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10.8
Factoring Cubic Polynomials
What are the dimensions of a terrarium?
Goal
Factor cubic polynomials.
In Example 6 you will factor a
cubic polynomial to determine the
dimensions of a terrarium, which
is an enclosed space for keeping
small animals indoors.
Key Words
• prime polynomial
• factor a polynomial
completely
You have already been using the distributive property to factor out constants that
are common to the terms of a polynomial.
9x2 15 3(3x2 5)
Factor out common factor.
You can also use the distributive property to factor out variable factors that are
common to the terms of a polynomial. When factoring a cubic polynomial, you
should factor out the greatest common factor (GCF) first and then look for
other patterns.
EXAMPLE
1
Find the Greatest Common Factor
Factor the greatest common factor out of 14x 3 21x 2.
Student Help
SKILLS REVIEW
For help with finding
the GCF, see p. 761.
Solution
First find the greatest common factor of 14x3 and 21x2.
14x 3 2 p 7 p x p x p x
21x 2 3 p 7 p x p x
GCF 7 p x p x 7x2
Then use the distributive property to factor out the greatest common factor
from each term.
ANSWER 14x 3 21x 2 7x2(2x 3).
Find the Greatest Common Factor
Factor out the greatest common factor.
616
Chapter 10
1. 11x 22
2. 6x2 12x 18
3. 8x3 16x
4. 3n3 36n2 12n
5. 4y3 10y2
6. 9x3 6x2 18x
Polynomials and Factoring
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A polynomial is prime if it cannot be factored using integer
coefficients. To factor a polynomial completely, write it as the product of
monomial and prime factors.
PRIME FACTORS
EXAMPLE
2
Factor Completely
Factor 4x3 20x2 24x completely.
Solution
4x3 20x2 24x 4x(x2 5x 6)
Factor out GCF.
4x(x 2)(x 3)
Monomial factor
Factor trinomial.
Prime factors
Factor Completely
Factor the expression completely.
7. 2n3 4n2 2n
10. x3 4x2 4x
8. 3x3 12x
9. 5m3 45m
11. 2x3 10x2 8x
12. 6p3 21p2 9p
FACTORING BY GROUPING Another use of the distributive property is in
factoring polynomials that have four terms. Sometimes you can factor the
polynomial by grouping the terms into two groups and factoring the greatest
common factor out of each term.
EXAMPLE
3
Factor by Grouping
Factor x3 2x2 9x 18 completely.
Solution
x3 2x2 9x 18 (x3 2x2) (9x 18)
Group terms.
x2(x 2) (9)(x 2)
Factor each group.
(x 2)(x 9)
Use distributive
property.
(x 2)(x 3)(x 3)
Factor difference of
two squares.
2
Factor by Grouping
Use grouping to factor the expression completely.
13. 2x3 8x2 3x 12 14. x3 5x2 4x 20
10.8
15. x3 4x2 9x 36
Factoring Cubic Polynomials
617
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Student Help
MORE EXAMPLES
NE
ER T
INT
More examples
are available at
www.mcdougallittell.com
SUM OR DIFFERENCE OF TWO CUBES In Lessons 10.3 and 10.7, you used the
difference property to study the special product pattern of the difference of two
squares. You can also use the distributive property to confirm the following
special product patterns for the sum or difference of two cubes.
FACTORING MORE SPECIAL PRODUCTS
Sum of Two Cubes Pattern
a3 b3 (a b)(a2 ab b2) Example: (x3 1) (x 1)(x 2 x + 1)
Difference of Two Cubes Pattern
a3 b3 (a b)(a2 ab b2) Example: (x3 8) (x 2)(x 2 2x 4)
4
EXAMPLE
Factor the Sum of Two Cubes
Factor x3 27.
Solution
x3 27 x3 33
(x 3)(x2 3x 9)
Write as sum of cubes.
Use special product pattern. Notice that
x 2 3x 9 is prime and does not factor.
Factor the Sum of Two Cubes
Factor the expression.
16. x3 125
17. n3 8
5
EXAMPLE
18. 2m3 2
19. 4x3 32
Factor the Difference of Two Cubes
Factor n3 64.
Solution
n3 64 n3 43
(n 4)(n2 4n 16)
Write as difference of cubes.
Use special product pattern. Notice that
n2 4n 16 is prime and does not factor.
Factor the Difference of Two Cubes
Factor the expression.
20. x3 27
618
Chapter 10
Polynomials and Factoring
21. p3 216
22. 2n3 250
23. 4z3 32
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6
EXAMPLE
Science
Write and Use a Polynomial Model
A terrarium
has a volume of 12 cubic feet. Find the
dimensions of the terrarium. Do the
dimensions meet the space requirements
of an adult bearded dragon lizard?
SPACE REQUIREMENTS
x ft
(x 4) ft
(x 1) ft
Solution
V height p width p length
Write volume model for a prism.
12 x(x 1)(x 4)
Substitute for height, width and length.
12 x3 3x2 4x
Multiply.
SPACE REQUIREMENTS
0 (x 3x ) (4x 12)
Write in standard form and group terms.
Generally, an adult bearded
dragon lizard will need a
terrarium or cage that is at
least 4 to 6 feet in length,
2 to 3.5 feet in height and
2 to 3.5 feet in depth.
0 x2(x 3) (4)(x 3)
Factor each group of terms.
0 (x 3)(x 4)
Use distributive property.
0 (x 3)(x 2)(x 2)
Factor difference of two squares.
3
2
2
By setting each factor equal to zero, you can see that the solutions are 3, 2,
and 2. The only positive solution is x 2.
ANSWER The dimensions of the terrarium are 2 feet by 1 foot by 6 feet.
Because the height must be between 2 and 3.5 feet, the dimensions
do not meet the space requirements of an adult bearded dragon lizard.
SUMMARY
Patterns Used to Solve Polynomial Equations
Can be used to solve any equation, but gives only
approximate solutions. Examples 2 and 3, pp. 527–528
GRAPHING:
THE QUADRATIC FORMULA: Can be used to solve any quadratic
equation. Examples 1–3, pp. 533–534
Can be used with the zero-product property to solve
an equation that is in standard form and whose polynomial is
factorable.
FACTORING:
• Factoring x 2 bx c: Examples 1–7, pp. 595–598
• Factoring ax 2 bx c: Examples 1–5, pp. 603–605
• Special Products: Examples 1–6, pp. 610–612 and Examples 4
and 5, p. 618
a b (a b)(a b)
2
2
a2 2ab b2 (a b)2
a2 2ab b2 (a b)2
a3 b3 (a b)(a2 ab b2)
a3 b3 (a b)(a2 ab b2)
• Factoring Completely: Examples 1–3, pp. 616–617
10.8
Factoring Cubic Polynomials
619
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10.8 Exercises
Guided Practice
Vocabulary Check
Skill Check
1. What does it mean to say that a polynomial is prime?
ERROR ANALYSIS Find and correct the error.
2.
3.
4x3 + 36x
–2b3 + 12b2 – 14b
= 4x(x2 + 9)
= –2b(b2 + 6b – 7)
= –4x(x + 3)(x – 3)
= –2b(b + 7)(b – 1)
Find the greatest common factor of the terms and factor it out of
the expression.
4. 5n3 20n
5. 6x2 3x4
6. 6y4 14y3 10y2
Factor the expression.
7. x3 1
8. x3 64
9. 27x3 1
10. 125x3 1
Factor the expression completely.
11. 2b3 18b
12. 7a3 14a2 21a
13. 3t3 18t2 27t
14. y3 6y2 5y
15. x3 16x
16. 5b3 25b2 70b
Practice and Applications
FACTORING THE GCF Find the greatest common factor of the terms and
factor it out of the expression.
17. 6v3 18v
18. 4q4 12q
19. 3x 9x2
20. 10x2 15x3
21. 4a2 8a5
22. 24t5 6t3
23. 15x3 5x2 10x
24. 4a5 8a3 2a2
25. 18d6 6d2 3d
FACTOR BY GROUPING Factor the expression.
Student Help
HOMEWORK HELP
Example 1: Exs. 17–25
Example 2: Exs. 36–44
Example 3: Exs. 26–31
Example 4: Exs. 32–35
Example 5: Exs. 32–35
Example 6: Exs. 59–61
620
Chapter 10
26. x2 2x xy 2y
27. a2 3a ab 3b
28. 2x3 3x2 4x 6
29. 10x2 15x 2x 3
30. 8x2 3x 8x 3
31. 10x2 7x 10x 7
SUM AND DIFFERENCE OF TWO CUBES Factor the expression.
32. m3 1
Polynomials and Factoring
33. c3 8
34. r3 64
35. m3 125
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FACTORING COMPLETELY Factor the expression completely.
36. 24x 3 18x 2
37. 2y 3 10y 2 12y
38. 5s3 30s2 40s
39. 4t 3 144t
40. 12z3 3z2
41. c4 c3 12c 12
42. x 3 3x 2 x 3
43. 3x 3 3000
44. 2x 3 6750
SOLVING EQUATIONS Solve the equation. Tell which method you used.
Student Help
45. y 2 7y 12 0
46. x 2 3x 4 0
47. 27 6w w 2 0
48. 5x 4 80x 2 0
49. 16x 3 4x 0
50. 10x 3 290x 2 620x 0
FINDING ROOTS OF POLYNOMIALS Use the quadratic formula or
LOOK BACK
factoring to find the roots of the polynomial. Write your solutions in
simplest form.
For help with finding
roots, see p. 534.
51. 4x2 9x 9
52. 5x2 2x 3
53. 2x2 5x 1
54. 3x2 4x 1
55. 6x2 2x 7
56. 3x2 8x 2
In Exercises 57 and 58, use the vertical motion models,
where h is the height (in feet), v is the initial upward velocity
(in feet per second), s is the initial height (in feet), and t is the time
(in seconds) the object spends aloft.
Vertical motion model for Earth:
h 16t 2 vt s
Vertical motion model for the moon:
16
6
h t 2 vt s
Note: the two equations are different because the acceleration due to gravity on
the moon’s surface is about one-sixth that of Earth.
57. EARTH On Earth, you toss a tennis ball from a height of 96 feet with an
Careers
initial upward velocity of 16 feet per second. How long will it take the tennis
ball to reach the ground?
58. MOON On the moon, you toss a tennis ball from a height of 96 feet with an
initial upward velocity of 16 feet per second. How long will it take the tennis
ball to reach the surface of the moon?
PACKAGING In Exercises 59–61, use the following information. Refer to
the diagram of the box.
The length l of a box is 3 inches less than the
height h. The width w is 9 inches less than the
height. The box has a volume of 324 cubic inches.
PACKAGE DESIGNERS
consider the function of a
package to determine the
appropriate size, shape,
weight, color and materials
to use.
h
59. Copy and complete the diagram by labeling
the dimensions.
60. Write a model that you can solve to find
the length, height, and width of the box.
l
w
61. What are the dimensions of the box?
10.8
Factoring Cubic Polynomials
621
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Standardized Test
Practice
62. MULTIPLE CHOICE Which of the following is the complete factorization of
x3 5x2 4x 20?
A
C
B
D
(x 2)(x 2)(x 5)
(x2 4)(x 5)
(x 2)(x 2)(x 5)
(x 4)(x 1)(x 20)
63. MULTIPLE CHOICE Solve x3 4x 0.
F
Mixed Review
G
0 and 2
H
0, 2, and 2
2 and 2
J
2 and 0
SOLVING INEQUALITIES Solve the inequality. (Lesson 6.3)
64. 7 x ≤ 9
65. 3 2x 5
66. x 6 ≤ 12
SOLVING ABSOLUTE-VALUE EQUATIONS Solve the equation. (Lesson 6.6)
67. x 3
68. x 5 7
69. x 6 13 70. 4x 3 9
GRAPHING INEQUALITIES Graph the inequality. (Lesson 6.8)
72. y 3x ≥ 2
71. x y 9
Maintaining Skills
73. y 4x ≤ 10
RECIPROCALS Find the reciprocal. (Skills Review p. 763)
74. 18
75. 7
2
76. 9
3
77. 1
4
5
78. 6
5
79. 2
8
7
80. 9
10
3
81. 8
4
Quiz 3
Factor the expression. Tell which special product factoring pattern you
used. (Lesson 10.7)
1. 49x2 64
2. 121 9x2
3. 4t2 20t 25
4. 72 50y2
5. 9y2 42y 49
6. 3n2 36n 108
Solve the equation by factoring. (Lesson 10.7)
7. x2 8x 16 0
8. 4x2 32x 64 0
9. x3 9x2 36x 0
Find the greatest common factor and factor it out of the expression.
(Lesson 10.8)
10. 3x3 12x2
11. 6x2 3x
12. 18x4 9x3
13. 8x5 4x2 2x
Factor the expression completely. (Lesson 10.8)
14. 2x3 6x2 4x
15. x3 3x2 4x 12
16. 4x3 500
Solve the equation by factoring. (Lesson 10.8)
17. 108y3 75y 0
622
Chapter 10
Polynomials and Factoring
18. 3x3 6x2 5x 10