Question:
Find the value of $a$ for which $(x-4)$ is a factor of $\left(2 x^{3}-3 x^{2}-18 x+a\right)$.
Solution:
Let:
$f(x)=2 x^{3}-3 x^{2}-18 x+a$
$(x-4)$ is a factor of $f(x)=2 x^{3}-3 x^{2}-18 x+a$.
$\Rightarrow f(4)=0$
$\Rightarrow 2 \times 4^{3}-3 \times 4^{2}-18 \times 4+a=0$
$\Rightarrow 8+a=0$
$\Rightarrow a=-8$
Hence, the required value of a is −8.
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