25. If the dimensions (length and breadth) of a rectangle are represented by x+2 and x-1, what represents the area? 26. We know (see Example 18, page 21) that the area of a triangle equals one half the product of its base times its altitude (or height). What represents the areas of the triangles having the following bases and altitudes?

Base

Altitude

FIG. 21.

27. What is the area of the triangle in Ex. 26 (c) in case r=3, s=2?

Upper Base

28. A trapezoid is a four-sided figure whose upper and lower sides (called bases) are parallel. The area of a trapezoid is equal to one half the sum of its bases multiplied by its altitude. What formula, therefore,

represents the area

of the trapezoid

Altitude

Lower Base

FIG. 22.-TRAPEZOID.

whose bases are a and b and whose altitude is h?

29. What represents the area of the trapezoid whose bases are 2 x and 3 y and whose altitude is x+y?

30. What is the area of the trapezoid in Ex. 29 if x=2, y=4?

31. If the radius of a circle is represented by x+3 what represents its area?

54. Further Study of Factoring. We saw in § 52 how such an expression as 3 ax3+6 x2+12 bx could be factored. In fact, since 3x is a factor in each term, the answer here is 3 x(ax2+2x+4b). Note that in all such cases the factor common to the terms (in this case 3 x), is a monomial. We shall now consider some examples in factoring in which the common term is not a monomial.

EXAMPLE 1. Factor the expression 2(a+b)+x(a+b).

SOLUTION. Since a+b is a common factor of both terms, as in § 52 we have

CHECK. When we multiply a+b by 2+x (in the way shown in § 53) we get 2a+2b+xa+xb, and this is the same as 2(a+b)+x(a+b). EXAMPLE 2. Factor the expression x(x+y)+y(x+y)+ 2(x+y).

SOLUTION. Here x+y is a common factor of each term. Hence

x(x+y)+y(x+y)+2(x+y)=(x+y)(x+y+2).

Ans.

CHECK. Multiplying x+y by x+y+2 as in § 53 gives

x2+2 xy+y2+2x+2 y.

But this is also the result one gets when he performs the multiplications and simplifies the expression

x(x+y)+y(x+y)+2(x+y).

EXAMPLE 3. Factor the expression ax+ay+bx+by.

SOLUTION. First observe that the expression may be written in the form (ax+ay)+(bx+by), thus grouping terms having a common factor. This form may be changed to a(x+y)+b(x+y). Now proceed as in Examples 1 and 2, and obtain the answer (x+y) (a+b).

EXAMPLE 4. Factor cx+y-dy+cy-dx+x.

SOLUTION. By rearranging the terms, the expression may be written in the form

cx-dx+x+cy-dy+y=(c-d+1)x+(c-d+1)y= (c-d+1)(x+y). Ans.

EXERCISES

Factor each of the following expressions.

1. a(x-y)+b(x−y).

2. 3x(2y-3 z) +4 y(2 y−3 z).
3. m(a+b-c)+n(a+b−c)+q(a+b−c).
4. 6 a(a-b)+5 b(a−b)-2(a−b).

5. a(a-b)+3(b-a).

[HINT. Write 3(b-a) in the form -3(a-b).]

6. am-an+mx−nx.

55. Arrangement of Terms before Multiplying. We have already seen in § 33 how the addition and subtraction of polynomials is best carried out by first arranging them in the ascending or descending powers of some one letter. This is also true in the multiplication of two polynomials. For example, in multiplying x−1−3 x3+2 x4 by 2+x, we first arrange both in descending powers of x, thus giving them the forms 2x4-3 x3+x-1 and x+2. Then we multiply (in the manner explained in § 53), the work appearing as below:

WRITTEN EXERCISES

1. Multiply x-1-3 x3+2x4 by 2+x.

[HINT. First arrange the polynomials in ascending powers of x. Is your result the same as was obtained in § 55?]

Carry out each of the following indicated multiplications, first rearranging where desirable.

2. (3 a2-2 a-1) (a-1).

Check for a = 2.

3. (x-3+x2) (2+x). Check for x=2. (x−3+x2)(2+x).

4. (3 a2+2a-4) (5-a).

5. (5 n−4+6 n2) (8+n2−4 n).

6. (2+3x2-x+x3)(x2-2x+4).

7. (x2+2 xy+y2)(x+y).

[HINT. Here both polynomials are already arranged according to descending powers of x.]

8. (a2-ab+b2)(a+b).

9. (a2+ab+b2) (a−b).

10. (m2-mn+n2) (m2+mn+n2). Check for m=1, n = 1. 11. (8 r2-2 s2+4 rs) (2 rs+3 s2+4 r2).

[HINT. Arrange in descending powers of r.]

12. (A2+B2−2 AB) (A2+B2+2 AB).

13. (x2y2+xy+1)(1−xy+x2y2).

14. (4 x3-3x2y+5 xy2 −6 y3)(5 x+6 y).

15. Expand (x+r)2. [HINT. (x+r)2 means (x+r)(x+r).]

Expand each of the following expressions.

23. (x2+2x+1)2. Check for x=2.

[HINT. Find (x+1)2 and multiply it by x+1.]

22. (2 a-5)2.

24. (x+1)3.

25. (x+1)4.

26. (x+1)2(x+2)2.

27. (a+b+c+d)2.

28. (x−y)(x+y)(x2+x2y2+y1).

WRITTEN EXERCISES

COMBINATIONS OF ADDITION, SUBTRACTION, AND
MULTIPLICATION

1. Simplify a2+a(b-a)-b(3 b-a).

SOLUTION. This expression means the sum of a2, a(b-a), and -b(3 b-a). Now, a (b-a)=ab-a2 and —b(3 b—a) = −3 b2+ab. Therefore, writing the terms in their order with proper signs, the expression becomes a2+ab-a2-3 b2+ab=2 ab−3 b2. Ans.

13. (3 r-1)(r+2)−3 r(r+3)+2(r+1).

14. (x-y)2-2(x2-y2) − 2 x (− x − y) — 4 y2.

15. (x+y)2(x− y)2 — (x2 + y2)2.

16. n*+(m_mntn?) (m+n) — (m3n3)(m+2n).

17. (x+y)(x2-xy+y2)+(x−y) (x2+xy+y2).

56. Further Study of the Equation. We frequently meet with equations containing such multiplications as those just studied. For example, consider the equation 5x+x(x-3)=12+x2. The root (or value of the unknown number x) is found as follows:

5x+x(x−3)=12+x.