Question

Find the value of k for which (x − 1) is a factor of (2x³ + 9x² + x + k).

Answer 1

Let:
$f\left(x\right)=2x^3+9x^2+x+k$
$$\begin{gathered} \left(x-1\right)\text{is a factor of}f\left(x\right)=2x^3+9x^2+x+k. \\ \Rightarrow f\left(1\right)=0 \\ \Rightarrow 2\times 1^3+9\times 1^2+1+k=0 \\ \Rightarrow 12+k=0 \\ \Rightarrow k=-12 \end{gathered}$$
Hence, the required value of k is $-$12.