The factor theorem

The factor theorem states that:

If is a polynomial and for some constant , then is a factor of the polynomial .

This looks incredibly complicated, but you’ve already seen it before.

When you solve an equation that looks something like, for example, , you used the factor theorem without even knowing it!

From that equation, you know that must be a root of the polynomial (in other words, where , the polynomial equals zero). You can also see that is a factor of the polynomial. This is exactly what the factor theorem states.

Examples

Find the two roots of the polynomial using the factor theorem

• Factorise:
• Find the roots:
  • If is a factor of the polynomial, then .
  • We know that is a factor, so .
  • We also know that is a factor, so .
• So the roots the roots of the polynomial are and .

Given that is a factor of , find a root of

• The factor theorem states that if is a factor of , then .
• .
• is a root of .

Other uses of the factor theorem

The factor theorem can also be used to help factorise polynomials which we otherwise wouldn’t know how to factorise.

For example, a cubic polynomial like is not immediately factorable using simple methods. But if we know one of the roots, we also know a factor, which then makes this much easier to factorise.

We cover that in the polynomial division section.