# How do you use the factor theorem to determine whether x-5 is a factor of 3x^2 + 7x + 40?

Jan 10, 2016

Substitute $x = 5$ and see if the result is $0$.

#### Explanation:

If $f \left(x\right) = 3 {x}^{2} + 7 x + 40$ then $\left(x - 5\right)$ is a factor if and only if $f \left(5\right) = 0$

We find:

$f \left(5\right) = 3 \cdot {5}^{2} + 7 \cdot 5 + 40 = 75 + 35 + 40 = 150 \ne 0$

So $\left(x - 5\right)$ is not a factor.

In fact we can tell that $3 {x}^{2} + 7 x + 40$ has no factors with Real coefficients, since its discriminant is negative:

$\Delta = {b}^{2} - 4 a c = {7}^{2} - \left(4 \cdot 3 \cdot 40\right) = 49 - 480 = - 431$