# How do you use the remainder theorem and the factor theorem to determine whether (b+4) is a factor of (b^3 + 3b^2 − b + 12)?

Apr 14, 2016

$\left(b + 4\right)$ is a factor of ${b}^{3} + 3 {b}^{2} - b + 12$

#### Explanation:

Remainder theorem states that if we divide a polynomial function $f \left(x\right)$ by $\left(x - a\right)$, the remainder is $f \left(a\right)$.

Hence, if $\left(x - a\right)$ is a factor of $f \left(x\right)$, $f \left(a\right) = 0$ (This is factor theorem.)

As we have to determine whether $\left(b + 4\right)$ is a factor of $f \left(b\right) = {b}^{3} + 3 {b}^{2} - b + 12$ or not,

we should evaluate $f \left(- 4\right)$ which is equal to

${\left(- 4\right)}^{3} + 3 {\left(- 4\right)}^{2} - \left(- 4\right) + 12 = - 64 + 48 + 4 + 12 = 0$

Hence $\left(b + 4\right)$ is a factor of ${b}^{3} + 3 {b}^{2} - b + 12$.