Factor theorem

The factor theorem states that:

If f(x) is a polynomial and f(c) = 0 for some constant c, then (x - c) is a factor of the polynomial f(x).

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

When you solve an equation that looks something like, for example, (x-2)(x+3)=0, you used the factor theorem without even knowing it!

From that equation, you know that 2 must be a root of the polynomial (in other words, where x=2, the polynomial equals zero). You can also see that (x-2) is a factor of the polynomial. This is exactly what the factor theorem states.

Notation

We can write the factor theorem as: $$ \text{if} f(a)=0\text{, then } (x-a)\text{ is a factor of }f(x)$ $$

Examples

Find the two roots of the polynomial f(x) = x^2 - x - 6 using the factor theorem

• Factorise:
  • (x-3)(x+2)
• Find the roots:
  • If (x-a) is a factor of the polynomial, then f(a) = 0.
  • We know that (x-3) is a factor, so f(3) = 0.
  • We also know that (x+2) is a factor, so f(-2) = 0.
• So the roots the roots of the polynomial are x=3 and x=-2.

Given that (x-3) is a factor of f(x), find a root of f(x)=0

• The factor theorem states that if (x-3) is a factor of f(x), then f(3)=0.
• f(3)=0.
• 3 is a root of f(x)=0.

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 f(x) = x^3 - 6x^2 + 11x - 6 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.