Question

Determine whether $(2x+1)$ is a factor of $4x^3+4x^2-x-1$.

Answer 1

According to factor theorem, when $f\left(x\right)$ is divided by $(2x+1)$, $f\left(\frac{-1}{2}\right)=0$

$f\left(x\right)=4x^3+4x^2-x-1$

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

$=4\times \frac{-1}{8}+4\times \frac{1}{4}+\frac{1}{2}-1$

$=\frac{-1}{2}+1+\frac{1}{2}-1=0$

$\because f\left(\frac{-1}{2}\right)=0,(2x+1)$ is a factor of the polynomial.