Factoring
Polynomials
CALIFORNIA COMMON CORE
ESSENTIAL QUESTION
LESSON 13.1
Factoring Polynomials
How can you factor expressions of the
form ax2 + bx + cl
A.SSE.2, A.SSE.3
LESSON 13.2
Factoring x2 + 6x -f c
A.SSE.2, A.SSE.3
LESSON 13.3
Factoring
Factoring
Special Products
Ruling out common elements in a scientific
experiment is similar to removing common factors
in an equation: logically, whatever is common
to two samples can't be the cause of differences
between them.
ft
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519
YOU
A
e
Complete these exercises to review skills you will need for
this module.
Factors
EXAMPLE List the factors of 12.
1,12
3,4
2,6
Any whole number that can
be multiplied by another
whole number to get 1 2 is a
factor of 12.
List the factors of each number.
2. 10
1. 8
3. 30
Multiply Monomials and Polynomials
EXAMPLE Multiply.
6x(2x + 5)
Apply the Distributive Property.
+ 6x(5) = 12*2 + 30*
Multiply.
4. 100 - 5)
5. 3/z(/i2j + 2/z2)
6. y(7/ -4/-1)
Multiply Binomials
EXAMPLE Find the product.
First: # • JC = x
Outer: x • 8 = 8x
Inner: 3 • x — 3x
Last: 3 • 8 = 24
x2 + 8% + 3x + 24 =
Find each product.
7. & -
520
Unit 4
Use FOIL to multiply each term
in the first binomial by each term
in the second binomial.
+ 11*+ 24
Reading Start-Up
Review Words
• binomial (binomio)
Visualize Vocabulary
• constant (constante)
Fill in the missing information in the chart below.
Word
Vocabulary
Definition
factor
• factor (factor)
Examples
prime factor (factor primo)
• trinomial (trinomio)
12 = 3 - 4
3 and 4 are factors of 12.
Preview Words
greatest common factor
xy = x • y
(GCF)
x and y are factors of xy.
binomial
a polynomial with
terms
a polynomial with
'.
I I i!terms
'
constant
4,0, TT
Understand Vocabulary
To become familiar with some of the vocabulary in the module, consider
the following. You may refer to the module, the glossary, or a dictionary.
1. The largest common factor of two or more given numbers is the
2. The
of monomials is the product of
the greatest integer and the greatest power of each variable that divide evenly
into each monomial.
Active Reading
Four-Corner Fold Before beginning the module, create a
Four-Corner Fold to help you organize what you learn. Use one
flap for each lesson in the module. As you study the module, note
important facts, examples, and formulas on the flaps. Look for
similarities and differences between the lessons. Use your
FoldNote to complete assignments and to study for tests.
Module 15
521
GETTING READY FOR
2^niU:H5iMiiiilL
CALIFORNIA
nderstanding the standards and the vocabulary terms in the standards
will help you know exactly what you are expected to learn in this module.
CACC A.SSE.2
Use the structure of an
expression to identify ways to
rewrite it. For example, see
What It Means to You
recognizing it as a difference of
squares that can be factored as
EXAMPLE A.SSE.2
You can rewrite expressions by factoring out common factors and
working FOIL in reverse.
Martown Park has an area of (x2 — 3x — 18) feet.
If the width is (x + 3) feet, what is the length?
Key Vocabulary
Iw — area
greatest common
factor (maxima comun divisor
de una expresion)
(length?! (x + 3) = (x2 - 3x
Factors that are shared by two or
more whole numbers are called
common factors. The greatest
of these common factors is the
greatest common factor.
think about FOIL in reverse:
(x + or -?)!(* + 3)= (x
2
18)
- 3x
18)
The missing value and 3 need to have a sum
of —3, which means the binomial needs to be
The length is (x - 6} feet.
23CACC A.SSE.3
Choose and produce an
equivalent form of an
expression to reveal and explain
properties of the quantity
represented by the expression.
What It Means to You
You can use patterns to recognize and rewrite expressions to
reveal properties of the expression.
EXAMPLE A.SSE.3
Factor 25m2 - 16n2.
25m2 - 16n2
perfect
square
difference
perfect
square
This binomial is the difference of two squares, so it factors as
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Common Core
Standards
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522
Unit 4
(5m — 4n)(5m + 4n)
A.SSE.2
LESSON
15*1 Factoring Polynomials
Use the structure of an
expression to identify ways to
rewrite it. AlsoA.SSE.3
ESSENTIAL QUESTION
! How can you use the greatest common factor to factor polynomials?
A.SSE.Z
EXPLORE ACTIVITY
Factoring and Greatest Common Factor
Factors that are shared by two or more whole numbers are called common factors.
The greatest of these common factors is called the greatest common factor, or GCF.
Factors of 12:
1, 2, 3, 4, 6, 12
Factors of 32:
1, 2, 4, 8, 16, 32
Common Factors: 1, 2, 4
The greatest common factor is 4.
Use the greatest common factor (GCF) and the Distributive Property to factor the
expression 30x + 18.
I A .' Write out the prime factors of each term.
30* + 18 = 2 •
•
• x + 2-
•
. Circle the common factors.
30* + 18 =
2
• x
(Jjjp Write the expression as the product of the GCF and a sum.
30* + 18 = (
)(
x+
)
REFLECT
1. Will you get a completely factored expression if you factor out a common
factor that is not the GCF? Explain.
2.
Is the expression 2(3x — 4x) completely factored? Explain.
Lesson 15.1
523
Factoring Out a Common Binomial Factor
Sometimes the GCF of the terms in an expression is a binomial. Such a GCF is
called a common binomial factor. You factor out a common binomial factor the
same way you factor out a monomial factor.
(^) my.hrw.com
Factor each expression.
(A) 7(x - 3) - 2x(x - 3)
My Notes
(?)
7(x - 3) - 2x(x - 3)
(x —3) is a common binomial factor.
(*-3)(7-2x)
Factor out (x— 5).
-£(** 4- 4) 4- (** 4- 4)
-*(** 4- 4) + (f 2 4- 4)
(t +4) is a common binomial factor.
-t(& 4- 4) 4- 1't*2 4- 4)
(t 2 4-4) = 1
4)
Factor out (t 2 + 4).
C) 5*0 + 3) - 4(3 + x)
5x(x + 3) - 4(3 4- x)
(3 + x) = (x -f 3), so (x + 3) is a common
binomial factor.
5x(x + 3) - 4(# + 3)
Factor out (x+ 3),
(p) -3x2(x 4- 2) 4- 4(x - 7)
-Sx2^ 4- 2) 4- 4(x - 7)
There are no common factors.
The expression cannot be factored.
YOUR TURN
Factor each expression, if possible.
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526
Unit 4
12. 7x(2x 4- 3) 4- (2x +3)
13.
14. 7(3* - 2) 4- 2*2(2* - 3)
15.
9(^:4-2)
-8(64-*)
Factoring by Grouping
Some polynomials can be factored by grouping. When a polynomial has four
terms, you may be able to make two groups and factor the GCF from each.
QCACC A.SSE.2
EXAMPLE 4
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Factor each polynomial by grouping. Check your answer.
(A) 12a3 - 9a2 + 20a - 15
(12a3-9a2) + (20a- 15)
Group terms that have a common
number or variable as a factor.
3a2(4a - 3) + 5(4a - 3)
Factor out the GCF of each group.
3a2(4a - 3) + 5(4a - 3)
(A-a — 3) is a common factor.
(4a-3)(3a 2 + 5)
Factor out (Aa— 3).
Check:
(4a - 3)(3a2 + 5)
Multiply using FOIL.
4a(302) + 4a(5) - 3(3a2) - 3(5)
12a3 + 20a - 9a2 - 15
12a3 - 9a2 + 20a - 15
The product is the original polynomial.
(2/-f 10/
Group terms.
Factor out the GCF of each group.
(g + 5) is a common factor.
Factor out (g + 5).
Check:
Multiply using FOIL.
+*(!) + 5(2^) + 5(1)
The product is the original polynomial.
YOUR
u,
Factor each polynomial. Check your answer.
16. 6b3 + Sb1 + 9b + 1.2
17. 4r* + 24r + r2 + 6
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Lesson 15.1
527
Factoring with Opposites
Recognizing opposite binomials can help you factor polynomials. The binomials
(5 — x) and (x — 5) are opposites, because (5 — x) = — l(x — 5).
EXAMPLE 5
A.SSE.2
Factor the polynomial by grouping and using opposites. Check your answer.
3x3 - 15X2 + 10 - 2x
My Notes
(3x3 - 15*2) + (10 - 2x)
Group terms.
3x2(x - 5) + 2(5 - x)
Factor out the GCf of each group.
3x2(x - 5) + 2(-l)(x - 5)
Write (5 - x) as -1 (x- 5).
3x2(x - 5) - 2(x - 5)
Simplify.
(x-5) (3x2-2)
Factor out (x — 5).
Check:
(X-5)(3x2-2)
Multiply using FOIL
x(3x2) - x(2) - 5(3^) - 5(-2)
3x3 -2x-
ISx2 + 10
3x3 — ISx2 + 10 — 2x /
The product is the original polynomial.
REFLECT
18. Critique Reasoning Inara thinks that the opposite of (a — b) is (a + b),
since addition and subtraction are opposites. Is she correct? Explain.
YOUR
TURN
..-.
--.'Factor each polynomial. Check your answer.
Personal
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528
Unit 4
19. ISx2 - Wx3 -\-Sx- 12
20. 8y-S
21. 48n6 - I8n5 - 56n + 21
22. 8t4 - 4813 -3t+I8
-x
Guided Practice
Write the expression as a product of the greatest common factor and a sum. (Explore Activity)
1. 15/ + 20y
a.
Write out the prime factors of each term.
15/ + 2Qy = 3 •
b.
•
Circle the common factors.
ISy3 + 20;> = 3 •
c.
•
• y
•.
Write the product of the GCF and a sum.
Find the GCF of each pair of monomials. (Example 1 )
14y3 +'
2. 95 and 63s3
9s
=3
63s3 = 3
D
Q'O
The GCF of 9s and 63s3 is
14/ =
The GCF of -14/ and 28/ is
Factor each polynomial. (Example 2)
5. 9d2 - 18
4. -18/-7/-7
X
(d2-
~Vi)
6. 6jc4 - 2x3
7. 36^
Factor each expression. (Example 3)
8. 4s(s + 6) - 5(s + 6)
10. (6z)(z+8)
9. -3(2 + b) + 4b(b + 2)
11. 8w(5 - w) 4- 3(w - 5)
Lesson 15.1
529
Factor each polynomial. (Exampie 4)
12. 9x3
13. 2m3 + 4m2 + 6m + 12
c
+ 4m2) + (
(m
(*
r
(m
-\
+
J
)({
2(m
14. lO^3 - 40*2 + 14x - 56
15. 2n5 - 2n4 + 7n2 - In
Factor each polynomial. (Example 5)
16. 2 r 2 - 6 r + 1 2 - 4 r
17.
+ 6 - 4q
-3)
J)
2r(r - 3) +
18. 6c - 48 + 40C2 - 5c3
+2
19. 3x3 - 27X2 + 45 - 5x
ESSENTIAL QUESTION CHECK-IN
20.
530
How can you use the greatest common factor to factor polynomials?
Unit 4
Class
Name.
Date
Personal
Math Trainer
.1 Independent Practice
.•",-' •' ?•;-.•• ..-:.£••-:..,•..-: .-,•":-,'.V-' .-" "^Ai^*kdj«fef«&^tai*i4J.'t«fcL*^';-i:.r
Online Practice
and Help
A.SSE.2, A.SSE.3
21.
I
Find the GCF of -64n4 and 24n
29.
Factor each expression or state if it cannot be
factored.
22.
After t years, the amount of money in a
savings account that earns simple interest
is P + Prt, where P is the starting amount
and r is the yearly interest rate. Factor this
expression.
30. Communicate Mathematical Ideas
Explain how you can show that (x - a) and
(a - x) are opposites.
23.
14n3 + In + 7n2
24.
3)
3)
25. 4(* - 3) - x (y + 2)
26. 7r3 - BSr2 + 6r - 30
27.
28.
Explain how to check that a polynomial has
been factored correctly.
31. The solar panel on Mandy s calculator
has an area of (Ix2 + x) cm2. Factor this
polynomial ,to find possible expressions for
the dimensions of the solar panel.
32.
A model rocket is fired vertically into the
air at 320 ft/s. The expression -let2 + 320t
gives the rockets height after t seconds.
Factor this expression.
33.
The area of a triangle is ^(x3 — 2x + 2X2 — 4).
The height hisx + 2. Write an expression
for the base b of the triangle. (Hint: Area of a
triangle = ^bh)
Explain the Error Billie says the factored
form of 18^:8 - 9x4 - 6x3 is 3x(6x7 - 3x3 - 2X2).
Explain her error and give the correct factored
form.
Lesson 15.1
531
34.
35.
Raspberries come in a container with a square bottom
whose bottom side length is x. An expression, for its
volume is x3 - 2x2. Blueberries come in a container with
a square bottom whose bottom side length is (x - 2). An
expression for its volume is x3 - 4x2 + 4x. Factor both
expressions.
o
The area of a rectangle is represented by the polynomial
x2 + 3x-6x- 18.
a.
Find possible expressions for the length and width of the rectangle.
b.
Use your answers from part a to find the length, width, and area of the
rectangle if x = 12.
FOCUS DIM HIGHER ORDER THINKING
36.
Critical Thinking Show two methods of factoring the expression
ax — bx — ay + by. Is the result the same?
37. Explain the Error Audrey and Owen
came up with two different answers
when they factored the expression
3n3 - n2. Who was correct? Explain the
error.
_
Owen
Audrey
3n3 - n2
3n3 - n2
n2(3n) - n2 (0)
n 2 (3n)-n 2 (1)
n2(3n - 0)
n 2 (3n-1)
38. Communicating Mathematical Ideas
Describe how to find the area of the figure. Show
each step and write your answer in factored form.
532
Work Area
Unit 4
i
2x +
x+8
A.SSE.2
LESSON
:
15*2 Factoring x2 + bx + c
Use the structure of an
expression to identify ways to
rewrite it. AlsoA.SSE.3
How can you factor expressions of the form x2 + bx + c?
EXPLORE ACTIVITY 1
A.SSE.2
Exploring Factors of x2 + bx + c
when c is Positive
You know how to multiply binomials using FOIL. In this lesson, you will learn how
to reverse this process and factor trinomials into two binomials.
Use algebra tiles to factor x1 + 7x + 6.
/—\} Identify the tiles you need to model the expression.
jc-tile(s), and
f
unit tile(s)
X
B ) Arrange the algebra tiles on the grid. Place the
n
x
the upper left corner, and arrange the
unit tiles in two
rows and three columns in the lower right corner.
Fill in the empty spaces on the grid with x-tiles. Only
jc-tiles fit on the grid, so this arrangement is not correct.
X
Rearrange the unit tiles into a rectangle with different dimensions.
|" I i ' •
!'
:
i
What is the length and width of the new rectangle?
E,' Fill in the empty spaces on the grid with #-tiles.
All
is correct.
x+6
:t-tiles were used, so this arrangement
REFLECT
1. Finn checks the answer by multiplying and gets x2 + lx + 6x + 6. He believes
he must have made a multiplication error. Is he correct? Explain.
Lesson 15.2
533
EXPLORE ACTIVITY 1 (confd)
2. What If? Suppose the second arrangement of unit tiles was a rectangle
1 tile high and 6 tiles wide. Could the arrangement have used a rectangle
6 tiles high and 1 tile wide? Explain.
3. Critical Thinking Are there any other ways to factor the polynomial
x2 + 7x + 6 besides (x + l)(x + 6)? Explain.
EXPLORE ACTIVITY 2
A.SSE.2
Exploring Factors of x2 + bx + c
when c is Negative
When using algebra tiles to factor polynomials, you may have to use both negative
and positive tiles.
KEY
I '£2$ = positive variable
Animated
Math
MflB = negative variable
£3 = 1 B = - i
my.hrw.com
Use algebra tiles to factor x1 + x — 2.
(A) Identify the tiles you need to model the expression.
positive x2-tile
positive x-tile
negative unit tiles
( B) The unit tiles will be placed on a grid to form a rectangle. List all the factor
pairs for 2:
Arrange the algebra tiles on the grid. Place the
positive A^-tile in the upper left
corner, and arrange the
unit tiles in the lower right corner.
534
Unit 4
negative
X
n
Fill in the empty spaces on the grid with
X
positive
x-tiles. There is
x-tile to place on the grid, so there will
be
x-tiles.
f
empty places for
\ E ) Fill the empty places with zero pairs.
A zero pair is two tiles that add to 0.
Add 1 positive
1 negative
and
to the grid.
. F ) The empty spaces on the grid were
completely filled by zero pairs, so this
arrangement is correct.
REFLECT
4. Analyze Relationships Why were the unit tiles not rearranged when the
x-tile did not complete the grid?
5. Why were both positive x-tiles placed in the top row?
Factoring Trinomials
When factoring a polynomial in the form x2 + bx + c, you are looking for two
binomials in the form (x + n) and (x + m}, where n and m are a pair of numbers
whose product is c and whose sum is b.
The first step is to find factor pairs of c. Always pay attention to the sign of c.
If c is positive, find factors of c that both have the same sign. If c is negative, find
one positive factor and one negative factor.
Lesson 15.2
535
EXAMPLE 1
C.^CACC A.SSE.2
f A ) Factor x2 - 7x + 12.
STEP 1
th Talk
List factor pairs of c and find the sum of each pair. Since c = 12,
use factor pairs where both factors have the same sign.
Factors of 12
Mathematical Practices
Sum of Factors
1 and 12
If c is positive, how can
you use the sign of b to
decide whether to choose
positive or negative
factors of c?
1 +12-13
2 and 6
3 and 4
3 + 4= 7
-1 and-12
-2_and -6
-j
f - 3 and -4
STEP 2
(-3) + (-4) = -7
-7 is the sum that
you're looking for.
Use the factor pair whose sum equals b to factor the polynomial.
x2 - 7x + 12 = (x - 3)0 - 4)
Factor x2 -\-4x - 45.
STEP 1
List factor pairs of c and find the sum of each pair. Since
c = —45, use factor pairs where one factor is positive and the
other factor is negative.
Factors of —45 j
1 and -45
3and-15
5 and-9
I 9 and-5
STEP 2
Sum of Factors
1 + (-45) - -44
j 3 + (-15) = -12
5 + (-9) = -4
9 + (-5) = 4
4 is the sum that
you're looking for.
You can stop here.
Use the factor pair whose sum equals b to factor the polynomial.
x2 + 4x - 45 = (x + 9)(x - 5)
REFLECT
6. When factoring a trinomial of the form x2 + fox + c where c is negative, one
binomial factor contains a positive factor of c and one contains a negative
factor of c. How do you know which factor of c should be positive and which
should be negative?
536
Unit 4
YOUR TURN
Factor each trinomial.
7.
x2
+ 5x + 4
8.
9. x2 - 5x - 14
10.
12.
11.
1. Use algebra tiles to factor x2 + 6jc + 8. (Explore Activity 1)
a. Identify the tiles you will need to model the expression.
x -tile
x-tiles
unit tiles
b. This arrangement does not model the correct factors
because it needs
x-tiles to fill the grid.
c. This arrangement models the correct factors because it
needs
d.
2
X
X
I ' !
X
x-tiles to fill the grid.
I £•
i o
I
i uX ~ t ~ O — \^X '
>(*+(")
**'*XX^-**X:-=-#K'-
2.
Use algebra tiles to factor x2 — 4x — 5. (Explore Activity 2)
a. Identify the tiles you will need to model the expression.
positive %2-tile
negative unit tiles
negative x-tiles
b. This arrangement has space for
have to add a
c. x2 - 4x - 5 = (x - {
x-tiles. You will
X
of x-tiles.
j) (x + f
^
Lesson 15.2
537
3. Factor the polynomial x2 — Wx + 9. (Example 1)
o
a. Complete the table with factor pairs of c.
.
.
Factors of 9
Sum of Factors
1 +9=10
1 and
3 and
.
3+
i
-1 and
and
.+
The factor pair whose sum equals b is
D
and
C
\-
I'.
.--
i)
Factor each trinomial. (Example 1)
4. x2 + 6x + 9
5. x1 - 5x + 4
6. x2 - 3x - 18
7. jc2 + 14* + 40
- 36
8.
10. oC - 7x-30
9. x2 - 2x - 35
11.
12. The length of a rectangular porch is (x -f 7) feet. The area of the porch is
(x2 + 9x + 14) square feet. Factor the expression for the area in order
to find an expression for the width of the porch.
ESSENTIAL QUESTION CHECK-IN
538
Unit 4
\. How can you factor expressions of the form x2 -f bx -f c?
Date.
Class
Name.
5.2 Independent Practice
S3bfc*- ?-•;-. v.viucJfcun&fetii&»Mft* •
»fe«M»«WBrihi|
A.SSE.2, A.SSE.3
Factor each trinomial.
22.
14. x2 - 2x- 15
15.
9x+18
A homeowner wants to enlarge a rectangular
closet that has an area of (x2 -\- 3x + 2) ft2.
The length is (x -f 2) ft. After construction,
the area will be (x2 + 8x + 15) ft2 with a
length of (x + 3) ft.
a.
Find the dimensions of the closet before
construction.
b.
Find the dimensions of the closet after
construction.
c.
By how many feet will the length and
width increase after construction?
16.
17.
18. x2 - 10*-24
19. x 2 - 12%+ 32
20.
Write the polynomial modeled and then
factor it.
21. The area of a rectangle in square feet can
be represented by x2 + 8x -f- 12. The length
is (x + 6) ft. What is the width of the
rectangle?
23.
Can all trinomials of the form x2 + bx + c
be factored? Explain and defend your answer
with an example.
24.
Give a value of b that would make
x2 + bx — 36 factorable. Show the
factorization.
Lesson 15.2
539
25.
Represent Real-World Problems The area of a rectangular
fountain is (x2 + I2x + 20) ft2. The width is (x + 2) ft.
a.
Find the length of the fountain.
b. A 2-foot wide walkway is built around the fountain. Find the
dimensions of the outside border of the walkway.
c.
26.
Find the total area covered by the fountain and walkway.
Give a value of b that would not make x2 + bx — 36 factorable. Show
that it cannot be factored.
FOCUS ON HIGHER ORDER THINKING
27. Justify Reasoning The area of a rectangle is x2 + 6x + 8. The length is
(x + 4). Find the width of the rectangle. Is the rectangle a square? Explain.
28.
Communicate Mathematical Ideas Rico says the expression x2 -f bx -f c is
factorable when b — c = 4. Are there any other values where b — c that make
the expression factorable? Explain.
29.
Critical Thinking Explain how to find all the possible positive values of b
such that x2 + bx + 6 can be factored into binomial factors. Write the possible
trinomials.
540
Unit 4
Work Area
LESSON
15«3 Factoring ax1 + bx + c
Use the structure of an
expression to identify ways to
rewrite it. AlsoA.SSE.3
• . ...... .':.
How can you factor expressions of the form ox2 + bx -\- cl
Factoring ax2 + bx + c where c > 0
When you factor a polynomial in the form ax2 + bx + c, the result will be the
product of two binomial factors, in the form (H-X + 91) (3^ + B)- The product
of the two coefficients of x will be a, and the product of the two constant terms will
be c. The sum of the products of the inner and outer terms will be bx.
Q5)my.ht
Product =
Product =
Sum of outer and inner products = b
EXAMPLE
CACC A.SSE.2
) Factor 4x2 + 26* + 42.
STEP1
Factor out any common factors of 4, 26, and 42.
4X2 + 26x + 42 = 2(2x2 + 13* + 21)
Make a table that lists the factor pairs for a and c. Find the
value of b that results from each combination of factor pairs.
STEP 3
Factors of a
Factors of c
1 and 2
1 and 21
1 and 2
3 and 7
1 and 2
7 and 3
1 and 2
21 and 1
= 23
13 is the sum that
you're looking for.
(1X1) + (2(21) = 43
Use the combination of factor pairs that yields the correct value
of b to factor the polynomial.
(Ix + 3)(2x + 7) = (x + 3)(2x + 7)
4X2 + 26* + 42 = 2(x + 3)(2x + 7)
Factor 3x2 - 26x + 35.
STEP 1
Factor out any common factors of 3, -26, and 35.
3, -26, and 35 share no common factors other than 1.
Lesson 15.3
541
STEP 2
Make a table that lists the factor pairs for a and c. Find the value
of b that results from each combination of factor pairs.
Factors of a
a =3
My Notes
1and3
STEP 3
_
Factors of c
c = 35
Outer Product +
Inner Product
— 1 and —35
(l)(-35) + (3)(-l) = -38
1 and 3
-5 and -7
~(1j(-7) + (3X-5) = -22
1 and 3
—7 and —5
(1)(-5) + (3)(-7) = -26 x
land 3
—35 and —1
(1)(-1) + (3)(-35) = -106N
Use the combination of factor pairs that yields
the correct value of b to factor the polynomial.
— 26 is the sum that
you're looking for.
3X2 - 26x + 35 = (Ix - 7)(3x - 5) = (x - 7)(3x - 5)
REFLECT
1. Critical Thinking When factoring 3x2 - 26x + 35, why should both factors
of c be negative?
2.
What If? If none of the factor pairs for a and c result in the correct value
for b, what do you know about the polynomial?
YOUR TURN
Factor each polynomial.
3.
5.
Personal
Math Trainer
Online Practice
and Help
imy.hrw.com
542
Unit 4
7.
Bx2 - 14* + 8
- 48x + 45
I4x2 + 33* + 18
4.
6.
8.
3x2+llx+6
62x + 70
50*2- 165x+ 135
Factoring ax2 + bx + c where c < 0
When factoring ax1 -f bx + c, if the value of c is negative, you know that one of
the factors of c must be negative and one must be positive. Apply what you already
know about factoring trinomials to this new situation.
(p> my.hrw.com
EXAMPLE 2
(A) Factor 6X2 - 2lx - 45.
STEP 1
Factor out any common factors of 6, -21, and -45.
6x2 - 2lx -45 = 3(2x2 - 7x - 15)
STEP 2
Make a table that lists the factor pairs for a and c. Since c
is negative, one factor will be positive and the other will be
negative.
Factors of a
o=2
STEP 3
Factors of c
c=-15
Outer Product +
Inner Product
land 2
1 and -15
land 2
3 and -5
(1)(-5) + (2)(3) = 1
1 and 2
5 and -3
(1)(-3) + (2)(5) = 7
land 2
15 and —1
(1)(-1) + (2)(15) = 29
1 and 2
-1 and 15
(1)(15) + (2)(-1) = 13
land 2
-3 and 5
(1)(5) + (2)(-3) = -1
land 2
—Sand 3
(1)(3) + (2)(-5) = -7>-
1 and 2
-15 and 1
(1)(-15) + (2)(1) = -13
7 is the sum that
u're looking for.
(1)(1) + (2)(-15)=-29
Use the combination of factor pairs that yields the correct value
of b to factor the polynomial.
(Ix - 5)(2x + 3) = (x - 5)(2x + 3)
6X2 - 2lx - 45 = 3(x - 5)(2x + 3)
4x - 35.
Factor out any common factors for 4, 4, and -35.
4, 4, and -35 share no common factors other than 1.
H
©
STEP 2
Make a table that lists the factor pairs for a and c. Since c
is negative, one factor will be positive and the other will be
negative.
Lesson 15.3
543
Factors of a
o =4
How does the sign of c help
you choose the correct
factor pair for c?
Factors of c
c=-35
Outer Product +
Inner Product
1 and 4
1 and -35
1 and 4
5 and -7
(1)(_7) + (4)(5) = 13
1 and 4
7 and —5
(1)(_5) + (4)(7) = 23
(1)(_35) + (4)(1) = -31
1 and 4 _ . -i
1 and 4
35 and -1
(1)(-1) + (4)(35) = 139
-land 35
(1)(35) + (4)(-1) = 31
1 and 4
-5 and 7
(1)(7) + (4)(-5)=-13
land 4
-7 and 5
(1)(5) + (4)(-7)=-23
1 and 4
-35 and 1
(1)(1) + (4)(-35)=-139
2 and 2
1 and —35
2 and 2
5 and -7
(2)(-7) + (2)(5)=-4
2 and 2
7 and -5
(2)(-5) + (2)(7) = 4
2 and 2
35 and -1
(2)(-35) + (2)(1) = -68
(2)~(-1 ) + (2)(35) = 68\P 3
the correct value of b to factor the polynomial.
4 is the sum that
you're looking for.
4X2 + 4x - 35 = (2x + 7)(2x - 5)
REFLECT
9.
What If? Suppose a is a negative number. What would be the first step in
factoring ax2 + bx + c? Explain.
10. Make a Conjecture Using the information in the tables in Example 2,
make a conjecture about what happens to b when you swap the positions of
the plus and minus signs in the binomial factors.
YOUR TURN
Factor each polynomial.
11. 24x2 + 32x- 6
544
Unit 4
12.
2lx-8
Use th
1. Factor 3x2 + 13* + 12. (Example 1)
Complete the table for all factors of a and c.
Factors of a
a =3
1 and 3
Factors of c
c=12
.
1 and 12
1 and 3
2 and
1 and 3
Outer Product +
inner Product
(1)(12) + (3)(1) =
(1)(
)+ (
V2^ = 12
and
(IX
) + f3V
1 and 3
and
MX
)+ (
1 and 3
and
1 and 3
and 1
)(4> =
)+ (
v
mm + (
V
mr
)) = 37
o
The factored form of Sx2 + 13* + 12 is (x +
v».~—^:
2.
)=
V-
Factor 8*2 — 2x — 6. (Example 2)
8, —2, and —6 have a common factor of
., so 8x — 2x — 6 —
- x - 3)
Complete the table for all factors of a and c.
Factors of a
Factors of c
c= -3
Outer Product +
Inner Product
1 and 4
1 and -3
1 and
3 and
1 and
and
(IV
) +(
V
1 and
and
OX
)+ (
/ L
2 and
and
and
3 and
(1)(-3) + (4)(1) = 1
OH
(2V
\ / \
) + (4}(
\\
V
(2V
)(
) + (/ \
}=
)=
)=
!
VI) =
) + (2V3) = 4
The factored form of Sx2— 2x — 6 is
[| ESSENTIAL QUESTION CHECK-IN
3. How can you factor expressions of the form ax2 + bx + c?
Lesson 15.3
545
Class.
Name
Date
A.SSE.2, A.SSE.3
Factor each trinomial, if possible.
4. 30*2 + 35* - 15
5. 6X2 -29x
13. The area of a soccer field is
(24x2 + 100* + 100) m2. The width of the
field is (4x + 10) m. What is the length?
14. Find all the possible values of b such that
3X2 + bx — 2 can be factored.
82x + 56
6.
15. Write the polynomial modeled, and then
factor it.
7.
8. 30c/2
9. 2/
14
16. Representing Real-World Problems The
10.
attendance at a team's basketball game can
be approximated with the polynomial
Sx2 + 80* + 285, where x is the number of
wins the team had in the previous month.
11.
a.
12. How is factoring a trinomial in the form
ax2 -f fot + c similar to factoring a trinomial
in the form x2 + bx + c* How is it different?
Factor the polynomial completely.
b. Estimate the attendance if the team won
4 games in the previous month.
17. Kyle stood on a bridge and threw a
rock up and over the side. The height of the
rock, in meters, can be approximated by
—5t 2 + 5t + 24, where t is the time in
seconds after Kyle threw it. Completely
factor the expression.
546
Unit 4
18. A triangle has an area of |(4x2 + 29x + 30) ft2. If the base of the triangle is
(x + 6) ft, find the height of the triangle.
19. Draw Conclusions If a polynomial in the form ax2 4- bx -f c has a = b = c=l,
can the expression be factored? Explain.
20.
Counterexamples Marc thinks the only time a polynomial in the form
ax2 -\- bx, + c cannot be factored is when at least one of the values for a, b, or c is
a prime number. Find a counterexample to Marc's statement.
21. Shruti has a rectangular picture frame with an area of 30x2 -f 5x — 75 cm2.
.i1
a.
Find the width of the frame if the height is (3x + 5) cm.
b.
Find the width of the frame if the height is (2x — 3) cm.
c.
Find the width of the frame when the height is 5 cm.
22.
Communicate Mathematical Ideas Has the expression (3x + 7)(6x -\- 3
been factored completely? Explain.
23.
Explain the Error Luna performed the work shown below to factor the
polynomial 24X2 + 18x + 3. Explain her error, and find the correctly factored
form.
24x2 + 18;c + 3 =
+ 6x + 0)
+ 6x)
Lesson 15.3
547
24.
25.
The length of Rebeccas rectangular garden was two times the width, w. Rebecca
increased the length and width of the garden so that the area of the new garden
is (2w2 + 7w + 6) square yards. By how much did Rebecca increase the length
and the width?
The height in feet above the ground of a football that has been thrown or
kicked can be described by the expression — 16^ + vt + h where t is the time
in seconds, v is the initial upward velocity in feet per second, and h is the initial
height in feet.
a.
Write an expression for the height of a football at time t when the initial
upward velocity is 20 feet per second and the initial height is 6 feet.
b.
Factor your expression from part a.
c.
Find the height of the football after 1 second.
FOCUS ON HIGHER ORDER THINKING
26.
Critical Thinking Is there a value of m that will make x2 + mx + 80
factorable? If so, how many? Explain and give all the possible values.
27.
Explain the Error Frank has factored the polynomial I2x2 + 5x — 2 as
(3x — l}(4x + 2). Explain his error. Give the correct factorization.
28. Communicate Mathematical Ideas Can the polynomial 4X2 + Ox - 25 be
factored? Explain.
548
o
Unit 4
Work Area
LESSON
Factoring Special
15.4 Products
A.SSE.2
Use the structure of an
expression to identify ways to
rewrite it. AlsoA.SSE.3
: How can you use special products to aid in factoring?
EXPLORE ACTIVITY
A.SSE.2
Factoring a Perfect-Square Trinomial
j
When you use algebra tiles to factor a polynomial, you must arrange the unit
tiles on the grid in a rectangle. Sometimes, you can arrange the unit tiles to form
a square. Trinomials of this type are called perfect-square trinomials.
Use algebra tiles to factor x2 + 6x + 9.
( A j Identify the tiles you need to model the expression.
x2-tile;
:
x-tiles;
unit tiles
( B j The unit tiles will be placed on a grid to form a square. Which
x
factor pair for 9 will arrange the tiles in a square?
Arrange the algebra tiles on the grid. Place the
x2-tile in the upper left corner, and arrange the
unit tiles in the lower right corner.
( D) Fill in the empty spaces on the grid with x- tiles.
All
is correct.
X
. x-tiles were used, so this arrangement
>—-s,
x2 + 6x + 9 = (x +
(
j)
+ 1+ + +
Now, use algebra tiles to factor x2 — Sx + 16.
( E) Identify the tiles you need to model the expression.
positive ar-tile
negative x-tiles
positive unit tiles
The unit tiles will be placed on a grid to form a square. Which factor
pair for 16 mil arrange the tiles in a square?
Lesson 15.4
549
EXPLORE ACTIVITY (confc
\.G/ Arrange the algebra tiles on the grid.
o
X
positive
Place the
'-tile in the upper left corner, and
positive unit tiles
arrange
in the lower right corner.
(H • Fill in the empty spaces on the grid
X
with x-tiles. All
. negative
x-tiles were used, so this arrangement is
correct.
x2 - Sx + 16 =
-fl
REFLECT
How would the algebra
tile grid change if
the trinomial was
x2 + 8x + 16?
1. What If? Suppose that the middle term in x2 -j- 6x + 9 was changed from
6x to lOx. How would this affect the way you factor the polynomial?
2. If the unit tiles are arranged in a square when factoring with algebra tiles,
what will be true about the binomial factors?
Factoring Perfect-Square Trinomials
m
Math On the Spot
) my.hrw.com
A trinomial is a perfect-square trinomial if the first and last terms are perfect
squares and the middle term is 2 times one factor of the first term times one
factor of the last term. This can be represented algebraically in either the form
a2 + 2ab + b2 or the form a2 — 2ab + b2. Factor perfect-square trinomials
according to the rules below.
Perfect-Squat
lomiais
Perfect-Square Trinomial
a2 + 2ab + b2 = (a + b)(a + b)
Example
3)(x+3)
1 = (x-1)(jf-1)
550
Unit 4
HCACC A.SSE.2
EXAMPLE 1
Factor each perfect-square trinomial.
My Notes
#2 + 2(*)(6) +6 2
Rewrite in the form a2 + 2af7+b 2 .
(x + 6) (jc + 6)
Rewrite in the form (a + b) (a 4- b).
......... .................................... ..
The factored form of x1 + 12* + 36 is (x + 6)(* + 6), or (x + 6)2.
(§) 4x2 - 12* + 9
(2#)2 - 2(2#)(3) -I- 32
Rewrite in the form a2 - 2^+ b2,
(2jc — 3)(2AC — 3)
Rewrite in the form (a — b)(a — b).
The factored form of 4x2 - I2x + 9 is (2x - 3)(2x - 3), or (2x - 3)2.
(C) 36X2 + 180* + 225
9(4*2 + 20x + 25)
Factor out the GCF of the terms.
9[(2x)2 + 2(2x)(5) + 52]
Rewrite in the form a2 + 2ab+ b2,
9(2x + 5)(2x + 5)
Rewriteintheform(a + ^)(a + /?).
...................... ............ ............
The factored form of 36X2 + 180* + 225 is 9(2* + 5)(2x + 5), or 9(2x + 5)2.
REFLECT
3. Lee says that the trinomial 4X2 + I5x + 9 is a perfect- square trinomial,
because 4x and 9 are both perfect squares. Is Lee correct? Explain.
4.
.............
......... ...... ... ......
Wendy checked the answer to Example 1A. Her work is shown below.
Explain her error.
(x + 6)2 = x2 + 62 = x2 + 36
5. Perfect-square trinomials can be in the form a2 + lab + b2 or
a2 — 2ab + b2. Why is the b2 term always positive?
Lesson 15.4
551
BBBMBMI
YOUR TURN
Factor each perfect-square trinomial.
6. x2 + 16* + 64
7. 25.T + 60* + 36
8. 36x2 - 12* + 1
9. 16X2 - 16* + 4
10. 9x2-l8x
11.
Factoring a Difference of Squares
What does "difference of two squares" mean?
x2 - 100
Math On the Spot
/ I \e difference square
imy.hrw.com
A polynomial is a difference of two squares if:
• It has two terms, one subtracted from the other.
• Both terms are perfect squares.
4x2
A
- 9
A
2x 2 x - 3 3
The difference of two squares can be written algebraically as a2 — b2 and factored
as (a + b)(a - b).
Difference of Two Squares
Difference of Two Squares
Example
x2- 9 = (x+3)(x-3)
552
Unit 4
EXAMPLE 2
r*]CACC
A.SSE.2
Factor each difference of squares.
th Talk
(A) x2 - si
Mathematical Practices
x2 — 92
Rewrite in the form a2 — b2.
(x -\- 9) (x — 9)
Rewrite in the form (a + b) (a — b).
Why isn't there a
fc> term in a difference of
two squares?
The factored form of x2 - 81 is (x + 9)(x - 9).
(B) 16<?2 - 9p
Remember^"7)" = *'""•
(4g)2 — (3p2)2
Rewrite in the form a2 — b2.
(4q + 3p2) (4q — 3p2)
Rewrite in the form (a + b) (a — b}.
The factored form of I6q2 - 9p4 is (4q + 3p2)(4q - 3p2).
(C} 4/ - 25>'2
y2(4y2 — 25)
Factor out the <3CF of the terms.
y2[(2y)2 - 52}
Rewrite in the form a2 - b2.
y2(2y + 5)(2j — 5)
Rewrite in theform (a + b)(a — b}.
The factored form of 4/ - 25y2 is y2(2y + 5)(2y - 5).
REFLECT
12. Explain the Error Sam factored 9x6 - 25x4 as (3x3 + 5x2)(3*3
Explain his error.
YOUR TURN
Factor each difference of squares.
13. 1 -4x2
14.
15.
Personal
Math Trainer
(o) my.hrw.com
Lesson 15.4
553
uided Practice
For each trinomial, draw algebra tiles to show the factored form, then write the
factored form. (Explore Activity)
1. xl - 10*+ 25
2. x2 + 8x + 16
X
X
Factor each perfect-square trinomial. (Example 1)
3. 9:xr + 18* + 9
4.
6. 4JT + 24JC
7.
8. 25X2 + 10* + 1
Factor each difference of two squares. (Example 2)
9. r - 16
12. 400*4 - 4S4*2
10. 81 - 144x4
11. x* - 49
13.
14. 25^-64
)| ESSENTIAL QUESTION CHECK-IN
15. How can you use the rules for special products to aid in factoring?
554
Unit 4
o
Class
Name.
Date.
.4 Independent Practice
te£y»a^fct>Mft^%^Jfai»i^v
1
«, ,v?.,*iNii,yAi*;*to&&' !•?• :<> v, ''SkSi/Ui&i*
A.SSE.2, A.SSE.3
Determine whether each polynomial is a perfectsquare trinomial or the difference of two squares.
Then factor each expression.
22.
16. 4x 2 -20x + 25
17.
Represent Real-World Problems You are
given a sheet of paper and asked to cut
out a square piece with an area of
(4x2 - 44% +121) mm2. The dimensions
of the square have the form ax — b,
where a and b are whole numbers.
a.
Find the length of one of the sides of the
square you cut out.
b.
Find an expression for the perimeter of
the square you cut out.
c.
Find the perimeter when x — 41.
18. 49Af + 140% + 100
19. 4% 2 -36
20.
23.
Critique Reasoning Michelle factored
x2 - 6x + 9 as follows:
x2- 6x + 9 =
x2 - 2(x • 3) + 32 =
(x-3)(x + 3)
Is she correct? If not, explain and correct
her error.
24.
A square poster has an area of x2 +16% + 64
square inches. Find the length of one side of
the square.
An architect is designing square windows
with an area of (x2 + 20* + 100) ft2. The
dimensions of the windows are of the form
ax + by where a and b are whole numbers.
a.
Find the dimensions of each square
window.
b.
Find an expression for the perimeter of
a window.
c.
_
Find the perimeter of a window when
21. Explain the Error Ed factored 16%2 — Sxy
+ y2 as (4x + y}2. What was his error?
Lesson 15.4
555
25.
An artist framed a picture. The dimensions
of the picture and frame are shown.
Completely factor the expression for the area of
the frame.
4x
26.
Explain how to find the value of z if you know
that WOx2 + 120* + z is a perfect square
trinomial.
2y
27. Multi-step The area of a square is (36d2 - 36d + 9) in2.
a.
What expression represents the length of a side of the square?
b.
What expression represents the perimeter of the square?
c.
What are the length of a side, the perimeter, and the area of the square
when d = 2?
KL8L7A
FOCUS ON HIGHER ORDER THINKING
28.
Critical Thinking Sinea thinks that the fully factored form of the expression
(x4 - 1) is (x2 - 1)02 + 1). Is she correct? Explain.
29.
Explain the Error When Jeremy factored I44x2 — 100, he first got
(I2x + 10)(12x — 10). Then he noticed a common factor, so he factored it
further to get 2(6x + 5)(6.x — 5). What was his error, and what is the correct
factorization?
30.
Communicate Mathematical Ideas Explain how to fully factor the
expression x4 — 2x2y2 + y4.
556
Unit 4
Work Area
Ready
Personal
Math Trainer I
Online Practice
and Help
15.1 Factoring Pol)
Factor each expression.
1.
-Ux-Ux2
2. 3x(x + 6) - 5(6 + x)
3. 2x3- Ux2 + 18 - 3x
15.2 Factoring x2 + bx + c
Factor each trinomial.
4. x2 - 20* + 19
5. n2 4- 13n + 36
6. x2-4x-2l
153 Factoring ax2 -f bx + c
Factor each trinomial.
7. 3n2 - 26n + 35
8.
+ 11 + 4
15*4 Factoring Special Products
Determine whether each polynomial is a perfect-square trinomial or
the difference of two squares. Then factor each polynomial.
11.
10.
12.
ESSENTIAL QUESTION
"^
How can you factor expressions of the form ax -\-bx-\- c?
9. -4X2 -lQx-6
""I!
CALIFORNIA
Personal
Math Trainer
MODULE 15
MIXED REVIEW
Online Practice
and Help
Assessment Readiness
1. Look at each polynomial below. Does the polynomial have a factor of (x - 4)?
Select Yes or No for polynomials A-C.
A. 6x- 10
B. x2-x-'\2
O Yes
O No
O Yes Q N°
C. 2x2 + 7 x - 4
O
Yes
O
No
2. Consider the polynomial 4X2 + 40x -f 36.
Choose True or False for each statement.
A. The GCF of the terms is 4.
B. The factored form has a factor of
(x+1).
C. The polynomial is a perfect-square
trinomial.
Q
True
O
False
O True
Q False
O
O
True
False
3. A school plans to expand its rectangular vegetable garden. Both the length and
the width will be increased by x feet. After the expansion, the garden will have an
area of (x2 -j- 12x + 32) square feet. What are the original length and width of the
garden? Explain how you used factoring to determine your answer.
4. Emily can spend no more than $40.00 on a raincoat. Sales tax is 8.25% of the
marked price. Can Emily afford a raincoat priced at $37.95? Write and solve an
inequality to support your answer.
558
Unit 4