Question

Find the value of a for which (x + 2a) is a factor of (x⁵ – 4a² x³ + 2x + 2a +3).

Answer 1

Let f(x) = x⁵ – 4a² x³ + 2x + 2a +3

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

Using factor theorem, we have

f(−2a) = 0
$$\begin{gathered} \Rightarrow {\left(-2a\right)}^5-4a^2\times {\left(-2a\right)}^3+2\times \left(-2a\right)+2a+3=0 \\ \Rightarrow -32a^5-4a^2\times \left(-8a^3\right)+2\times \left(-2a\right)+2a+3=0 \\ \Rightarrow -32a^5+32a^5-4a+2a+3=0 \\ \Rightarrow -2a+3=0 \end{gathered}$$
$$\begin{gathered} \Rightarrow 2a=3 \\ \Rightarrow a=\frac{3}{2} \end{gathered}$$
Thus, the value of a is $\frac{3}{2}$.