# How do you use the factor theorem to determine whether x+1/3 is a factor of f(x) = 3x^4 + x^3 - 3x + 1?

##### 1 Answer
Dec 5, 2015

It is not a factor since $f \left(- \frac{1}{3}\right) \ne 0$
So it will leave a remainder when long divided in.

#### Explanation:

If $\left(x + \frac{1}{3}\right)$ is a factor of $f \left(x\right)$, then when divided into f it must leave no remainder, ie. $f \left(- \frac{1}{3}\right)$ must be zero.

But : $f \left(- \frac{1}{3}\right) = 3 {\left(- \frac{1}{3}\right)}^{4} + {\left(- \frac{1}{3}\right)}^{3} - 3 \left(- \frac{1}{3}\right) + 1$

$= 2$

$\ne 0$

$\therefore \left(x + \frac{1}{3}\right)$ is not a factor of $f \left(x\right)$.