Remainder and factor theorems

Name: Remainder and factor theorems
Uploaded: 2011-02-08
Description: Is $(x-1)$ a factor of $f(x) = x^4 - 2x^2 + 1$?

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:

$$ f(2) = 2^3 - 2 \cdot 2 + 6 = 8 - 4 + 6 = 10 $$

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

Video lesson

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