Lesson Objectives

- Demonstrate an understanding of how to find the GCF of a polynomial
- Learn how to factor out the GCF from a polynomial

## How to Factor out the Greatest Common Factor (GCF) from a Polynomial

In our last lesson, we learned how to find the GCF or greatest common factor for a group of monomials. In this lesson, we will use that knowledge and learn how to factor out the GCF from a polynomial. Let’s begin by thinking about the distributive property:

2(3 + 5) = 2 • 3 + 2 • 5

4(3 - 5) = 4 • 3 - 4 • 5

At this point, we should know that the distributive property allows us to distribute multiplication over addition or subtraction. Since we are dealing with an equality, the reverse is also true. When we factor, we are reversing the distributive property:

Since 2 is common to both terms, we can move this outside of the parentheses:

2 • 3 + 2 • 5 = 2(3 + 5)

Since 4 is common to both terms, we can move this outside of the parentheses:

4 • 3 - 4 • 5 = 4(3 - 5)

Let's think about factoring a simple polynomial. Suppose we come across the following polynomial:

3x

First, let's think about the GCF of the two terms. The GCF of 3x

3x • x + 3x • 4

Since 3x is common to each term, we can factor this out. This means we pull 3x out from each term and place it outside of a set of parentheses:

3x • x + 3x • 4 = 3x(x + 4)

3x

To factor out the GCF from a polynomial, we can use the following steps:

Example 1: Factor out the GCF.

16x

Step 1) Find the GCF of the polynomial:

Our polynomial has terms of:

16x

GCF(16x

Step 2) Wrap the polynomial inside of parentheses:

(16x

Steps 3 & 4) Factor each term and remove the GCF, write the GCF outside of the parentheses:

(2x • 8x

2x(8x

Example 2: Factor out the GCF.

25x

Step 1) Find the GCF of the polynomial:

GCF(25x

Step 2) Wrap the polynomial inside of parentheses:

(25x

Steps 3 & 4) Factor each term and remove the GCF, write the GCF outside of the parentheses:

(5xy • 5x

5xy(5x

Example 3: Factor out the GCF.

22x

Step 1) Find the GCF of the polynomial:

GCF(22x

Step 2) Wrap the polynomial inside of parentheses:

(22x

Steps 3 & 4) Factor each term and remove the GCF, write the GCF outside of the parentheses:

(11x • 2x

11x(2x

2(3 + 5) = 2 • 3 + 2 • 5

4(3 - 5) = 4 • 3 - 4 • 5

At this point, we should know that the distributive property allows us to distribute multiplication over addition or subtraction. Since we are dealing with an equality, the reverse is also true. When we factor, we are reversing the distributive property:

Since 2 is common to both terms, we can move this outside of the parentheses:

2 • 3 + 2 • 5 = 2(3 + 5)

Since 4 is common to both terms, we can move this outside of the parentheses:

4 • 3 - 4 • 5 = 4(3 - 5)

Let's think about factoring a simple polynomial. Suppose we come across the following polynomial:

3x

^{2}+ 12xFirst, let's think about the GCF of the two terms. The GCF of 3x

^{2}and 12x is 3x. To make our next step crystal clear, let's rewrite our polynomial as:3x • x + 3x • 4

Since 3x is common to each term, we can factor this out. This means we pull 3x out from each term and place it outside of a set of parentheses:

3x • x + 3x • 4 = 3x(x + 4)

3x

^{2}+ 12x = 3x(x + 4)To factor out the GCF from a polynomial, we can use the following steps:

- Find the GCF of the polynomial
- Wrap the polynomial inside of parentheses
- Inside of the parentheses, pull out the GCF from each term
- We can do this by dividing each term by the GCF or we can factor each term and remove the GCF

- Write the GCF outside of the parentheses

Example 1: Factor out the GCF.

16x

^{5}- 4x^{3}+ 2xStep 1) Find the GCF of the polynomial:

Our polynomial has terms of:

16x

^{5}, 4x^{3}, and 2xGCF(16x

^{5}, 4x^{3}, 2x) = 2xStep 2) Wrap the polynomial inside of parentheses:

(16x

^{5}- 4x^{3}+ 2x)Steps 3 & 4) Factor each term and remove the GCF, write the GCF outside of the parentheses:

(2x • 8x

^{4}- 2x • 2x^{2}+ 2x • 1) =2x(8x

^{4}- 2x^{2}+ 1)Example 2: Factor out the GCF.

25x

^{4}y^{2}+ 50x^{2}y^{2}- 10xyStep 1) Find the GCF of the polynomial:

GCF(25x

^{4}y^{2}, 50x^{2}y^{2}, 10xy) = 5xyStep 2) Wrap the polynomial inside of parentheses:

(25x

^{4}y^{2}+ 50x^{2}y^{2}- 10xy)Steps 3 & 4) Factor each term and remove the GCF, write the GCF outside of the parentheses:

(5xy • 5x

^{3}y + 5xy • 10xy - 5xy • 2) =5xy(5x

^{3}y + 10xy - 2)Example 3: Factor out the GCF.

22x

^{3}y^{2}+ 132x^{2}y + 121xStep 1) Find the GCF of the polynomial:

GCF(22x

^{3}y^{2}, 132x^{2}y, 121x) = 11xStep 2) Wrap the polynomial inside of parentheses:

(22x

^{3}y^{2}+ 132x^{2}y + 121x)Steps 3 & 4) Factor each term and remove the GCF, write the GCF outside of the parentheses:

(11x • 2x

^{2}y^{2}+ 11x • 12xy + 11x • 11) =11x(2x

^{2}y^{2}+ 12xy + 11)#### Skills Check:

Example #1

Factor out the GCF. $$10x^{3}y^{2}- 15x^{2}y + 35xy$$

Please choose the best answer.

A

$$5xy(2x^{2}y - 3x + 7)$$

B

$$10x(2x^{2}y -5 xy + 15)$$

C

$$7(5xy + 2x + 4y)$$

D

$$5x^{2}y(2xy - 5y - 7)$$

E

$$7x^{2}y^{2}(5x^{2}y^{2}+ 2xy - y)$$

Example #2

Factor out the GCF. $$4x^{2}y^{2}- 8xyz - 6xy$$

Please choose the best answer.

A

$$3xy(2xy + 4z - 2)$$

B

$$8x^{2}y^{2}(2xy + 4x - 3)$$

C

$$2x^{2}(y + z + 6)$$

D

$$2xy(2xy - 4z - 3)$$

E

$$4y(8xy + 2z + 3)$$

Example #3

Factor out the GCF. $$6x^{3}y^{5}- 9x^{2}y^{5}- 3xy^{5}$$

Please choose the best answer.

A

$$2xy(3x^{2}- x - 1)$$

B

$$3x^{5}y(2xy^{5}- x - 1)$$

C

$$3xy^{5}(2x^{2}- 3x - 1)$$

D

$$xy^{5}(6x^{2}- 3x - 3)$$

E

$$6(x^{3}y^{5}- 2x^{2}y^{5}- 3)$$

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