Question

Find the value of m for which (2x – 1) is a factor of (8x⁴ + 4x³ – 16x² + 10x + m).

Answer 1

Let f(x) = 8x⁴ + 4x³ – 16x² + 10x + m

It is given that $\left(2x-1\right)=2\left(x-\frac{1}{2}\right)$ is a factor of f(x).

Using factor theorem, we have

$$\begin{gathered} f\left(\frac{1}{2}\right)=0 \\ \Rightarrow 8\times {\left(\frac{1}{2}\right)}^4+4\times {\left(\frac{1}{2}\right)}^3-16\times {\left(\frac{1}{2}\right)}^2+10\times \frac{1}{2}+m=0 \\ \Rightarrow \frac{1}{2}+\frac{1}{2}-4+5+m=0 \end{gathered}$$
$$\begin{gathered} \Rightarrow 2+m=0 \\ \Rightarrow m=-2 \end{gathered}$$
Thus, the value of m is −2.