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Similarly, if we wish to multiply a2-6 ab+8 b2 by 2 a2b the work is arranged as follows:

a2-6 ab+8 b2

2 a2b

2 a1b-12 a3b2+16 a2b3. Ans.

From these illustrations, we have the following rule.

RULE FOR MULTIPLYING A POLYNOMIAL BY A MoNOMIAL. To multiply a polynomial by a monomial, multiply each term of the polynomial separately and combine results. This rule is stated in simple form in the following formula. Formula III. a(b+c)=ab+ac.

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8. 3x2-3 xy+5y by -2xy.

9. a2-10 ab+15 b2 by 4 a2b2.

10. Simplify (-2ac+4 ax) (-5acx).

11. Subtract a(b+c) from b(a+c).

12. Show that the relation a(b+c) = ab+ac is illustrated by the figure below.

[graphic]

[HINT.

FIG. 20.

ab is the area of the rectangle whose dimensions (length

and breadth) are a and b.]

WRITTEN EXERCISES

Carry out the following indicated multiplications.

1. mn(m-n+mn). Check when m=1, n=2. SOLUTION. Multiplying in the way shown in § 51, the work appears as follows:

m―n+mn

mn

m2n- mn2+ m2 n2. Ans.

CHECK. Suppose m=2 and n=1.

Then m-n+mn=2−1+2=3,

and mn=2 and the product of these is 3×2=6. At the same time, our answer (which is m2n— mn2+m2n2) becomes 4×1-2×1+4X1=6. Since both results are 6, the work checks.

2.

2m(m+mn-n). Check when m=1, n=2.

3. -6 ab(-2 a2+4 ab−3 b2). Check when a=1, b=1. 4. (ab+bc+ac) abc. Check when a=2, b=2, c=1.

5. -3x2(-2x3+3 x2+4 x). Check when x=3.

6. 3(-33+4r2+2 r). Check when r=3.

7. rs(1+2r-3 s).

8. 3 xy(-2xy3+4 x2y2 — x3y).

9. 8 ab3c2(-2 a2bc3 — ab1c2+1).

10. xy2(x3-2 x2y2+y3).

11. (L2-3 L3+4)2 L2.

Simplify the following expressions.

12. 2(3 a+4 b)-2(2 a-b).

13. 2x(3x-2y)+3 x(x+2y).

14. 2 b(b2-b)-2 b2.

15. ab3-ab(a+b3). Check for a=1, b=2.

16. Show that a(d+e)+b(d+e)+c(d+e)−e(a+b+c)= (a+b+c)d.

For other exercises on this topic, see review exercises, p. 114, and Appendix, p. 298.

52. Factoring an Expression. In the preceding exercises certain expressions were given us that had been separated into their factors, and we were asked to multiply the factors together. Thus, in Ex. 1, we were given the expression mn(m-n+mn) and we multiplied this, getting the product m2n-mn2+m2n2.

Suppose now we try to reverse this process; that is, suppose we start with a product itself and ask what the factors are which when multiplied together give that product. We can often answer such questions in the manner illustrated below.

EXAMPLE 1. Factor the expression ab+ac-ad.

SOLUTION. The letter a is here contained as a factor in every Hence we may write

term.

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CHECK. When we simplify the answer by multiplying it out (as in § 51) we get ab+ac-ad.

EXAMPLE 2. Factor the expression a2x+ax2+a2x2.

SOLUTION.

out,

Here az is a factor of every term. Hence, taking it

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CHECK. Multiplying out in ax (a+x+ax) gives a2x+ax2+a2x2.

EXAMPLE 3. Factor 3 x3y3-3x2y2+12 xy.

SOLUTION.

Here 3 xy is a factor of every term.

3 x3y3-3x2y2+12 xy=3 xy(x2y2-xy+4).

WRITTEN EXERCISES

Hence,
Ans.

Factor each of the following expressions, and check your

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For further exercises on this topic, see the review exercises, p. 114, and Appendix, p. 298.

53. Multiplication of a Polynomial by a Polynomial. In multiplying a polynomial by a polynomial we multiply the multiplicand by each of the monomials in the multiplier, and then combine these partial results. For example, in multiplying 2x-3y by 3 x+4y the work is as follows:

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Adding gives 6x2-xy-12 y2 Ans.

NOTE. In this work the expressions 6 x2-9 xy and 8 xy—12 y2 are called partial products. The adding of the partial products always gives the answer.

Again, let us multiply x-y+3 z by 2x+3y-z. The work, which should be examined carefully, follows.

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2x2+ xy+5 xz-3 y2+10 yz-3 z2.

CHECK. Suppose x=1, y=1, z=1. Then x-y+3 z=1−1+3=3, and 2 x+3 y―z=2+3−1=4, and the product of these is 3×4=12. At the same time, when x=1, y=1, z=1, our answer (which is 2 x2+xy+5 xz-3 y2+10 yz-3 z2) becomes 2+1+5-3+10-3=12. Since both are 12, the work checks.

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4. (a-2)(a-2). 5. (a+2)(a−2). 6. (m+n) (m+n). 7. (m2+n2) (m2-n2). 8. (m2 — n2) (m2-n2). 9. (ab+c) (ab-c). 10. (ab-5) (ab+4). 11. (2 r+7)(3r+5). 12. (3r+s-2 t) (r−5 s). 13. (x2y2-6)(x2y2-2). 14. (3 a2-2) (4 a+1).

15. (5 y2+322) (2 y-z).

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Check when m=2, n=2.
Check when m=2, n=2.
Check when m=1, n=2.
Check when a=1, b=2, c=1.
Check when a = 2, b=3.

16. (4 a−10 b+1) (2 a−b+2).
17. (11+a+b2) (4-5 a-b2).
18. (82-5 r+1) (3 r2+2 r−2).
19. (3x2-2 y2) (x2 — 3 y2).
20. (A+B-C)(A−B+C).

APPLIED PROBLEMS

21. If the side of a square is represented by 2x+3, what represents its area?

SOLUTION. The area will be represented by (2x+3)2, which, when multiplied out, becomes 4 x2+12 x+9. Ans.

22. If the side of a square is represented by 5x-2, what represents its area?

23. If the edge of a cube is represented by 3x+2, what represents its volume?

24. If the edge of a cube is represented by 2x+y, what represents its volume?

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