Question

Use the Factor Theorem to determine whether $g\left(x\right)$ is factor of $f\left(x\right)$ or not.
$f\left(x\right)=3x^3+x^2-20x+12,g\left(x\right)=3x-2$

Answer 1

Given $f\left(x\right)=3x^3+x^2-20x+12$ the $g\left(x\right)=3x-2$ is the factor

If $g\left(x\right)=3x-2$ is factor then $g\left(x\right)=3x-2=0$ or $x=\frac{2}{3}$

Replace x by $\frac{2}{3}$ in f(x) we get

$f\left(x\right)=3x^3+x^2-20x+12$

$f\left(\frac{2}{3}\right)=3{\left(\frac{2}{3}\right)}^3+{\left(\frac{2}{3}\right)}^2-20\left(\frac{2}{3}\right)+12$

$\Rightarrow f\left(\frac{2}{3}\right)=3\times \frac{8}{27}+\frac{4}{9}-20\times \frac{2}{3}+12$

$f\left(\frac{2}{3}\right)=\frac{8}{9}+\frac{4}{9}-\frac{40}{3}+12$

$f\left(\frac{2}{3}\right)=\frac{8+4-120+108}{9}=\frac{120-120}{9}=0$

So f(x)is zero by g(x) then g(x) =3x-2 is factor of $f\left(x\right)=3x^3+x^2-20x+12$