Factors and Greatest

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Factors and Greatest Common Factors
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Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 1
Holt McDougal Algebra 1
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Warm Up 1. 50, 6 2. 105, 7 3.
List the factors of 28. Tell whether each number
is prime or composite. If the number is
composite, write it as the product of two
numbers.
Tell whether the second number is a factor of the
first number
no
yes
1, 2, 4, 7,
14, 28
prime
composite 49 ? 2
4. 11
5. 98
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Objectives
Write the prime factorization of numbers. Find
the GCF of monomials.
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Vocabulary
prime factorization greatest common factor
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The whole numbers that are multiplied to find a
product are called factors of that product. A
number is divisible by its factors.
You can use the factors of a number to write the
number as a product. The number 12 can be
factored several ways.
Factorizations of 12
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The order of factors does not change the product,
but there is only one example below that cannot
be factored further. The circled factorization is
the prime factorization because all the factors
are prime numbers. The prime factors can be
written in any order, and except for changes in
the order, there is only one way to write the
prime factorization of a number.
Factorizations of 12
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Example 1 Writing Prime Factorizations
Write the prime factorization of 98.
Method 1 Factor tree
Method 2 Ladder diagram
Choose any two factors of 98 to begin. Keep
finding factors until each branch ends in a prime
factor.
Choose a prime factor of 98 to begin. Keep
dividing by prime factors until the quotient is 1.
The prime factorization of 98 is 2 ? 7 ? 7 or 2
? 72.
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Check It Out! Example 1
Write the prime factorization of each number.
a. 40
b. 33
40 23 ? 5
33 3 ? 11
The prime factorization of 40 is 2 ? 2 ? 2 ? 5
or 23 ? 5.
The prime factorization of 33 is 3 ? 11.
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Check It Out! Example 1
Write the prime factorization of each number.
c. 49
d. 19
49 7 ? 7
19 1 ? 19
The prime factorization of 49 is 7 ? 7 or 72.
The prime factorization of 19 is 1 ? 19.
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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 of the common factors is 4.
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Example 2A Finding the GCF of Numbers
Find the GCF of each pair of numbers.
100 and 60
Method 1 List the factors.
factors of 100 1, 2, 4, 5, 10, 20, 25, 50, 100
List all the factors.
factors of 60 1, 2, 3, 4, 5, 6, 10, 12, 15, 20,
30, 60
Circle the GCF.
The GCF of 100 and 60 is 20.
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Example 2B Finding the GCF of Numbers
Find the GCF of each pair of numbers.
26 and 52
Method 2 Prime factorization.
Write the prime factorization of each number.
26 2 ? 13
52 2 ? 2 ? 13
Align the common factors.
2 ? 13 26
The GCF of 26 and 52 is 26.
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Check It Out! Example 2a
Find the GCF of each pair of numbers.
12 and 16
Method 1 List the factors.
List all the factors.
factors of 12 1, 2, 3, 4, 6, 12
Circle the GCF.
factors of 16 1, 2, 4, 8, 16
The GCF of 12 and 16 is 4.
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Check It Out! Example 2b
Find the GCF of each pair of numbers.
15 and 25
Method 2 Prime factorization.
Write the prime factorization of each number.
15 1 ? 3 ? 5
25 1 ? 5 ? 5
Align the common factors.
1 ? 5 5
The GCF of 15 and 25 is 5.
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You can also find the GCF of monomials that
include variables. To find the GCF of monomials,
write the prime factorization of each coefficient
and write all powers of variables as products.
Then find the product of the common factors.
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Example 3A Finding the GCF of Monomials
Find the GCF of each pair of monomials.
15x3 and 9x2
Write the prime factorization of each coefficient
and write powers as products.
15x3 3 ? 5 ? x ? x ? x
9x2 3 ? 3 ? x ? x
Align the common factors.
3 ? x ? x 3x2
Find the product of the common factors.
The GCF of 15x3 and 9x2 is 3x2.
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Example 3B Finding the GCF of Monomials
Find the GCF of each pair of monomials.
8x2 and 7y3
Write the prime factorization of each coefficient
and write powers as products.
8x2 2 ? 2 ? 2 ? x ? x
7y3 7 ? y ? y ? y
Align the common factors.
There are no common factors other than 1.
The GCF 8x2 and 7y3 is 1.
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Check It Out! Example 3a
Find the GCF of each pair of monomials.
18g2 and 27g3
Write the prime factorization of each coefficient
and write powers as products.
18g2 2 ? 3 ? 3 ? g ? g
27g3 3 ? 3 ? 3 ? g ? g ? g
Align the common factors.
3 ? 3 ? g ? g
Find the product of the common factors.
The GCF of 18g2 and 27g3 is 9g2.
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Check It Out! Example 3b
Find the GCF of each pair of monomials.
Write the prime factorization of each coefficient
and write powers as products.
16a6 and 9b
16a6 2 ? 2 ? 2 ? 2 ? a ? a ? a ? a ? a ? a
9b
3 ? 3 ? b
Align the common factors.
The GCF of 16a6 and 9b is 1.
There are no common factors other than 1.
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Check It Out! Example 3c
Find the GCF of each pair of monomials.
8x and 7v2
Write the prime factorization of each coefficient
and write powers as products.
8x 2 ? 2 ? 2 ? x
7v2 7 ? v ? v
Align the common factors.
There are no common factors other than 1.
The GCF of 8x and 7v2 is 1.
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Example 4 Application
A cafeteria has 18 chocolate-milk cartons and 24
regular-milk cartons. The cook wants to arrange
the cartons with the same number of cartons in
each row. Chocolate and regular milk will not be
in the same row. How many rows will there be if
the cook puts the greatest possible number of
cartons in each row?
The 18 chocolate and 24 regular milk cartons must
be divided into groups of equal size. The number
of cartons in each row must be a common factor of
18 and 24.
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Example 4 Continued
Find the common factors of 18 and 24.
Factors of 18 1, 2, 3, 6, 9, 18
Factors of 24 1, 2, 3, 4, 6, 8, 12, 24
The GCF of 18 and 24 is 6.
The greatest possible number of milk cartons in
each row is 6. Find the number of rows of each
type of milk when the cook puts the greatest
number of cartons in each row.
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Example 4 Continued
When the greatest possible number of types of
milk is in each row, there are 7 rows in total.
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Check It Out! Example 4
Adrianne is shopping for a CD storage unit. She
has 36 CDs by pop music artists and 48 CDs by
country music artists. She wants to put the same
number of CDs on each shelf without putting pop
music and country music CDs on the same shelf. If
Adrianne puts the greatest possible number of CDs
on each shelf, how many shelves does her storage
unit need?
The 36 pop and 48 country CDs must be divided
into groups of equal size. The number of CDs in
each row must be a common factor of 36 and 48.
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Check It Out! Example 4 Continued
Find the common factors of 36 and 48.
Factors of 36 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 48 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
The GCF of 36 and 48 is 12.
The greatest possible number of CDs on each shelf
is 12. Find the number of shelves of each type of
CDs when Adrianne puts the greatest number of CDs
on each shelf.
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When the greatest possible number of CD types are
on each shelf, there are 7 shelves in total.
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Lesson Quiz Part I
Write the prime factorization of each number. 1.
50 2. 84 Find the GCF of each pair of
numbers. 3. 18 and 75 4. 20 and 36
2 ? 52
22 ? 3 ? 7
3
4
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Lesson Quiz Part II
Find the GCF of each pair of monomials. 5. 12x
and 28x3 6. 27x2 and 45x3y2 7. Cindi is
planting a rectangular flower bed with 40 orange
flower and 28 yellow flowers. She wants to plant
them so that each row will have the same number
of plants but of only one color. How many rows
will Cindi need if she puts the greatest possible
number of plants in each row?
4x
9x2

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7-2
Factoring by GCF
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 1
Holt McDougal Algebra 1
32
Warm Up 1. 2(w 1) 2. 3x(x2
4)
Simplify.
2w 2
3x3 12x
Find the GCF of each pair of monomials.
2h
3. 4h2 and 6h
13p
4. 13p and 26p5
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Objective
Factor polynomials by using the greatest common
factor.
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Recall that the Distributive Property states that
ab ac a(b c). The Distributive Property
allows you to factor out the GCF of the terms
in a polynomial to write a factored form of the
polynomial.
A polynomial is in its factored form when it is
written as a product of monomials and polynomials
that cannot be factored further. The polynomial
2(3x 4x) is not fully factored because the
terms in the parentheses have a common factor of
x.
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Example 1A Factoring by Using the GCF
Factor each polynomial. Check your answer.
2x2 4
2x2 2 ? x ? x
Find the GCF.
4 2 ? 2
The GCF of 2x2 and 4 is 2.
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Write terms as products using the GCF as a factor.
2x2 (2 ? 2)
2(x2 2)
Use the Distributive Property to factor out the
GCF.
Multiply to check your answer.
Check 2(x2 2)
The product is the original polynomial.
?
2x2 4
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Example 1B Factoring by Using the GCF
Factor each polynomial. Check your answer.
8x3 4x2 16x
8x3 2 ? 2 ? 2 ? x ? x ? x
Find the GCF.
4x2 2 ? 2 ? x ? x
16x 2 ? 2 ? 2 ? 2 ? x
The GCF of 8x3, 4x2, and 16x is 4x.
2 ? 2 ? x 4x
Write terms as products using the GCF as a factor.
2x2(4x) x(4x) 4(4x)
Use the Distributive Property to factor out the
GCF.
4x(2x2 x 4)
Check
4x(2x2 x 4)
Multiply to check your answer.
The product is the original polynomials.
8x3 4x2 16x
?
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Example 1C Factoring by Using the GCF
Factor each polynomial. Check your answer.
14x 12x2
Both coefficients are negative. Factor out 1.
1(14x 12x2)
14x 2 ? 7 ? x
12x2 2 ? 2 ? 3 ? x ? x
Find the GCF.
The GCF of 14x and 12x2 is 2x.
2 ? x 2x
17(2x) 6x(2x)
Write each term as a product using the GCF.
12x(7 6x)
Use the Distributive Property to factor out the
GCF.
2x(7 6x)
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Example 1C Continued
Factor each polynomial. Check your answer.
14x 12x2
Check 2x(7 6x)
Multiply to check your answer.
?
14x 12x2
The product is the original polynomial.
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Example 1D Factoring by Using the GCF
Factor each polynomial. Check your answer.
3x3 2x2 10
3x3 3 ? x ? x ? x
Find the GCF.
2x2 2 ? x ? x
10 2 ? 5
There are no common factors other than 1.
3x3 2x2 10
The polynomial cannot be factored further.
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Check It Out! Example 1a
Factor each polynomial. Check your answer.
5b 9b3
5b 5 ? b
Find the GCF.
9b 3 ? 3 ? b ? b ? b
The GCF of 5b and 9b3 is b.
b
Write terms as products using the GCF as a factor.
5(b) 9b2(b)
Use the Distributive Property to factor out the
GCF.
b(5 9b2)
Multiply to check your answer.
The product is the original polynomial.
5b 9b3
?
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Check It Out! Example 1b
Factor each polynomial. Check your answer.
9d2 82
Find the GCF.
9d2 3 ? 3 ? d ? d
82 2 ? 2 ? 2 ? 2 ? 2 ? 2
There are no common factors other than 1.
9d2 82
The polynomial cannot be factored further.
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Check It Out! Example 1c
Factor each polynomial. Check your answer.
18y3 7y2
1(18y3 7y2)
Both coefficients are negative. Factor out 1.
18y3 2 ? 3 ? 3 ? y ? y ? y
Find the GCF.
7y2 7 ? y ? y
The GCF of 18y3 and 7y2 is y2.
y ? y y2
Write each term as a product using the GCF.
118y(y2) 7(y2)
1y2(18y 7)
Use the Distributive Property to factor out the
GCF..
y2(18y 7)
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Check It Out! Example 1d
Factor each polynomial. Check your answer.
8x4 4x3 2x2
8x4 2 ? 2 ? 2 ? x ? x ? x ? x
4x3 2 ? 2 ? x ? x ? x
Find the GCF.
2x2 2 ? x ? x
2 ? x ? x 2x2
The GCF of 8x4, 4x3 and 2x2 is 2x2.
Write terms as products using the GCF as a factor.
4x2(2x2) 2x(2x2) 1(2x2)
2x2(4x2 2x 1)
Use the Distributive Property to factor out the
GCF.
Check 2x2(4x2 2x 1)
Multiply to check your answer.
8x4 4x3 2x2
The product is the original polynomial.
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To write expressions for the length and width of
a rectangle with area expressed by a polynomial,
you need to write the polynomial as a product.
You can write a polynomial as a product by
factoring it.
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Example 2 Application
The area of a court for the game squash is (9x2
6x) square meters. Factor this polynomial to find
possible expressions for the dimensions of the
squash court.
A 9x2 6x
The GCF of 9x2 and 6x is 3x.
Write each term as a product using the GCF as a
factor.
3x(3x) 2(3x)
3x(3x 2)
Use the Distributive Property to factor out the
GCF.
Possible expressions for the dimensions of the
squash court are 3x m and (3x 2) m.
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Check It Out! Example 2
What if? The area of the solar panel on another
calculator is (2x2 4x) cm2. Factor this
polynomial to find possible expressions for the
dimensions of the solar panel.
A 2x2 4x
The GCF of 2x2 and 4x is 2x.
Write each term as a product using the GCF as a
factor.
x(2x) 2(2x)
2x(x 2)
Use the Distributive Property to factor out the
GCF.
Possible expressions for the dimensions of the
solar panel are 2x cm, and (x 2) cm.
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Sometimes the GCF of terms is a binomial. This
GCF is called a common binomial factor. You
factor out a common binomial factor the same way
you factor out a monomial factor.
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Example 3 Factoring Out a Common Binomial Factor
Factor each expression.
A. 5(x 2) 3x(x 2)
The terms have a common binomial factor of (x
2).
5(x 2) 3x(x 2)
(x 2)(5 3x)
Factor out (x 2).
B. 2b(b2 1) (b2 1)
The terms have a common binomial factor of (b2
1).
2b(b2 1) (b2 1)
2b(b2 1) 1(b2 1)
(b2 1) 1(b2 1)
(b2 1)(2b 1)
Factor out (b2 1).
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Example 3 Factoring Out a Common Binomial Factor
Factor each expression.
C. 4z(z2 7) 9(2z3 1)
There are no common factors.
4z(z2 7) 9(2z3 1)
The expression cannot be factored.
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Check It Out! Example 3
Factor each expression.
a. 4s(s 6) 5(s 6)
The terms have a common binomial factor of (s
6).
4s(s 6) 5(s 6)
(4s 5)(s 6)
Factor out (s 6).
b. 7x(2x 3) (2x 3)
7x(2x 3) (2x 3)
The terms have a common binomial factor of (2x
3).
7x(2x 3) 1(2x 3)
(2x 3) 1(2x 3)
(2x 3)(7x 1)
Factor out (2x 3).
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Check It Out! Example 3 Continued
Factor each expression.
c. 3x(y 4) 2y(x 4)
There are no common factors.
3x(y 4) 2y(x 4)
The expression cannot be factored.
d. 5x(5x 2) 2(5x 2)
The terms have a common binomial factor of (5x
2 ).
5x(5x 2) 2(5x 2)
(5x 2)(5x 2)
(5x 2)2
(5x 2)(5x 2) (5x 2)2
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You may be able to factor a polynomial by
grouping. When a polynomial has four terms, you
can make two groups and factor out the GCF from
each group.
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Example 4A Factoring by Grouping
Factor each polynomial by grouping. Check your
answer.
6h4 4h3 12h 8
Group terms that have a common number or variable
as a factor.
(6h4 4h3) (12h 8)
2h3(3h 2) 4(3h 2)
Factor out the GCF of each group.
2h3(3h 2) 4(3h 2)
(3h 2) is another common factor.
(3h 2)(2h3 4)
Factor out (3h 2).
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Example 4A Continued
Factor each polynomial by grouping. Check your
answer.
Check (3h 2)(2h3 4)
Multiply to check your solution.
3h(2h3) 3h(4) 2(2h3) 2(4)
6h4 12h 4h3 8
The product is the original polynomial.
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Example 4B Factoring by Grouping
Factor each polynomial by grouping. Check your
answer.
5y4 15y3 y2 3y
(5y4 15y3) (y2 3y)
Group terms.
Factor out the GCF of each group.
5y3(y 3) y(y 3)
(y 3) is a common factor.
5y3(y 3) y(y 3)
(y 3)(5y3 y)
Factor out (y 3).
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Example 4B Continued
Factor each polynomial by grouping. Check your
answer.
5y4 15y3 y2 3y
Check (y 3)(5y3 y)
Multiply to check your solution.
y(5y3) y(y) 3(5y3) 3(y)
5y4 y2 15y3 3y
The product is the original polynomial.
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Check It Out! Example 4a
Factor each polynomial by grouping. Check your
answer.
6b3 8b2 9b 12
(6b3 8b2) (9b 12)
Group terms.
Factor out the GCF of each group.
2b2(3b 4) 3(3b 4)
(3b 4) is a common factor.
2b2(3b 4) 3(3b 4)
Factor out (3b 4).
(3b 4)(2b2 3)
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Check It Out! Example 4a Continued
Factor each polynomial by grouping. Check your
answer.
6b3 8b2 9b 12
Multiply to check your solution.
Check (3b 4)(2b2 3)
3b(2b2) 3b(3) (4)(2b2) (4)(3)
6b3 9b 8b2 12
The product is the original polynomial.
6b3 8b2 9b 12
?
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Check It Out! Example 4b
Factor each polynomial by grouping. Check your
answer.
4r3 24r r2 6
(4r3 24r) (r2 6)
Group terms.
4r(r2 6) 1(r2 6)
Factor out the GCF of each group.
4r(r2 6) 1(r2 6)
(r2 6) is a common factor.
(r2 6)(4r 1)
Factor out (r2 6).
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Check It Out! Example 4b Continued
Factor each polynomial by grouping. Check your
answer.
Check (4r 1)(r2 6)
Multiply to check your solution.
4r(r2) 4r(6) 1(r2) 1(6)
4r3 24r r2 6
The product is the original polynomial.
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Recognizing opposite binomials can help you
factor polynomials. The binomials (5 x) and (x
5) are opposites. Notice (5 x) can be written
as 1(x 5).
1(x 5) (1)(x) (1)(5)
Distributive Property.
Simplify.
x 5
5 x
Commutative Property of Addition.
So, (5 x) 1(x 5)
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Example 5 Factoring with Opposites
Factor 2x3 12x2 18 3x by grouping.
2x3 12x2 18 3x
(2x3 12x2) (18 3x)
Group terms.
2x2(x 6) 3(6 x)
Factor out the GCF of each group.
2x2(x 6) 3(1)(x 6)
Write (6 x) as 1(x 6).
2x2(x 6) 3(x 6)
Simplify. (x 6) is a common factor.
(x 6)(2x2 3)
Factor out (x 6).
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Check It Out! Example 5a
Factor each polynomial by grouping.
15x2 10x3 8x 12
(15x2 10x3) (8x 12)
Group terms.
Factor out the GCF of each group.
5x2(3 2x) 4(2x 3)
5x2(3 2x) 4(1)(3 2x)
Write (2x 3) as 1(3 2x).
Simplify. (3 2x) is a common factor.
5x2(3 2x) 4(3 2x)
(3 2x)(5x2 4)
Factor out (3 2x).
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Check It Out! Example 5b
Factor each polynomial by grouping.
8y 8 x xy
(8y 8) (x xy)
Group terms.
8(y 1) (x)(1 y)
Factor out the GCF of each group.
8(y 1) (x)(y 1)
(y 1) is a common factor.
(y 1)(8 x)
Factor out (y 1) .
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Lesson Quiz Part I
Factor each polynomial. Check your answer. 1. 16x
20x3 2. 4m4 12m2 8m Factor each
expression. 3. 7k(k 3) 4(k 3) 4. 3y(2y 3)
5(2y 3)

4x(4 5x2)
4m(m3 3m 2)
(k 3)(7k 4)
(2y 3)(3y 5)
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Lesson Quiz Part II
Factor each polynomial by grouping. Check your
answer. 5. 2x3 x2 6x 3 6. 7p4 2p3
63p 18 7. A rocket is fired vertically into
the air at 40 m/s. The expression 5t2 40t
20 gives the rockets height after t seconds.
Factor this expression.

(2x 1)(x2 3)
(7p 2)(p3 9)
5(t2 8t 4)