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Introduction
A variety of concepts, systems, and processes can be
described using polynomial expressions. Some of these
include familiar geometrical ideas such as area and
perimeter. Other real-world applications involve more
complex expressions, like the paths followed by
baseballs, or satellites in orbit. Factoring, or writing the
expressions as the product of their factors, can make the
expressions easier to work with and apply to different
scenarios. One of the most common forms of factoring is
by the greatest common factor, or GCF.
1
3A.2.2: Factoring Expressions by the Greatest Common Factor
Key Concepts
• Recall that a monomial is an expression with one
term, consisting of a number, a variable, or the
product of a number and variable(s), whereas a
polynomial is an expression with two or more terms,
consisting of a number, a variable, or the product of a
number and variable(s).
• The factors of a given number or monomial are
numbers or expressions which multiply together to
equal that given number, or monomial. For example,
6 and 3 are factors of 18, and x and x2 are factors of
x3. Two possible factors of the monomial 15x4y are
5xy and 3x3.
3A.2.2: Factoring Expressions by the Greatest Common Factor
2
Key Concepts, continued
• A common factor is a factor that two or more numbers
share, or the same number or variable that appears in
all terms of a polynomial. You can find common
factors by listing the factors for each term and then
comparing the results. For example, consider the
polynomial 36x4 + 27x3y. One common factor of 36
and 27 is 3, since 3 • 12 = 36 and 3 • 9 = 27. As for
the variables, x is a common factor found in both
terms. Notice that the variable y does not appear in
both terms of the polynomial, so it is not a common
factor. Thus, a common factor of 36x4 + 27x3y is 3x.
3
3A.2.2: Factoring Expressions by the Greatest Common Factor
Key Concepts, continued
• The greatest common factor (GCF) of a polynomial
is the common factor with the largest constant number
and power of each variable. In other words, the GCF
is the largest factor that two or more terms share.
• The GCF for the polynomial 36x4 + 27x3y can be
found by first breaking down each term by listing all of
its prime factors. Prime factors are factors that are
prime numbers, which are whole numbers that can
only be divided evenly by 1 and the number itself.
4
3A.2.2: Factoring Expressions by the Greatest Common Factor
Key Concepts, continued
• For instance, the prime factors of the terms of the
polynomial 36x4 + 27x3y are as follows:
36x4 = 2 • 2 • 3 • 3 • x • x • x • x
27x3y = 3 • 3 • 3 • x • x • x • y
• To determine the GCF of this polynomial, determine
which factors each term has in common. Each term
has two prime factors of 3 in common, and each term
also has three factors of x:
36x 4 = 2 • 2 • 3 • 3 • x • x • x • x
27x 3 y = 3 • 3 • 3 • x • x • x • y
5
3A.2.2: Factoring Expressions by the Greatest Common Factor
Key Concepts, continued
• Therefore, the GCF of the polynomial 36x4 + 27x3y is
3 • 3 • x • x • x = 9x3.
• Factoring is the process of rewriting a given
expression as the product of its factors. Once the
GCF is identified, then the remaining factors are
determined by multiplying the GCF by the constants
and/or variables necessary to obtain the original term:
9x3 • 4x = 36x4
9x3 • 3y = 27x3y
6
3A.2.2: Factoring Expressions by the Greatest Common Factor
Key Concepts, continued
• The completed factoring for the polynomial expression
36x4 + 27x3y is 9x3(4x + 3y).
• To verify that the factors have been correctly
identified, apply the Distributive Property to the
factored expression. The result should be the
original polynomial.
• Distributing 9x3 over (4x + 3y) yields
9x3 • 4x + 9x3 • 3y = 36x4 + 27x3y, which verifies that
the identified factors are correct.
• An expression that cannot be factored is said to
be prime.
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3A.2.2: Factoring Expressions by the Greatest Common Factor
Common Errors/Misconceptions
• misidentifying the greatest common factor
• incorrectly applying a negative sign when factoring or
applying the Distributive Property
• incorrectly applying the rules of exponents; e.g.,
x2 • x3 ≠ x6
• including factors that are common to some terms but
not to all terms
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3A.2.2: Factoring Expressions by the Greatest Common Factor
Guided Practice
Example 1
Factor the polynomial 12x3 + 30x2 + 42x by finding the
greatest common factor (GCF). Verify your results using
the Distributive Property.
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3A.2.2: Factoring Expressions by the Greatest Common Factor
Guided Practice: Example 1, continued
1. Determine the common factors of the
terms’ coefficients and variables.
First factor the coefficients (12, 30, and 42) into their
prime factors.
The prime factors are as follows:
12 = 2 • 2 • 3
30 = 2 • 3 • 5
42 = 2 • 3 • 7
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3A.2.2: Factoring Expressions by the Greatest Common Factor
Guided Practice: Example 1, continued
Each coefficient’s factors include 2 and 3, so 2 • 3 = 6 is
common to all three terms.
Next, determine the greatest power of x common to all
three terms. The first term has x3, the second term has
x2, and the third term has x (or x1). Expand these terms:
x3 = x • x • x
x2 = x • x
x1 = x
Only x (or x to the first power) is common to all three
terms. Combining this with the greatest numerical factor
of 6, the resulting GCF is 6x.
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3A.2.2: Factoring Expressions by the Greatest Common Factor
Guided Practice: Example 1, continued
2. Write expressions for the remaining
non-common factors.
Write out the prime factors of each term:
12x3 = 2 • 2 • 3 • x • x • x
30x2 = 2 • 3 • 5 • x • x
42x = 2 • 3 • 7 • x
12
3A.2.2: Factoring Expressions by the Greatest Common Factor
Guided Practice: Example 1, continued
Eliminate the common factors of each term and
determine which non-common factors are left:
12x3 = 2 • 2 • 3 • x • x • x ; the remaining factors
are 2 • x • x, or 2x2
30x 2 = 2 • 3 • 5 • x • x ; the remaining factors are
5 • x, or 5x
42x = 2 • 3 • 7 • x ; the remaining factor is 7
The expressions for the remaining non-common
factors are 2x2, 5x, and 7.
13
3A.2.2: Factoring Expressions by the Greatest Common Factor
Guided Practice: Example 1, continued
3. Factor the original polynomial by writing it
as the product of the GCF and a
polynomial whose terms are the
expressions found in the previous step.
The GCF is 6x. The remaining terms are 2x2, 5x, and
7, which can be summed to produce the polynomial
2x2 + 5x + 7.
The product of the GCF and this polynomial is
6x(2x2 + 5x + 7).
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3A.2.2: Factoring Expressions by the Greatest Common Factor
Guided Practice: Example 1, continued
4. Verify the result using the Distributive
Property.
To verify the result, apply the Distributive Property by
multiplying 6x by each term of the polynomial.
6x(2x2 + 5x + 7)
= 6x • 2x2 + 6x • 5x + 6x • 7
= 12x3 + 30x2 + 42x
The result of applying the Distributive Property is the
original expression, 12x3 + 30x2 – 42x.
Thus, the polynomial is factored correctly
and 12x3 + 30x2 + 42x = 6x(2x2 + 5x + 7).
✔
15
3A.2.2: Factoring Expressions by the Greatest Common Factor
Guided Practice: Example 1, continued
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3A.2.2: Factoring Expressions by the Greatest Common Factor
Guided Practice
Example 2
Factor the polynomial 35xy4 – 14x4y2z + 56x3y3z3 by
finding the GCF. Verify your results using the
Distributive Property.
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3A.2.2: Factoring Expressions by the Greatest Common Factor
Guided Practice: Example 2, continued
1. Determine the common factors of the
coefficients for each term.
First factor the coefficients (35, –14, and 56) into their
prime factors:
35 = 5 • 7
–14 = –1 • 2 • 7
56 = 2 • 2 • 2 • 7
Note that a negative number, like –14, will always have
a negative factor that can be represented by –1.
The only numerical factor that is common to all three
terms is 7.
3A.2.2: Factoring Expressions by the Greatest Common Factor
18
Guided Practice: Example 2, continued
2. Determine the greatest power of each
variable common to all terms, and multiply
this by the common factor from step 1 to
complete the GCF.
The first term has the variables xy4, the second has
variables x4y2z, and the third has variables x3y3z3.
Expand these terms:
xy4 = x • y • y • y • y
x4y2z = x • x • x • x • y • y • z
x3y3z3 = x • x • x • y • y • y • z • z • z
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3A.2.2: Factoring Expressions by the Greatest Common Factor
Guided Practice: Example 2, continued
One factor of x and two factors of y, or y2, are
common to all three terms. Since z is not a factor of
the first term, it is not a common factor.
Multiplying these common variables by the result of
step 1 yields a GCF of 7xy2.
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3A.2.2: Factoring Expressions by the Greatest Common Factor
Guided Practice: Example 2, continued
3. Write expressions for the remaining
non-common factors.
Write out the prime factors of each term:
35xy4 = 5 • 7 • x • y • y • y • y
–14x4y2z = –1 • 2 • 7 • x • x • x • x • y • y • z
56x3y3z3 = 2 • 2 • 2 • 7 • x • x • x • y • y • y • z • z • z
Eliminate the common factors of each term and
determine which non-common factors are left:
35xy 4 = 5 • 7 • x • y • y • y • y
The remaining factors are 5 • y • y, or 5y2.
3A.2.2: Factoring Expressions by the Greatest Common Factor
21
Guided Practice: Example 2, continued
–14x 4 y 2z = –1 • 2 • 7 • x • x • x • x • y • y • z
The remaining factors are –1 • 2 • x • x • x • z, which
simplifies to –2x3z.
56x3 y 3z3 = 2 • 2 • 2 • 7 • x • x • x • y • y • y • z • z • z
The remaining factors are 2 • 2 • 2 • x • x • y • z • z • z,
which simplifies to 8x2yz3.
The expressions for the remaining non-common
factors are 5y2, –2x3z, and 8x2yz3.
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3A.2.2: Factoring Expressions by the Greatest Common Factor
Guided Practice: Example 2, continued
4. Factor the original polynomial by writing it
as the product of the GCF and a
polynomial whose terms are the
expressions found in the previous step.
The GCF is 7xy2. The remaining terms are
5y2, –2x3z, and 8x2yz3, which can be summed to
produce the polynomial 5y2 – 2x3z + 8x2yz3.
The product of the GCF and this polynomial
is 7xy2(5y2 – 2x3z + 8x2yz3).
23
3A.2.2: Factoring Expressions by the Greatest Common Factor
Guided Practice: Example 2, continued
5. Verify the result using the Distributive
Property.
7xy2(5y2 – 2x3z + 8x2yz3)
= 7xy2 • 5y2 – 7xy2 • 2x3z + 7xy2 • 8x2yz3)
= 35xy4 – 14x4y2z + 56x3y3z3
The result of applying the Distributive Property is the
original expression, 35xy4 – 14x4y2z + 56x3y3z3. Thus,
the polynomial is factored correctly and
35xy4 – 14x4y2z + 56x3y3z3 =
7xy2(5y2 – 2x3z + 8x2yz3).
✔
24
3A.2.2: Factoring Expressions by the Greatest Common Factor
Guided Practice: Example 2, continued
25
3A.2.2: Factoring Expressions by the Greatest Common Factor