Question

For what value of k is the polynomial p(x) = 2x³ − kx + 3x + 10 exactly divisible by (x + 2)?
(a) $-\frac{1}{3}$
(b) $\frac{1}{3}$
(c) 3
(d) −3

Answer 1

(d) −3

Let:
$p\left(x\right)=2x^3-k{x}^2+3x+10$
Now,
$x+2=0\Rightarrow x=-2$
$$\begin{gathered} p\left(x\right)\text{is completely divisible by}\left(x+2\right)\text{.} \\ \therefore p\left(-2\right)=0 \\ \Rightarrow 2\times {\left(-2\right)}^3-k\times {\left(-2\right)}^2+3\times \left(-2\right)+10=0 \\ \Rightarrow -16-4k-6+10=0 \\ \Rightarrow -12-4k=0 \\ \Rightarrow 4k=-12 \\ \Rightarrow k=\frac{-12}{4} \\ \Rightarrow k=-3 \end{gathered}$$