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, [latex]12[/latex] as [latex]2\cdot 6\text{or}3\cdot 4\text{),}[/latex] 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:
[latex-display]\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}[/latex-display]
Here, we will start with a product, like [latex]2x+14[/latex], and end with its factors, [latex]2\left(x+7\right)[/latex]. To do this we apply the Distributive Property "in reverse".
Distributive Property
If [latex]a,b,c[/latex] are real numbers, then
[latex-display]a\left(b+c\right)=ab+ac\text{ and }ab+ac=a\left(b+c\right)[/latex-display]
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: [latex]2x+14[/latex].
Solution
| Step 1: Find the GCF of all the terms of the polynomial. |
Find the GCF of [latex]2x[/latex] and [latex]14[/latex]. |
 |
| Step 2: Rewrite each term as a product using the GCF. |
Rewrite [latex]2x[/latex] and [latex]14[/latex] as products of their GCF, [latex]2[/latex].
[latex-display]2x=2\cdot x[/latex-display]
[latex]14=2\cdot 7[/latex] |
[latex]2x+14[/latex]
[latex]\color{red}{2}\cdot x+\color{red}{2}\cdot7[/latex] |
| Step 3: Use the Distributive Property 'in reverse' to factor the expression. |
|
[latex]2\left(x+7\right)[/latex] |
| Step 4: Check by multiplying the factors. |
|
Check:
[latex-display]2(x+7)[/latex-display]
[latex-display]2\cdot{x}+2\cdot{7}[/latex-display]
[latex]2x+14\quad\checkmark[/latex] |
try it
[ohm_question]146330[/ohm_question]
Notice that in the example, we used the word factor as both a noun and a verb:
[latex-display]\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}[/latex-display]
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: [latex]3a+3[/latex].
Answer:
Solution
 |
|
|
[latex]3a+3[/latex] |
| Rewrite each term as a product using the GCF. |
[latex]\color{red}{3}\cdot a+\color{red}{3}\cdot1[/latex] |
| Use the Distributive Property 'in reverse' to factor the GCF. |
[latex]3(a+1)[/latex] |
| Check by multiplying the factors to get the original polynomial. |
|
| [latex]3(a+1)[/latex]
[latex-display]3\cdot{a}+3\cdot{1}[/latex-display]
[latex]3a+3\quad\checkmark[/latex] |
|
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: [latex]12x - 60[/latex].
Answer:
Solution
 |
|
|
[latex]12x-60[/latex] |
| Rewrite each term as a product using the GCF. |
[latex]\color{red}{12}\cdot x-\color{red}{12}\cdot 5[/latex] |
| Factor the GCF. |
[latex]12(x-5)[/latex] |
| Check by multiplying the factors. |
|
| [latex]12(x-5)[/latex]
[latex-display]12\cdot{x}-12\cdot{5}[/latex-display]
[latex]12x-60\quad\checkmark[/latex] |
|
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: [latex]3{y}^{2}+6y+9[/latex].
Answer:
Solution
 |
|
|
[latex]3y^2+6y+9[/latex] |
| Rewrite each term as a product using the GCF. |
[latex]\color{red}{3}\cdot{y}^{2}+\color{red}{3}\cdot 2y+\color{red}{3}\cdot 3[/latex] |
| Factor the GCF. |
[latex]3(y^{2}+2y+3)[/latex] |
| Check by multiplying. |
|
| [latex]3(y^{2}+2y+3)[/latex]
[latex-display]3\cdot{y^2}+3\cdot{2y}+3\cdot{3}[/latex-display]
[latex]3y^{2}+6y+9\quad\checkmark[/latex] |
|
try it
[ohm_question]146333[/ohm_question]
In the next example, we factor a variable from a binomial.
example
Factor: [latex]6{x}^{2}+5x[/latex].
Answer:
Solution
|
[latex]6{x}^{2}+5x[/latex] |
| Find the GCF of [latex]6{x}^{2}[/latex] and [latex]5x[/latex] and the math that goes with it. |
 |
| Rewrite each term as a product. |
[latex]\color{red}{x}\cdot{6x}+\color{red}{x}\cdot{5}[/latex] |
| Factor the GCF. |
[latex]x\left(6x+5\right)[/latex] |
| Check by multiplying. |
|
| [latex]x\left(6x+5\right)[/latex]
[latex-display]x\cdot 6x+x\cdot 5[/latex-display]
[latex]6{x}^{2}+5x[/latex] |
|
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: [latex]4{x}^{3}-20{x}^{2}[/latex].
Answer:
Solution
 |
|
|
|
[latex]4x^3-20x^2[/latex] |
| Rewrite each term. |
|
[latex]\color{red}{4{x}^{2}}\cdot x - \color{red}{4{x}^{2}}\cdot 5[/latex] |
| Factor the GCF. |
|
[latex]4x^2(x-5)[/latex] |
| Check. |
 |
|
try it
[ohm_question]146337[/ohm_question]
example
Factor: [latex]21{y}^{2}+35y[/latex].
Answer:
Solution
| Find the GCF of [latex]21{y}^{2}[/latex] and [latex]35y[/latex] |
 |
|
|
|
[latex]21y^2+35y[/latex] |
| Rewrite each term. |
|
[latex]\color{red}{7y}\cdot 3y + \color{red}{7y}\cdot 5[/latex] |
| Factor the GCF. |
|
[latex]7y(3y+5)[/latex] |
try it
[ohm_question]146338[/ohm_question]
example
Factor: [latex]14{x}^{3}+8{x}^{2}-10x[/latex].
Answer:
Solution
Previously, we found the GCF of [latex]14{x}^{3},8{x}^{2},\text{and}10x[/latex] to be [latex]2x[/latex].
|
[latex]14{x}^{3}+8{x}^{2}-10x[/latex] |
| Rewrite each term using the GCF, 2x. |
[latex]\color{red}{2x}\cdot 7{x}^{2}+\color{red}{2x}\cdot4x-\color{red}{2x}\cdot 5[/latex] |
| Factor the GCF. |
[latex]2x\left(7{x}^{2}+4x - 5\right)[/latex] |
 |
|
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: [latex]-9y - 27[/latex].
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 [latex]9y[/latex] and [latex]27[/latex] is [latex]9[/latex]. |
 |
| Since the expression [latex]−9y−27[/latex] has a negative leading coefficient, we use [latex]−9[/latex] as the GCF. |
|
[latex]-9y - 27[/latex] |
| Rewrite each term using the GCF. |
[latex]\color{red}{-9}\cdot y + \color{red}{(-9)}\cdot 3[/latex] |
| Factor the GCF. |
[latex]-9\left(y+3\right)[/latex] |
| Check.
[latex-display]-9(y+3)[/latex-display]
[latex-display]-9\cdot{y}+(-9)\cdot{3}[/latex-display]
[latex]-9y-27\quad\checkmark[/latex] |
|
try it
[ohm_question]146340[/ohm_question]
Pay close attention to the signs of the terms in the next example.
example
Factor: [latex]-4{a}^{2}+16a[/latex].
Answer:
Solution
| The leading coefficient is negative, so the GCF will be negative. |
|
 |
| Since the leading coefficient is negative, the GCF is negative, [latex]−4a[/latex]. |
|
[latex]-4{a}^{2}+16a[/latex] |
| Rewrite each term. |
[latex]\color{red}{-4a}\cdot{a}-\color{red}{(-4a)}\cdot{4}[/latex] |
| Factor the GCF. |
[latex]-4a\left(a - 4\right)[/latex] |
| Check on your own by multiplying. |
|
TRY IT
[ohm_question]146341[/ohm_question]
VIDEO REQUESTLicenses & Attributions
CC licensed content, Original
- Question ID 146341, 146340, 146339, 146338, 146337, 146335, 146333, 146331, 146330. Authored by: Lumen Learning. License: CC BY: Attribution.
CC licensed content, Shared previously
- Ex: Factor a Binomial - Greatest Common Factor (Basic). Authored by: James Sousa (mathispower4u.com). License: CC BY: Attribution.
CC licensed content, Specific attribution
