Download Factoring Polynomials

LESSON
6.4
Name
Factoring Polynomials
Class
6.4
Date
Factoring Polynomials
Essential Question: What are some ways to factor a polynomial, and how is factoring useful?
Common Core Math Standards
The student is expected to:
Resource
Locker
A-SSE.2
Explore
Use the structure of an expression to identify ways to rewrite it. Also
N-CN.8(+), A-SSE.1a, A-APR.3, A-CED.1
Factoring a polynomial of degree n involves finding factors of a lesser degree that can be multiplied together to
produce the polynomial. When a polynomial has degree 3, for example, you can think of it as a rectangular prism
whose dimensions you need to determine.
Mathematical Practices
MP.8 Patterns
A
Language Objective
R
Yellow(Y): V = 8x
Total volume: V = x 3 + 6x 2 + 8x
x+ 2
B
The volume of the red piece is found by cubing the length of one edge. What is the height of
this piece?
© Houghton Mifflin Harcourt Publishing Company
x
C
The volume of a rectangular prism is V = lwh, where l is the length, w is the width, and
h is the height of the prism. Notice that the green prism shares two dimensions with the
cube. What are these dimensions?
x and x
D
What is the length of the third edge of the green prism?
2x
Divide by both of the known dimensions to find the missing edge length. ____
x· x =2
2
The length of the last edge is 2.
E
x and the width of the green prism
You showed that the width of the cube is
2
is
. What is the width of the entire prism?
x+2
Module 6
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
HARDCOVER PAGES 223230
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Lesson 4
309
Module 6
L4 309
4_U3M06
SE38589
A2_MNLE
Lesson 6.4
x+ 4
Volume = x3 + 6x2 + 8x
A2_MNLESE385894_U3M06L4 309
309
G
x
Blue(B): V = 4x 2
Essential Question: What are some
ways to factor a polynomial, and how is
factoring useful?
View the Engage section online. Discuss the photo
and how the volume of the flower bed is related to its
outer and inner dimensions. Then preview the
Lesson Performance Task.
Y
B
Green(G): V = 2x 2
ENGAGE
PREVIEW: LESSON
PERFORMANCE TASK
The volumes of the parts of the rectangular prism are as follows:
Red(R): V = x 3
Work with a partner to complete a chart detailing how factoring can
be used.
Possible answer: Methods include factoring out the
greatest common monomial factor, recognizing
special factoring patterns, and factoring by
grouping. Factoring is useful when solving
polynomial equations because the zero-product
property can be applied to the factored polynomial
on one side of the equation as long as the other
side is 0.
Analyzing a Visual Model
for Polynomial Factorization
6/8/15
11:43 PM
6/8/15 11:41 PM

You determined that the length of the green piece is x. Use the volume of the yellow piece
and the information you have derived to find the length of the prism.
EXPLORE
Since the volume of the yellow piece is 8x, the height of the yellow piece is x, and the width
of the yellow piece is 2, simply divide out to find that the last remaining edge length is 4.
Analyzing a Visual Model for
Polynomial Factorization
INTEGRATE TECHNOLOGY
x
Students have the option of completing the
polynomial factorization activity either in the
book or online.
x+ 4
x+ 2
Volume = x3 + 6x2 + 8x

Since the dimensions of the overall prism are x, x + 2, and x + 4, the volume of the
overall prism can be rewritten in factored form as V = (x)(x + 2)(x + 4). Multiply these
polynomials together to verify that this is equal to the original given expression for the
volume of the overall figure.
QUESTIONING STRATEGIES
How is factoring useful when determining the
variable dimensions of a polynomial model
like a rectangular prism? Factoring is useful when
the volumes of parts of a three-dimensional model
are known and can be added to get the variable
dimensions of the figure. Since factoring is the
opposite of multiplying, the polynomial can then be
written in factored form and multiplied to give the
volume of the whole figure.
V = x 3 + 6x 2 + 8x
Reflect
1.
Discussion What is one way to double the volume of the prism?
Sample Answer: One way to double the volume of the prism is to double the height of
the prism.
Explain 1
Factoring Out the Greatest Common Monomial First
Factor each polynomial over the integers.
Example 1

6x 3 + 15x 2 + 6x
6x + 15x + 6x
Write out the polynomial.
x(6x + 15x + 6)
Factor out a common monomial, an x.
3
2
2
3x(2x + 5x + 2)
2
3x(2x + 1)(x + 2)
© Houghton Mifflin Harcourt Publishing Company
Most polynomials cannot be factored over the integers, which means to find factors that use only integer coefficients.
But when a polynomial can be factored, each factor has a degree less than the polynomial’s degree. While the goal is
to write the polynomial as a product of linear factors, this is not always possible. When a factor of degree 2 or greater
cannot be factored further, it is called an irreducible factor.
EXPLAIN 1
Factoring Out the Greatest Common
Monomial First
Factor out a common monomial, a 3.
AVOID COMMON ERRORS
Factor into simplest terms.
Students often forget to factor out the greatest
common monomial as the first step in factoring.
Point out that if the terms of a polynomial have a
greatest common factor, it almost always should be
factored first, before analyzing the remainder of the
terms for factoring. Emphasize that sometimes
finding the greatest common factor is the only way to
factor some polynomials.
Note: The second and third steps can be combined into one step by factoring out the greatest
common monomial.
Module 6
310
Lesson 4
PROFESSIONAL DEVELOPMENT
A2_MNLESE385894_U3M06L4 310
Integrate Mathematical Practices
This lesson provides an opportunity to address Mathematical Practice MP.8,
which calls for students to “look for and express regularity in repeated reasoning.”
Students are already familiar with multiplying polynomials but, in this lesson, they
must analyze the conditions that help them factor a polynomial, or rewrite it as
the product of individual factors of lesser degree. These factors, when multiplied,
give the original polynomial. Many methods of factoring, including special
factoring patterns, are presented so that students can analyze the polynomial and
explain which method gives the factorization more easily. Factoring polynomials
is a useful tool for solving polynomial equations by using the zero-product
property.
6/27/14 2:37 PM
Factoring Polynomials 310

QUESTIONING STRATEGIES
How can you tell if the terms of a polynomial
have a greatest common factor? Each of the
terms has a number that divides all of the terms
and/or a power of a variable that divides each term.
The common monomial factor is a product of this
number and the power of one or more of the
variables.
2x 3 - 20x
2x 3 - 20 x
Write out the polynomial.
2x (x 2 - 10)
Factor out the greatest common monomial.
Reflect
2.
Why wasn’t the factor x 2 - 10 further factored?
Since 10 does not have any two factors that sum to 0, (x 2 - 10) is irreducible
over the integers.
3.
What does it mean for a polynomial to be
“factored completely”? The greatest
monomial factor is factored out and the remaining
polynomial is factored into factors that cannot be
factored further: they are irreducible.
Consider what happens when you factor x 2 - 10 over the real numbers and not merely the integers. Find a
such that x 2 - 10 = (x - a)(x + a).
a = √10
―
Your Turn
4.
3x 3 + 7x 2 + 4x
x(3x 2 + 7x + 4)
Factor out a common monomial, an x.
x(3x + 4)(x + 1)
EXPLAIN 2
Explain 2
Factor into simplest terms.
Recognizing Special Factoring Patterns
Remember the factoring patterns you already know:
Recognizing Special Factoring
Patterns
Difference of two squares: a 2 - b 2 = (a + b)(a - b)
Perfect square trinomials: a 2 + 2ab + b 2 = (a + b) and a 2 - 2ab + b 2 = (a - b)
2
2
There are two other factoring patterns that will prove useful:
they learned in Algebra 1, including: difference of
two squares: a 2 - b 2 = (a + b)(a - b); and perfect
2
square trinomials: a 2 + 2ab + b 2 = (a + b) and
2
a 2 - 2ab + b 2 = (a - b) . Emphasize that there are
two more useful factoring patterns in this lesson: sum
of two cubes: a 3 + b 3 = (a + b)(a 2 - ab + b 2) and
difference of two cubes:
a 3 - b 3 = (a - b)(a 2 + ab + b 2). Students should see
that each of the factors is irreducible.
Sum of two cubes : a 3 + b 3 = (a + b)(a 2 - ab + b 2)
© Houghton Mifflin Harcourt Publishing Company
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Patterns
MP.8 Students should review the factoring patterns
Difference of two cubes : a 3 - b 3 = (a - b)(a 2 + ab + b 2)
Notice that in each of the new factoring patterns, the quadratic factor is irreducible over the integers.
Example 2

Factor the polynomial using a factoring pattern.
27x 3 + 64
27x 3 + 64
Write out the polynomial.
27x 3 = (3x)
64 = (4)
3
Check if 27x 3 is a perfect cube.
3
Check if 64 is a perfect cube.
a 3 + b 3 = (a + b)(a 2 - ab + b 2)
Use the sum of two cubes formula to factor.
(3x) + 4 3 = (3x +4)((3x) - (3x)(4) + 4 2)
3
2
27x 3 + 64 = (3x + 4)(9x 2 - 12x + 16)
Module 6
311
Lesson 4
COLLABORATIVE LEARNING
A2_MNLESE385894_U3M06L4.indd 311
Small Group Activity
Help groups of students review the factoring patterns they learned in previous
courses along with how to factor the sum or difference of two cubes. Provide each
student with an example of one type of factoring, and ask them to show and
explain the first step in factoring their problem. Then they pass the problem to
another student, who writes the next step and explains it. They continue to pass
the problem on until each problem is completely factored and all steps are
explained. Encourage them to use these as examples of the types of factoring
when they write about factoring in their journals.
311
Lesson 6.4
3/19/14 1:37 PM
B
8x 3 - 27
3
8 x - 27
QUESTIONING STRATEGIES
Write out the polynomial.
8x 3 = ( 2 x) 3
Check if 8x 3 is a perfect cube.
27 = ( 3 ) 3
Check if 27 is a perfect cube.
a 3 - b 3 = (a - b)(a 2 + ab + b 2)
What are the steps for using the sum or
difference of cubes formulas to factor? Factor
out the greatest monomial factor, rewrite as the
sum or difference of perfect cubes, and then apply
the sum or difference of cubes formula.
Use the difference of two cubes formula to factor.
8x 3 - 27 = ( 2 x - 3 )( 4 x 2 + 6 x + 9 )
Reflect
5.
The equation 8x 3 - 27 = 0 has three roots. How many of them are real, what are they, and how
many are nonreal?
To find the roots of the given expression, notice that the expression is the difference of
cubes. Use the difference of cubes formula to factor 8x 3 - 27
8x 3 - 27 = (2x-3)(4x 2 + 6x + 9)
To find the first root, set 2x - 3 equal to 0 and solve for x.
2x - 3 = 0
2x = 3
3
x = __
2
3
is a real root. Next, look at 4x 2 + 6x + 9 Notice that because the polynomial is
Therefore __
2
irreducible over the integers, the quadratic formula must be used to solve for the roots.
―――――
-6± √6 2 - 4(4)(9)
x = ________________
2(4)
―――
-6± √36 - 144
= _____________
8
――
-6± √-108
= ___________
8
Your Turn
6.
40x 4 + 5x
40x 4 + 5x
Write out the polynomial.
5x(8x + 1)
3
Factor out 5x.
8x = (2x)
3
1 = (1)
3
Check if 8x 3 is a perfect cube.
3
Check if 1 is a perfect cube.
a + b = (a + b)(a - ab + b
3
3
2
2
)
© Houghton Mifflin Harcourt Publishing Company
As the number under the square root symbol is negative, both of the values of x will be
3
and two roots are nonreal.
nonreal numbers. Therefore, the one real root is __
2
Use the formula to factor.
8x + 1 = (2x + 1)(4x - 2x + 1)
3
2
5x(8x 3 + 1) = 5x(2x + 1)(4x 2 - 2x + 1)
40x 4 + 5x = 5x(2x + 1)(4x 2 - 2x + 1)
Module 6
312
Lesson 4
DIFFERENTIATE INSTRUCTION
A2_MNLESE385894_U3M06L4 312
Communicating Math
6/27/14 2:40 PM
Give groups of students several different polynomials to factor. Have all students
explain their reasoning or describe the procedures they used to factor the
polynomial completely. Encourage them to make visual aids or graphic organizers
to help them with their explanations. Ask for a volunteer from each group to
summarize to the class how to factor a polynomial.
Factoring Polynomials 312
Explain 3
EXPLAIN 3
Another technique for factoring a polynomial is grouping. If the polynomial has pairs of terms
with common factors, factor by grouping terms with common factors and then factoring out the
common factor from each group. Then look for a common factor of the groups in order to
complete the factorization of the polynomial.
Factoring by Grouping
Example 3
AVOID COMMON ERRORS

Students may not have a systematic approach to
factoring. They will find it easier to factor by
grouping if they use a strategy of looking for a
common monomial factor, checking the factoring
patterns, checking if factoring by grouping applies to
their problem, and so on.

QUESTIONING STRATEGIES
product property in the context of factoring and
solving a real-world polynomial equation. Tell them
that once a polynomial equation is established that
models the real-world situation, then they apply the
known information to the polynomial and rewrite it
in a form (with zero on one side) that makes it
possible to factor and solve the polynomial using the
zero-product property.
313
Lesson 6.4
Write out the polynomial.
x3 - x 2 + x - 1
Group by common factor.
(x3 - x 2) + (x - 1)
Factor.
x 2(x - 1) + 1(x - 1)
Regroup.
(x 2 + 1)(x - 1)
x4 + x3 + x + 1
Write out the polynomial.
x4 + x3 + x + 1
Group by common factor.
4
3
( x + x ) + (x + 1)
x 3 (x + 1) + 1 (x + 1)
3
( x + 1 )(x + 1)
2
( x - x + 1)(x + 1)(x + 1)
Apply sum of two cubes to the first term.
2
( x + 1 ) 2( x - x + 1)
Substitute this into the expression and simplify.
Your Turn
x 3 + 3x 2 + 3x + 2
x 3 + 3x 2 + 3x + 2
© Houghton Mifflin Harcourt Publishing Company
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Math Connections
MP.1 Discuss with students how to use the zero-
x3 + x 2 + x + 1
Regroup.
7.
Solving a Real-World Problem by
Factoring a Polynomial
Factor the polynomial by grouping.
Factor.
How do you factor a polynomial by
grouping? 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.
EXPLAIN 4
Factoring by Grouping
(x
3
+ 2x
) + (x
2
3
+3x + 3)
x (x +2) + (x + 1)(x + 2)
2
(x 2 + x + 1)(x + 2)
Explain 4
Write out the polynomial.
Group by common factor.
Factor.
Regroup.
Solving a Real-World Problem
by Factoring a Polynomial
Remember that the zero-product property is used in solving factorable quadratic equations.
It can also be used in solving factorable polynomial equations.
Module 6
A2_MNLESE385894_U3M06L4 313
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Lesson 4
6/27/14 2:41 PM
Example 4

Write and solve a polynomial equation for the situation described.
QUESTIONING STRATEGIES
A water park is designing a new pool in the shape of a rectangular prism.
The sides and bottom of the pool are made of material 5 feet thick.
The length must be twice the height (depth), and the interior width must
be three times the interior height. The volume of the box must be 6000
cubic feet. What are the exterior dimensions of the pool?
How is the zero-product property used to
solve a polynomial equation? The polynomial
equation is rewritten so that 0 is on one side, and
then the equation is factored. The zero-product
property states that if each factor is set equal to
zero and the resulting equation solved, then the
values obtained are solutions to the original
polynomial equation.
The dimensions of the interior of the pool, as described by the problem, are
the following:
h=x-5
w = 3x - 15
l = 2x - 10
The formula for volume of a rectangular prism is V = lwh. Plug the values into the volume equation.
V = (x - 5)(3x - 15)(2x - 10)
V = (x - 5)(6x 2 - 60x + 150)
V = 6x 3 - 90x 2 + 450x - 750
Now solve for V = 6000.
6000 = 6x 3 - 90x 2 + 450x - 750
0 = 6x 3 - 90x 2 + 450x - 6750
Factor the resulting new polynomial.
6x 3 - 90x 2 + 450x - 6750
= 6x 2(x - 15) + 450(x - 15)
= (6x 2 + 450)(x - 15)
The only real root is x = 15.

Engineering To build a hefty wooden feeding trough for a zoo, its sides and bottom should be 2 feet
thick, and its outer length should be twice its outer width and height.
What should the outer dimensions of the trough be if it is to hold 288 cubic feet of water?
Volume = Interior Length(feet) ⋅ Interior Width(feet) ⋅ Interior Height(feet)
288 = ( 2x - 4)( x - 4)( x - 2)
288 = 2 x 3 - 16 x 2 + 40 x - 32
0 = 2 x 3 - 16 x 2 + 40 x - 320
2
0 = 2x (x - 8 ) + 40 (x - 8 )
0= 2
(x 2 + 20 )(x - 8
© Houghton Mifflin Harcourt Publishing Company • Image Credits:
©morrison77/Shutterstock
The interior height of the pool will be 10 feet, the interior width 30 feet, and the interior length 20 feet.
Therefore, the exterior height is 15 feet, the exterior length is 30 feet, and the exterior width is 40 feet.
)
The only real solution is x = 8 . The trough is 16 feet long, 8 feet wide, and 8 feet high.
Module 6
A2_MNLESE385894_U3M06L4.indd 314
314
Lesson 4
3/19/14 1:37 PM
Factoring Polynomials 314
Your Turn
ELABORATE
8.
QUESTIONING STRATEGIES
Volume = Interior Length(feet) ⋅ Interior Width(feet) ⋅ Interior Height(feet)
How do you determine whether a polynomial
is not factorable? First determine that the
terms do not have a common factor, that no
factoring pattern applies to the terms of the
polynomial, and that there is no other way to
rewrite the polynomial as the product of
irreducible factors.
972 = (2x − 6)(x − 6)(x − 3)
972 = 2x 3 - 24x 2 + 90x - 108
0 = 2x 3 - 24x 2 + 90x - 1080
0 = 2x 2(x - 12) + 90(x - 12)
0 = 2(x 2 + 45)(x - 12)
The only real solution is x = 12. The shed is 24 feet long, 12 feet wide, and 12 feet high.
Elaborate
9.
COGNITIVE STRATEGIES
Describe how the method of grouping incorporates the method of factoring out the greatest
common monomial.
After grouping an expression into separate parts, you factor out the greatest
common monomial of each part to reveal a polynomial factor that is common to all the
To help students remember the formulas for the sum
and difference of cubes, you may wish to use the
mnemonic device SOPPS for the order of the terms
in the second factor:
separate parts.
10. How do you decide if an equation fits in the sum of two cubes pattern?
If there are two terms in the equation being added and these terms are perfect cubes, then
it fits the sum of two cubes pattern.
Square Opposite-sign Product Plus Square.
11. How can factoring be used to solve a polynomial equation of the form p(x) = a, where a is a nonzero
constant?
Subtract a to get p(x) - a = 0. Factor out common monomials or use grouping to factor
SUMMARIZE THE LESSON
the polynomial.
© Houghton Mifflin Harcourt Publishing Company
Have students make a graphic organizer that
lists their own methods of factoring a
polynomial completely. The lists should include
factoring out the greatest monomial factor, applying
rules for factoring the sum and difference of two
cubes, and previously learned rules such as factoring
the difference of two squares.
Engineering A new shed is being designed in the shape of a rectangular prism. The shed’s side and
bottom should be 3 feet thick. Its outer length should be twice its outer width and height.
What should the outer dimensions of the shed be if it is to have 972 cubic feet of space?
12. Essential Question Check-In What are two ways to factor a polynomial?
Recognizing special factoring patterns and factoring by grouping
Module 6
Lesson 4
315
LANGUAGE SUPPORT
A2_MNLESE385894_U3M06L4 315
16/10/14 9:31 AM
Connect Vocabulary
Have students work together to complete a table like the one shown.
Useful Ways to Factor Polynomials
Perfect square trinomials:
Sum of two cubes:
Difference of two cubes:
315
Lesson 6.4
a 2 + 2ab + b 2 = (a + b) and
2
a 2 - 2ab + b 2 = (a - b)
2
a 3 + b 3 = (a + b)(a 2 - ab + b 2)
a 3 - b 3 = (a - b)(a 2 + ab + b 2)
EVALUATE
Evaluate: Homework and Practice
• Online Homework
• Hints and Help
• Extra Practice
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 sides 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, 16 feet wide, and 16 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
6/8/15 11:42 PM
21. Engineering A new rectangular holding tank is being built. The tank’s sides
and bottom should be 1 foot thick. Its outer length should be twice its outer
width and height.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Patterns
MP.8 Point out that students should review the
What should the outer dimensions of the tank be if it is to have a volume of 36 cubic feet?
36 = (2x − 2)(x − 2)(x − 1)
36 = 2x 3 − 8x 2 + 10x − 4
factoring patterns they learned in Algebra 1 as well as
the two useful factoring patterns in this lesson:
0 = 2x 3 − 8x 2 + 10x − 40
0 = 2x (x − 4) + 10(x − 4)
0 = 2(x 2 + 5)(x − 4)
2
Sum of two cubes:
The only real solution is x = 4 . The tank is 8 feet long, 4 feet wide, and 4 feet high.
a 3 + b 3 = (a + b)(a 2 - ab + b 2);
22. Construction A piece of granite is being cut for a
building foundation. You want its length to be 8 times
its height and its width to be 3 times its height. If you
want the granite to have a volume of 648 cubic yards,
what will its length, width, and height be?
Difference of two cubes:
a 3 - b 3 = (a - b)(a 2 + ab + b 2)
V = (8x)(3x)(x)
Emphasize that the exercises may go beyond
factoring over the integers and may include factoring
over the real numbers or complex numbers.
V = 24x 3
648 = 24x 3
27 = x 3
3=x
The length of the slab will be 24 yards, the width 9 yards, and the height 3 yards.
AVOID COMMON ERRORS
b. 3x 3 + 5
none
c. 4x + 25
none
2
d. 27x 3 + 1000
sum of two cubes
e. 64x - x + 1
none
3
2
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Gennadiy
Iotkovskiy/Alamy
23. State which, if any, special factoring pattern each of the following polynomial
functions follows:
difference of two squares
a. x 2 - 4
H.O.T. Focus on Higher Order Thinking
24. Communicate Mathematical Ideas What is the relationship between the degree of a
polynomial and the degree of its factors?
The degree of a polynomial is always at least 1 larger than the degree of
any of its factors.
Some students may not be sure of the point at which
a polynomial has been factored completely. Discuss
ways to determine whether the factoring is complete,
checking that the greatest monomial factor has been
factored out, and making sure that each factor is itself
irreducible.
25. Critical Thinking Why is there no sum-of-two-squares factoring pattern?
There is no sum-of-two-squares factoring pattern because any sum
of two squares will only have complex roots as an answer.
Module 6
A2_MNLESE385894_U3M06L4 318
318
Lesson 4
6/27/14 2:57 PM
Factoring Polynomials 318
26. Explain the Error Jim was trying to factor a polynomial and produced the
following result:
PEER-TO-PEER DISCUSSION
Instruct one student in each pair to write a
polynomial while the other student gives verbal
instructions for factoring the polynomial. Then have
students switch roles, and repeat the exercise, giving
instructions for factoring a different polynomial.
3x 3 + x 2 + 3x + 1
Write out the polynomial.
3x (x + 1) + 3(x + 1)
Group by common factor.
2
3(x + 1)(x + 1)
Regroup.
2
Explain Jim’s error.
Jim misgrouped the polynomial. He should have grouped it like this:
3x 3 + x 2 + 3x + 1
Write out the polynomial.
(x 2 + 1)(3x + 1)
Regroup.
3x(x 2 + 1) + 1(x 2 + 1)
JOURNAL
Group by common factor.
27. Factoring can also be done over the complex numbers. This allows you to find all the
roots of an equation, not just the real ones.
Have students make a table describing the methods
and patterns for factoring polynomials. Give
examples of using the sum and difference of two
cubes and of factoring polynomials by grouping.
Complete the steps showing how to use a special factor identity to factor x 2 + 4 over
the complex numbers.
x2 + 4
Write out the polynomial.
Rewrite as a difference of two squares.
x - (-4)
√
√
(x + -4 ) (x - -4 )
――
2
――
Factor.
(x + 2i) (x - 2i )
Simplify.
28. Find all the complex roots of the equation x 4 - 16 = 0.
(x 2 + 4)(x 2 - 4) = 0
(x 2 + 4)(x - 2)(x + 2) = 0
Factors (x - 2) and (x + 2) will yield real roots. factor (x 2 + 4) will yield complex roots.
© Houghton Mifflin Harcourt Publishing Company
x2 + 4
x 2 -(-4)
――
(x + √――
-4 )(x - √-4 )
(x + 2i)(x - 2i)
x + 2i = 0
or
x = -2i
x - 2i = 0
x = 2i
The complex roots of the equation x 4 - 16 = 0 are x = -2i and x = 2i.
29. Factor x 3 + x 2 + x + 1 over the complex numbers.
x3 + x2 + x + 1
x 2(x + 1) + 1(x + 1)
(x + 1)(x 2 + 1)
(x + 1)⎡⎣x 2 - (-1)⎤⎦
――
――
(x + 1)(x + √-1 )(x - √-1 )
(x + 1)(x + i)(x - i)
Therefore, x 3 + x 2 + x + 1 = (x + 1)(x + i)(x - i)
Module 6
A2_MNLESE385894_U3M06L4 319
319
Lesson 6.4
319
Lesson 4
6/8/15 11:42 PM
Lesson Performance Task
LANGUAGE SUPPORT
Some students may not be familiar with the term
flower bed. Explain that a bed can refer to an area for
a garden. It may be enclosed by fencing, boards, or
logs, and is sometimes built from the ground to a
height of several inches to make it easier for the
gardener to work in the soil. Have them look at the
photo of the flower bed and describe any similarities
they see between it and a bed that a person might
sleep on. Both are rectangular, flat, and low to the
ground.
Sabrina is building a rectangular raised flower bed. The boards
on the two shorter sides are 6 inches thick, and the boards on
the two longer sides are 4 inches thick. Sabrina wants the outer
length of her bed to be 4 times its height and the outer width to
be 2 times its height. She also wants the boards to rise 4 inches
above the level of the soil in the bed. What should the outer
dimensions of the bed be if she wants it to hold 3136 cubic
inches of soil?
Let x = the external height
Volume = Interior length(inches) · Interior Width(inches) · Interior Height (inches)
3136 = (4x - 12)(2x - 8)(x - 4)
3136 = 8x 3 - 88x 2 + 320x - 384
0 = 8x 3 - 88x 2 + 320x - 3520
0 = 8x 2(x - 11) + 320(x - 11)
QUESTIONING STRATEGIES
x = 11 is the only real solution
Why is it important to move all the terms of
the equation to one side? The other side will
then be zero, and you can use the zero-product
property to solve for the solution.
0 = (8x 2 + 320)(x - 11)
Thus the external dimensions must be 44 inches long by 22 inches wide by 11 inches high.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Gary K
Smith/Alamy
If the final factoring gives more than one
solution, how can you determine the correct
one? The answer must make sense for the problem.
If the question is asking for a length of a board, then
a negative answer would not be realistic. The
answer must be positive.
Module 6
320
Lesson 4
EXTENSION ACTIVITY
A2_MNLESE385894_U3M06L4 320
Have students design a circular flower bed using a tire or other doughnut-shaped
object. Have them discuss the constraints they would put on the dimensions of the
bed and on the volume of the soil. Then ask them to set up a polynomial equation
for the interior volume of the bed and solve for the unknown dimension. Have
students discuss what shape would be best for a garden bed and why. A
rectangular bed that is long and narrow allows the gardener to easily reach all
plants without stepping on the soil.
6/27/14 2:59 PM
Scoring Rubric
2 points: Student correctly solves the problem and explains his/her reasoning.
1 point: Student shows good understanding of the problem but does not fully
solve or explain his/her reasoning.
0 points: Student does not demonstrate understanding of the problem.
Factoring Polynomials 320