Factoring out the GCF and factoring by grouping

Multiplying in reverse

So far we've been multiplying: starting from \(4x(x - 3)\) you get \(4x^2 - 12x\). Factoring goes the other way. You start with \(4x^2 - 12x\) and rewrite it as a product:

You already do this with numbers: \(90 = 2 \cdot 3^2 \cdot 5\) is the prime factorization of 90. With polynomials, the goal is the same: break the expression into factors that can't be broken down any further.

Why bother? Factoring is how you simplify rational expressions, find their common denominators and solve equations of degree two and higher. It's one of the most-used skills in all of high school math.

Factoring out the GCF

This is always the first thing to try, before anything else.

When the leading term is negative, it's customary to factor out a negative GCF so that the parentheses start with a positive term:

The common factor can also be a whole binomial.

Factoring by grouping

Sometimes no factor is shared by every term, but one is shared within pairs of terms. That's typical of four-term polynomials.

Practice