Find the value of a for which (x + 2a) is a factor of


Find the value of $a$ for which $(x+2 a)$ is a factor of $\left(x^{5}-4 a^{2} x^{3}+2 x+2 a+3\right)$.



Let $f(x)=x^{5}-4 a^{2} x^{3}+2 x+2 a+3$

It is given that $(x+2 a)$ is a factor of $f(x)$

Using factor theorem, we have

$f(-2 a)=0$

$\Rightarrow(-2 a)^{5}-4 a^{2} \times(-2 a)^{3}+2 \times(-2 a)+2 a+3=0$

$\Rightarrow-32 a^{5}-4 a^{2} \times\left(-8 a^{3}\right)+2 \times(-2 a)+2 a+3=0$

$\Rightarrow-32 a^{5}+32 a^{5}-4 a+2 a+3=0$

$\Rightarrow-2 a+3=0$

$\Rightarrow 2 a=3$

$\Rightarrow a=\frac{3}{2}$

Thus, the value of $a$ is $\frac{3}{2}$.


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