Question

Use the Factor Theorem to determine whether $g\left(x\right)$ is factor of $f\left(x\right)$ in each of the following cases:
$f\left(x\right)=x^3-4x^2+x+6,g\left(x\right)=x-2$

Answer 1

Given polynomial $f\left(x\right)=x^3-4x^2+x+6$ the factor of $g\left(x\right)=x-2$

If x-2 is factor then x-2=0 or x=2

Replace x in p(x) by 2 we get

$f\left(x\right)=x^3-4x^2+x+6$

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

$\Rightarrow f\left(2\right)=8-16+2+6$

$\Rightarrow f\left(2\right)=0$

So f(x) is zero by g(x) =x-2 then g(x)=(x-2) is factor of $f\left(x\right)=x^3-4x^2-x+6$