Question

If x + α is a factor of the polynomial 2x² + 2αx + 4x + 12, then α = _______.

Answer 1

Let f(x) = 2x² + 2αx + 4x + 12

It is given that one factor of f(x) is (x + α).

Therefore, $f\left(x\right)=0\mathrm{when}x=-\alpha$.

On putting x = –α in f(x) = 0, we get

$$\begin{gathered} 2{\left(-\alpha \right)}^2+2\alpha \left(-\alpha \right)+4\left(-\alpha \right)+12=0 \\ \Rightarrow 2{\left(\alpha \right)}^2-2{\alpha }^2-4\alpha +12=0 \\ \Rightarrow -4\alpha +12=0 \\ \Rightarrow -4\alpha =-12 \\ \Rightarrow \alpha =3 \end{gathered}$$

Hence, if x + α is a factor of the polynomial 2x² + 2αx + 4x + 12, then α = 3.