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Section 7.8 Factoring Polynomials Completely 403
Essential QuestionEssential Question How can you factor a polynomial completely?
Writing a Product of Linear Factors
Work with a partner. Write the product represented by the algebra tiles. Then multiply to write the polynomial in standard form. a.
b.
c.
d.
e.
f.
Matching Standard and Factored Forms
Work with a partner. Match the standard form of the polynomial with the equivalent factored form. Explain your strategy.
a. x^3 + x^2 A. x ( x + 1)( x − 1)
b. x^3 − x B. x ( x − 1)^2
c. x^3 + x^2 − 2 x C. x ( x + 1)^2
d. x^3 − 4 x^2 + 4 x D. x ( x + 2)( x − 1)
e. x^3 − 2 x^2 − 3 x E. x ( x − 1)( x − 2)
f. x^3 − 2 x^2 + x F. x ( x + 2)( x − 2)
g. x^3 − 4 x G. x ( x − 2)^2
h. x^3 + 2 x^2 H. x ( x + 2)^2
i. x^3 − x^2 I. x^2 ( x − 1)
j. x^3 − 3 x^2 + 2 x J. x^2 ( x + 1)
k. x^3 + 2 x^2 − 3 x K. x^2 ( x − 2)
l. x^3 − 4 x^2 + 3 x L. x^2 ( x + 2)
m. x^3 − 2 x^2 M. x ( x + 3)( x − 1)
n. x^3 + 4 x^2 + 4 x N. x ( x + 1)( x − 3)
o. x^3 + 2 x^2 + x O. x ( x − 1)( x − 3)
Communicate Your AnswerCommunicate Your Answer
3. How can you factor a polynomial completely? 4. Use your answer to Question 3 to factor each polynomial completely. a. x^3 + 4 x^2 + 3 x b. x^3 − 6 x^2 + 9 x c. x^3 + 6 x^2 + 9 x
REASONING
ABSTRACTLY
To be proficient in math, you need to know and fl exibly use different properties of operations and objects.
Factoring Polynomials Completely
404 Chapter 7 Polynomial Equations and Factoring
7.8 Lesson^ What You Will LearnWhat You Will Learn
Factor polynomials by grouping. Factor polynomials completely. Use factoring to solve real-life problems.
Factoring Polynomials by Grouping You have used the Distributive Property to factor out a greatest common monomial from a polynomial. Sometimes, you can factor out a common binomial. You may be able to use the Distributive Property to factor polynomials with four terms, as described below.
Factoring by Grouping
Factor each polynomial by grouping. a. x^3 + 3 x^2 + 2 x + 6 b. x^2 + y + x + xy
SOLUTION
a. x^3 + 3 x^2 + 2 x + 6 = ( x^3 + 3 x^2 ) + (2 x + 6) Group terms with common factors. = x^2 ( x + 3) + 2( x + 3) Factor out GCF of each pair of terms. = ( x + 3)( x^2 + 2) Factor out ( x + 3).
So, x^3 + 3 x^2 + 2 x + 6 = ( x + 3)( x^2 + 2).
b. x^2 + y + x + xy = x^2 + x + xy + y Rewrite polynomial. = ( x^2 + x ) + ( xy + y ) Group terms with common factors. = x ( x + 1) + y ( x + 1) Factor out GCF of each pair of terms. = ( x + 1)( x + y ) Factor out ( x + 1).
So, x^2 + y + x + xy = ( x + 1)( x + y ).
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Factor the polynomial by grouping.
1. a^3 + 3 a^2 + a + 3 2. y^2 + 2 x + yx + 2 y
Factoring Polynomials Completely You have seen that the polynomial x^2 − 1 can be factored as ( x + 1)( x − 1). This polynomial is factorable. Notice that the polynomial x^2 + 1 cannot be written as the product of polynomials with integer coefficients. This polynomial is unfactorable. A factorable polynomial with integer coefficients is factored completely when it is written as a product of unfactorable polynomials with integer coefficients.
Common binomial factor is x + 3.
Common binomial factor is x + 1.
factoring by grouping, p. 404 factored completely, p. 404 Previous polynomial binomial
Core VocabularyCore Vocabullarry
CoreCore ConceptConcept
Factoring by Grouping
To factor a polynomial with four terms, group the terms into pairs. Factor the GCF out of each pair of terms. Look for and factor out the common binomial factor. This process is called factoring by grouping.
406 Chapter 7 Polynomial Equations and Factoring
Solving Real-Life Problems
Modeling with Mathematics
A terrarium in the shape of a rectangular prism has a volume of 4608 cubic inches. Its length is more than 10 inches. The dimensions of the terrarium in terms of its width are shown. Find the length, width, and height of the terrarium.
SOLUTION
1. Understand the Problem You are given the volume of a terrarium in the shape of a rectangular prism and a description of the length. The dimensions are written in terms of its width. You are asked to find the length, width, and height of the terrarium. 2. Make a Plan Use the formula for the volume of a rectangular prism to write and solve an equation for the width of the terrarium. Then substitute that value in the expressions for the length and height of the terrarium. 3. Solve the Problem Volume = length (^) ⋅ width (^) ⋅ height Volume of a rectangular prism 4608 = (36 − w )( w )( w + 4) Write equation. 4608 = 32 w^2 + 144 w − w^3 Multiply. 0 = 32 w^2 + 144 w − w^3 − 4608 Subtract 4608 from each side. 0 = (− w^3 + 32 w^2 ) + (144 w − 4608) Group terms with common factors. 0 = − w^2 ( w − 32) + 144( w − 32) Factor out GCF of each pair of terms. 0 = ( w − 32)(− w^2 + 144) Factor out ( w − 32). 0 = −1( w − 32)( w^2 − 144) Factor −1 from − w^2 + 144. 0 = −1( w − 32)( w − 12)( w + 12) Difference of two squares pattern w − 32 = 0 or w − 12 = 0 or w + 12 = 0 Zero-Product Property w = 32 or w = 12 or w = − 12 Solve forw. Disregard w = −12 because a negative width does not make sense. You know that the length is more than 10 inches. Test the solutions of the equation, 12 and 32, in the expression for the length.
length = 36 − w = 36 − 12 = 24 ✓ or length = 36 − w = 36 − 32 = 4 ✗
The solution 12 gives a length of 24 inches, so 12 is the correct value of w. Use w = 12 to fi nd the height, as shown. height = w + 4 = 12 + 4 = 16
The width is 12 inches, the length is 24 inches, and the height is 16 inches.
4. Look Back Check your solution. Substitute the values for the length, width, and height when the width is 12 inches into the formula for volume. The volume of the terrarium should be 4608 cubic inches.
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9. A box in the shape of a rectangular prism has a volume of 72 cubic feet. The box has a length of x feet, a width of ( x − 1) feet, and a height of ( x + 9) feet. Find the dimensions of the box.
Check V = wh
4608 =
4608 = 4608 ✓
(36 − w ) in. w^ in.
( w + 4) in.
Section 7.8 Factoring Polynomials Completely 407
7.8 Exercises Dynamic Solutions available at BigIdeasMath.com
In Exercises 3–10, factor the polynomial by grouping. (See Example 1.)
3. x^3 + x^2 + 2 x + 2 4. y^3 − 9 y^2 + y − 9 5. 3 z^3 + 2 z − 12 z^2 − 8 6. 2 s^3 − 27 − 18 s + 3 s^2 7. x^2 + xy + 8 x + 8 y 8. q^2 + q + 5 pq + 5 p 9. m^2 − 3 m + mn − 3 n 10. 2 a^2 + 8 ab − 3 a − 12 b
In Exercises 11–22, factor the polynomial completely. (See Example 2.)
11. 2 x^3 − 2 x 12. 36 a^4 − 4 a^2 13. 2 c^2 − 7 c + 19 14. m^2 − 5 m − 35 15. 6 g^3 − 24 g^2 + 24 g 16. − 15 d^3 + 21 d^2 − 6 d 17. 3 r^5 + 3 r^4 − 90 r^3 18. 5 w^4 − 40 w^3 + 80 w^2 19. − 4 c^4 + 8 c^3 − 28 c^2 20. 8 t^2 + 8 t − 72 21. b^3 − 5 b^2 − 4 b + 20 22. h^3 + 4 h^2 − 25 h − 100
In Exercises 23–28, solve the equation. (See Example 3.)
23. 5 n^3 − 30 n^2 + 40 n = 0 24. k^4 − 100 k^2 = 0 25. x^3 + x^2 = 4 x + 4 26. 2 t^5 + 2 t^4 − 144 t^3 = 0 27. 12 s − 3 s^3 = 0 28. 4 y^3 − 7 y^2 + 28 = 16 y
In Exercises 29–32, find the x -coordinates of the points where the graph crosses the x -axis.
29.
x
y
−− 150150150
y = x^3 − 81 x
− 6 6
ERROR ANALYSIS In Exercises 33 and 34, describe and correct the error in factoring the polynomial completely.
33. a^3 + 8 a^2 − 6 a − 48 = a^2 (a + 8) + 6(a + 8) = (a + 8)(a^2 + 6)
x^3 − 6 x^2 − 9 x + 54 = x^2 (x − 6) − 9(x − 6) = (x − 6)(x^2 − 9)
35. MODELING WITH MATHEMATICS
You are building a birdhouse in the shape of a rectangular prism that has a volume of 128 cubic inches. The dimensions of the birdhouse in terms of its width are shown. (See Example 4.) a. Write a polynomial that represents the volume of the birdhouse. b. What are the dimensions of the birdhouse?
Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with Mathematics
1. VOCABULARY What does it mean for a polynomial to be factored completely? 2. WRITING Explain how to choose which terms to group together when factoring by grouping.
Vocabulary and Core Concept CheckVocabulary and Core Concept Check
x
y
− 1
y = − 2 x^4 + 16 x^3 − 32 x^2
5 1 3 5
x
y
− 4 4 − 140
420
− 140140
y = 4 x^3 + 25 x^2 − 56 x
x
y
− 4 − 2 1
y = − 3 x^4 − 24 x^3 − 45 x^2
30
− 45
ww in.in.
( w + 4) in.
4 in.
