Factoring the GCF from Polynomials

It is sometimes useful to represent a number as a product of factors, for example, 12 as $2·6$ or $3·4$. In algebra, it can also be useful to represent a polynomial in factored form. We will start with a product, such as $3x^2+15x$, and end with its factors, $3x(x+5)$. To do this, we apply the Distributive Property “in reverse.” We state the Distributive Property here just as you saw it in earlier chapters and “in reverse.”

So how do you use the Distributive Property to factor a polynomial? You just find the GCF of all the terms and write the polynomial as a product!

Example 1

Factor: $8m^3-12m^{2}n+20m{n}^2$.

How to factor the greatest common factor from a polynomial:

Step 1 - Find the GCF of all the terms of the polynomial. For this problem, find the GCF of $8m^3$, $12m^{2}n$, and $20m{n}^2$.

Step 2 - Rewrite each term as a product using the GCF. In this case, rewrite $8m^3$, $12m^{2}n$, and $20m{n}^2$ as products of their GCF, $4m$.

• $8m^3=4m·2m^2$
• $12m^{2}n=4m·3mn$
• $20m{n}^2=4m·5n^2$

$8m^3-12m^{2}n+20m{n}^2$

$4m·2m^2-4m·3mn+4m·5n^2$

Step 3 - Use the “reverse” Distributive Property to factor the expression.

$4m(2m^2-3mn+5n^2)$

Step 4 - Check by multiplying the factors.

$4m(2m^2-3mn+5n^2)$

$4m·2m^2-4m·3mn+4m·5n^2$

$8m^3-12m^{2}n+20m{n}^2$

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

Example 2

Factor: $-4a^3+36a^2-8a$.

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

Step 1 - Rewrite each term using the GCF, $-4a$.

$-4a·a^2-(-4a)·9a+(-4a)·2$

Step 2 - Factor the GCF.

$-4a(a^2-9a+2)$

Step 3 - Check.

$-4a(a^2-9a+2)$

$-4a·a^2-(-4a)·9a+(-4a)·2$

$-4a^3+36a^2-8a$

So far, our greatest common factors have been monomials. In the next example, the greatest common factor is a binomial.

Example 3

Factor: $3y(y+7)-4(y+7)$.

The GCF is the binomial $y+7$.

$3y(y+7)-4(y+7)$

Step 1 - Factor the GCF, $(y+7)$.

$(y+7)(3y-4)$

Step 2 - Check on your own by multiplying.