yio. (2) 4a2+ r2ab I 9b^ (3) x'^ + x-y' + y*. ^4) 9x*-4x-y"'' + i6xy .3 (5) 2X y" + y^ (6) a-b-' + c'-^da ~ a-c- >5) (7) a"- I. (8) ao + ^a-'b^ + b". (9) x'» l-y (11) 8ia^-256x^ (14) x^-a^ x''4-2x3y'^ + y''. x^ 4- yl — I 4. 3xy. (13) a''-2a^4- 1 (16) x8 4-x*a*+a«. EXERCISE XI. -(Page 16). A.— (i) x\ (2) x a^x3. (5) (4) 9a 9) (13) 6a^ (17) loy-x. 5x^3. (10) (6) -9a-'. (7) -8x«. 4ac' (11) -- sa'-b'^c'"' (3) 3a^ (8) xy^ (12) 5x*y. (16) -I. (20) B.— (i) 2x*4-3x'' + 4x4-5- (2) -a= + 5a + 6. (3) -5a2b2 4-2ab-4. (4) 9a-b- - I2ab2c i I5a'-^bc. (5) -a + b-c. (6) a"*-a''*b4-ab--b-^ (7) -2x'^y''^ + 3xy-4. 77a^b"*x'^ (14) 7a''. (15) 7a''^b'^c=^. (18) -6b-x. (19) 8abc«. (9) 27m''n'^ 28nT"*n*4-9mn-p. {i\) i9b-c"-^+ 1 2bc-'* - 7C'*. (8) 8x*y* - 5x-*y - 2x. (10) I3a2b-9ab2-7b. (12) I2x3-9x''^y + 8xy- - sy*^. EXERCISE XII.- (Page 17). A.-(i) x + 6. (2)x-io. (3) a -5. (4)x-24. (5)3a+i. (6)5a+i. (7) x + 5. (8) 4a-7. (9) 3a- 5- (10) 5x-a. (ii) 3a + 7C. (12) iox-9y. (13) 8x + 3y. (14) 3a2 4-2a4-i. (i5> x2-3x4-7. (i6)x-+x4-i. (i7)x- + 3y. (18) a3 4-3a-b4-3ab2+b-». (19) a* -4a3b + 6a2b2 -4ab3 + b*. (20) x'--2X+i. B.— (0 x2 -2x4-2. (2)a2-3a-i. (3) x^ 4- 5x4- 6 (4) 6a2-7a4-8. (5) ya^ 4-5ab4-2b2. (6)x2-2x4-3. (7) x2 4- 8x + 1 2. (8) m"^ 4- 7m - 5. (9) a" + 2ab 4- b- f a 4-b4-i. (10) b54-5b*c4-iob3c24-iob2c34-5bc* + c'^. (11) 3a4-2b4-c. (12) x* -x'^y4-x2y'- -xy3 4-y*. (13) a*4-a»b4-a'-^b-4-ab-M-b*. (14) x^ 4-x*y4-x'»y2 4-x2y3 ANSWERS. V + xy* + y''. (15) a'»-2a2b + 4ab2-8b' (i6j 27x^ - i8x--^y+ I2\y2 -8y«. (17) 8a''+ I2a-b+ i8ab- + 27b"«. (fS) x* + 3x^y4-9xy-+27y=». (19) x"-x''' + x'* -x-' + x'-^ ~x+i. (20) x"* + 2x''«y + 2xy24-y"*. (21) a'' -a" + 2a- -2. (22) x- + 2\y + y- - xz-yz + z-. (23)a + b-t-c. (24) a-f b -c-d. (25) x'-^+y-^+z- -xy-xz-yz. EXERCISE XIII.-(Page 18). A.-(i) a. (2) a4-b-c. (3) a- b. (4) 2x. (5) a + a**. (6) -2b + 2c. (7)3a-b-c. (8)a+3b-4c. (9) S^- (io)4a. (11) x. (12) -x-2y+6z. (13) -5a. (14) 2a + 4b. (i5)iix -36y. (16) 2ia + b. ([7) 2x- 3y+ I2z. (18) -a- + 8b'--9c". (19) -50c. (20) -a- iob + 2c. (2i)x + c. (22) x- -ax + b. (23) x'-^- (a- 2b)x-2ab. ^24) x^ -px + q. (25) x-+ax-2b. (26) x'^ + bx + a^. (27) x^ --ax- 4-bx. (28) x* + (p-q)x'-^+pq. (29) px2 + qx-|-r. (30) x'-^ -hCn-HOax-a"-^. IJ.- (i) (ax — bx -f- ex) — (ay - by 4- cy) ; (ax -ay) - (bK - by) + (ex - cy). (2) (ax» - dx») + (bx"-^ - dx") + (bx-cx-2x) +(7-c). (3) -(a^x-a'-^y) - (7a-l-ab) -(2X-3). (4) (ax^-|-3x*)-h(bx--8x*'2)4-(3bx-9x)-f-7. (5) (6ax=i ~ bx'») 4- (4bx''2 - 2x-'^) + (ex - 5x) + (ab - 8). (6) (loax'* - 8x'*) 4- (6ax'- - I2x-') -f- (9X - 3ex) + 4. (7) (3cxS -2a'fx''')-l-(3x^-4bx^)-f-5dx-4abe. (8) -(bx* -l-2a-\^) -(3bx'*-4x^)-(3x'''-ax-). (9) -(abx^ - 7x")-(abex'* -8x"*)-(3c"x-9ax). (lo) -(rx'* -a'-x^) -(bx--ax- -f-5x-\ (11) -(3ax*-6b-'x* + rx-^-|-7x*)-(2bx4-5e-x). (12) - (-5ax-'^-4cx"') - (-3ax^-f 6bx-- 7ex=') - (- 2ax + 7bx). (13) (2.1 3b)-(4e-5d)-(4e-3f); (2a- 3b -4c)-t.(5d- 46-^30. (f») -(b + 5c) + (6d-3e)4-(4f + ^0; -(b-|-5c-6d)-(3e 4f-g). ('5) -(3-^-4y) - (2z - 3a) 4- (2b - c) ; - (3x - 4y 4- 2z) 4- (3a 4- 2b - c). (16) (4c-2d) + (3e4 2x)-(y-|-5z); (4e - 2d-|-3c)4-(2x y -5z). (17) -(2m-3n)-|-(4a-61))-(5x-7y) ; -(-m -3n-4a) - (6b-l- 5x-7y). (18) (3p 4- 2q) - (4r+5m) -l-(3n - 2a) ; (3P+ ::q - 40- (5"^ - 3" + -O- EXERCISE XIV.— (Page 20). A.-(i) 3. (2) 2. (3)9. (4)7. (5)11. (6)7. (7)2. (8)2. (9)3. (10)4. (:i)9. (12)56^.(13)2::!. (.4)9- ('5) -4]. 78 EXKRCISKS IN ALGKBRA. (16) -3. (17)2. (18)7. (19)4- (20) 75- (21)8. (22) 7. (23) 10. (24) f (25) 4. B.-(i) 13. (2) 5- (3) 16. (4) 10. (5) 4- (6) 15. (7) - I. (8) I. (9) 2. (10) I. (11) I. (12) I. (13) 2. (14) I. («5) 20. (16) 3. (17) 12. (18) 19. (19) I. (20) 2. (21) 2. (22) 4. (23) 7. (24) -6i (25) 5*- EXERCISE XV. -(Page 21). A. -(7) x-(a + b). , , m-(ax + by) , , x (8) x + y + z. (9) na + x. (10) 4 —. (11) a + b liours ; ax and bx miles. a + b """ a + b B.— (i) abc ; 2ab + 2ac + 2bc ; 4a + 4b + 4c. (2) 432xy. (3) %-{ 2(a + b)c + ab )■ (4) (mp + nq) (5) 9CX5 * 100 ax + by + cz /^\ / , v . / , \ 1 -aTbTT' (6)('c + y)-i (3) (8) (12) 2X 3y' ax^z'^ 9bc 6x2z^ be (4) ^. (5) ax 3ax (9) :-7zr' ('o) (6) 3cd' 2b 3nz i5cy (•3) I. (14) (,6)1^5l<_m! ^ ' I2pq (17) y«z2 nx^ 2m a^b^c^ x'^y'^z"* (18) ^^ (7) (n) 9a 3 8b-c- 5mp2 2n , , 8a2c2 (^9) ' (20) 49a2cy^ 64bd2x« (21) ^ 2np*ci'''y'* 25m-x^ ax^y (^-''^) b^ • (23) 4a''*c 9c 8x , ^ io8x*y2z2 EXERCISE XX.— (Page 30). (i) I2a2b2. (2) 36a'*b''^c». (3) 24a2b2x»y=*. (4) I2a3x2. (5) I2x«y3. (6) Sa^b^c*. (7) 2a2bc. (8) 24abxy. (9) i5a*b*cS. (10) i2xVz- (11) 228a3b2x3y. (12) 8im8n2pci. (13) 24a-bc2. (14) a^b'-'c^. (15) i2xV- (•6)42x*y«. (17) 2ioa3b3c3. (18) i2oa2b2c-. U9) I32a*b*c3. % '%. iju 8o EXERCISKS IN AIXIKHRA. (20) 2 52oabcxyz. (21) 6omnpq. (23) 204xyz. (24) ii4oabx-y'z-. (22) 78oa2bac« EXERCISE XXI.-(P. — 6^;^ — EXERCISE XXVI.--(Page 34). (1) c- ; b 4- c. (2) 9 ; c + 3. (3) q-^ ; p + q. (4) 25q- ; p4-5q. (5) I ; X4-I. (6) 9; ab4-3. (?) 16; y -4. l«) 36; x-6. (0) 49b''* ; a -7b. (10) iooz- ; x-ioz. (n) gb*-* ; 2a4-3':>- (12) i6q2 ; 3p4-4q. (13) 9; 4x + 3- (14) 361 ; y 4- 19. (i5)i;6a+i. (16) 4b2 ; 5a4-2b. (17) c- ; 2a4-(:. («8)y2;9x-y. (19) 4b2 ; 3a - 2b. ^20) 9d- ; 2c-3d. (21) 9y2 ; 5x-3y. (22) a^ ; i-a. (23) 4x • ; a-2x. (^4) 490^ ; 2ax-7c. (?5) 25x-y2 ; 3am4-5xy. (26) i ; 4C-1. (27) 9; 2xy4-3. (28) i ; ax- I. (29) 4 ; by + 2. (30) 4 ; 13XZ-2. EXERCISE XXVII. -^Page 34). (21) x24-2xy + y--z'-. 1,22) a- + 2ab4-b--c''*. (23) r-» + 2lm4-m^ ANSWfc.ii.i. (24) 4u- 4 I2:il) + c>l)- - 1 6c-. + 9z - . (26) 9a - + (y.h: - 1 6b - + c - . 81 (25) 25X- 30XZ (27) x*4-x-+ I -n- -4V (28) a*x^ + a«x- + i. (29)xM-x-. - + y*. (30) a* +a'''b''' + b*. EXERCISE XXVIlI.-(Page 35). C— (i)a + b + (:) (a + b~c). (2) (x-y + z)(x-y-z). (3) (a-b + x) (a-b-x). (4) (a + b-c)(a-b + c). (5)(i4-m + n) ( r - m - n). (6) (x + y + 3xy) (x f y - 3xy)- (7 ) a + b + 2ab)(a + b-2ab). (8) (x + a + y) (x + a-y). (9)(y + c -x)(y-c + x). (10) (x-2y + 3\y) (x-2y-3xy). (m) (a-5b+i)(a-5b-i). (i2)(4x + a + 3b)(4x-a- 3b. (13) (2in4-p-q)'2m-p + q). (14^ (p f-2q + r) (p + 2q- r;. (k) (x + y-' z) (x-y + z). (16) (a + b + c-d) (a+b-c + cl). (17) (a-b + c + d)(a~b-c-cl). (18) (2a + 6b + 4c)(2a-4c). (19) (x-y + b-2c) (x-y-b + 2(). (20) (a + b 4- 3x + y) (a - b f 3x - y). (21) (x + m + 211 - i ) (x-m-2n-i). (22) (a + b + c + d)(a + b-c-d). (23) (a f b-c-d)(a-b--c + d*. (24) (a + x + y + z) (a + x - y -z). (25) (a-4b + 5c-i)(a-4b-5c+i). (26) (3ab - 4cd + X + y) (3ab + 4cd 4- x - y). (27) (x - y 4- z + w) Tx-y-z-w). (28) 2ab4-3(:d 4-4xy'i (2ab4-3cd-4xy . (29) (5ab-h6c-9d) (5ab-6c + 9d). (30) (2x-3a4-c + d) (2x~3a-c-d). EXERCISE XXIX.— (I 'age 36). (i) 35CXXX). (2) 136 00. (3) 30CXXX). (4) 2452000. (5) 336000. (6) 21320000. (7) 25930800. (8) 4000. (9) 81735. (10) 236880. EXERCISE XXX.- -(Page 36). (1) a2 4-b3 + c2 4-2ab4-2ac4-2bc. (2) x'- +y- 4-z" + 2xy + 2xz-f 2yz. (3) m''^4-n-4-p'^+2mn4-2 rp + 2np. (4) a- + b-+c"-' + 2ab-2ac-2bc, (3) x- +y- +z- - 2xy4- 2xz - 2yz. (6) a2 4-b2 + c'--2ab-2a(: + 2bc. (7) x^^ +4y''« +z-4-4xy 4-2xz + 4yz. (8) 9a- 4- i6b^4-4c^ -24ab-f I2ac- i6bc. (9) I +4x-4-9y- + 4x-6y- I2xy. (10) 25 4-64y- 4-36z'-^ - Soy + 60Z - 96yz. (11) 16- 40X - 23X- 4- 6ox"* 4- 36X*. (12) 9x* 4-42X"' -i-x^ - 112x4-64. (13) 25P"^4- i6q-4-8ir2 -4opqf9opr-72qr. (14) x* f y* 4-z* 4-2x''^y"'* 4-2x''^z^ 4-2y-z-. (15) a^ 4- ])''4-c* - 2a-b- - 2a-c'^4-2b-c2. M 82 EXERCISES IN ALGEBRA. (i6) x2y2+y2z2 4. 22x2 4. jxySz + 2\^y7. ■\- 2xyz^ . (17) 25X* -9ox-* + I nx" - 54x4-9. (18) i+a'-x- +b''^y- -2ax -2by + 2abxy (19) 4(x- +y2 + z-). (20). 4(a'^ + b'-^-fc2). (21) 2x''y- + 2x-z'-i + 2y-z--x*-y*-z*. (22) 2a-b'» + 2a2c2 4-2b-c--a*-b^-c*. (23) a"'+b'^ -f-c- -ab -ac-bc. (24) 6a- f-2b'''-f2c2-f 4ab + 4ac. (25) 5X — 6y - 7z. EXERCISE XXXI. -(Page 37). A.— (i) x^ + ax^y 4-3xy-+y'*. (2) a"* + 3a-b + 3ab- +b'^ (31 x'^ + ax'-^a + 3.\a-+a^. (4) x'»-3x'-'a + 3xa'-^-a'». (5) x'» -I- ix'-^z + 3XZ- + z"» • («^) x"» - 9X''' + 27X - 27. (7) a» + 1 2a'- + 48a + 64, (8) a-^ - I5a''^ + 75a- 125. (9) x-' h^jx^^y + I2xy=* -f8y^ (10) a'^-6a2bH-i2ab*-J-8b'*. (i i) 27x-» +27x-'y + 9xy'* + y ,3 (12) 64X'* - 24ox''^y 4- 3ooxy''^ - I25y (f3) a'M3-»4-3a2b2c + 3abc-4-c'\ (14) 8x''y»-36x*-^y-z + 54xyz--27z-^ (15) 2i6a«+ io8a2bc4- i8ab"'^c''^4-b'»c3. (16) 125X rt _ aoox^y-^ -f 240x''^y* 64y''. (17) 27a' -54a*b- + 36a'-'b'^-8b«. (18; I25x'»4-75x'' + i5x'»+ f. (19) 343a" - I47a'^ + 2ia^ - i. (20) a^x-^ - aa'-x'^^y'-* -f 3axy* -y". B.-(i) a'^+b'^-: c'^ + 3a2b + 3ab2 4-3a2c + 3ac2 4-3b2c 4-3br2 4.6al)c. (2) x"M-y^4z'> + 3x-y + 3xy- 4- 3x-z 4-3x7,'* + 3y'^z 4- Syz'-* 4- 6xyz. (3) a^4-b-'' -c"' 4-3a-b 4- 3ab2 - 3a'-c 4- 3ac-* - 3l)-c 4- 3bc- - 6abc. (4) x^ 4- y"' - z3 4- 3x"y 4- 3xy- - 3x2z4- 3xz- - 3y2z 4- 3yz'' - 6xyz. (5) x» - y» 4- r^ - 3x-y 4- 3xy''* 4- 3x-z 4- 3xz2 4-3y-z-3yz'' -6xyz. (6) a-' - b^fc* - 3\-b4-3ab"'' 4- 3a=*c 4- Sac* + 3b-c-3bc- ~6abc. (7) 8x"^ 4- 27y"^ 4- 64z-^ -f 36x-y 4- 54xy*-* 4- /-Sx-z 4-96x22 4- loSy^z 4- I44yz2 4- I44xyz. (8) 64a^ - 27b"* 4- 8c-* - I44a-b 4- loSab"-* 4- 96a2c 4- 48ac2 4-54b2c-36bc2 - i44abc. (9) I25x^ - 64y'' - 343Z* - 30ox-y 4-240xy-- 525X2Z 4- 735XZ- - 336y'-z - 588yz2 4- 84oxyz. (10) 2 i6a-' - 1 25b^ - 64c*'' - 54oa'-b 4-45oab2 - 432a-c 4- 288ac2 - 3oob2c - 24obc2 4- 72oabc. (11) 27p'* 4- 64q3 _ 5i2r» 4- io8p2q 4- I44j3q2 - 2i6p2r 4- 576pr2 -384q2r 4- ^68qr2 - 576pqr. (12) 27a-**~8b3 4- 125x3 - 54a -b 4- 36ab" 4- I35a2x 4- 225ax2 4. 6ob2x - i5obx2 - i8oabx. ^ (13) x" 4- y" 4-z"4-3x*y2 4- 3x2y*4-3x*z2 ANSWERS. «3 + 3X 27* 7* + 3y*z2 + yyii 3a"c-' + (14) a^-l)»-L« 3a'' ID-' + 3a 'M)" - 3a"c"' + 3a ^c" — 3b"r-' 3b'*c'' + 6a•Mr'♦c=^ (15) 8x" I25y« + lOz" -r)ox^y- + 1 50x-y* + 48x''z'-^ + 96X-Z' +3ooy*z- - 24oy-z^ - 24o\'-y"-z-. (i6) a-'x-' - Ir'y"* + c-'z-' - 3a-x-by -i- 3axb-y''* 4- 3a-x-c:z + 3axc"7'" + 3'5''y"cz - 3byc-z- - 6axl)ycz. ( 1 7) a"» 4- b'» + c-'* + d + 3a^b + 3ab- + 3a-c + 3ac'^ + 3a-d 4- ^ad"'* + 3b-c + 3bc-4-3b-d + 3bd- 4- 3c-d4-3cd- 4- d3c4-6abd4()acd 4-6bcd. (18) Work by substitution. EXERCISE XXXII. -(Page 3>). (1) x*'4-(a + b + c)x'-^ 4- (ab 4- ar 4- bc)x -f abc. (2) x"' 4- (a - b 4- c)-k'^ 4-(-ab4-ac-bc)x-abc. (3) x"'4-(a-b-c)x2 4-( -ab (5) (7) x**4- 15X' x-*4- Mx- -ac4-bc)x4-abc. (4) x'M-6x-4- nx4-6. 4-74x4-120. (6) a'* - 12a- 4 41a- 42. + 55x-f-42. (8) x"»-6x"'^- 67:4-360. (9) 8x-*4-6ox'^ 4-142x4-105. (ro) 8x"^-72x-4-2o8x- 192. (ri) 27x"^ 4- I I7x" 4- 141x4-35. (12) 27x^ - 63X- - 1 14X 4- 240. (13) 64a"^-64a- - 156a -54. (14) x'^ 4- ux'"* 4-7'^- 4- i54x-}-i20. ([5) a*4-2oa"' + i37a-4-358a4-240. (16) X* -2x» - 1 13X-4- 1 14x4-3024. EXERCISE XXXIII. -(P.ige 38). A.-(i) x^-xy 4-y^ (2) a--i -ab4-b-. (3) x- -xa4-a-. (4) b-' - by + y'-i. (5)x--x4-(. (6)a--a4-i. (7) i -c4-c-i. (8)i-z4-z2. (9) x'--- 2x4-4. (fo) a2-3a4-9. (n) \=^-2xy4-4v2. (12) 9a--3ab4-b-. (i3)4x--6xl) 4-9b-'^. (14) x2 4.xy4-y*-^. (15) a- 4-ab4-b-. (16) a'-^ 4-5a-l-25. (r7) 36f6x4-x2. (18) i6x- 4-2oxy4- 25y". X ([9) m* ~ m -ab-«4-b^ ^n 4- m-i (21) X - mn"» 4- n '^ (20) a* - a"'b 4- a-b- x'''y4-x*y- x-'iy-J + X- ,4 _ xy' 4-y«. (22) a''4-a-'b 4- a-*b'-^ 4- a-Mr'«+a-b* 4- ab'"' 4- b". (23) m''4-nv"'n4-ni'n-4-m«n-*^+-m-n* -(-mn^4-n". C24) X* -x*'*4-x'^ -x4- 1. (25) c^4-c-*4-c''^4-c4- 1. !'26) a* x^ 3x-"* -f 9x- - 27X + 81. 4- 2ab 4- b'** - ac-bc4-c-. - 2a** -I- 4a- - 8a 4- 16. (27) (28) X* -x-a4-a-. (29) a*- (30) x'^ -2xy-f y'-^4-xz-yz4-z-. B.— (i) x'* 4- x2y + xy2 4-y=^ (2) a" 4- a*b 4- a'^bs 4-a2b''4-ab*4-b". (3) m^ 4-m"n4-m'"'n'-4-ni^n-'» 4-m'hv* 4m-n-'''-|-mn"4-n'. (4) x'' 4-\* 4-x-"'4-x- -f x4- 1. (5) 84 KXF.RCISKS IM AT.r.EnRA. ifa + a-+ +a^ (6) 8x-' - i2x-y+ iSxy'-^ (7) i25a"*-icx)a='b + 8oab--64b«. (8) x^ x^y-'+xy^-y" 2a'-^b + 4ab''^-8b-» 27y* :''y (lo) x*y4-x''y2 (0 (x + y) (3) (a-b) (5) (I +a) (7)(a + 2b) (9)(a-c) (9) a (a + b)»-(a-hb)- + (a + b)- EXERCISE XXXIV. —(Page 39). (x=*-xy + y''^). (2) (x-y)(x- + xy + y-). (a2 + ab + b"). (4) (a 4- i ) (a*-^ - a + i ). ( I - a + a*-^;. (6) ( 2x - y) (4X- -\- 2xy 4- y -). (a--2ab4-4b2). (8) (3X + 2) (9X- -6x4-4) (a-»4-a'''c4-ac- fc^). (10) (a- i) (a" 4-a"+ 4-0- («0 (p- q) (p« 4- p^q 4- p'^q^ 4- p-q"* + pq' K|-')- (52) (x-3)(xM-3x*4-9x"'4-27x-4-8ix4-243). (13) (ab-fc) (a'-*b--abc + c'*). (14) (3x-2y) (9X- -|-6xv + 4y-)- (>5) (c-6)(c'-^+6c4-36V (16) (5 4-z)(25-5z + /;-) (17) (xy4-8) (x-y'-2-8xy4-64). (18) (7a- i) (49a- + 7a 4- 1). (19) (5x - ioy)(25x- 4- 5oxy 4- looy'-*). (20) (x - y) (x« + x»y4- +y«). (21) (a 4- b) (a* -a='b 4- a'-b-" -ab"^4-b'*). (22) (n+i)(n"-n''4- + 1). (24) (pq + 3^) (p-q' (23) 3pq«- (i - c)(i4-c 4- c- 4- c"'). 4-9r2). EXERCISE XXXV.— (Page 39). A.-(i) 5x(x--3). (2) a-(a- I). (3) 3v''(i - 2x). (4) 4x(2 - x). (5) 7ax(a-2x). (6)p(6p+i). (7) x(x4-y). (8) x'»(x -y*-*). (9) 3xy''^(x-3y-). (10) 8(2 4- 3x-). (i i) 4f3ab - 2). (12) i7(2 4-3x=^y). (13) 15a- ta*-'- 15). (14) 5x»(5y-2). (15) 3a(8a - 9b"). (r^) 3ab(4a- - 3b). (17) 3x-y (5x*-'-2y'-i). (i8)4m2n''p;2m4 5n). (u;) x(x2 +xy-y2). (20) 2x''*y" (2y - 34- 2x). (21) 5\^v3x- 2a''' - a'*x). (22) I9a-x''(2x- 4- 4ax4-3a-). (23) 3ab(a*-ab 4- b'-*). (24) 5a(4x- -8x4-9). (25) xX6-9x + 4x-»). (26) 35x2 (m^ - 2m 4- 3x). (27) 7a( i - 2a 4- a"'). (28) i i(2m"^ - 3mn 44n2). (29) I3p0pq''^ + 2q-4p'''). (30) ax>=* -bx + c). (31) 7a-b-c'J(2bc'" 4 ab'-^ - 3a'^c). B.-(i)(a4-b)(a4c). (2) (a4-b) (a -c). (3) (x + y) (x4-z). (4) (x-y)(x-z). (5) (ab4-c)(ab + d). (6) (ac-b)(at: -d). (7) (a4-b)(a + 4). (8) (x + y)(m-n). (9) (2x + 3y) (a - b). ( ro) (p - q) (r I s). ( 1 1 ) (x- + 1 ) (x4i). (12; (x4-.i)(in--n). (13) (3a - 2b} (5X 4y). ANSWERS. 8S (14) (2x-y)(x-5). (15) (3x + 2y)(ax + by). (c6) (xy-z)(a-bc). (17) (x-+y-)(a2+b''). («8) (y- + (y-i). (19) (x» + 3)(x+i). (2o)(a + b + c)(x-y). EXERCISE XXXVI. -(Page 40). A. -(1) (x + 2) (x+i). (2)(a + .3)( +4). (3)(x+i2)(x + 8). (4) (a 4- 6) (a+17). (5)(P+i-^)(P+iB). (6) (xy f 5)(xy + 26). (7) (11 -f-x)(7 + x). (8) (X + 6HX+7). (9) (x + 2y)(x + 3y). (io)(a + 2b)(a-l-7b). (1 i)(x + 24y)(x4-25y). (i2)(m+i3n) (m-H3on). (rj) (p-f 8q)(i3 + 274). (14) a(i + 2a) (i>3a). (IS) 3(^ + 7) (xf 8). (,6) (x- 5)(x -6). (17) (x-9)(x-io). (i8)(x 7)(x' H). (i9)(a-io)(a~ii). (20) (a - 3) (a 1 5). (2 1 ) (a 8) (a - 1 3). (22) (in - 711) (111-1511). (23)(xy-4)(xy-i3). (24) (xy- ii)(xy - 12). (25) (a --7b) (a -13b). (26) (a-6bc) (a- i -be). (27) (11 - x) ( 10 " x). (28) (ax - 1 1) (ax - 13). (29) 7(x - 7) (x-8). (30) (ax -7) (ax -14). B.-~(i)(x + i6)(x-5). (2) (x+ ii)(x-io). (3) (a + 26)(a-io). (4) (ax + 24) (ax- 10). (^ (a-l- I2y) (x-5y). (6) (a + 7b) (a -6b). (7) (ni + 1 3) (ni - 1 2). (8) (a + I5bx)(a - 3bx). (9) (x^ + ]8a-) (x-^ - i2a'-^). (10) 3a(x + I7y)(x- 5y). (i 1) (x - 15) (x + 6). (12) (x-i9)(x + 8). («3) (a-35l>)(a + 3^)). (14) (a-24b) (a + 4b). ( 1 5 » (in - 1 3) (m + 2). ( 1 6) (m - 8n) (m + 7n). ( 1 7) (xy - 5z) (xy + 2z). ( 1 8) ( 1 1 ax) ( r 5 + ax). (19) (,20 - a) (2 1 + a). (20) 5x(a - 1 3xy) (a + 3xy). EXERCISE XXXVII.— (Page 41). (i) (x + 9)^ (2) a +13)-. (3) (111+17)'. (4)(y+0^ (5)(z+io)2. (6) (x--^ \7r- (7) (x4-6y)'^ (8) (m+iin)^. (9) (x"'fi2)-. (10) (a -18)=^. (11) (p-ieq)'-!. (12) (a-i5b)"^. (13) (2a~3b)2. (i4)(a-2x)^ (i5)(2ax -7c)^ (16) (3ani + 5xyy^. (17) (4ax- -b2c»)5«. (f8) (4xy4-3z)2. (i9)(,-x3)2. (20) (3a- 2)'-^. (21) (a-bc^'-^. f22)(5-4n)2. (25) ->x(a2 + 2y2) [>!-•. (24) (7x«+3yz-'*)''^. (25)(a'^b*'^-|)2. (26) -{ x/3(ab-3)|-2. (.7) (a + b + ck (28) (x + y-z)^ (29) (a-b + 5)'^. (30) (2in-3n~4pr- EXERCISE XXXVIII. -(Page 4!). (i)(x2 + x+i) (x^-x-fi). (2) (a- fa+i)(a--a+i). (3) (x- 4-2x ^- 7) 1^ r 5! ,/i 86 exp:rcises in algkbra. (x2-2x + 7). (4) (x2+4.x+9)(x2-4x + 9)- (5)(a- + 5a + 25) (a3-5a + 25). (6) (1112 + 2m + 4) (m--2m + 4)- (7) (a2 4-4ab-b2)(a2-4ab-b-). (8) (x- 4- xy + >'-) {x--xy + y-). (9) (2a- 4- 2a- i) (2a- -2a- 1). (10) (9c-4-3cd + d''«)(9c--3cd-f-cl-). (11) (x-^ + 2xy + 2>yn {x^ -2xy A^3y-). (12) (3X*-* + 2xy 4- i)(3x--2xy + i). (13) (a-'+ay + y-) (a— ay + y^). (14) (x- + xy + 2y-) (x--xy4-2y-). (15) (a- 4-2ax + 3x-) (a- - 2ax + 3x-'). (16) (a- + 2ax + 4x2)(a--2ax + 4x-). (17) (x- + 3x + 5) (x--^-3x + 5). (18) (a2 + 3a + 6)(a2-3a + 6). (19) (2a2 + 3ub+4b2) (2a'- -3ab + 4b''^). (20) (x- +2xy + 2y-) v^x- -2xv + 2y''^). (21) (a- + 2abc 4- 2b'-< -) (a'''-2abc. 4-2b-5c-). (22) (x- 4-xy4 y") (x-' - xy 4- y'-) (x* - x'-^y'-^ 4-y*). (23) (x*-^4-x4-i)(x«-x4-i)(x*-x2 4-i). (24) (2a 4- b) (2a ~b) (a 4- 5b) (a -5b). (25) (2x4-3)') (2x-- 3y) (3x4-2y)(3x 2y). {2C ) (4111 + 511) (4111 - 511) (3m 4- 211) (3111-211). (27) (2C2 4-2C4-|)(2C--2C4-I). (28) (j)-|-C|) (p-q'(3p+q)(3P-q)- (29) (x" 4- iix4-4)(*''-"x4 4). (30) (3x-^4-7xy~4y')(3x'^-7^y-4y-). (30(a''^ 4- 3ac ^-7c') (a- -3ac4-7c-). (3^) (x- 4-3xy + 8y--) (x^ - 3xy + 8y^ (33)(m^4-5ni + 9)(nT-=- 5n' + 9)- (34)(x"4-3x4- 1) (x2 -3x + i). (35) (a- 4- 4a 4- i) (a- -4a 4- 1). (36) (x 4- 2y) (x 4- y) (x - 2y) (\ - y). EXERCISE XXXIX. -CPage 42). (0 ('f+y) (x-y) (x'-^ - xy 4- V-:) (x'-^ 4- xy 4- y-). (2) (a + 1)) (a - b) (a*-* - ab 4- b *') (a'-* 4- ab 4- b"^). ( 3) (ab 4- 1 ) (ab - i ) (a-b'-^ -ab4-i)(a-b=*4-;ib 4- 1). (4) (xy !- z) (x-y" -xy/4-z-). (5) (2x4-a-) (4x=^ - 2a2x 4- ^*). (6) (x- - 2a) (x* 4- 2ax- 4-4a-). (7) (x- fy-)(x-^-x2v2 4-y*). (8) (a-4-b2) (a* -a'-b«+b*). (9) ^3a 4b) (9a-4- i2ab+ i6b'<'). (10) (Pq - 30 (p-q- + 3Pqr4- yr^). (1 1) (x 4- 2y) (x' - 2x»y + 4x2y'S „ 8xy=» + l6y^). (12) (x y) (x-^ 4- xy 4- y-) (x"4-x»yM-y*'). (13) (m4-b) (m - 1)) (in*4-m''»b4-m-b2 4-mb« 4- b*) (m* - nr'M) 4- m-b^ - nib'* 4- b*). (14) (x'«H-yi«j(x«4-y'^)(x^4-y*)(x''^4-y-)fx4-y)(x-y). (fs (h - b) (a + b) (a2 4- b'-^) (a*-^ ~ ab + b'-') (a'-^ 4- ab 4- b*-^; (a* -a-i)- (b^). (i6)(2ab 4 y) 4a-l)- - loabc 4 25r-). (17) (xyz-i)(x-y"z2 4-xyz4-i. (18) (a4 b + c) (a2f2ab ANSWERS. 87 + b*'«-ac-bc4-c2). (19) (x-y -z) (x2-2xy + y- + xz -yz + Z-). (20) (a-b-i)(a*''-2ab + b--fa-b + i). 21) {i -x-y) (1 +x + y + x'--h2xy + y'^). (22) (a-b--c) a + b + c)(a''' + b--J + 2bc-c''^). (23) (a + b) (a^ -ab + b'-^ 4-1). (24) {K + y/K (25) (x - y)a. (26) 2c (c- + 3(1-). 27) ab (3H + b) (3a - b) (9a- + 3ab + b-) (9a- - 3ab + b'-^). 28) a- (ax-t-2y) (ax-2y) (a-x=* f 2axy + 4y=«) (a-'x--2axy + 4y-). (29) (a-f-b-c) (a- 4-2ab-f-b- + ac -f be + c'-*). 30) (x-y-z) (x--2xy + y- + xz-yz + z-). EXERCISE XL.-(Page 42). A.-(i) (3x + 2)(x + i). 2) (2a+i)(a + 2). (3) (3x+i)(x-f-3). (4) (4a+i) a + 2). (5) (3x + 5)(x+i). (6) (x+2)(5x + 4). (7) X + 2) (2x + 5). (8) (a f 2) (3a + 4). (9) (x 4- 2) (7x f 2). 10) (a + 5) (4a + 3). (11) (x + i3)(3x + 2). (i2)(4in + S) 2111 + 7). (13) (3x-5)(2x--7). (i4)(2x-5)(3x-4). '5)(3x-2)(2x-i). (i6)(2x-7y)(4x-3y). (17) (a -i) 4a + 9>. (r8) (i-in)(7 -3111). (19) 2(x-2)(2x -3). 20) (x - 7) (3x - 2). (2i)(8x-5z)(7x-4z). (22)(3b-c) 8b - 3c). <23) (2x - 3y)(i2x - 7y). (24) (111-411) 56m - 5n). B.-(i)(x + 3)(4x--i). (2)(a + 6)(^i-5). (3) (x + s) 3X-I). (4) (m + 8) (2111-1). (5) i^ + 3)(3c-2). (6) p + 2)<4p-?). (7) (4a+rK3a-i). (8) (r + 7y) 5-3y- (9' (3x + 7yH4x-5y'- (10) i3(a + b) C^a - 2b). (I) (2x-3)(3x+i). (i:) (a-3)(2af5). (/0(a-7) 3a^2). (14) (b-3)(i2b-|-5). (15) (3x^-5)^4x-7)• i6) (x-3y)(2x + y). (i7) (3a - 4b)''8a + b). 08) 4-5t^)(5-t-4c • ;i9; (i-:;n)<4+;n). (20) (3X f y) 1 5x - I r y). EXERCISE XLI. ^ Page 43). (1) (2x + 3y+4z) 3x+4y+5z). (2) (2X + 5y + 3z nx+2y + 2z), (3) 4a+5b + 6i>(3a-» 4b + 7(). (4) n^.ir^b+c) (3a \ 5b + 7c). (5) (2x + 5y + 6)(9x + 7y + 6). (6)(2a+3b + 4) 3a + 5b + 6). (7) (x + 3y-2z)(2x + y-3z). (8)(4x-5y + 3z) (7x-2y-4z). (9) (2a -b + 5c) (5a - 2b -3c). 101 (7x - 8y + 3) (5 x+ 2y - 4). (1 1) (9a - 7b -I- 4) 3a + 2b - 1 1 ). ( 1 2) (x + y + z) (x- + y- + z- - xy - xz - yz). 88 KXEkCISES IK ALC.KHRA. (13) (x + y-z)(x3 + y2 + z''*-xy + xz + yz). (i4)(x-y4-z) (x' + y^ + z'-'+xy-xz + yz). (15) (x-y-z) (x3 + y'^^-z■- + xy + xz - yz). EXERCISE XLII. -(Page 44). A.-(i)x4-y. (2) x + y. (3) a. (4)2x-3y. (5) a-^b-(a-b). (6) x + y. (7) x + 2y. (8) a(a -x). (9) b(a + b). fio) c-d. (ii)y(x-i). (12) a'-' t-x2. (r3)x-5. (i4)x-io. (I5)x-i2. (16) X4-3- (17) x + 2y. (18) a + 3. (19) x + 2. (20) x-3y. (2i) 3X+I. (22) x- 3. (23) a + 3b. (24) a(x + a). B.~(i) X--13X + 5. (2)a2-3a + 2. (3)x2-3x + 7. (4)x-3. (5)a+i. (6) 7x'-^+3x-i. (7)x-i. (8) a -3. (9) 3x-5a. (lo) None. (11) x'-^ + xy + y^ (12) X- - 2X+4. EXERCISE XLIIL — (Page 45)- A.— (i) (?) 2(x-y). X-+X+I (3) -r^ (4) 5x (5) ax — I 2(b + c) (6) (10) 2a + c ^" 3y '■" b^ , , x2-ax + a2 b . . x X+1 (n) 2y (12) 3X + 5 2x4-3 (13) (14) 4x''^ + 1 Sx'-^ + x+l • ( , 2a + 3b '5i 2a (18) xy 3az ('9) xy x-2 (22) x+? X+-2 (16) a 4- b - c a + b + c ' 3a be' (20) I -X i+y (17) (21) a + 2 x + 3 B.-(i) (4) x-2 X--2* (5) (2) x+ r x4-r x-3' ni4- r jj2 _ xy — V" (3) ;^ xy-y- (6) m- I , ^ X- 4- 2x4-4 (8) (n) (i4) x3 - 5a-x + 7a^ X- - 2X + 3' 2-2X 4-4X- 3-3X + 9X-' ANSWERS. X* + 3X - 2 89 (9) fi* — -Jv!* — } (10) 3X-5 x'-* (12): 2b a + 2b' ('5) (13) 5X + 6 a ♦-4 a-f5 X- - 7ax+ I2a'^ 2x''4- I4ax + 24a''^* (16) ^^ — '. 5x- -x-3 EXERCISE XLIV^ -(Page 47). (1) x-y X- (2) (3) (7) 5a -b ^^^ x(3a - 2) r- (5): x-3 X - I I 4 x + 3y X + 2 ^^ , . X-3 x" X2-7X + 6* ^^ 7-6' ^^^ X---7X+IO (6) -^. ^ ' 2a - I (10) xf I (11) . (12) y — ^ ^ X - 2 ^ ^ (X + 4)'' x + 4 (•3)(^~5)-'. ('4) I. ('5) (18) (22) (24) (a-b)» x + m -n X - m + n (lb) a - li + c a-» b-c' (19) X. (20) _5 I (x + a)(x + b)(x-b)(x-c) 2X - I . . I 07) (21) (23) x4-y4-z x-y-z* I (x y)'^ • (x + y) ^ x« + y- (25) (26) x-3. 2X-5 ' •" a + b EXERCISE XLV-— (Page 48). (i) x{k"- - i). (2) ab(a4-b). (3) xy(4x'-^ - i). (4) 6x(3x- i). (5) a3+b-». f6) ab(4a2 - i). (7) x(x2 -4). (8) (x+ 1) (x- i)*-^. (9) (x + 2)3 (x + 3). (10) (x-i)(x-2)(x-4). (11) (x-5)rx-6)(x + 7). (t2) (x + 3)(x + 4)(x4-5). (13) (X-2)(X 4- 2)(X - 11). (14) (X + l)(x - 2)(2X + l). (15) (x-sXx'-* + 3x + 9)(x-i2). (16) (x + 4)Mx*-4x + 16). (17) (x -4) (x + 5) (n 6). (18) (x+ (x + 2) ' ■ I t 90 EXERCISES IN ALGEHRA. (2x+iV (19) (x + 2)(x + 3)(5x4-i). (20) (x-2)rx + 2) (3X-7). (21) 6(x + 2)(2x+i)(4x-7). (22) (x4-y) (2x-7y)(4x-5y). (23)(H + 2b)(a-2b)(a2-b-) (24) (x + 3)(x + 4)(x-+9x + 2o). EXERCISE XLVI.-(Page 49)- A— (i) — ^ 4X+9 25X-16 . V I7X ,_, i9x-2or (2) — 7 • (3) — T, — • (4) (6) 5 — 12 12.Y- + 28x- 17 36 5X + 29 8X^ • ^''^ I02X (9) o. (10) 8o\3 + 64x-4-84x + 45 60X- (5) (8) (M) 3a -8b 8a 4y+3z yz a--» -b'^-c0 4-abc abc IC4-6C- (12) , ^ ija'^c + 2a-bc 4- 9ab - 27; (,3) : U__-5 , , iix'- 18X--27X-16 , .6x-4a . ,^ 2\ + 5 (m) —Ji • (»5) -TT-- ("6) 30X- ax x« + 5x + 6 7X + 3f / ox X~I 2X+F2 (17) ..-^ — -^^ — • (18) -7 ; — . (19) -^i • ^ '' X-+9X + 20 ^ '^x^-9x+2o ^^'x-*-4x-i2 ^-°^x2-(a-Hb)x + ab'' ^^'^\''+(a+ b)x + ab * (22) -5 4xy (24) 1_ x--y- ~" x--y^ "' X--OX4-I5 , . 20X 2x"^ , . 2ab 2x'* (25) -7; . (26) r. (27) -T-nr- (-^)— — 7-' ^ ^'x--2i; ' i-x* a-^-b'' ^ x--y'* o , . •» (29) 2b. (30) B.-(i) (4) x"+y xy(x- -y^) 2 . . 2a , , a*+6a2x''2 + x* (2) rrz-r- (3) - x + y' ' a" -X 9x^4-34x4-29 a*-x* (x4-i)(x4-2)(xf3)- (5)-^^ jx • 2 X- - I ANSWKRS. 9« (7) (9) (13) 38X+14 3(x--4) * 4x- 18 (8) I IX -56 (x-4)(x-5)(x-6)' (x-2)(x~3)(x-4)* 7x -f 76 -. (10)0. (11) 1. (.:) ^^—ry x-Hc (x + 2)(x + 3)(x+7)* (16) o. (17) o. 48a» ^''^^ (x-^Tii) (X -b) (18) o. (15) o. (20) (x^-a=*;(x''-9a''')* EXERCISE XLVII. -(Pa^-e 51), (3) 8. (4) 16. (5) 25. (6) 17. (9) 4. (10) -}. (n) bo. 12) 8. (ic) 120. ^6) 4. (17) I. .18) 24. ^21) 5. (22) 41. (23) 17. (24) 3 (27) 1. (28) 5. (29) 3f • (30) 8. (0 7f. (2) 15. (7) 13- V'^) 3i%- (>3) 45- <'4) 7. (19) 2. (20) I. (25) 3. (20) 2. EXERCISE XLVIIl (Page 53). A. (1) 72. (2) 48. (3) 480. (4) 720. (5) 3' 5- ((') 49 ; 50- (7) .144 ; 128. (8) 98 ; 99 ; »oo. ''9) $255 ; 204 ; 136. (10) 436. (11) 18; 24. (12) $724.80; 634.20. (13) 54. (14) 100; 12. (15) $346.20. 15- -(0 35- ^"^ 78 ; 82 ; 160; 40. (3) 64. (4) 23. (;) 36. (6) 54. ; 4if- (») 6i. (9) 25. (10) $9c» ; 750- ('0 39- ' 12 ; 6. EXERCISE XLlX.-(Page 56). A. —(1)2; 3. (2) 5 ; 6. (3) 6 ; 7. (4) 9; 7. (5) 12 ; 3. (6) 12 ; 4. (7) 13 ; 3- (8) 13 ; 9- (9) 7 ; 17. (10) 4 ; 3. B. -(1)6; 12. (2) 12; 8. (3) 18; 12. (4) 14', 15. (5) 3 ; 5- (6) 10 ; 5- (7) 7 ; 3- («) 2 ; 3. (9) 5 ; 5- 2 2 (10) ; : . ^ '^ a + b ; a-b M IMAGE EVALUATION TEST TARGET (MT-3) f • • # 1.0 I.I (50 '""= i?" 1!^ IIIIIM m H40 IIM Z2 ZO 1.8 ■I 1.25 1.4 1.6 < 6" — ► Photographic Sciences Corporation ^ 1S ^N^ \ :\ \ 6^^7+^77^ ' ^ ^ (3x + 4y)(4x+5y) 8-X* + I (3)2. (4)-B+x'^ + i (5; i_a^« V 2X (7) -• (8) 6x-9 (14) (16) (x-'^-i)(2x + 3)* , , }"ix-4y (12) 4x ^a- ''^' 84 x2+4x-65 (9): (13) (6)7- 2xy2 (10) ^^rry4 • a + 2* 3x2+29 x4-5 8 (x-3) (x-5)(x-8)' (15) (x + 2) (x+3) (x + 4) x2 + x-i68 3a / . _y_ ^Tb- ^^7^x + y- (^«) ^b- 0r^(x+7)(x + 8) I . . 4x (i9)7f:^.-(^°)^' 1 4-4X m P :) U^- c? BARD PLACES IN GRAMMAR HADE EAST. By A. B. 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