Question

Find the remainder when $f\left(x\right)=2x^3-6x^2+4x-2$ is divided by $g\left(x\right)=2x-1$

Answer 1

When $f\left(x\right)$ is divisible by $(x-a)$ then the remainder is $f\left(a\right)$

We have $g\left(x\right)=2x-1=2(x-\frac{1}{2})$

Therefore $a=\frac{1}{2}$

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

$=2\cdot \frac{1}{8}-6\cdot \frac{1}{4}+4\cdot \frac{1}{2}-2$

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

$=\frac{1}{4}-\frac{3}{2}$

$f\left(\frac{1}{2}\right)=-\frac{5}{4}$

Thus the remainder is $-\frac{5}{4}$