Question

For what value of $a$ the polynomial $2x^4+a{x}^3+4x^2+2x+1$ is divisible by $1-2x$?

• A. $a=-25$
• B. $a=-24$
• C. $a=-23$
• D. $a=-22$

Answer 1

Answer: (A)

$a=-25$

If $(1-2x)$ is a factor of the polynomial $2x^4+a{x}^3+4x^2+2x+1$,

Then, $(1-2x)$ must be the zero of the polynomial.

Putting $x=\frac{1}{2}$ in the polynomial we get,

$2{\left(\frac{1}{2}\right)}^4+a{\left(\frac{1}{2}\right)}^3+4{\left(\frac{1}{2}\right)}^2+2\left(\frac{1}{2}\right)+1=0$

$\Rightarrow 2\times \frac{1}{16}+a\frac{1}{8}+4\times \frac{1}{4}+1+1=0$

$\Rightarrow \frac{1}{8}+a\times \frac{1}{8}+3=0$

$\Rightarrow \frac{a}{8}=\frac{-1}{8}-3$

$\Rightarrow \frac{a}{8}=\frac{-1-24}{8}$

$\Rightarrow a=-25$

Hence, option A is the correct answer.