Finding the Greatest Common Factor of a Polynomial

Learning Outcomes

Factor the greatest common factor from a polynomial

Just like in arithmetic, where it is sometimes useful to represent a number in factored form (for example,

12

2\cdot 6\text{or}3\cdot 4\text{),}

in algebra it can be useful to represent a polynomial in factored form. One way to do this is by finding the greatest common factor of all the terms. Remember that you can multiply a polynomial by a monomial as follows:

\begin{array}{ccc}\hfill 2\left(x& +& 7\right)\text{factors}\hfill \\ \hfill 2\cdot x& +& 2\cdot 7\hfill \\ \hfill 2x& +& 14\text{product}\hfill \end{array}

Here, we will start with a product, like

2x+14

, and end with its factors,

2\left(x+7\right)

. To do this we apply the Distributive Property "in reverse".

Distributive Property

a,b,c

are real numbers, then

a\left(b+c\right)=ab+ac\text{ and }ab+ac=a\left(b+c\right)

The form on the left is used to multiply. The form on the right is used to factor. So how do we use the Distributive Property to factor a polynomial? We find the GCF of all the terms and write the polynomial as a product!

example

Factor:

2x+14

. Solution

Step 1: Find the GCF of all the terms of the polynomial.	Find the GCF of $2x$ and $14$ .
Step 2: Rewrite each term as a product using the GCF.	Rewrite $2x$ and $14$ as products of their GCF, $2$ . $2x=2\cdot x$ $14=2\cdot 7$	$2x+14$ $\color{red}{2}\cdot x+\color{red}{2}\cdot7$
Step 3: Use the Distributive Property 'in reverse' to factor the expression.		$2\left(x+7\right)$
Step 4: Check by multiplying the factors.		Check: $2(x+7)$ $2\cdot{x}+2\cdot{7}$ $2x+14\quad\checkmark$

try it

[ohm_question]146330[/ohm_question]

Notice that in the example, we used the word factor as both a noun and a verb:

\begin{array}{cccc}\text{Noun}\hfill & & & 7\text{ is a factor of }14\hfill \\ \text{Verb}\hfill & & & \text{factor }2\text{ from }2x+14\hfill \end{array}

Factor the greatest common factor from a polynomial

Find the GCF of all the terms of the polynomial.
Rewrite each term as a product using the GCF.
Use the Distributive Property ‘in reverse’ to factor the expression.
Check by multiplying the factors.

example

Factor:

3a+3

Answer: Solution


	$3a+3$
Rewrite each term as a product using the GCF.	$\color{red}{3}\cdot a+\color{red}{3}\cdot1$
Use the Distributive Property 'in reverse' to factor the GCF.	$3(a+1)$
Check by multiplying the factors to get the original polynomial.
$3(a+1)$ $3\cdot{a}+3\cdot{1}$ $3a+3\quad\checkmark$

try it

[ohm_question]146331[/ohm_question]

The expressions in the next example have several factors in common. Remember to write the GCF as the product of all the common factors.

example

Factor:

12x - 60

Answer: Solution


	$12x-60$
Rewrite each term as a product using the GCF.	$\color{red}{12}\cdot x-\color{red}{12}\cdot 5$
Factor the GCF.	$12(x-5)$
Check by multiplying the factors.
$12(x-5)$ $12\cdot{x}-12\cdot{5}$ $12x-60\quad\checkmark$

try it

[ohm_question]146332[/ohm_question]

Watch the following video to see more examples of factoring the GCF from a binomial. https://youtu.be/68M_AJNpAu4 Now we’ll factor the greatest common factor from a trinomial. We start by finding the GCF of all three terms.

example

Factor:

3{y}^{2}+6y+9

Answer: Solution


	$3y^2+6y+9$
Rewrite each term as a product using the GCF.	$\color{red}{3}\cdot{y}^{2}+\color{red}{3}\cdot 2y+\color{red}{3}\cdot 3$
Factor the GCF.	$3(y^{2}+2y+3)$
Check by multiplying.
$3(y^{2}+2y+3)$ $3\cdot{y^2}+3\cdot{2y}+3\cdot{3}$ $3y^{2}+6y+9\quad\checkmark$

try it

[ohm_question]146333[/ohm_question]

In the next example, we factor a variable from a binomial.

example

Factor:

6{x}^{2}+5x

Answer: Solution

	$6{x}^{2}+5x$
Find the GCF of $6{x}^{2}$ and $5x$ and the math that goes with it.
Rewrite each term as a product.	$\color{red}{x}\cdot{6x}+\color{red}{x}\cdot{5}$
Factor the GCF.	$x\left(6x+5\right)$
Check by multiplying.
$x\left(6x+5\right)$ $x\cdot 6x+x\cdot 5$ $6{x}^{2}+5x$

try it

[ohm_question]146335[/ohm_question]

When there are several common factors, as we’ll see in the next two examples, good organization and neat work helps!

example

Factor:

4{x}^{3}-20{x}^{2}

Answer: Solution


		$4x^3-20x^2$
Rewrite each term.		$\color{red}{4{x}^{2}}\cdot x - \color{red}{4{x}^{2}}\cdot 5$
Factor the GCF.		$4x^2(x-5)$
Check.

try it

[ohm_question]146337[/ohm_question]

example

Factor:

21{y}^{2}+35y

Answer: Solution

Find the GCF of $21{y}^{2}$ and $35y$
		$21y^2+35y$
Rewrite each term.		$\color{red}{7y}\cdot 3y + \color{red}{7y}\cdot 5$
Factor the GCF.		$7y(3y+5)$

try it

[ohm_question]146338[/ohm_question]

example

Factor:

14{x}^{3}+8{x}^{2}-10x

Answer: Solution Previously, we found the GCF of $14{x}^{3},8{x}^{2},\text{and}10x$ to be $2x$ .

	$14{x}^{3}+8{x}^{2}-10x$
Rewrite each term using the GCF, 2x.	$\color{red}{2x}\cdot 7{x}^{2}+\color{red}{2x}\cdot4x-\color{red}{2x}\cdot 5$
Factor the GCF.	$2x\left(7{x}^{2}+4x - 5\right)$

try it

[ohm_question]146339[/ohm_question]

When the leading coefficient, the coefficient of the first term, is negative, we factor the negative out as part of the GCF.

example

Factor:

-9y - 27

Answer: Solution

When the leading coefficient is negative, the GCF will be negative. Ignoring the signs of the terms, we first find the GCF of $9y$ and $27$ is $9$ .
Since the expression $−9y−27$ has a negative leading coefficient, we use $−9$ as the GCF.
	$-9y - 27$
Rewrite each term using the GCF.	$\color{red}{-9}\cdot y + \color{red}{(-9)}\cdot 3$
Factor the GCF.	$-9\left(y+3\right)$
Check. $-9(y+3)$ $-9\cdot{y}+(-9)\cdot{3}$ $-9y-27\quad\checkmark$

try it

[ohm_question]146340[/ohm_question]

Pay close attention to the signs of the terms in the next example.

example

Factor:

-4{a}^{2}+16a

Answer: Solution

The leading coefficient is negative, so the GCF will be negative.

Since the leading coefficient is negative, the GCF is negative, $−4a$ .
	$-4{a}^{2}+16a$
Rewrite each term.	$\color{red}{-4a}\cdot{a}-\color{red}{(-4a)}\cdot{4}$
Factor the GCF.	$-4a\left(a - 4\right)$
Check on your own by multiplying.

TRY IT

[ohm_question]146341[/ohm_question]

VIDEO REQUEST

Finding the Greatest Common Factor of a Polynomial

Learning Outcomes

Distributive Property

example

try it

Factor the greatest common factor from a polynomial

example

try it

example

try it

example

try it

example

try it

example

try it

example

try it

example

try it

example

try it

example

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Study Guides > Prealgebra

Finding the Greatest Common Factor of a Polynomial

Learning Outcomes

Distributive Property

example

try it

Factor the greatest common factor from a polynomial

example

try it

example

try it

example

try it

example

try it

example

try it

example

try it

example

try it

example

try it

example

TRY IT

Licenses & Attributions

CC licensed content, Original

CC licensed content, Shared previously

CC licensed content, Specific attribution