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, 1212 as 2⋅6 or 3⋅42⋅6 or 3⋅4) 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:
2(x+7)factors2⋅x+2⋅72x+14product2(x+7)factors2⋅x+2⋅72x+14product
Here, we will start with a product, like 2x+142x+14, and end with its factors, 2(x+7)2(x+7). To do this we apply the Distributive Property “in reverse”.
Distributive Property
If a,b,ca,b,c are real numbers, then
a(b+c)=ab+ac and ab+ac=a(b+c)a(b+c)=ab+ac and ab+ac=a(b+c)
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+142x+14
Solution
Step 1: Find the GCF of all the terms of the polynomial. | Find the GCF of 2x2x and 1414. | ![]() |
Step 2: Rewrite each term as a product using the GCF. | Rewrite 2x2x and 1414 as products of their GCF, 22.
2x=2⋅x2x=2⋅x 14=2⋅714=2⋅7 |
2x+142x+14
2⋅x+2⋅72⋅x+2⋅7 |
Step 3: Use the Distributive Property ‘in reverse’ to factor the expression. | 2(x+7)2(x+7) | |
Step 4: Check by multiplying the factors. | Check:
2(x+7)2(x+7) 2⋅x+2⋅72⋅x+2⋅7 2x+14✓2x+14✓ |
try it
Notice that in the example, we used the word factor as both a noun and a verb:
Noun7 is a factor of 14Verbfactor 2 from 2x+14Noun7 is a factor of 14Verbfactor 2 from 2x+14
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+33a+3
try it
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−6012x−60
try it
Watch the following video to see more examples of factoring the GCF from a binomial.
Now we’ll factor the greatest common factor from a trinomial. We start by finding the GCF of all three terms.
example
Factor: 3y2+6y+93y2+6y+9
try it
In the next example, we factor a variable from a binomial.
example
Factor: 6x2+5x6x2+5x
try it
When there are several common factors, as we’ll see in the next two examples, good organization and neat work helps!
example
Factor: 4x3−20x24x3−20x2
try it
example
Factor: 21y2+35y21y2+35y
try it
example
Factor: 14x3+8x2−10x14x3+8x2−10x
try it
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−27−9y−27
try it
Pay close attention to the signs of the terms in the next example.
example
Factor: −4a2+16a−4a2+16a
TRY IT
Candela Citations
- Question ID 146341, 146340, 146339, 146338, 146337, 146335, 146333, 146331, 146330. Authored by: Lumen Learning. License: CC BY: Attribution
- Ex: Factor a Binomial - Greatest Common Factor (Basic). Authored by: James Sousa (mathispower4u.com). Located at: https://youtu.be/68M_AJNpAu4. License: CC BY: Attribution
- Prealgebra. Provided by: OpenStax. License: CC BY: Attribution. License Terms: Download for free at http://cnx.org/contents/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757