Lesson 4: Factoring Polynomials

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 there is no common factor of all the terms of a polynomial, look for a common factor in just some of the terms. When there are four terms, a good way to start is by separating the polynomial into two parts with two terms in each part. Then look for the GCF in each part. If the polynomial can be factored, you will find a common factor emerges from both parts.

HOW TO: Factor by grouping.
Step 1. Group terms with common factors.
Step 2. Factor out the common factor in each group.
Step 3. Factor the common factor from the expression.
Step 4. Check by multiplying the factors.

Examples

• Factor $ab + 5a + 3b + 15$. Group the terms: $(ab + 5a) + (3b + 15)$. Factor the GCF from each group: $a(b+5) + 3(b+5)$. Factor out the common binomial: $(b+5)(a+3)$.
• Factor $x^2 + 2x - 5x - 10$. Group the terms: $(x^2 + 2x) + (-5x - 10)$. Factor GCFs: $x(x+2) - 5(x+2)$. Factor out the common binomial: $(x+2)(x-5)$.
• Factor $mn - 8m + 4n - 32$. Group the terms: $(mn - 8m) + (4n - 32)$. Factor GCFs: $m(n-8) + 4(n-8)$. Factor out the common binomial: $(n-8)(m+4)$.