Question

Find the value of a for which the polynomial (x⁴ − x³ − 11x² − x + a) is divisible by (x + 3).

Answer 1

Let:
$f\left(x\right)=x^4-x^3-11x^2-x+a$
Now,
$x+3=0\Rightarrow x=-3$
By the factor theorem, $f\left(x\right)\text{is exactly divisible by}\left(x+3\right)\text{if}f\left(-3\right)=0$.
Thus, we have:
$$\begin{gathered} f\left(-3\right)=\left[{\left(-3\right)}^4-{\left(-3\right)}^3-11\times {\left(-3\right)}^2-\left(-3\right)+a\right] \\ =\left(81+27-99+3+a\right) \\ =12+a \end{gathered}$$
Also,
$$\begin{gathered} f\left(-3\right)=0 \\ \Rightarrow 12+a=0 \\ \Rightarrow a=-12 \end{gathered}$$
Hence, $f\left(x\right)\text{is exactly divisible by}\left(x+3\right)\text{when}a\mathrm{is}-12.$