If x + 2a is a factor of x^5 – 4a^2 x^3 + 2x + 2a + 3, find a. - Vidyadip

Question

If $x + 2a$ is a factor of $x^5 – 4a^2 x^3 + 2x + 2a + 3,$ find a.

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Answer

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

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