Find the value of m,
Question:

Find the value of m, so that 2x -1 be a factor of

8x4 +4x3 -16x2 +10x+07.

Solution:

Let $p(x)=8 x^{4}+4 x^{3}-16 x^{2}+10 x+m$

Since, $2 x-1$ is a factor of $p(x)$, then put $p\left(\frac{1}{2}\right)=0$

$\therefore \quad 8\left(\frac{1}{2}\right)^{4}+4\left(\frac{1}{2}\right)^{3}-16\left(\frac{1}{2}\right)^{2}+10\left(\frac{1}{2}\right)+m=0$

$\Rightarrow \quad 8 \times \frac{1}{16}+4 \times \frac{1}{8}-16 \times \frac{1}{4}+10\left(\frac{1}{2}\right)+m=0$

$\Rightarrow \quad \frac{1}{2}+\frac{1}{2}-4+5+m=0$

$\Rightarrow \quad 1+1+m=0$

$\therefore \quad m=-2$

Hence, the value of $m$ is $-2$.