Question

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

Answer 1

Step 1: Find the root of the given polynomial equation.

Since, it is given that $x+2a$ is a factor of $x^5-4a^2{x}^3+2x+2a+3$.

$$\begin{gathered} \Rightarrow x+2a=0 \\ \Rightarrow x=-2a \end{gathered}$$

Therefore, $x=-2a$ is one of the roots of the given polynomial equation.

Step 2: Compute the required value.

Assume that, $p\left(x\right)=x^5-4a^2{x}^3+2x+2a+3$

Since, $x=-2a$ is one of the roots of the given polynomial equation.

Therefore, $p(-2a)=0$.

$$\begin{gathered} \Rightarrow {(-2a)}^5-4a^2{(-2a)}^3+2(-2a)+2a+3=0 \\ \Rightarrow -32a^5+32a^5-4a+2a+3=0 \\ \Rightarrow -2a+3=0 \\ \Rightarrow -2a=-3 \\ \Rightarrow a=\frac{3}{2} \end{gathered}$$

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