Lesson 6: Factoring ax² + bx + c

Property

We use the Distributive Property in reverse to factor a polynomial. Find the GCF of all the terms and write the polynomial as a product.

Distributive Property:
If $a$, $b$, and $c$ are real numbers, then $a(b + c) = ab + ac$ and $ab + ac = a(b + c)$. The form on the right is used to factor.

To factor the GCF from a polynomial:
Step 1. Find the GCF of all terms.
Step 2. Rewrite each term as a product using the GCF.
Step 3. Use the “reverse” Distributive Property to factor the expression.
Step 4. Check by multiplying the factors.

Property

When the leading coefficient is negative, we factor the negative out as part of the GCF. This changes the signs of the terms inside the parentheses.

For example, to factor $-4a^3 + 36a^2 - 8a$, the GCF is taken as $-4a$. We rewrite each term using the GCF:
$-4a^3 = -4a \cdot a^2$
$36a^2 = -4a \cdot (-9a)$
$-8a = -4a \cdot 2$
The factored expression is $-4a(a^2 - 9a + 2)$.

Examples

• Factor $-7x - 21$. The GCF is $-7$. Factoring it out gives $-7(x + 3)$. Notice the sign inside the parenthesis flipped.