Factoring trinomials and using the Factor Theorem

Where the pattern comes from

Multiply two binomials of the form \((x + p)(x + q)\):

The middle coefficient is the sum of \(p\) and \(q\); the constant term is their product. Read backwards, that's a factoring method.

Start from the product: only a few integer pairs multiply to a given number, and among those you look for the one with the right sum.

Getting the signs right

The signs of \(b\) and \(c\) tell you the signs of the two numbers before you even start searching.

When the leading coefficient isn't 1: the AC method

For \(ax^2 + bx + c\) with \(a \neq 1\), use a variation of the same idea.

Cubics and beyond: the Factor Theorem

When none of the patterns fit a higher-degree polynomial, use the Factor Theorem from the synthetic division lesson.

Practice