Question

Find the remainder when the polynomial $f\left(x\right)=4x^3-12x^2+14x-3$ is divided by $g\left(x\right)=2x-1$.

Answer 1

Given, $f\left(x\right)=4x^3-12x^2+14x-3$.

Here, $f\left(x\right)$ is divided by $2x-1$.

Then, $2x-1=0\Rightarrow x=\frac{1}{2}$.

By the remainder theorem, we know, when $f\left(x\right)$ is by $(2x-1)$, the remainder is $f\left(\frac{1}{2}\right)$.

Now,

$f\left(\frac{1}{2}\right)=[4\times {\left(\frac{1}{2}\right)}^3-12\times {\left(\frac{1}{2}\right)}^2+14\times \frac{1}{2}-3]$
$=(\frac{1}{2}-3+7-3)$

$=(\frac{1}{2}+1)=\frac{3}{2}$.

Hence, the remainder obtained is $\frac{3}{2}$.