Question

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

Answer 1

Step 1:Write the given information.

Polynomial is $x^5-4a^2{x}^3+2x+2a+3$.

The factor of the polynomial is $x+2a$.

Step 2: Calculating the value of $x$:

Factor theorem:- $p\left(x\right)$ be a polynomial of degree greater than or equal to 1 and a be a real number. $p\left(a\right)=0$, then $(x-a)$ is a factor of $p\left(x\right)$.

Let $p\left(x\right)=x^5-4a^2{x}^3+2x+2a+3$ and

Now. $g\left(x\right)=x+2a$

From Factor's theorem,

$$\begin{gathered} g\left(x\right)=x+2a \\ x+2a=0 \\ x=-2a \end{gathered}$$

Step 3: Calculating the value of $a$:

If $g\left(x\right)$ is a factor of $p\left(x\right)$, then $p(-2a)$ must be equal to $0$

Thus, $p(-2a)=0$

$$\begin{gathered} \Rightarrow {(-2a)}^5-4a²{(-2a)}^3+2(-2a)+2a+3=0 \\ \Rightarrow -32a^5+32a^5-4a+2a+3=0 \\ \Rightarrow -2a+3=0 \\ \Rightarrow -2a=-3 \\ \Rightarrow 2a=3 \\ \Rightarrow a=\frac{3}{2} \end{gathered}$$

Hence, if $x+2a$ is a factor of $x^5-4a^2{x}^3+2x+2a+3$ then the value of $a$ is $\frac{3}{2}$.