Question

In each of the following, using the remainder theorem, find the remainder when $f\left(x\right)$ is divided by $g\left(x\right)$ and verify the result by actual division:

5. $f\left(x\right)=x^3-6x^2+2x-4,g\left(x\right)=1-2x$

Answer 1

Given: $f\left(x\right)=x^3-6x^2+2x-4$,is divided by $g\left(x\right)=1-2x$.

By remainder theorem, if polynomial $f\left(x\right)$ is divided by the linear polynomial $(x-a)$; then remainder is equal to $f\left(a\right)$.

$\therefore f\left(\frac{1}{2}\right)={\left(\frac{1}{2}\right)}^3-6\times {\left(\frac{1}{2}\right)}^2+2\times \left(\frac{1}{2}\right)-4$

$=\frac{1}{8}-\frac{3}{2}+1-4=-\frac{35}{8}$

Now, verify the result by long division method.


Hence, $-\frac{35}{8}$ is the remainder verified.