The same result will be obtained by arranging the dividend and divisor according to the ascending powers of x. Thus, 69. The operation of division can often be shortened in certain cases by the use of parentheses. Divide (a - b) x3 + (b3 — a3) x + ab (a2 — b2) by (a — b) x — a2 — b2. + (a - b) x3 + (b3 — a3) x + ab (a2 — b2) | (a - b) x + a2-b2 Divisor. (a - b) x3 + (a2 — b2) x2 | x2 — (a + b) x + ab Quotient. - (a2—b2)x2 + (l3 — a3) x + ab (a2 — b2) 70. It may happen as in Arithmetic, that the division can not be exactly performed. by a - b: Thus, for example, if a2 — 2ab+ 3b2 is divided The result is expressed in a manner similar to that in Arithmetic: algebraic fractions will be further considered. If we multiply both members of the equation above by a- b, we have Hence, if the division of one polynomial by another is not exact, and if the dividend, quotient, divisor, and remainder are respectively represented by D, q, d, R, we have the formula: D= qd + R. That is, the dividend is equal to product of the quotient, at any stage, by the divisor, plus the remainder at this stage. Thus, NOTE. It is very important to arrange the terms of the dividend and divisor in the ascending powers or descending powers of some letter, and to keep this order through out the operation. 7. 8. 9. 11. 13. 14. 16. 18. 19. 20. 21. 22. 23. 24. 25. (ac-ad+bc — bd) ÷ (c−d). 5. (mm-mx — m + x) ÷ (m − 1). (6am-9an − 4 bm +6 bn) ÷ (3 a − 2 b). (6ac2ad4 af-9bc+3bd-6bf) + (2a-36). (2 ax-6 bx+8cx-ay+3 by-4 cy) (a2 — l2 + 2 bc — c2) ÷ (a+b− c). (3a2-4ab8ac-4b28be-3 c2) ÷ (a-2b+3c). (x2-2xz-4y2+8 yz-32)+(x-2y+2). (16x24a2+9 a2b2 — 36 b2x2) (3 ab-2a+6bx-4x). (329 bx — 208 ax + 87 ab — 153x2— 156 b2+ 153 a2) ÷ (17 a−13b+9x). 28. (6x-21.38.xy-6xz+18.5y+1.64 yz-36 z2)+(2.5x-3.7y+52). (0.06 m3 +0.01 mn-0.18 mp-18.2 n2+13.57 np-2.4p3) + 29. 30. 31. 32. (1.5p-5.2n+0.3m). (2b+c6a2+ab+18 ac-337 bc)+(2a-4c+3b). 60 (24x2-15 yo - 622-78 xy-32.xz+189 yz) + (fz-10y+3x). ({ a2 - 18} ab+2} } ac+} b2 − 2} bc − 21⁄2 c2) ÷ (} a− } b— {c). C 3b 9 25 3 10 5 x2y+y3+x3+5xy by 4 xy + y2+xo. 15+2a-3a2+a3+2a-a5 by 5+4a-a3. x-6x+9x-4 by x2-1. ХБ 39. Divide the product of x3. 12x+16 and x3- 12x - 16 by x2 – 16. 40. Divide the product of a3-2x+1 and x3-3x+2 41. Divide the product of 2- x − 1, 2 x2 + 3, x2+x+1 and x 42. Divide the product of a2+ax+a2 and a3+x3 by a1+a2x2+xa. 43. a3+ab+ a2c — abc — b3c — bc2 by a2 — bc. 44. xy3+2y3z — xy3z + xyz2 — x3y — 2 yx3 + x3z — xz3 by y + z− x. 45. a3+b3c3+3 abc by a+b-c. 46. x+y+3xy-1 by x + y −1. 47. x+y+x3-x3y2-2xy + y3 by x2+xy — y3. 48. (xy)-2(x+y) z+22 by x+y-z. 49. arab+b2x-x3 by (x+b) (x − a). 50. (bc) a3+ (c− a) b3 + (a - b) c3 by a-a(b+c)+bc. (a-b+c) x2+(acab-be)x-abe by x+c. 51. 52. 53. 54. (3b)x+(c-3b-2)2+(2b+3c)x-2c by x+3x-2. (a2-3 ab) x2+(2 a2+4 ab+3 b2)x — (2 ab+5b2) by ax —b. +5x+16 by +2x+1. 55. ax+bx3 + cx2+dx+e by x+ax2 + bx+c. 56. 24-b by x2+ax+b. 57. 33-3 by 2+x+}. 58. 1x3++1⁄2x — 11⁄2 by {x—}. 59. x-y3 by x2+xy+ { y2. 62. 6a5-11 a1n +23 a3n +13 a2-3a" +2 by 3an +2. 63. x2y2+2xym+nz+2xymr+ y2n z+2y1zr+r2 by xym+z yn +r. 64. 32-52 by 3-5o. 65. 32x-9xзym+12x2 y2m-18.3m-52 ym by x-ym. Find the remainder in each of the following indicated divisions, and verify the work by applying the principle in 270. 71. +3+3x+1 by x+x+1. 72. 2-3 ax-2a2 by 1+2 ax. 73. 18x3-5x+1 by 6x+2x+1. 74. (xy)-2(x-y)2+1 by (x − y)2+2(x-y) — 1. 75. 4x3-13x3y2+14x4nyn - 2x by xy3n — 2x2nyn + x3n. 26. 27. (3.9x-4.1xy-113 y) + (1)x-3.5 y). (2 a2 - fax - 11} x2) ÷ (3.5x+1.5 a). 28. (6x2-21.38xy-6xz+18.5 y2+1.64 yz -- 36 z2) ÷ (2.5 x −3.7y+52). (0.06m+0.01 mn -0.18 mp-18.2n+13.57 np -2.4p2) + (1.5p-5.2+0.3 m). 60 (2b+c2 6a2+13 ab+18 ac-237 bc) (2a-} c+b). 39. Divide the product of 3 — 12x+16 and x3- 12x-16 by x2 – 16. 40. Divide the product of a3-2x+1 and x3-3x+2 by x3-3x2+3x-1. 41. Divide the product of 2 — x − 1, 2 x2 + 3, x2+x+1 and x by a1-3a2+1. 42. Divide the product of a2 + ar+a2 and a3 +x3 by aa +a2x2 + xa. 43. a3+a2b+ a2c — abc — b2c — bc2 by a2 · bc. 44. xy3+2y3z — xy2z + xyz2 — x3y — 2 y.x3 + x3z — xz3 by y + z — x. --- 45. a3+b3 — c3+3 abc by a+b-c. 46. x3+y+3xy-1 by x + y −1. 47. x+x+y+x3-x3y2-2xy+y3 by + xy-y3. 48. (xy)-2(x+y) z+2 by x+y-z. 49. axab2+b2x-x3 by (x+b) (x − a). 50. (bc)+(ca) b3 + (a - b) c3 by a2-a (b+c)+bc. 51. (a-b+c)x+(acab-be)x-abe by x+c. 52. x+(3b) x3 + (c−3 b − 2) x2+(2b+3c)x-2c by x2+3x-2. 53. (a2-3 ab) x2 + (2 a2 +4 ab+3b2)x− (2 ab+5b2) by ax-b. 54. 3+5x+16 by x2+2x+1. 55. x+x+bx3+cx2+dx+e by x2+ax2+ bx + c. |