Overview

# Factor Theorem

## Quick Reference

The following result, which is an immediate consequence of the Remainder Theorem:

Theorem

Let *f*(*x*) be a polynomial. Then *x*−*h* is a factor of *f*(*x*) if and only if *f*(*h*)=0.

The theorem is valuable for finding factors of polynomials. For example, to factorize 2*x* ^{3}+3*x* ^{2}−12*x*−20, look first for possible factors *x*−*h*, where *h* is an integer. Here *h* must divide 20. Try possible values for *h*, and calculate *f*(*h*). It is found that *f*(−2)=−16+12 +24−20=0, and so *x*+2 is a factor. Now divide the polynomial by this factor to obtain a quadratic which it may be possible to factorize further.