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)=x3-2x-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)=x3-2x-6


Video lesson

Is (x-1) a factor of f(x)=x4-2x2+1?