Question:
Find the value of $k$ for which $(x-1)$ is a factor of $\left(2 x^{3}+9 x^{2}+x+k\right)$.
Solution:
Let:
$f(x)=2 x^{3}+9 x^{2}+x+k$
$(x-1)$ is a factor of $f(x)=2 x^{3}+9 x^{2}+x+k$.
$\Rightarrow f(1)=0$
$\Rightarrow 2 \times 1^{3}+9 \times 1^{2}+1+k=0$
$\Rightarrow 12+k=0$
$\Rightarrow k=-12$
Hence, the required value of k is −12.
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