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Therefore we say in algebraic language that + multiplied by t, or — multiplied by -, gives +; – multiplied by t, or + 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. 3a? + 4 ab + b2 by 2a +56.

3a + 4 ab + b2
2a + 5b
6a3 + 8a2b + 2 ab?

+ 15a2b + 20 ab? + 563
6a +23 ab + 22 ab? + 563

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. 2? + 2ax + a2 by x+a. Ans. 2 + 3 ax? + 3 a2x + a'. 3. 2 + yo by x+y.

Ans. ** + xys + xoy + y4. 4. 3ab? +6a’cby 3 ab? + 3 aʼc?.

Ans. 9a2b4 + 27 a?bc? + 18 a*c*.

5. ab? + c'd by a +b. Ans. ab? +acd+a?b% + bcod. 6. 3ax? +9ab3 + cd by 6 aʼc.

Ans. 18a%cox2 + 54 a*c+63 + 6a2cd5. 7. 64a928 +27 a2x + 9 ab by 8a'cd.

Ans. 512acdx+ 216 aʼcdx + 72 a:bcd. 8. a* + 3 aʼx + 3 ax? + x3 by a +x.

Ans. a* + 4aRx + 6 aʼx? + 4 a.x3 + x*. 9. x2 + y by x+y.

Ans. 22 + xy + xy + yo. 10. 208 + xy + 7ax by ax + 5ax.

Ans. 6 axe + 6 axʻy® + 42 aʼx”. 11. a* + 3a2b + 3 ab? + 33 by a + b.

Ans. at +4 aob + 6a?b? +4 ab3 +64. 12. 2 + xy + xy + yg by x+y.

Ans. 2+ + 2xy + 2x+y+ + 2 xy + y4. 13. 23 + 2x2 + x + 3 by 3x + 1.

Ans. 3x4 + 7 x + 5x2 + 10x + 3.

FOR REVIEW.
Multiply the following: -
1. 2ax 3 ab by 3x b.

2 ax 3 ab
3x - 6
6 ax? — 9 abx

- 2 abx + 3 ab?

6 ax? — 11 abx + 3 ab? 2. a* - 263 by a -6.

Ans. a— 2ab ab +264. 3. x2 – 3x — 7 by x — 2. Ans. 2* — 5x? — x + 14. 4. 3 a? — 5 ab +262 by a’ 7ab.

Ans. 3a* – 26 a'b + 37 a62 – 14 ab. 5. 62 +86 +by 6? -- 1.

Ans. 68 ? 6. ** – 2x®y + 4x?j— 8 xy + 16 y* by x +2y.

Ans. 20 + 32 y. 7. 4x2 – 2y by 2y.

Ans. 8xy 4y?. 8. 2x+4y by 2x – 4y.

Ans. 4x2 – 16 y. 9. 2 + xy + xy + y by x y.

Ans. we — y". 10. x2 + xy + y by x? xy + 3%. Ans. x* + xyj? + y*. 11. 2a’ — 3 ax + 4x2 by 5 a? — 6 ax — 2x”.

Ans. 10 a* — 27 a®x + 34 a*r? — 18 ax: — 8x*. 12. 3x2 – 2xy + 5 by x2 + 2xy – 3.

Ans. 32+ + 4x®y 4x2 — 4x*y? + 16 xy 15. 13. 3.08 + 2x+y2 + 3 ya by 233 – 3x?y? + 5 . Ans. 6.26 — 5 x®yja – 6x*y* + 6 x®y+ 15x*y* — 9x?yt

+10x +15 g. 14. 8ax — 6 ab c by 2 ax + ab +c.

Ans. 16 aʼx? — 4abx — 6 aʼb? +6 acx 7 abc c?. 15. 3a? – 562 +3 c? by a?.

Ans. 3a- 8a2b2 + 3 aʼc? +564 3 boc?. 16. 3a 5bd+ cf

- 5a” +4 bd 8cf

- 15a +37abd — 29 aof -- 206d? +44 bcdf - 8c?f? 17. ax ab? by aʻx”.

Ans. am+2+a*b*x. 18. am + b1 by am 6".

Ans. a2m 32m. 19. am + bn by am + 6".

Ans. a2m + 2 amb+ b?n.

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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46. To Divide One Monomial by Another.
Divide 72a' by 8a'.
The division is indicated thus : 729°

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 a?; for these quantities multiplied by 8 as will give 72a. Hence

72a5

=9a.

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 t, 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 t.

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

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