# How do you use the factor theorem to determine whether 3x+1 is a factor of f(x)= 3x^4 -11x^3 - 55x^2 + 163x + 60?

Jun 13, 2018

$3 x + 1$ is a factor of $f \left(x\right)$

#### Explanation:

Let's define $f \left(c\right)$ to be equal to $3 x + 1$. Thus, the factor theorem tells us if $f \left(c\right) = 0$, then $3 x + 1$ is a factor of our polynomial $f \left(x\right)$.

Let's set $f \left(c\right)$ equal to zero. We get

$3 x + 1 = 0$

$\implies 3 x = - 1$

$\implies \textcolor{b l u e}{x = - \frac{1}{3}}$

Now, let's plug this value into our polynomial. This is a hairy problem, so let's not be ashamed to use a calculator. I did but had issues uploading the image, but this evaluates to zero.

Since it evaluated to zero, $3 x + 1$ is a factor of $f \left(x\right)$.

Hope this helps!