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22. 75 axyz by 5a2bcdx2y2. 23. 64 a'm'x'yz by 8ab2c3. 24. 9ab2c2d by 12a3b*c.

25. 216 ab'c'd by 3ab2c".

26. 70a3b1c*d2fx by 12a1b3c3dx2y3.

43. Multiplication of Polynomials.

(1) Multiply ab by c.

Ans. 375 abcdx3y3z. Ans. 512ab2cm3x*yz.

Ans. 108abc8d3.

Ans. 648 ab°c3d®. Ans. 840a5b12cd3ƒx3y3.

It is required to take the difference between a and b, c times; or to take c, a – b times.

As we cannot subtract b from a, we begin by taking a, c times, which is ac; but this product is too large by b taken c times, which is bc: hence the true product is ac – bc.

8, b = 3,

If a, b, and c denote numbers, as a = and c=7, the operation may be written in figures.

(2) Multiply ab by c-d.

It is required to take a

bas many times as

there are units in c d.

If we take a bc times, we have ac - bc; but this product is too large by a-b taken d times. ab taken d times is ad - db. Subtracting this product from the preceding by changing the signs of its terms (§ 37, rule), we have (a - b) + (a — c) — ab — bc — ad + bd.

=

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From the preceding examples we deduce the following rule:

When the factors have like signs, the sign of their product will be +.

When the factors have unlike signs, the sign of their product will be

-.

Therefore we say in algebraic language that + multiplied multiplied by +, or

by, or multiplied by -, gives +;

by,

+multiplied by -, gives

Hence for the multiplication of polynomials we have the following rule

Multiply every term of the multiplicand by each term of the multiplier, observing that like signs give +, and unlike signs -; then reduce the result to its simplest form.

Exercises.

NOTE. All the terms in the exercises below are positive.

Multiply the following:

1. 3a2+4ab+b2 by 2a+5b.

-

3a2+ 4ab + b2

2a + 5b

6a3 + 8a2b+ 2ab2

+15a2b+20 ab2 + 5b3

6 a3 +23 a2b+22 ab2 + 5 b3

44. NOTE. It will be found convenient to arrange the terms of the polynomials with reference to the ascending or descending powers of some letter: that is, to write them down so that the highest or lowest power of that letter shall enter the first term; the next highest or lowest, the second term; and so on to the last term.

The letter with reference to which the arrangement is made is called the leading letter. In the above example the leading letter is a. The leading letter of the product will always be the same as that of the factors.

2. x2+2ax+ a2 by x + a. 3. xy' by x+y.

4. 3 ab'+6a2c by 3ab2+3a2c2.

Ans. x+3ax2+3a2x + a3.
Ans. x + xy3 + x3y + yʻ.

Ans. 9 ab+27 a3b2c2+18a*c*.

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5. b2+b* + bo by b2 - 1.

6. 22x3y+4x2y2 — 8xy3 +16y by x+2y.

7. 4x-2y by 2y.

[ocr errors]

8. 2x+4y by 2x-4y.

9. x2+x2y+xy2+y3 by x − y.

10. x2+xy + y2 by x2- xy + y2.

11. 2a2 - 3 ax+4x2 by 5a2 – 6 ax − 2 x2.

Ans. b8 — b2.

Ans. x5+32y3.

Ans. 8x'y-4y2.

Ans. 4x2-16y2.

Ans. xy.

Ans. x + x2y2+y'.

Ans. 10a-27 a3x+34a2x2-18 ax3- 8x*.

12. 3x2-2xy +5 by x2+2xy - 3.

Ans. 3x+4x3y — 4x2 — 4x2y2 + 16 xy — 15.

13. 3x3+2x2y2+3y2 by 2x3 – 3x2y2+5y3.

Ans. 6x6 - 5x3у2 — 6x*y* + 6x3y2 + 15x3y3 — 9 x2y* +10x2+15y3.

[blocks in formation]

Ans. 16a2x2 - 4a2bx — 6a2b2 + 6 acx — 7 abc — c2.

15. 3a2-5b+3c2 by a2b2.

Ans. 3a8a2b2+3a2c2+5b1 — 3 b2c2.

16.

3a2-5bd of
5a2+4bd-8cf

17. amx-a2b2 by a2x2.

18. am+b" by am-b".

- 15 a* +37 a2bd - 29 a'cf — 20 b2d2 + 44bcdf - 8 c2ƒ2

Ans. am+2+1 — a+b2x.

Ans. a2m-b2n.

19. amb by am+b”.

Ans. a2m+2ambn + b2n.

DIVISION.

45. Division is the operation of finding from two quantities

a third, which, being multiplied by the second, will produce the first.

The first is called the dividend; the second, the divisor; and

the third, the quotient.

Division is the converse of multiplication.

In it we have given the product and one factor to find the other. The rules for division are just the converse of those for multiplication.

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The quotient must be such a monomial as, being multiplied by the divisor, will give the dividend: hence the coefficient of the quotient must be 9, and the literal part a2; for these quantities multiplied by 8 a3 will give 72a5. Hence

72a51
8a3

9a2.

The coefficient 9 is obtained by dividing 72 by 8; and the literal part is found by giving to a an exponent equal to 5 minus 3.

For dividing one monomial by another, then, we deduce from the above example the following rule:

Divide the coefficient of the dividend by the coefficient of the divisor, for a new coefficient.

After this coefficient write all the letters of the dividend, giving to each an exponent equal to the excess of its exponent in the dividend over that in the divisor.

47. Signs in Division. Since the quotient multiplied by the divisor must produce the dividend, and since the product of two factors having the same sign will be +, and the product of two factors having different signs will be, we conclude,

(1) When the signs of the dividend and divisor are like, the sign of the quotient will be +.

(2) When the signs of the dividend and divisor are unlike, the sign of the quotient will be -.

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