# Remainder and factor theorems

If we divide a polynomial by (*x − r*), we obtain a result of the form:

*f*(*x*) *=* (*x − r*) *q*(*x*) *+ f(r)*

where *q*(*x*) is a polynomial with one degree less than the degree of f(x) and f(r) is the remainder. This is called the remainder theorem.

If the remainder *f*(*r*) *=* 0, then (*x − r*) is a factor of *f*(*x*).

**Example**

Is (x-2) a factor of f(x)=x^{3}-2x-6?

We identify r as 2 and plug that value into our function:

$$f(2)=(2)^{3}-2(2)+6=8-4+6=10$$

(x-2) is not a factor of f(x)=x^{3}-2x-6

**Video lesson**

Is (x-1) a factor of f(x)=x^{4}-2x^{2}+1?