# How to Use the Roots of a Polynomial to Find Its Factors

The *factor theorem* states that you can go back and forth between the roots of a polynomial and the factors of a polynomial. In other words, if you know one, you know the other. At times, your teacher or your textbook may ask you to factor a polynomial with a degree higher than two. If you can find its roots, you can find its factors.

In symbols, the factor theorem states that if *x* – *c* is a factor of the polynomial *f*(*x*),* *then *f*(*c*) = 0. The variable *c* is a zero or a root or a solution — whatever you want to call it (the terms all mean the same thing).

Here’s an example. Say you have to look for the roots of the polynomial *f*(*x*) = 2*x*^{4} – 9*x*^{3} – 21*x*^{2} + 88*x* + 48. You find that they are *x* = –1/2, *x* = –3, and *x* = 4 (multiplicity two). How do you use those roots to find the factors of the polynomial?

The factor theorem states that if *x* = *c* is a root, (*x* – *c*) is a factor. For example, look at the following roots:

If

*x*= –1/2, (*x*– (–1/2)) is your factor, which you write as (*x*+ 1/2).If

*x*= –3 is a root, (*x*– (–3)) is a factor, which you write as (*x*+ 3).If

*x*= 4 is a root, (*x*– 4) is a factor with multiplicity two.

You can now factor *f*(*x*) = 2*x*^{4} – 9*x*^{3} – 21*x*^{2} + 88*x* + 48 to get *f*(*x*) = 2(*x* + 1/2)(*x* + 3)(*x* – 4)^{2}. Observe that 2 is a factor because 2 is the leading coefficient (the coefficient of the term with the highest exponent.)