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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,

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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.

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7.

8.

9.

(acad+bc- bd) ÷ (c—d).

·4 bm +6 bn) ÷ (3 a — 2b).

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-b

· mx — m + x)÷ (m − 1).

(6 ac-2 ad+4 af-9bc+3 bd — 6 bf) ÷ (2 a-3b).

(2 ax-6 bx+8cx-ay+3 by-4 cy)

(a2 + ab — 2 b2) ÷ (a — b). 11. (a3-a2b+2b3) ÷ (a+b).

13.

(2x-y).

10. (3aab-2b2)+(3a-2b).
12. (6.32-29x+21) + (2x-3).

(2x-2x2-6x+71) + (2x-3).

14. (a3 — b3) ÷ (a — b).

15. (a3+b3) ÷ (a+b).

16. (81 a1- 16 b1) ÷ (3 a − 2 b). 17. (a+b)÷(a+b).
(9 a22-4a2c2+4 abc2 — b2c2) ÷ (3 ab −2 ac + bc).

18.

19.

20.

21.

22.

23.

24.

25.

(a2 2+2bc-c2) ÷ (a+b−c).

(3 a2-4ab+8 ac-4b28bc-3 c)

(a-2b+3c).

(x2-2xz-4y2+8 yz-3 2o)+(x −2y+ z).

(16 x2-4a2+9 a2b2 — 36 b2x2) ÷ (3 ab − 2a +6 bx − 4 x).

(329 bx — 208 ax + 87 ab — 153 x2— 156 b2+ 153 a2) ÷ (17 a — 13b+9x).

(0.4x2+1.47 x — 8.5) ÷ (0.8 x — 2.5).

(2.21 n2 - 1.8 np — 1.61 p2) + (0.7p+1.3 n).

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(3.9 x2-4.1xy — 11} y2) ÷ (1} x — 3.5 y).

(2a-ax-114) (3.5x+1.5a).

(6x-21.38 xy-6xz+18.5 y2+1.64 yz-362°)(2.5x-3.7y+52).

(0.06 m2 +0.01 mn — 0.18 mp — 18.2 n2+13.57 np — 2.4 p3) ÷ (1.5p-5.2n+0.3m).

69

(2 b2+c2 −6a2+43 ab + 18 ac — 257 bc) ÷ (2 a — § c +{b).
(24.xo — 15 y2 — 6 z2 — 72 xy — 32 xz+1§9 yz) ÷ (}z — 1o y +3x).
({ a2-18} ab+2} } ac+} b2 −23 bc — 21 c2) ÷ (} a − } b − { c).

(bc − 1 + 16 2 - 2 b3) + (" +

25

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5

3

3b

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2

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5 x2y+y3+x3+5xy2 by 4 xy + y2+x2.

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

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35.

36.

37.

38.

39. Divide the product of a3. 12x+16 and a3 40. Divide the product of

-12-16 by x2 — 16.

x-6x9x-4 by x-1.

by x3-3x2+3x-1.

-2x+1 and x3-3x+2

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

by x-3x2+1.

4

42. Divide the product of a2 + ax + x2 and a3 +x3 by aa+a2x2+x^. 43. a3+a2b+ a2c — abc — b3c — be2 by a2 — bc.

44. xy3+2 y3z — xy2z+xyz2 — x3y — 2 yx3 + x3z − xz3 by y + z — x. 45. a3+b3—c3+3 abc by a+b-c.

46. x+y+3xy-1 by x+y-1.

47. x+x+y+x3 — x3y2 — 2 xy2+y3 by x2+xy-y2.

48. (xy)-2(x+y) z+22 by x+y-z.

49. ax2-ab2+b2x − x3 by (x+b) (x − a).

50. (bc)+(ca) b3+ (a - b) c3 by a-a (b+c)+bc. (a-b+c) x2+(ac-ab-be) x-abe by x+c.

51.

52.

53.

54.

55.

(3b)x+(c-3b-2)x+(2b+3c)x-2c by x2+3x-2.

(a2-3 ab) x2 + (2 a2+4 ab+362)x-(2ab+562) by ax-b.
+5x+16 by x+2x+1.

+ax+b.x2+cx2+dx+e by a3+ax2+ bx + c.

56. x-b by x2+ax+b.

57. 3-3 by 1+1+1.

x2 x

58. 4x3+71⁄2 x − 11⁄2 by i̟x− }.

59. x3- by x2+xy+} y2.

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62. 6q5”—11a12 + 23 a3 +13a2 −3 a1+2 by 3ɑ"+2.

63. x2y2+2xym+nz+2xyTMr+ y2nz+2y^zr+r2 by xym+zy" +r. 64. 3-5 by 3*-5o.

65. 32x-931ym +12.x21 y2m —– 18.x” y3m — 52 y1m by xa — ym.

Carry the division to five terms in the following:

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Find the remainder in each of the following indicated divisions, and verify the work by applying the principle in 270.

71. x3+3x2+3x+1 by x2+x+1.

72. 2-3 ax-2a2 by 1+2ax.

73. 18-5x+1 by 6x+2x+1.

74. (xy)-2(x-y)2+1 by (x-y)2+2(x − y) - 1.

75. 4.-13.xy2+14x4"уn-25 by x" y2n — 2x2nyn +x3n.

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a

b

d

72. Equation 1, 63, has a numerical interpretation only when and are numbers. The numerical definition of a quotient gives it such narrow limits as to make division an unimportant operation as compared with addition, multiplication, and subtraction as discussed in Chapter III.

This restriction on division can be removed in the same way as subtraction was generalized.

We accept as the quotient of a by any number b, which is not 0. (i. e., b0) the symbol defined by the equation

b = a

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which simply declares, that the symbol () is equivalent to the symbol a, and that either may be substituted for the other in any reckoning.

not.

By this definition it does not matter whether is a number or

a

b

When is used as a symbol it is called a fraction, and on the contrary a symbol a is called an integral symbol.

Definitions of the addition, subtraction, multiplication, and division of this symbol have been given. Moreover they are definitions which are consistent with the corresponding numerical definitions and with one another, as soon as 1, 63, 2, 167, 3 and 4, 71 are assumed to hold as symbolic statements as well as numerical statements.

a

b

The purely symbolical character of and its operations detracts nothing from the right to use them, and they establish division on a footing of at least formal equality with the addition, subtraction, and multiplication of Arithmetic.

The complete discussion of the fraction and its properties will be given in Chapter X.

73. The Indeterminateness of Division by Zero.-Division by 0 does not conform to the law of determinateness (262, XI); Equation 1, 63, and 1, 2, of index law of division, 64, are therefore not valid when 0 is one of the divisors.

The symbols

0 a

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of which but little use is made in mathe" 0 0 matics, are indeterminate.

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fore may be any number whatever.

a

0

0

ахо

a

2. is indeterminate. Because, by definition, ()0= a. If were determinate, since then by 63, 1, ()0 would be equal to axo, or to o, therefore, the number a would be equal to indeterminate expression. Therefore, division by zero is not an admissible operation.

0

74.

0'

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an

Determinateness of Symbolic Division.-The exception to the determinateness of division just pointed out might seem to raise an objection to the right to assume that symbolic division is determinate, as is done when the demonstrations 1, 63; 2, 67; 3 and 4, 71, are made to apply to symbolic quotients.

It must be observed that

a

are indeterminate in the numeri

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