Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

The same result will be obtained by arranging the dividend and divisor according to the ascending powers of x. Thus,

[blocks in formation]

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)

[blocks in formation]

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

[merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small]

The result is expressed in a manner similar to that in Arithmetic:

[blocks in formation]

algebraic fractions will be further considered.

If we multiply both members of the equation above by a- b, we have

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small]

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,

[merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors]

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.

[blocks in formation]

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)

[blocks in formation]

(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).

[merged small][merged small][merged small][ocr errors][merged small]
[blocks in formation]

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).

[blocks in formation]

C 3b

[ocr errors]

9 25

[merged small][ocr errors]
[ocr errors]
[merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small]
[merged small][ocr errors][subsumed][merged small]
[ocr errors]

3 10

5 x2y+y3+x3+5xy by 4 xy + y2+xo.

15+2a-3a2+a3+2a-a5 by 5+4a-a3.
x-2x+1 by x2-2x+1.

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

[ocr errors]

41. Divide the product of 2- x − 1, 2 x2 + 3, x2+x+1 and x

[ocr errors][merged small]

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.

[merged small][merged small][merged small][merged small][ocr errors]

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.

[merged small][merged small][merged small][subsumed][ocr errors][merged small]

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.

[merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

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.

[merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

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).

[merged small][merged small][merged small][merged small][ocr errors]

(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).
(24-15y-622-7xy-32xz+182 yz) + (12-10y+3x).
(fa2-11 ab+231ac +36-23 bc-21)+(ab-c).

[blocks in formation]

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.

[ocr errors][merged small]

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).

[ocr errors]

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.

[merged small][merged small][merged small][ocr errors][merged small]
« ΠροηγούμενηΣυνέχεια »