Lesson Objectives

- Demonstrate an understanding of how to find the GCF for a group of numbers
- Demonstrate an understanding of the distributive property of multiplication
- Learn how to find the GCF for a group of monomials
- Learn how to factor out the GCF from a polynomial

## How to Factor out the GCF from a Polynomial

In our pre-algebra course, we learned how to find the Greatest Common Factor or GCF for a group of numbers. To find the GCF for a group of numbers, we factor each number completely and then build a list of prime factors that are common to all numbers. The GCF is the product of the numbers in the list.

GCF(12,18,90)

12 = 2 • 2 • 3

18 = 2 • 3 • 3

90 = 2 • 3 • 3 • 5

What's common to everything? Let's organize our factors in a table.

One factor of 2 and one factor of 3 is common to every number.

GCF List: 2, 3

To find the GCF, we multiply the numbers in the GCF list (2,3):

GCF(12,18,90) = 2 • 3 = 6

When we find the GCF for a group of variables, the procedure is the same, however, we can use a shortcut. First and foremost, the variable must appear in each term, secondly, we use the smallest exponent that appears on that variable in any term.

GCF(x

We can quickly identify that x is in each term, x

GCF(x

Let's take a look at a few examples.

Example 1: Find the GCF.

GCF(3x

Let's begin by looking at the number parts:

3 = 3

9 = 3

24 = 3 • 2

For the number part, our GCF is 3.

If we look at our variable parts:

x

xyz

x

We can see that x and y are in each term, whereas z is not. The smallest exponent on both x and y is 1.

For the variable part, our GCF is xy.

Our GCF is the product of the number part (3) and the variable part (xy).

GCF(3x

Example 2: Find the GCF.

GCF(121x

Let's begin by looking at the number parts:

121 = 11

132 = 11 • 3 • 2

209 = 11 • 19

For the number part, our GCF is 11.

If we look at our variable parts:

x

y

y

We can see that only y is in each term and the smallest exponent is 1.

For the variable part, our GCF is y. Our GCF is the product of the number part (11) and the variable part (y).

GCF(121x

12(3 + 4) = 12 • 3 + 12 • 4 = 36 + 48

Since we have an equality, we can reverse the process:

36 + 48 = 12 • 3 + 12 • 4 = 12(3 + 4)

When we factor out the GCF from a polynomial, we first identify the GCF of all terms of the polynomial. We can then pull this GCF out from each term and place it outside of a set of parentheses. Let's look at a few examples.

Example 3: Factor out the GCF.

65x

First, we want to find the GCF of all terms.

GCF(65x

Second, we will rewrite each term as the product of 13x and another factor.

13x • 5x

Third, we will use our distributive property to factor out the GCF.

13x(5x

Example 4: Factor out the GCF.

95x

First, we want to find the GCF of all terms.

GCF(95x

Second, we will rewrite each term as the product of 19y and another factor.

19y • 5x

Third, we will use our distributive property to factor out the GCF.

19y(5x

Example 5: Factor out the GCF.

(x - 7)(x - 12) + (x + 5)(x - 7)

In this case, we can see a common binomial factor of (x - 7). This can be factored in the same way:

(x - 7)(x - 12) + (x + 5)(x - 7)

(x - 7)[(x - 12) + (x + 5)]

We can combine like terms inside of the brackets:

(x - 7)[x - 12 + x + 5]

(x - 7)(2x - 7)

GCF(12,18,90)

12 = 2 • 2 • 3

18 = 2 • 3 • 3

90 = 2 • 3 • 3 • 5

What's common to everything? Let's organize our factors in a table.

Number | Prime Factors | ||||
---|---|---|---|---|---|

12 | 2 | 2 | 3 | ||

18 | 2 | 3 | 3 | ||

90 | 2 | 3 | 3 | 5 |

GCF List: 2, 3

To find the GCF, we multiply the numbers in the GCF list (2,3):

GCF(12,18,90) = 2 • 3 = 6

When we find the GCF for a group of variables, the procedure is the same, however, we can use a shortcut. First and foremost, the variable must appear in each term, secondly, we use the smallest exponent that appears on that variable in any term.

GCF(x

^{5}, x^{3}, x^{2}y)We can quickly identify that x is in each term, x

^{5}, x^{3}, and x^{2}y, however, y is only in one of the terms. The variable x will be in the GCF and the variable y will not be in the GCF. The smallest exponent on x in any of the terms is a 2. This means our GCF will be x^{2}.GCF(x

^{5}, x^{3}, x^{2}y) = x^{2}Let's take a look at a few examples.

Example 1: Find the GCF.

GCF(3x

^{2}y, 9xyz, 24x^{5}y^{2})Let's begin by looking at the number parts:

3 = 3

9 = 3

^{2}24 = 3 • 2

^{3}For the number part, our GCF is 3.

If we look at our variable parts:

x

^{2}yxyz

x

^{5}y^{2}We can see that x and y are in each term, whereas z is not. The smallest exponent on both x and y is 1.

For the variable part, our GCF is xy.

Our GCF is the product of the number part (3) and the variable part (xy).

GCF(3x

^{2}y, 9xyz, 24x^{5}y^{2}) = 3xyExample 2: Find the GCF.

GCF(121x

^{9}yz, 132y^{3}, 209y)Let's begin by looking at the number parts:

121 = 11

^{2}132 = 11 • 3 • 2

^{2}209 = 11 • 19

For the number part, our GCF is 11.

If we look at our variable parts:

x

^{9}yzy

^{3}y

We can see that only y is in each term and the smallest exponent is 1.

For the variable part, our GCF is y. Our GCF is the product of the number part (11) and the variable part (y).

GCF(121x

^{9}yz, 132y^{3}, 209y) = 11y### Factoring out the GCF

When we factor a polynomial, we are writing the polynomial as the product of two or more polynomials. When we factor, we are just reversing the distributive property of multiplication that we learned in pre-algebra.12(3 + 4) = 12 • 3 + 12 • 4 = 36 + 48

Since we have an equality, we can reverse the process:

36 + 48 = 12 • 3 + 12 • 4 = 12(3 + 4)

When we factor out the GCF from a polynomial, we first identify the GCF of all terms of the polynomial. We can then pull this GCF out from each term and place it outside of a set of parentheses. Let's look at a few examples.

Example 3: Factor out the GCF.

65x

^{5}- 39x^{2}+ 13xFirst, we want to find the GCF of all terms.

GCF(65x

^{5}, 39x^{2}, 13x) = 13xSecond, we will rewrite each term as the product of 13x and another factor.

13x • 5x

^{4}- 13x • 3x + 13x • 1Third, we will use our distributive property to factor out the GCF.

13x(5x

^{4}- 3x + 1)Example 4: Factor out the GCF.

95x

^{2}y^{2}- 228xy + 133y^{2}First, we want to find the GCF of all terms.

GCF(95x

^{2}y^{2}, 228xy, 133y^{2}) = 19ySecond, we will rewrite each term as the product of 19y and another factor.

19y • 5x

^{2}y - 19y • 12x + 19y • 7yThird, we will use our distributive property to factor out the GCF.

19y(5x

^{2}y - 12x + 7y)### Factoring out a Common Binomial Factor

In some cases, we will need to factor out a common binomial factor. Let's look at an example.Example 5: Factor out the GCF.

(x - 7)(x - 12) + (x + 5)(x - 7)

In this case, we can see a common binomial factor of (x - 7). This can be factored in the same way:

(x - 7)(x - 12) + (x + 5)(x - 7)

(x - 7)[(x - 12) + (x + 5)]

We can combine like terms inside of the brackets:

(x - 7)[x - 12 + x + 5]

(x - 7)(2x - 7)

#### Skills Check:

Example #1

Factor out the Greatest Common Factor (GCF). $$14x^{3}+ 21x^{2}+ 7$$

Please choose the best answer.

A

$$14(x^{3}+ 7x^{2}+ 1)$$

B

$$7(2x^{3}+ 3x^{2}+ 1)$$

C

$$-7(2x^{3}- 3x^{2}- 1)$$

D

$$\frac{1}{7}(x^{3}+ 2x^{2}+ 1)$$

E

$$7x(2x^{2}+ 3x + 1)$$

Example #2

Factor out the Greatest Common Factor (GCF). $$18x^{2}+ 15x - 21$$

Please choose the best answer.

A

$$3(6x^{2}+ 5x - 7)$$

B

$$3x(6x - 7)$$

C

$$-3(6x^{2}- 5x - 7)$$

D

$$\frac{1}{3}(2x^{2}+ 3x - 1)$$

E

$$6(x^{2}+ 3x - 12)$$

Example #3

Factor out the Greatest Common Factor (GCF). $$(x - 1)(x + 2) + (x - 1)(x + 5)$$

Please choose the best answer.

A

$$(x - 1)(3x + 2)$$

B

$$(x - 1)(x + 3)$$

C

$$(x - 1)(2x + 7)$$

D

$$(x - 1)(x + 1)$$

E

$$(x - 1)(5x + 12)$$

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