Question
If $x+2 a$ is a factor of $x^5-4 a^2 x^3+2 x+2 a+3$, find $a$.
✓
Answer
Let $p(x)=x^5-4 a^2 x^3+2 x+2 a+3$
If $x-(-2 a)$ is a factor of $p(x)$, then $p(-2 a)=0 $
$\therefore p(-2 a)=(-2 a)^5-4 a^2(-2 a)^3+2(-2 a)+2 a+3$
$=-32 a^5+32 a^5-4 a+2 a+3=-2 a+3$ Now, $p(-2 a)=0 $
$\Rightarrow-2 a+3=0 $
$\Rightarrow a=\frac{3}{2}$
If $x-(-2 a)$ is a factor of $p(x)$, then $p(-2 a)=0 $
$\therefore p(-2 a)=(-2 a)^5-4 a^2(-2 a)^3+2(-2 a)+2 a+3$
$=-32 a^5+32 a^5-4 a+2 a+3=-2 a+3$ Now, $p(-2 a)=0 $
$\Rightarrow-2 a+3=0 $
$\Rightarrow a=\frac{3}{2}$
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