Question:
Determine which of the following polynomial has $x-2$ a factor
(i) $3 x^{2}+6 x-24$
(ii) $4 x^{2}+x-2$
Solution:
Let $p(x)=x^{5}-4 a^{2} x^{3}+2 x+2 a+3$
Since, $x+2 a$ is a factor of $p(x)$, then put $p(-2 a)=0$
$\therefore \quad(-2 a)^{5}-4 a^{2}(-2 a)^{3}+2(-2 a)+2 a+3=0$
$\Rightarrow \quad-32 a^{5}+32 a^{5}-4 a+2 a+3=0$
$\Rightarrow \quad-2 a+3=0$
$\Rightarrow \quad 2 a=3$
$\therefore$ $a=\frac{3}{2}$
Hence, the value of $a$ is $\frac{3}{2}$.