Evaluate: Homework and Practice
EVALUATE
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Factor the polynomial, or identify it as irreducible.
1.
x 3 + x 2 - 12x
2.
x(x + x - 12)
2
x3 + 5
Irreducible.
x(x + 4)(x - 3)
3.
x 3 - 125
4.
x = (1x)
3
125 = (5)
(
8x 3 + 125
125 = (5)
216x 3 + 64
8.
8(27x 3 + 8)
27x = (3x)
3
3
8 = (2)
2x 3 + 6x
Example 1
Factoring Out the Greatest Common
Monomial Factor First
Exercises 1–2, 4,
6, 10–12
8x 3 - 64
Example 2
Recognizing Special Factoring
Patterns
Exercises 3, 5,
7–9
8(x - 2)(x 2 - 2x + 4)
Example 3
Factoring by Grouping
Exercises 13–18
Example 4
Solving a Real-World Problem by
Factoring a Polynomial
Exercises 19–22
2x(x 2 + 3)
8x 3 + 125 = (2x + 5)(4x 2 - 10x + 25)
8(x 3 - 8)
3
a 3 + b 3 = (a + b)(a 2 - ab + b 2)
216x 3 + 64 = 8(3x + 2)(9x 2 - 6x + 4)
10x 3 - 80
10. 2x 4 + 7x 3 + 5x 2
10(x - 8)
x 2(2x 2 + 7x + 5)
3
x 3 = (1x)
3
8 = (2)
3
2
2
10x - 80 = 10(x - 2)(x 2 + 2x + 4)
11. x + 10x + 16x
x (x + 1)(2x + 5)
2
12. x + 9769
2
x(x + 10x + 16)
3
2
(
)
x (x(2x + 5) + 1(2x + 5))
)
2
3
3
(
x 2 (2x 2 + 5x) + (2x + 5)
3
a - b = (a - b)(a + ab + b
3
© Houghton Mifflin Harcourt Publishing Company
9.
Irreducible.
)
x (x + 2x) + (8x + 16)
2
x(x(x + 2) + 8(x + 2))
Practice
Explore
Analyzing a Visual Model for
Polynomial Factorization
2x 3 + 6x
3
a 3 + b 3 = (a + b)(a 2 - ab + b 2)
7.
Concepts and Skills
x(x + 2)(x + 3)
6.
3
)
x(x(x + 2) + 3(x + 2))
x 3 - 125 = (x - 5)(x 2 + 5x + 25)
8x 3 = (2x)
ASSIGNMENT GUIDE
x (x 2 + 2x) + (3x + 6)
3
a 3 - b 3 = (a - b)(a 2 + ab + b 2)
5.
x 3 + 5x 2 + 6x
x(x 2 + 5x + 6)
3
INTEGRATE TECHNOLOGY
Emphasize that students should use caution
when checking answers on a graphing
calculator. The calculator provides support that the
answer is correct, but it cannot be used to prove
correctness.
x(x + 2)(x + 8)
Module 6
Lesson 4
316
Exercise
A2_MNLESE385894_U3M06L4.indd 316
Depth of Knowledge (D.O.K.)
Mathematical Practices
1–18
1 Recall of Information
MP.5 Using Tools
19–22
2 Skills/Concepts
MP.4 Modeling
23
1 Recall of Information
MP.2 Reasoning
24
1 Recall of Information
MP.3 Logic
25–26
2 Skills/Concepts
MP.3 Logic
27–29
3 Strategic Thinking
MP.2 Reasoning
3/19/14 1:37 PM
Factoring Polynomials 316
Factor the polynomial by grouping.
AVOID COMMON ERRORS
13. x 3 + 8x 2 + 6x + 48
x (x + 8) + 6(x + 8)
14. x3 + 4x 2 - x - 4
x 2(x + 4) - 1(x + 4)
2
Students may not recognize that a polynomial can
sometimes be factored if they regroup the terms. Give
students a pattern they can follow to test if factoring
by grouping applies to a polynomial: first, rearrange
the terms so that when they are grouped, they will
have common factors; group the terms; factor each
group, using factoring patterns if necessary; then,
rearrange and assemble the factors using the
distributive property
(x 2 + 6)(x + 8)
15. 8x + 8x + 27x + 27
4
3
(x 2 - 1)(x + 4)
(x - 1)(x + 1)(x + 4)
16. 27x 4 + 54x 3 - 64x - 128
8x 3(x + 1) + 27(x + 1)
27x 3(x + 2) - 64(x + 2)
(8x 3 + 27)(x + 1)
(2x + 3)(4x 2 - 6x + 9)(x + 1)
17. x 3 + 2x 2 + 3x +6
x (x + 2) + 3(x + 2)
(27x 3 - 64)(x + 2)
(3x - 4)(9x 2 + 12x + 16)(x + 2)
18. 4x 4 - 4x 3 - x + 1
4x 4 - 4x 3 - x + 1
2
(x 2 + 3)(x + 2)
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Math Connections
MP.1 After students have solved a polynomial
19. Engineering A new rectangular outbuilding for a
farm is being designed. The outbuilding’s side and
bottom should be 4 feet thick. Its outer length should
be twice its outer width and height. What should the
outer dimensions of the outbuilding be if it is to have a
volume of 2304 cubic feet?
2304 = (2x − 8)(x − 8)(x − 4)
2304 = 2x 3 - 32x 2 + 160x - 256
0 = 2x 3 - 32x 2 + 160x - 2560
0 = 2x 2(x - 16) + 160(x - 16)
0 = 2(x 2 + 80)(x - 16)
The only real solution is x = 16. The outbuilding is 32 feet long, 24 feet wide, and 24 feet high.
20. Arts A piece of rectangular crafting supply is being cut for a new sculpture. You want its
length to be 4 times its height and its width to be 2 times its height. If you want the wood
to have a volume of 64 cubic centimeters, what will its length, width, and height be?
V = (4x)(2x)(x)
V = 8x 3
64 = 8x 3
8 = x3
2=x
The length of the piece of crafting supply will be 8 cm, the width 4 cm,
and the height 2 cm.
Module 6
A2_MNLESE385894_U3M06L4 317
317
Lesson 6.4
(4x 3 - 1)(x - 1)
Write and solve a polynomial equation for the situation described.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Alex
Ramsay/Alamy
equation using the zero-product property, help them
understand and recall that the zeros of the
polynomial function f(x) associated with the
polynomial equation are the values of x where the
graph of the polynomial function crosses the x-axis.
The zeros of a function f(x) are also equivalent to the
solutions of the equation f(x) = 0 and are related to
the factors of the polynomial.
4x 3(x - 1) - 1(x - 1)
317
Lesson 4
6/27/14 2:50 PM