Factoring Special Polynomials

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423

Factoring Special Polynomials

6.6

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.

a2_{ b}2_{ (a  b)(a  b) (1)}

In words: The product of the sum and difference of two terms gives the difference of two squares.

Rules and Properties: 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

(a) Factor x2 25.

Note that our example has two terms—a clue to try factoring as the difference of two squares.  (x)2_{ (5)}2  (x  5)(x  5) (b) Factor 9a2_{ 16.}  (3a)2_{ (4)}2  (3a  4)(3a  4) (c) Factor 25m4_{ 49n}2_.  (5m2₎2_{ (7n)}2  (5m2_{ 7n)(5m}2_{ 7n)} 25m4 49n2 9a2 16 x2 25 C A U T I O N

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)}

NOTEWe are looking for perfect squares—the exponents must be multiples of 2 and the coefficients perfect squares—1, 4, 9, 16, and so on.

C H E C K Y O U R S E L F 1

Factor each of the following binomials.

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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 16ab2 a(a2 16b2)

We now see that the binomial factor is a difference of squares, and we can continue to factor as before. So

a3 16ab2 a(a  4b)(a  4b)

C H E C K Y O U R S E L F 2

Factor 2x3_{ 18xy}2_.

C H E C K Y O U R S E L F 3

Factor x4  16y4_.

Factoring the Difference of Two Squares Factor m4 81n4.

 (m2_{ 9n}2_)(m2_{ 9n}2₎

Do you see that we are not done in this case? Because m2 9n2is still factorable, we can continue to factor as follows.

 (m2_{ 9n}2_{)(m 3n)(m  3n)}

m4 81n4 m4 81n4

a3_{ b}3_{ (a  b)(a}2_{ ab  b}2_{) (2)}

a3_{ b}3_{ (a  b)(a}2_{ ab  b}2_{) (3)}

Rules and Properties: The Sum or Difference of Two Cubes

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 3

You may also have to apply the difference of two squares method more than once to completely factor a polynomial.

Two additional patterns for factoring certain binomials include the sum or difference of two cubes.

NOTEThe other binomial factor, m2_{ 9n}2_{, is a sum of two}

squares, which cannot be factored further.

NOTEBe sure you take the time to expand the product on the right-hand side to confirm the identity.

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Example 4

Example 5

Factoring the Sum or Difference of Two Cubes

(a) Factor x3 27.

The first term is the cube of x, and the second is the cube of 3, so we can apply equa-tion (2). Letting a x and b  3, we have

 (x  3)(x2_{ 3x  9)} (b) Factor 8w3 27z3.

This is a difference of cubes, so use equation (3). 8w3_{ 27z}3_{ (2w  3z)[(2w)}2_{ (2w)(3z)  (3z)}2_]

 (2w  3z)(4w2_{ 6wz  9z}2₎ (c) Factor 5a3b 40b4_.

First note the common factor of 5b. The binomial is the difference of cubes, so use equa-tion (3).

5a3_{b 40b}4_{ 5b(a}3_{ 8b}3₎

 5b(a  2b)(a2_{ 2ab  4b}2₎

x3 27

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.

NOTEWe 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.

NOTEAgain, looking for a common factor should be your first step.

NOTERemember to write the GCF as a part of the final factored form.

C H E C K Y O U R S E L F 4

Factor completely.

(a) 27x3_{ 8y}3 _{(b) 3a}4_{ 24ab}3

Factoring a Polynomial Function

Given the function f (x) 9x2_{ 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)

426 CHAPTER6 POLYNOMIALS ANDPOLYNOMIALFUNCTIONS

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(c) Find f (1) from the factored form of f (x). f (1) 3(1)(3(1)  5)

 3(8)  24

C H E C K Y O U R S E L F 5

Given the function f (x) 16x5_{ 10x}2_{, complete the following.}

(a) Find f (1). (b) Factor f (x).

(c) Find f (1) from the factored form of f (x).

C H E C K Y O U R S E L F A N S W E R S

1. (a) ( y 6)( y  6); (b) (5m  n)(5m  n); (c) (4a2 3b)(4a2 3b) 2. 2x(x 3y)( x  3y) 3. (x2 4y2)(x 2y)( x  2y)

4. (a) (3x 2y)( 9x2 6xy  4y2); (b) 3a(a 2b)(a2_{ 2ab  4b}2₎ 5. (a) 26; (b) 2x2(8x3_{ 5); (c) 26}

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Exercises

For each of the following binomials, state whether the binomial is a difference of squares.

1. 3x2_{ 2y}2 _{2. 5x}2_{ 7y}2

3. 16a2_{ 25b}2 _{4. 9n}2_{ 16m}2

5. 16r2_{ 4 6. p}2_{ 45}

7. 16a2_{ 12b}3 _{8. 9a}2_b2_{ 16c}2_d2

9. a2_b2_{ 25 10. 4a}3_{ b}3

Factor the following binomials.

11. x2_{ 49 12. m}2_{ 64} 13. a2_{ 81 14. b}2_{ 36} 15. 9p2_{ 1 16. 4x}2_{ 9} 17. 25a2_{ 16 18. 16m}2_{ 49} 19. x2_y2_{ 25 20. m}2_n2_{ 9} 21. 4c2_{ 25d}2 _{22. 9a}2_{ 49b}2 23. 49p2_{ 64q}2 _{24. 25x}2_{ 36y}2 25. x4_{ 16y}2 _{26. a}2_{ 25b}4

6.6

© 2001 McGraw-Hill Companies 27. a3 4ab2 28. 9p2q q3 29. a4 16b4 30. 81x4 y4 31. x3 64 32. y3 8 33. m3 125 34. b3 27 35. a3b3 27 36. p3q3 64 37. 8w3_{ z}3 _{38. c}3_{ 27d}3 39. r3_{ 64s}3 _{40. 125x}3_{ y}3 41. 8x3_{ 27y}3 _{42. 64m}3_{ 27n}3 43. 8x3_{ y}6 _{44. m}6_{ 27n}3 45. 4x3_{ 32y}3 _{46. 3a}3_{ 81b}3 47. 18x3_{ 2xy}2 _{48. 50a}2_{b 2b}3 49. 12m3_{n 75mn}3 _{50. 63p}4_{ 7p}2_q2 ANSWERS 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 428

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. f (x) 12x5 21x2 52. f (x) 6x3 10x

53. f (x) 8x5_{ 20x 54. f (x) 5x}5_{ 35x}3

55. f (x) x5 3x2 56. f (x) 6x6 16x5

Factor each expression.

57. x2(x y)  y2(x y)

58. a2(b c)  16b2(b c)

59. 2m2(m 2n)  18n2(m 2n)

60. 3a3(2a b)  27ab2(2a b)

61. Find a value for k so that kx2 25 will have the factors 2x  5 and 2x  5.

62. Find a value for k so that 9m2 kn2will have the factors 3m 7n and 3m  7n.

63. Find a value for k so that 2x3 kxy2will have the factors 2x, x 3y, and x  3y.

64. Find a value for k so that 20a3b kab3will have the factors 5ab, 2a 3b, and 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 means . . . .”

67. Verify the formula for factoring the sum of two cubes by finding the product (a b)(a2_{ ab  b}2_).

68. Verify the formula for factoring the difference of two cubes by finding the product (a b)(a2 ab  b2). © 2001 McGraw-Hill Companies 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 429

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69. What are the characteristics of a monomial that is a perfect cube?

70. Suppose you factored the polynomial 4x2 16 as follows: 4x2_{ 16  (2x  4)(2x  4)}

Would this be in completely factored form? If not, what would be the final form?

Answers

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)

33. (m 5)(m2 5m  25) 35. (ab 3)(a2b2 3ab  9) 37. (2w z)(4w2 2wz  z2) 39. (r 4s)(r2 4rs  16s2)

41. (2x 3y)(4x2 6xy  9y2) 43. (2x y2)(4x2 2xy2 y4)

45. 4(x 2y)(x2 2xy  4y2) 47. 2x(3x y)(3x  y)

49. 3mn(2m 5n)(2m  5n) 51. (a)33; (b)3x2(4x3 7); (c)33 53. (a)12; (b)4x(2x4 5); (c)12 55. (a)4; (b)x2(x3 3); (c)4 57. (x y)2(x y) 59. 2(m 2n)(m  3n)(m  3n) 61. 4 63. 18 65. 67. 69. ANSWERS 69. 70. 430