# How do you use the factor theorem to determine whether x+3 is a factor of 2x^3 + x^2 – 13x + 6?

Jan 6, 2016

For $f \left(x\right) = 2 {x}^{3} + {x}^{2} - 13 x + 6$
If $\left(x + 3\right)$ is a factor, $x = - 3$ should be a root.
So for $x = - 3$ to be a root, $f \left(- 3\right) = 0$
If $f \left(- 3\right) \ne 0$, $\left(x + 3\right)$ is not a factor.

#### Explanation:

Factor theorem basically boils down to testing potential roots in the equation.

So for some equation $f \left(x\right)$

A root is an $x$ value which satisfies $f \left(x\right) = 0$

So if a factor of $f \left(x\right)$ is $\left(x - a\right)$,
$x = a$ is a root of $f \left(x\right)$.

$f \left(x\right) = 2 {x}^{3} + {x}^{2} - 13 x + 6$,
we want to see if $\left(x + 3\right)$ is a factor.
So we test $x = - 3$
$f \left(- 3\right) = - 54 + 9 + 39 + 6 = 0$
$f \left(- 3\right) = 0$ so $x = - 3$ is a root.
Therefore $\left(x + 3\right)$ is indeed a factor.