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5. Multiply a2 + b2 — c2 by a2 + c2.

Ans. aa b2 + b2 c2 — c1.

6. Multiply 2-2xy + y2 by x2 + y2.

Ans. x2xy + 2 x2 y2 - 2 x y3 + y^.

x2 b2

7. Multiply 4 a1 — 2 a3 b + 3 a2 b2 by 2 a2-2 b2.

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Ans. 8a4a3 b —- 2 a1 b2 + 4 a3 b3 — 6 a2 ba.

8. Multiply x + 2 x3 + 3 x2 + 2x + 1 by x2 2x + 1. 9. Multiply x2+ y2+ z2-xy-xz-y z by x + y + z. Ans. 23+y+23 — 3 x y z.

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10. Multiply 4x57x1+ 10 x 13 x2 by 3x-2. 11. Multiply a3+a2-1 by a2-1.

12. Multiply x2+7a x 14 a3 by x-7 a.

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Ans. 49 a2 x — 14 a3 x + 98 a1.

13. Multiply x+ya by xy + a.

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Ans. a m+ n + ambr an bm -fm+n. 15. Multiply 7xy-14x2 y2+ 21 x3 3 by 6 x y — 3. Ans.21 xy + 84 x2 y2 — 147 x3 y3 + 126 x* y*.

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16. Multiply 6 a2b-9 a b2 - 12 a2b2 by 2 a b — 3 b2. Ans. 12 a3 b2-36 a2 b3-24 a3 b3 + 27 a b1 + 36 a2 b1.

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17. Multiply - x23 + x2 −x + 1 by x + 1.

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(19. Multiply x2 + x23 + x2 + x + 1 by x + 1.

20. Multiply + + x23 + x2 + x + 1 by x-1.

1.

SECTION VII.

DIVISION.

53. DIVISION is finding a quotient which, multiplied by the divisor, will produce the dividend.

In accordance with this definition and the Rule in Art. 48, the sign of the quotient must be when the divisor and the dividend have like signs; when the divisor and the dividend have unlike signs; i. e. in division as in multiplication we have for the signs the following

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54. When the divisor and dividend are both monomials.

1. Divide 6 ab by 2b.

OPERATION.

6ab2b= 3 a

The coefficient of the quotient must be a number which, multiplied by 2, the coefficient of the divisor, will give

6, the coefficient of the dividend; i. e. 3: and the literal part of the quotient must be a quantity which, multiplied by b, will give a b; i. e. a: the quotient required, therefore, is 3 a.

Hence, for division of monomials,

RULE.

Annex the quotient of the literal quantities to the quotient of their coefficients, remembering that like signs give and unlike,

2. Divide a by a2.

OPERATION.

a3 ÷ a2 = a3

For a3 X a2= a3. (Art. 50.) Hence,

Powers of the same quantity are divided by each other by subtracting the exponent of the divisor from that of the

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18. Divide 747 a3 b* c5 d by 83 a2 b c do.
19. Divide 12 (x + y)2 by 4 (x + y).
20. Divide
21. Divide (bc)5 by (bc)2.

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27 (a - b) by 9 (a - b)2.

22. Divide 14 (xy) by 7 (x — y)2.

Ans. xm-n

2 am -N

Ans. 3 (x + y).

Ans. (be)3.

CASE II.

55. When the divisor only is a monomial. 1. Divide a x + ay + az by a.

OPERATION.

a) a x +ay + a z

x + y +

z

In the multiplication of a polynomial by a monomial, each term of the multiplicand is multiplied by the multiplier; and therefore

we divide each term of the dividend a x + ay + az by the divisor a, and connect the partial quotients by their proper signs. Hence,

RULE.

Divide each term of the dividend by the divisor, and connect the several results by their proper signs.

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4. Divide 12 a x3- 24 a x2+42 a xy by 3 ax.

5. Divide 4 a2 x + 8 a b x-4b2x by - 4x.

6. Divide 6 a2x2 - 12 a3 x3 + 15 at x by 3 a2x2.
Ans. 2 x24 a x + 5 a2x2.
7. Divide 12 at yo — 16 a3 y5 + 20 a y* — 28 a y3 by

4 at y3.

8. Divide 5 x 10 x2-15 x by 5x.

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9. Divide 273 (a + x)2 → 91 (a + x) by 91 (a + x). Ans. 3 (ax) 13 a + 3 x — 1.

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10. Divide 20 abc-4ac8acd-12 a2c2 by — 4 a c.
11. Divide 16 a2x2 32 a2 x3 +48 at x by 16 a2x2.
12. Divide 72 x2 y2 — 36 x3 y3 — 54 x3 y2 2 by — 18 x2 y2.
Ans. 42xy + 3x z3.
13. Divide 18 a x3 y — 54 x2 y2 + 108 c x5 y3 by 9x2y.
14. Divide 40 a1 b2 + 8 a2 b2 — 96 a3 b3 x3 by 8 a2 b2.
15. Divide 39 x 25 65 a x3 z6+ 13 x3 25 by

13 x3 25.

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CASE III.

56. When the divisor and dividend are both polynomials.

1. Divide 3-3x2y + 3 x y2-y by x2-2xy + y2.

OPERATION.

x2-2xy + y2) x3-3x2y + 3 x y2 — y3 (x — y
23-2x2y + xy2

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x3

x2

=x must be the

The divisor and dividend are arranged in the order of the powers of x, beginning with the highest power. 3, the highest power of x in the dividend, must be the product of the highest power of x in the quotient and in the divisor; therefore, highest power of x in the quotient. The divisor x2 2xy + y2 multiplied by x must give several of the partial products which would be produced were the divisor multiplied by the whole quotient. When (2 x y + y2) x = x3- 2 x2y + xy is subtracted from the dividend, the remainder must be the product of the divisor and the remaining terms of the quotient; therefore we treat the remainder as a new dividend, and so continue until the dividend is exhausted.

Hence, for the division of polynomials we have the following

RULE.

Arrange the divisor and dividend in the order of the powers of one of the letters.

Divide the first term of the dividend by the first term of the divisor; the result will be the first term of the quotient.

Multiply the whole divisor by this quotient, and subtract the product from the dividend.

Consider the remainder as a new dividend, and proceed as before until the dividend is exhausted.

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