3.5
Special Products
3.5
OBJECTIVES
1. Square a binomial
2. Find the product of two binomials that differ
only in their sign
Certain products occur frequently enough in algebra that it is worth learning special formulas for dealing with them. First, let’s look at the square of a binomial, which is the product of two equal binomial factors.
(x y)2 (x y) (x y)
x 2 2xy y 2
(x y)2 (x y) (x y)
x2 2xy y 2
The patterns above lead us to the following rule.
Step by Step:
Step 1
Step 2
Step 3
To Square a Binomial
Find the first term of the square by squaring the first term of the
binomial.
Find the middle term of the square as twice the product of the two
terms of the binomial.
Find the last term of the square by squaring the last term of the
binomial.
Example 1
Squaring a Binomial
(a) (x 3)2 x2 2 · x · 3 32
C A U TI O N
A very common mistake in
squaring binomials is to forget
the middle term.
Square of
first term
Twice the
product of
the two terms
Square of
the last term
x2 6x 9
© 2001 McGraw-Hill Companies
(b) (3a 4b)2 (3a)2 2(3a)(4b) (4b)2
9a2 24ab 16b2
(c) (y 5)2 y2 2 · y · (5) ( 5)2
y2 10y 25
(d) (5c 3d)2 (5c)2 2(5c)(3d) (3d)2
25c2 30cd 9d 2
Again we have shown all the steps. With practice you can write just the square.
293
294
CHAPTER 3
POLYNOMIALS
CHECK YOURSELF 1
Multiply.
(a) (2x 1)2
(b) (4x 3y)2
Example 2
Squaring a Binomial
Find ( y 4)2.
2
2
( y 4)2
is not equal to
y2 42 or y2 16
The correct square is
( y 4)2 y2 8y 16
The middle term is twice the product of y and 4.
CHECK YOURSELF 2
Multiply.
(a) (x 5)2
(b) (3a 2)2
(c) (y 7)2
(d) (5x 2y)2
A second special product will be very important in the next chapter, which deals with
factoring. Suppose the form of a product is
(x y)(x y)
The two terms differ
only in sign.
Let’s see what happens when we multiply.
(x y)(x y)
x2 xy xy y2
x2 y2
0
Because the middle term becomes 0, we have the following rule.
Rules and Properties: Special Product
The product of two binomials that differ only in the sign between the terms is
the square of the first term minus the square of the second term.
© 2001 McGraw-Hill Companies
(2 3) 2 3 because
52 4 9
2
NOTE You should see that
SPECIAL PRODUCTS
SECTION 3.5
295
Let’s look at the application of this rule in Example 3.
Example 3
Multiplying Polynomials
Multiply each pair of binomials.
(a) (x 5)(x 5) x2 52
Square of
the first term
Square of
the second term
x2 25
NOTE
(b) (x 2y)(x 2y) x2 (2y)2
(2y) (2y)(2y)
2
4y 2
Square of
the first term
Square of
the second term
x2 4y2
(c) (3m n)(3m n) 9m2 n2
(d) (4a 3b)(4a 3b) 16a2 9b2
CHECK YOURSELF 3
Find the products.
(a) (a 6)(a 6)
(c) (5n 2p)(5n 2p)
(b) (x 3y)(x 3y)
(d) (7b 3c)(7b 3c)
© 2001 McGraw-Hill Companies
When finding the product of three or more factors, it is useful to first look for the pattern
in which two binomials differ only in their sign. Finding this product first will make it
easier to find the product of all the factors.
Example 4
Multiplying Polynomials
(a) x (x 3)(x 3)
x(x2 9)
x3 9x
These binomials differ only in the sign.
CHAPTER 3
POLYNOMIALS
(b) (x 1) (x 5)(x 5)
(x 1)(x2 25)
These binomials differ only in the sign.
With two binomials, use the FOIL method.
x3 x2 25x 25
(c) (2x 1) (x 3) (2x 1)
(x 3)(2x 1)(2x 1)
These two binomials differ only in the sign of
the second term. We can use the commutative
property to rearrange the terms.
(x 3)(4x2 1)
4x3 12x2 x 3
CHECK YOURSELF 4
Multiply.
(a) 3x(x 5)(x 5)
(c) (x 7)(3x 1)(x 7)
(b) (x 4)(2x 3)(2x 3)
CHECK YOURSELF ANSWERS
1.
2.
3.
4.
(a) 4x 2 4x 1; (b) 16x 2 24xy 9y 2
(a) x 2 10x 25; (b) 9a 2 12a 4; (c) y 2 14y 49; (d) 25x 2 20xy 4y 2
(a) a 2 36; (b) x 2 9y 2; (c) 25n 2 4p 2; (d) 49b 2 9c 2
(a) 3x 3 75x; (b) 4x 3 16x 2 9x 36; (c) 3x 3 x 2 147x 49
© 2001 McGraw-Hill Companies
296
Name
3.5
Exercises
Section
Date
Find each of the following squares.
ANSWERS
1. (x 5)2
2. (y 9)2
1.
3. (w 6)2
2.
4. (a 8)2
3.
5. (z 12)2
6. ( p 20)2
7. (2a 1)
8. (3x 2)
4.
5.
2
2
9. (6m 1)2
6.
7.
10. (7b 2)2
8.
11. (3x y)2
9.
12. (5m n)2
10.
13. (2r 5s)2
14. (3a 4b)2
15. (8a 9b)
16. (7p 6q)
11.
12.
2
2
13.
14.
1
17. x 2
2
1
18. w 4
2
15.
16.
17.
Find each of the following products.
19. (x 6)(x 6)
20. ( y 8)( y 8)
18.
© 2001 McGraw-Hill Companies
19.
21. (m 12)(m 12)
22. (w 10)(w 10)
20.
21.
1
23. x 2
1
x
2
2
24. x 3
2
x
3
22.
23.
24.
297
ANSWERS
25.
25. (p 0.4)(p 0.4)
26. (m 0.6)(m 0.6)
27. (a 3b)(a 3b)
28. (p 4q)(p 4q)
29. (4r s)(4r s)
30. (7x y)(7x y)
31. (8w 5z)(8w 5z)
32. (7c 2d)(7c 2d)
33. (5x 9y)(5x 9y)
34. (6s 5t)(6s 5t)
35. x(x 2)(x 2)
36. a(a 5)(a 5)
37. 2s(s 3r)(s 3r)
38. 5w(2w z)(2w z)
39. 5r(r 3)2
40. 3x(x 2)2
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
For each of the following problems, let x represent the number, then write an expression
for the product.
39.
40.
41. The product of 6 more than a number and 6 less than that number
41.
42. The square of 5 more than a number
42.
43.
43. The square of 4 less than a number
44.
45.
44. The product of 5 less than a number and 5 more than that number
46.
48.
49.
45. (49)(51)
46. (27)(33)
47. (34)(26)
48. (98)(102)
49. (55)(65)
50. (64)(56)
50.
298
© 2001 McGraw-Hill Companies
Note that (28)(32) (30 2)(30 2) 900 4 896. Use this pattern to find each of
the following products.
47.
ANSWERS
51. Tree planting. Suppose an orchard is planted with trees in straight rows. If there are
5x 4 rows with 5x 4 trees in each row, how many trees are there in the orchard?
51.
52.
53.
54.
52. Area of a square. A square has sides of length 3x 2 centimeters (cm). Express the
55.
area of the square as a polynomial.
3x 2 cm
3x 2 cm
53. Complete the following statement: (a b)2 is not equal to a2 b2 because. . . . But,
wait! Isn’t (a b)2 sometimes equal to a2 b2 ? What do you think?
54. Is (a b)3 ever equal to a3 b3? Explain.
55. In the following figures, identify the length, width, and area of the square:
a
b
Length a
Width b
Area a
3
a
Width Area © 2001 McGraw-Hill Companies
3
x
x
Length x2
2x
Length 2x
Width Area 299
ANSWERS
56. The square below is x units on a side. The area is
56.
.
a.
Draw a picture of what happens when the sides are
doubled. The area is
.
b.
Continue the picture to show what happens when the sides are tripled.
The area is
.
c.
If the sides are quadrupled, the area is
d.
In general, if the sides are multiplied by n, the area is
.
e.
If each side is increased by 3, the area is increased by
.
f.
If each side is decreased by 2, the area is decreased by
g.
In general, if each side is increased by n, the area is increased by
each side is decreased by n, the area is decreased by
.
.
.
, and if
h.
x
x
Getting Ready for Section 3.6 [Section 1.7]
Divide.
(a)
2x2
2x
(b)
3a3
3a
(c)
6p3
2p2
(d)
10m4
5m2
(e)
20a3
5a3
(f)
6x2y
3xy
(g)
12r3s2
4rs
(h)
49c4d 6
7cd3
Answers
3. w2 12w 36
11. 9x2 6xy y2
15. 64a2 144ab 81b2
19. x2 36
21. m2 144
1
25. p2 0.16
29. 16r2 s2
4
64w2 25z2
33. 25x2 81y2
35. x3 4x
37. 2s3 18r2s
3
2
2
2
5r 30r 45r
41. x 36
43. x 8x 16
45. 2499
884
49. 3575
51. 25x2 40x 16
55.
a. x
b. a2
c. 3p
d. 2m2
f. 2x
300
1
4
2
27. a 9b2
17. x2 x 23. x2 31.
39.
47.
53.
5. z2 24z 144
7. 4a2 4a 1
2
13. 4r 20rs 25s2
g. 3r2s
h. 7c3d 3
e. 4
© 2001 McGraw-Hill Companies
1. x2 10x 25
9. 36m2 12m 1