Question:
Find the value of $a$ for which the polynomial $\left(x^{4}-x^{3}-11 x^{2}-x+a\right)$ is divisible by $(x+3)$.
Solution:
Let:
$f(x)=x^{4}-x^{3}-11 x^{2}-x+a$
Now,
$x+3=0 \Rightarrow x=-3$
By the factor theorem, $f(x)$ is exactly divisible by $(x+3)$ if $f(-3)=0$.
Thus, we have:
$f(-3)=\left[(-3)^{4}-(-3)^{3}-11 \times(-3)^{2}-(-3)+a\right]$
$=(81+27-99+3+a)$
$=12+a$
Also
$f(-3)=0$
$\Rightarrow 12+a=0$
$\Rightarrow a=-12$
Hence, $f(x)$ is exactly divisible by $(x+3)$ when $a$ is $-12$.