If x+2 a is a factor of x^{5}-4 a^{2} x^{3}+2 x+2 a+3, find a

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2. Polynomials

hard

If $x+2 a$ is a factor of $x^{5}-4 a^{2} x^{3}+2 x+2 a+3,$ find $a$

A

$\frac{3}{2}$

B

$\frac{1}{2}$

C

$-\frac{3}{2}$

D

$1$

Solution

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, $\quad p(-2 a)=0$

$\Rightarrow \quad-2 a+3=0$

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

Standard 9

Mathematics

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