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3 Factoring the Trinomial ax[Sup(2)] + bx + c with a = 1

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Chapter 5 Factoring

The only integers that have a product of 6 and a sum of 5 are 2 and 3. Now replace

5x with 2x ϩ 3x and factor by grouping:

x 2 ϩ 5x ϩ 6 ϭ x 2 ϩ 2x ϩ 3x ϩ 6

Replace 5x by 2x ϩ 3x.

ϭ x(x ϩ 2) ϩ 3(x ϩ 2)

Factor out x and 3.

ϭ (x ϩ 3)(x ϩ 2)

Factor out x ϩ 2.

Check by FOIL: (x ϩ 3)(x ϩ 2) ϭ x2 ϩ 5x ϩ 6.

b) To factor x 2 ϩ 8x ϩ 12, we need two integers that have a product of 12 and a sum

of 8. Since the product and sum are both positive, both integers are positive.

Product

Sum

12 ϭ 1 и 12

12 ϭ 2 и 6

12 ϭ 3 и 4

1 ϩ 12 ϭ 13

2ϩ6ϭ8

3ϩ4ϭ7

The only integers that have a product of 12 and a sum of 8 are 2 and 6. Now

replace 8x by 2x ϩ 6x and factor by grouping:

x 2 ϩ 8x ϩ 12 ϭ x 2 ϩ 2x ϩ 6x ϩ 12

Replace 8x by 2x ϩ 6x.

ϭ x(x ϩ 2) ϩ 6(x ϩ 2) Factor out x and 6.

ϭ (x ϩ 6)(x ϩ 2)

Factor out x ϩ 2.

Check by FOIL: (x ϩ 6)(x ϩ 2) ϭ x 2 ϩ 8x ϩ 12.

c) To factor a2 Ϫ 9a ϩ 20, we need two integers that have a product of 20 and a sum

of Ϫ9. Since the product is positive and the sum is negative, both integers must be

negative.

Product

Sum

20 ϭ (Ϫ1)(Ϫ20)

20 ϭ (Ϫ2)(Ϫ10)

20 ϭ (Ϫ4)(Ϫ5)

Ϫ1 ϩ (Ϫ20) ϭ Ϫ21

Ϫ2 ϩ (Ϫ10) ϭ Ϫ12

Ϫ4 ϩ (Ϫ5) ϭ Ϫ9

Only Ϫ4 and Ϫ5 have a product of 20 and a sum of Ϫ9. Now replace Ϫ9a by

Ϫ4a ϩ (Ϫ5a) or Ϫ4a Ϫ 5a and factor by grouping:

a2 Ϫ 9a ϩ 20 ϭ a2 Ϫ 4a Ϫ 5a ϩ 20

Replace Ϫ9a by Ϫ4a Ϫ 5a.

ϭ a(a Ϫ 4) Ϫ 5(a Ϫ 4) Factor out a and Ϫ5.

ϭ (a Ϫ 5)(a Ϫ 4)

Factor out a Ϫ 4.

Check by FOIL: (a Ϫ 5)(a Ϫ 4) ϭ a2 Ϫ 9a ϩ 20.

Now do Exercises 1–14

We usually do not write out all of the steps shown in Example 1. We saw prior to

Example 1 that

x2 ϩ (m ϩ n)x ϩ mn ϭ (x ϩ m)(x ϩ n).

So once you know m and n, you can simply write the factors, as shown in Example 2.

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E X A M P L E

5.3

2

Factoring the Trinomial ax2 ϩ bx ϩ c with a ϭ 1

341

Factoring trinomials more efficiently

Factor.

a) x 2 ϩ 5x ϩ 4

b) y2 ϩ 6y Ϫ 16

c) w2 Ϫ 5w Ϫ 24

Solution

a) To factor x2 ϩ 5x ϩ 4 we need two integers with a product of 4 and a sum of 5.

The only possibilities for a product of 4 are

(1)(4), (Ϫ1)(Ϫ4), (2)(2), and (Ϫ2)(Ϫ2).

Only 1 and 4 have a sum of 5. So,

x 2 ϩ 5x ϩ 4 ϭ (x ϩ 1)(x ϩ 4).

Check by using FOIL on (x ϩ 1)(x ϩ 4) to get x2 ϩ 5x ϩ 4.

b) To factor y 2 ϩ 6y Ϫ 16 we need two integers with a product of Ϫ16 and a sum of 6.

The only possibilities for a product of Ϫ16 are

(Ϫ1)(16), (1)(Ϫ16), (Ϫ2)(8), (2)(Ϫ8), and (Ϫ4)(4).

Only Ϫ2 and 8 have a sum of 6. So,

y 2 ϩ 6y Ϫ 16 ϭ (y ϩ 8)( y Ϫ 2).

Check by using FOIL on ( y ϩ 8)( y Ϫ 2) to get y 2 ϩ 6y Ϫ 16.

c) To factor w2 Ϫ 5w Ϫ 24 we need two integers with a product of Ϫ24 and a sum

of Ϫ5. The only possibilities for a product of Ϫ24 are

(Ϫ1)(24), (1)(Ϫ24), (Ϫ2)(12), (2)(Ϫ12), (Ϫ3)(8), (3)(Ϫ8), (Ϫ4)(6), and (4)(Ϫ6).

Only Ϫ8 and 3 have a sum of Ϫ5. So,

w2 Ϫ 5w Ϫ 24 ϭ (w Ϫ 8)(w ϩ 3).

Check by using FOIL on (w Ϫ 8)(w ϩ 3) to get w2 Ϫ 5w Ϫ 24.

Now do Exercises 15–22

Polynomials are easiest to factor when they are in the form ax 2 ϩ bx ϩ c. So if a

polynomial can be rewritten into that form, rewrite it before attempting to factor it. In

Example 3, we factor polynomials that need to be rewritten.

E X A M P L E

3

Factoring trinomials

Factor.

a) 2x Ϫ 8 ϩ x2

b) Ϫ36 ϩ t 2 Ϫ 9t

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Solution

a) Before factoring, write the trinomial as x 2 ϩ 2x Ϫ 8. Now, to get a product of Ϫ8

and a sum of 2, use Ϫ2 and 4:

2x Ϫ 8 ϩ x 2 ϭ x 2 ϩ 2x Ϫ 8

Write in ax2 ϩ bx ϩ c form.

ϭ (x ϩ 4)(x Ϫ 2) Factor and check by multiplying.

b) Before factoring, write the trinomial as t 2 Ϫ 9t Ϫ 36. Now, to get a product of Ϫ36

and a sum of Ϫ9, use Ϫ12 and 3:

Ϫ36 ϩ t 2 Ϫ 9t ϭ t 2 Ϫ 9t Ϫ 36

Write in ax2 ϩ bx ϩ c form.

ϭ (t Ϫ 12)(t ϩ 3) Factor and check by multiplying.

Now do Exercises 23–24

To factor x 2 ϩ bx ϩ c, we search through all pairs of integers that have a product

of c until we find a pair that has a sum of b. If there is no such pair of integers, then

the polynomial cannot be factored and it is a prime polynomial. Before you can conclude that a polynomial is prime, be sure that you have tried all possibilities.

E X A M P L E

4

Prime polynomials

Factor.

a) x 2 ϩ 7x Ϫ 6

b) x 2 ϩ 9

Solution

a) Because the last term is Ϫ6, we want a positive integer and a negative integer

that have a product of Ϫ6 and a sum of 7. Check all possible pairs of integers:

Product

Sum

Ϫ6 ϭ (Ϫ1)(6)

Ϫ1 ϩ 6 ϭ 5

Ϫ6 ϭ (1)(Ϫ6)

1 ϩ (Ϫ6) ϭ Ϫ5

Ϫ6 ϭ (2)(Ϫ3)

2 ϩ (Ϫ3) ϭ Ϫ1

Ϫ6 ϭ (Ϫ2)(3)

Don’t confuse a2 ϩ b2 with the difference of two squares a2 Ϫ b2 which is

not a prime polynomial:

a2 Ϫ b2 ϭ (a ϩ b)(a Ϫ b)

Ϫ2 ϩ 3 ϭ 1

None of these possible factors of Ϫ6 have a sum of 7, so we can be certain that

x2 ϩ 7x Ϫ 6 cannot be factored. It is a prime polynomial.

b) Because the x-term is missing in x2 ϩ 9, its coefficient is 0. That is, x2 ϩ 9 ϭ

x2 ϩ 0x ϩ 9. So we seek two positive integers or two negative integers that have

a product of 9 and a sum of 0. Check all possibilities:

Product

9 ϭ (1)(9)

9 ϭ (Ϫ1)(Ϫ9)

9 ϭ (3)(3)

9 ϭ (Ϫ3)(Ϫ3)

Sum

1 ϩ 9 ϭ 10

Ϫ1 ϩ (Ϫ9) ϭ Ϫ10

3ϩ3ϭ6

Ϫ3 ϩ (Ϫ3) ϭ Ϫ6

None of these pairs of integers have a sum of 0, so we can conclude that x 2 ϩ 9 is

a prime polynomial. Note that x 2 ϩ 9 does not factor as (x ϩ 3)2 because

(x ϩ 3)2 has a middle term: (x ϩ 3)2 ϭ x2 ϩ 6x ϩ 9.

Now do Exercises 25–52

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5.3

343

The prime polynomial x 2 ϩ 9 in Example 4(b) is a sum of two squares. There are

many other sums of squares that are prime. For example,

x2 ϩ 1,

a2 ϩ 4,

b2 ϩ 9, and 4y2 ϩ 25

are prime. However, not every sum of two squares is prime. For example, 4x2 ϩ 16 is

a sum of two squares that is not prime because 4x2 ϩ 16 ϭ 4(x2 ϩ 4).

Sum of Two Squares

The sum of two squares a2 ϩ b2 is prime, but not every sum of two squares is prime.

U2V Factoring with Two Variables

In Example 5, we factor polynomials that have two variables using the same technique

that we used for one variable.

E X A M P L E

5

Polynomials with two variables

Factor.

a) x 2 ϩ 2xy Ϫ 8y 2

b) a 2 Ϫ 7ab ϩ 10b2

c) 1 Ϫ 2xy Ϫ 8x2y2

Solution

a) To factor x2 ϩ 2xy Ϫ 8y2 we need two integers with a product of Ϫ8 and a sum

of 2. The only possibilities for a product of Ϫ8 are

(Ϫ1)(8), (1)(Ϫ8), (Ϫ2)(4), and (2)(Ϫ4).

Only Ϫ2 and 4 have a sum of 2. Since (Ϫ2y)(4y) ϭ Ϫ8y 2, we have

x 2 ϩ 2xy Ϫ 8y 2 ϭ (x Ϫ 2y)(x ϩ 4y).

Check by using FOIL on (x Ϫ 2y)(x ϩ 4y) to get x2 ϩ 2xy Ϫ 8y2.

b) To factor a2 Ϫ 7ab ϩ 10b2 we need two integers with a product of 10 and a sum

of Ϫ7. The only possibilities for a product of 10 are

(Ϫ1)(Ϫ10), (1)(10), (Ϫ2)(Ϫ5), and (2)(5).

Only Ϫ2 and Ϫ5 have a sum of Ϫ7. Since (Ϫ2b)(Ϫ5b) ϭ 10b2, we have

a2 Ϫ 7ab ϩ 10b2 ϭ (a Ϫ 5b)(a Ϫ 2b).

Check by using FOIL on (a Ϫ 2b)(a Ϫ 5b) to get a2 Ϫ 7ab ϩ 10b2.

c) As in part (a), we need two integers with a product of Ϫ8 and a sum of Ϫ2. The

integers are Ϫ4 and 2. Since 1 factors as 1 и 1 and Ϫ8x2y2 ϭ (Ϫ4xy)(2xy), we have

1 Ϫ 2xy Ϫ 8x2y2 ϭ (1 ϩ 2xy)(1 Ϫ 4xy).

Check by using FOIL.

Now do Exercises 53–64

U3V Factoring Completely

In Section 5.2 you learned that binomials such as 3x Ϫ 5 (with no common factor) are

prime polynomials. In Example 4 of this section we saw a trinomial that is a prime

polynomial. There are infinitely many prime trinomials. When factoring a polynomial

completely, we could have a factor that is a prime trinomial.

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E X A M P L E

6

Factoring completely

Factor each polynomial completely.

a) x3 Ϫ 6x2 Ϫ 16x

b) 4x3 ϩ 4x2 ϩ 4x

Solution

a) x3 Ϫ 6x 2 Ϫ 16x ϭ x (x 2 Ϫ 6x Ϫ 16) Factor out the GCF.

ϭ x(x Ϫ 8)(x ϩ 2)

Factor x2 Ϫ 6x Ϫ 16.

b) First factor out 4x, the greatest common factor:

4x3 ϩ 4x2 ϩ 4x ϭ 4x (x2 ϩ x ϩ 1)

To factor x2 ϩ x ϩ 1, we would need two integers with a product of 1 and a sum

of 1. Because there are no such integers, x2 ϩ x ϩ 1 is prime, and the factorization

is complete.

Now do Exercises 65–106

Warm-Ups

Fill in the blank.

5.3

1. If there are no two integers that have a

of c and

2

a

of b, then x ϩ bx ϩ c is prime.

2. We can check all factoring by

the factors.

3. The sum of two squares a2 ϩ b2 is

.

4. Always factor out the

first.

True or false?

5.

6.

7.

8.

9.

10.

11.

x2 Ϫ 6x ϩ 9 ϭ (x Ϫ 3)2

x2 ϩ 6x ϩ 9 ϭ (x ϩ 3)2

x2 ϩ 10x ϩ 9 ϭ (x Ϫ 9)(x Ϫ 1)

x2 ϩ 8x Ϫ 9 ϭ (x Ϫ 1)(x ϩ 9)

x2 Ϫ 10xy ϩ 9y2 ϭ (x Ϫ y)(x Ϫ 9y)

x2 ϩ 1 ϭ (x ϩ 1)(x ϩ 1)

x2 ϩ x ϩ1 ϭ (x ϩ 1)(x ϩ 1)

Exercises

U Study Tips V

• Put important facts on note cards. Work on memorizing the note cards when you have a few spare minutes.

• Post some note cards on your refrigerator door. Make this course a part of your life.

U1V Factoring ax2 ؉ bx ؉ c with a ‫ ؍‬1

Factor each trinomial. Write out all of the steps as shown in

Example 1. See the Strategy for Factoring x2 ϩ bx ϩ c by

Grouping on page 339.

1. x 2 ϩ 4x ϩ 3

2. y 2 ϩ 6y ϩ 5

3. x 2 ϩ 9x ϩ 18

4. w 2 ϩ 6w ϩ 8

5. a2 ϩ 7a ϩ 10

6. b2 ϩ 7b ϩ 12

7. a2 Ϫ 7a ϩ 12

8. m 2 Ϫ 9m ϩ 14

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5.3

9. b2 Ϫ 5b Ϫ 6

10. a 2 ϩ 5a Ϫ 6

11. x2 ϩ 3x Ϫ 10

12. x2 Ϫ x Ϫ 12

13. x2 ϩ 5x Ϫ 24

14. a2 Ϫ 5a Ϫ 50

Factoring the Trinomial ax2 ϩ bx ϩ c with a ϭ 1

345

49. x2 Ϫ 5x Ϫ 150

50. x2 Ϫ 25x ϩ 150

51. 13y ϩ 30 ϩ y2

52. 18z ϩ 45 ϩ z2

U2V Factoring with Two Variables

Factor each polynomial. If the polynomial is prime, say so.

See Examples 2– 4.

Factor each polynomial. See Example 5.

15. y 2 ϩ 7y ϩ 10

54. a2 ϩ 7ab ϩ 10b2

16. x 2 ϩ 8x ϩ 15

55. x2 Ϫ 4xy Ϫ 12y 2

17. a 2 Ϫ 6a ϩ 8

56. y 2 ϩ yt Ϫ 12t 2

18. b2 Ϫ 8b ϩ 15

57. x 2 Ϫ 13xy ϩ 12y2

19. m Ϫ 10m ϩ 16

58. h2 Ϫ 9hs ϩ 9s 2

20. m 2 Ϫ 17m ϩ 16

59. x 2 ϩ 4xz Ϫ 33z2

21. w 2 ϩ 9w Ϫ 10

60. x 2 Ϫ 5xs Ϫ 24s2

22. m ϩ 6m Ϫ 16

61. 1 ϩ 3ab Ϫ 28a2b2

23. w Ϫ 8 Ϫ 2w

62. 1 Ϫ xy Ϫ 20x2y2

24. Ϫ16 ϩ m 2 Ϫ 6m

63. 15a2b2 ϩ 8ab ϩ 1

25. a 2 Ϫ 2a Ϫ 12

64. 12m2n2 Ϫ 8mn ϩ 1

2

2

2

53. x2 ϩ 5ax ϩ 6a2

26. x ϩ 3x ϩ 3

2

27. 15m Ϫ 16 ϩ m2

28. 3y ϩ y Ϫ 10

2

29. a 2 Ϫ 4a ϩ 12

30. y 2 Ϫ 6y Ϫ 8

31. z 2 Ϫ 25

32. p2 Ϫ 1

33. h2 ϩ 49

34. q2 ϩ 4

U3V Factoring Completely

Factor each polynomial completely. Use the methods discussed

in Sections 5.1 through 5.3. If the polynomial is prime say so.

See Example 6.

65. 5x3 ϩ 5x

66. b3 ϩ 49b

67. w2 Ϫ 8w

68. x4 Ϫ x3

35. m2 ϩ 12m ϩ 20

69. 2w 2 Ϫ 162

36. m2 ϩ 21m ϩ 20

70. 6w4 Ϫ 54w2

37. t2 Ϫ 3t ϩ 10

71. Ϫ2b2 Ϫ 98

38. x2 Ϫ 5x Ϫ 3

72. Ϫa3 Ϫ 100a

39. m2 Ϫ 18 Ϫ 17m

73. x3 Ϫ 2x2 Ϫ 9x ϩ 18

40. h2 Ϫ 36 ϩ 5h

74. x3 ϩ 7x2 Ϫ x Ϫ 7

41. m2 Ϫ 23m ϩ 24

75. 4r2 ϩ 9

42. m2 ϩ 23m ϩ 24

76. t2 ϩ 4z2

43. 5t Ϫ 24 ϩ t 2

77. x 2w 2 ϩ 9x2

44. t2 Ϫ 24 Ϫ 10t

78. a4b ϩ a2b3

45. t2 Ϫ 2t Ϫ 24

79. w2 Ϫ 18w ϩ 81

46. t2 ϩ 14t ϩ 24

80. w2 ϩ 30w ϩ 81

47. t2 Ϫ 10t Ϫ 200

81. 6w2 Ϫ 12w Ϫ 18

48. t2 ϩ 30t ϩ 200

82. 9w Ϫ w3

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Chapter 5 Factoring

83. 3y2 ϩ 75

108. Area of a sail. The area in square meters for a triangular

sail is given by A(x) ϭ x2 ϩ 5x ϩ 6.

84. 5x2 ϩ 500

a) Find A(5).

b) If the height of the sail is x ϩ 3 meters, then what is

the length of the base of the sail?

85. ax ϩ ay ϩ cx ϩ cy

86. y3 ϩ y2 Ϫ 4y Ϫ 4

87. Ϫ2x2 Ϫ 10x Ϫ 12

88. Ϫa3 Ϫ 2a2 Ϫ a

89. 32x2 Ϫ 2x4

90. 20w 2 ϩ 100w ϩ 40

91. 3w2 ϩ 27w ϩ 54

xϩ3m

92. w3 Ϫ 3w2 Ϫ 18w

93. 18w2 ϩ w3 ϩ 36w

94. 18a2 ϩ 3a3 ϩ 36a

Base

Area ϭ x 2 ϩ 5x ϩ 6 m 2

95. 9y2 ϩ 1 ϩ 6y

96. 2a2 ϩ 1 ϩ 3a

Figure for Exercise 108

97. 8vw2 ϩ 32vw ϩ 32v

98. 3h2t ϩ 6ht ϩ 3t

109. Volume of a cube. Hector designed a cubic box with

volume x 3 cubic feet. After increasing the dimensions

of the bottom, the box has a volume of x 3 ϩ 8x 2 ϩ 15x

cubic feet. If each of the dimensions of the bottom

was increased by a whole number of feet, then how

much was each increase?

99. 6x 3y ϩ 30x 2 y 2 ϩ 36xy3

100. 3x 3y 2 Ϫ 3x 2y 2 ϩ 3xy 2

101. 5 ϩ 8w ϩ 3w2

102. Ϫ3 ϩ 2y ϩ 21y2

103. Ϫ3y3 ϩ 6y2 Ϫ 3y

104. Ϫ4w3 Ϫ 16w2 ϩ 20w

105. a3 ϩ ab ϩ 3b ϩ 3a2

106. ac ϩ xc ϩ aw2 ϩ xw2

Applications

Use factoring to solve each problem.

107. Area of a deck. The area in square feet for a rectangular

deck is given by A(x) ϭ x 2 ϩ 6x ϩ 8.

a) Find A(6).

b) If the width of the deck is x ϩ 2 feet, then what is the

length?

110. Volume of a container. A cubic shipping container

had a volume of a3 cubic meters. The height was

decreased by a whole number of meters and the

width was increased by a whole number of meters so

that the volume of the container is now a3 ϩ 2a2 Ϫ 3a

cubic meters. By how many meters were the height

and width changed?

Getting More Involved

111. Discussion

Which of the following products is not equivalent

a) (2x Ϫ 4)(x ϩ 3)

c) 2(x Ϫ 2)(x ϩ 3)

b) (x Ϫ 2)(2x ϩ 6)

d) (2x Ϫ 4)(2x ϩ 6)

112. Discussion

L

Area ϭ x 2 ϩ 6x ϩ 8 ft 2

Figure for Exercise 107

x ϩ 2 ft

When asked to factor completely a certain polynomial,

four students gave the following answers. Only one

student gave the correct answer. Which one must it be?

a) 3(x 2 Ϫ 2x Ϫ 15)

c) 3(x Ϫ 5)(x Ϫ 3)

b) (3x Ϫ 5)(5x Ϫ 15)

d) (3x Ϫ 15)(x Ϫ 3)

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5.4

Mid-Chapter Quiz

Sections 5.1 through 5.3

Find the greatest common factor for each group of integers.

Factor each expression by factoring out the greatest common

factor.

6. 12x Ϫ 30x

3

2

2 2

13. 10x3 Ϫ 250x

14. Ϫ6x2 Ϫ 36x Ϫ 54

15. aw Ϫ 3w ϩ 6a Ϫ 18

7. 15ab Ϫ 25a b ϩ 35a b

3

11. 4h2 ϩ 12h ϩ 9

12. w2 Ϫ 16w ϩ 64

4. 60, 144, 240

5. 8w Ϫ 6y

Chapter 5

10. 4y2 Ϫ 9w2

2. 140

3. 36, 45

347

1

Factor completely.

Find the prime factorization of each integer.

1. 48

Factoring the Trinomial ax2 ϩ bx ϩ c with a

3

16. bx Ϫ 5b Ϫ 6x ϩ 30

Factor each expression.

17. ax2 Ϫ a ϩ x2 Ϫ 1

8. (x ϩ 3)x Ϫ (x ϩ 3)5

18. x3 Ϫ 5x Ϫ 4x2

9. m(m Ϫ 9) Ϫ 6(m Ϫ 9)

19. 2x3 ϩ 18x

20. a2 Ϫ 12as ϩ 32s2

5.4

In This Section

U1V The ac Method

U2V Trial and Error

U3V Factoring Completely

Factoring the Trinomial ax2 ؉ bx ؉ c with a

1

In Section 5.3, we used grouping to factor trinomials with a leading coefficient of 1.

In this section we will also use grouping to factor trinomials with a leading

coefficient that is not equal to 1.

U1V The ac Method

The first step in factoring ax2 ϩ bx ϩ c with a ϭ 1 is to find two numbers with a product of c and a sum of b. If a 1, then the first step is to find two numbers with a

product of ac and a sum of b. This method is called the ac method. The strategy for

factoring by the ac method follows. Note that this strategy works whether or not the

Strategy for Factoring ax 2 ؉ bx ؉ c by the ac Method

To factor the trinomial ax2 ϩ bx ϩ c:

1. Find two numbers that have a product equal to ac and a sum equal to b.

2. Replace bx by the sum of two terms whose coefficients are the two numbers

found in (1).

3. Factor the resulting four-term polynomial by grouping.

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E X A M P L E

1

The ac method

Factor each trinomial.

a) 2x2 ϩ 7x ϩ 6

b) 2x2 ϩ x Ϫ 6

c) 10x2 ϩ 13x Ϫ 3

Solution

a) In 2x2 ϩ 7x ϩ 6 we have a ϭ 2, b ϭ 7, and c ϭ 6. So,

ac ϭ 2 и 6 ϭ 12.

Now we need two integers with a product of 12 and a sum of 7. The pairs of

integers with a product of 12 are 1 and 12, 2 and 6, and 3 and 4. Only 3 and 4 have

a sum of 7. Replace 7x by 3x ϩ 4x and factor by grouping:

2x2 ϩ 7x ϩ 6 ϭ 2x2 ϩ 3x ϩ 4x ϩ 6

Replace 7x by 3x ϩ 4x.

ϭ (2x ϩ 3)x ϩ (2x ϩ 3)2 Factor out the common factors.

ϭ (2x ϩ 3)(x ϩ 2)

Factor out 2x ϩ 3.

Check by FOIL.

b) In 2x2 ϩ x Ϫ 6 we have a ϭ 2, b ϭ 1, and c ϭ Ϫ6. So,

ac ϭ 2(Ϫ6) ϭ Ϫ12.

Now we need two integers with a product of Ϫ12 and a sum of 1. We can list the

possible pairs of integers with a product of Ϫ12 as follows:

1 and Ϫ12

Ϫ1 and 12

2 and Ϫ6

3 and Ϫ4

Ϫ2 and 6

Ϫ3 and 4

Only Ϫ3 and 4 have a sum of 1. Replace x by Ϫ3x ϩ 4x and factor by grouping:

2x2 ϩ x Ϫ 6 ϭ 2x2 Ϫ 3x ϩ 4x Ϫ 6

Replace x by Ϫ3x ϩ 4x.

ϭ (2x Ϫ 3)x ϩ (2x Ϫ 3)2 Factor out the common factors.

ϭ (2x Ϫ 3)(x ϩ 2)

Factor out 2x Ϫ 3.

Check by FOIL.

c) Because ac ϭ 10(Ϫ3) ϭ Ϫ30, we need two integers with a product of Ϫ30 and a

sum of 13. The product is negative, so the integers must have opposite signs. We

can list all pairs of factors of Ϫ30 as follows:

1 and Ϫ30

Ϫ1 and 30

2 and Ϫ15

Ϫ2 and 15

3 and Ϫ10

Ϫ3 and 10

5 and Ϫ6

Ϫ5 and 6

The only pair that has a sum of 13 is Ϫ2 and 15:

10x2 ϩ 13x Ϫ 3 ϭ 10x2 Ϫ 2x ϩ 15x Ϫ 3

Replace 13x by Ϫ2x ϩ 15x.

ϭ (5x Ϫ 1)2x ϩ (5x Ϫ 1)3 Factor out the common factors.

ϭ (5x Ϫ 1)(2x ϩ 3)

Factor out 5x Ϫ 1.

Check by FOIL.

Now do Exercises 1–38

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3 Factoring the Trinomial ax[Sup(2)] + bx + c with a = 1

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