6.6
Factoring Special Polynomials
6.6
OBJECTIVES
1. Factor the difference of two squares
2. Factor the sum or difference of two cubes
In this section, we will look at several special polynomials. These polynomials are special
because they fit a recognizable pattern. Pattern recognition is an important element of
mathematics. Many mathematical discoveries were made because somebody recognized a
pattern.
The first pattern, which we saw in Section 6.4, is called the difference of two squares.
CAUTION
What about the sum of two
squares, such as
x2 25
In general, it is not possible to
factor (using real numbers) a
sum of two squares. So
(x2 25) (x 5)(x 5)
Rules and Properties:
The Difference of Two Squares
a2 b2 (a b)(a b)
(1)
In words: The product of the sum and difference of two terms gives the
difference of two squares.
Equation (1) is easy to apply in factoring. It is just a matter of recognizing a binomial as the
difference of two squares.
To confirm this identity, use the FOIL method to multiply
(a b)(a b)
Example 1
Factoring the Difference of Two Squares
NOTE We are looking for
perfect squares—the exponents
must be multiples of 2 and the
coefficients perfect squares—1,
4, 9, 16, and so on.
(a) Factor x2 25.
Note that our example has two terms—a clue to try factoring as the difference of two
squares.
x2 25 (x)2 (5)2
(x 5)(x 5)
(b) Factor 9a2 16.
9a2 16 (3a)2 (4)2
(3a 4)(3a 4)
© 2001 McGraw-Hill Companies
(c) Factor 25m 4 49n2.
25m4 49n2 (5m 2)2 (7n)2
(5m 2 7n)(5m 2 7n)
CHECK YOURSELF 1
Factor each of the following binomials.
(a) y 2 36
(b) 25m2 n2
(c) 16a 4 9b 2
423
424
CHAPTER 6
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
We mentioned earlier that factoring out a common factor should always be considered
your first step. Then other steps become obvious. Consider Example 2.
Example 2
Factoring the Difference of Two Squares
Factor a3 16ab 2.
First note the common factor of a. Removing that factor, we have
a3 16ab 2 a(a 2 16b 2)
We now see that the binomial factor is a difference of squares, and we can continue to
factor as before. So
a3 16ab 2 a(a 4b)(a 4b)
CHECK YOURSELF 2
Factor 2x 3 18xy 2.
You may also have to apply the difference of two squares method more than once to
completely factor a polynomial.
Example 3
Factoring the Difference of Two Squares
Factor m4 81n4.
m4 81n4 (m2 9n2)(m2 9n2)
Do you see that we are not done in this case? Because m2 9n2 is still factorable, we can
continue to factor as follows.
NOTE The other binomial
factor, m 9n , is a sum of two
squares, which cannot be
factored further.
2
2
m4 81n4 (m2 9n2)(m 3n)(m 3n)
CHECK YOURSELF 3
Two additional patterns for factoring certain binomials include the sum or difference of two
cubes.
Rules and Properties:
The Sum or Difference of Two Cubes
NOTE Be sure you take the
a3 b3 (a b)(a2 ab b2)
(2)
time to expand the product on
the right-hand side to confirm
the identity.
a3 b3 (a b)(a2 ab b2)
(3)
© 2001 McGraw-Hill Companies
Factor x 4 16y 4.
FACTORING SPECIAL POLYNOMIALS
SECTION 6.6
425
Example 4
Factoring the Sum or Difference of Two Cubes
NOTE We are now looking for
perfect cubes—the exponents
must be multiples of 3 and the
coefficients perfect cubes—1, 8,
27, 64, and so on.
(a) Factor x3 27.
The first term is the cube of x, and the second is the cube of 3, so we can apply equation (2). Letting a x and b 3, we have
x3 27 (x 3)(x 2 3x 9)
(b) Factor 8w 3 27z3.
This is a difference of cubes, so use equation (3).
8w 3 27z3 (2w 3z)[(2w)2 (2w)(3z) (3z)2]
(2w 3z)(4w 2 6wz 9z2)
NOTE Again, looking for a
common factor should be your
first step.
(c) Factor 5a3b 40b4.
First note the common factor of 5b. The binomial is the difference of cubes, so use equation (3).
5a3b 40b4 5b(a3 8b3)
5b(a 2b)(a2 2ab 4b2)
NOTE Remember to write the
GCF as a part of the final
factored form.
CHECK YOURSELF 4
Factor completely.
(a) 27x 3 8y 3
(b) 3a4 24ab3
In each example in this section, we factored a polynomial expression. If we are given a
polynomial function to factor, there is no change in the ordered pairs represented by the
function after it is factored.
Example 5
Factoring a Polynomial Function
© 2001 McGraw-Hill Companies
Given the function f(x) 9x 2 15x, complete the following.
(a) Find f(1).
f(1) 9(1)2 15(1)
9 15
24
(b) Factor f(x).
f(x) 9x2 15x
3x(3x 5)
CHAPTER 6
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
(c) Find f(1) from the factored form of f(x).
f(1) 3(1)(3(1) 5)
3(8)
24
CHECK YOURSELF 5
Given the function f(x) 16x 5 10x 2, complete the following.
(a) Find f(1).
(b) Factor f(x).
(c) Find f(1) from the factored form of f(x).
CHECK YOURSELF ANSWERS
1.
2.
4.
5.
(a) ( y 6)( y 6); (b) (5m n)(5m n); (c) (4a2 3b)(4a2 3b)
2x(x 3y)( x 3y)
3. (x 2 4y 2)(x 2y)( x 2y)
(a) (3x 2y)( 9x2 6xy 4y2); (b) 3a(a 2b)(a 2 2ab 4b2)
(a) 26; (b) 2x2(8x3 5); (c) 26
© 2001 McGraw-Hill Companies
426
Name
6.6 Exercises
Section
Date
For each of the following binomials, state whether the binomial is a difference of squares.
ANSWERS
1. 3x 2 2y 2
2. 5x 2 7y 2
1.
3. 16a2 25b2
4. 9n2 16m2
2.
3.
4.
5. 16r 2 4
6. p 2 45
5.
6.
7. 16a2 12b3
8. 9a 2b2 16c2d 2
7.
8.
9. a 2b2 25
10. 4a 3 b3
9.
10.
11.
Factor the following binomials.
11. x 2 49
12. m2 64
12.
13.
13. a 2 81
14. b 2 36
14.
15.
15. 9p2 1
16. 4x2 9
16.
17.
17. 25a2 16
18. 16m2 49
18.
19.
19. x 2y2 25
20. m2n2 9
20.
© 2001 McGraw-Hill Companies
21.
21. 4c2 25d 2
22. 9a2 49b2
22.
23.
23. 49p2 64q2
24. 25x 2 36y 2
24.
25.
25. x4 16y2
26. a2 25b4
26.
427
ANSWERS
27.
27. a3 4ab2
28. 9p2q q3
29. a4 16b4
30. 81x4 y4
31. x3 64
32. y 3 8
33. m3 125
34. b3 27
35. a3b3 27
36. p 3q3 64
37. 8w3 z3
38. c3 27d 3
39. r3 64s3
40. 125x3 y3
41. 8x3 27y3
42. 64m3 27n3
43. 8x3 y6
44. m6 27n3
45. 4x3 32y3
46. 3a3 81b3
47. 18x3 2xy 2
48. 50a2b 2b3
49. 12m3n 75mn3
50. 63p4 7p2q2
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
49.
50.
428
© 2001 McGraw-Hill Companies
48.
ANSWERS
For each of the functions in exercises 51 to 56, (a) find f(1), (b) factor f(x), and
(c) find f(1) from the factored form of f(x).
51.
51. f(x) 12x 5 21x 2
52. f(x) 6x 3 10x
52.
53. f(x) 8x 5 20x
54. f(x) 5x 5 35x 3
53.
55. f(x) x 5 3x 2
56. f(x) 6x 6 16x 5
54.
55.
Factor each expression.
57. x 2(x y) y 2(x y)
56.
57.
58. a (b c) 16b (b c)
2
2
59. 2m2(m 2n) 18n2(m 2n)
60. 3a3(2a b) 27ab2(2a b)
58.
59.
60.
61.
62.
61. Find a value for k so that kx 2 25 will have the factors 2x 5 and 2x 5.
63.
62. Find a value for k so that 9m2 kn2 will have the factors 3m 7n and 3m 7n.
64.
63. Find a value for k so that 2x 3 kxy2 will have the factors 2x, x 3y, and x 3y.
65.
64. Find a value for k so that 20a3b kab3 will have the factors 5ab, 2a 3b, and
66.
2a 3b.
65. Complete the following statement in complete sentences: “To factor a number
you . . . .”
66. Complete this statement: “To factor an algebraic expression into prime factors
© 2001 McGraw-Hill Companies
means . . . .”
67.
68.
67. Verify the formula for factoring the sum of two cubes by finding the product
(a b)(a2 ab b2).
68. Verify the formula for factoring the difference of two cubes by finding the product
(a b)(a2 ab b2).
429
ANSWERS
69. What are the characteristics of a monomial that is a perfect cube?
69.
70. Suppose you factored the polynomial 4x2 16 as follows:
70.
4x2 16 (2x 4)(2x 4)
Would this be in completely factored form? If not, what would be the final form?
Answers
© 2001 McGraw-Hill Companies
1. No
3. Yes
5. No
7. No
9. Yes
11. (x 7)(x 7)
13. (a 9)(a 9)
15. (3p 1)(3p 1)
17. (5a 4)(5a 4)
19. (xy 5)(xy 5)
21. (2c 5d)(2c 5d)
23. (7p 8q)(7p 8q)
25. (x2 4y)(x2 4y)
27. a(a 2b)(a 2b)
29. (a2 4b2)(a 2b)(a 2b)
31. (x 4)(x2 4x 16)
2
33. (m 5)(m 5m 25)
35. (ab 3)(a2b2 3ab 9)
2
2
37. (2w z)(4w 2wz z )
39. (r 4s)(r 2 4rs 16s2)
2
2
41. (2x 3y)(4x 6xy 9y )
43. (2x y2)(4x2 2xy 2 y4)
2
2
45. 4(x 2y)(x 2xy 4y )
47. 2x(3x y)(3x y)
49. 3mn(2m 5n)(2m 5n)
51. (a) 33; (b) 3x 2(4x 3 7); (c) 33
53. (a) 12; (b) 4x(2x4 5); (c) 12
55. (a) 4; (b) x 2(x3 3); (c) 4
2
57. (x y) (x y)
59. 2(m 2n)(m 3n)(m 3n)
61. 4
63. 18
65.
67.
69.
430