Question

Use the Factor Theorem to determine whether $g\left(x\right)$ is factor of $f\left(x\right)$
$f\left(x\right)=5x^3+x^2-5x-1,g\left(x\right)=x+1$

Answer 1

Given polynomial $f\left(x\right)=5x^3+x^2-5x-1$ the factor of $g\left(x\right)=x+1$

If $x+1$ is factor then $x+1=0$ or $x=-1$

Replace $x$ in $f\left(x\right)$ by $-1$ we get

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

$f(-1)=5(-1{)}^3+(-1{)}^2-5(-1)-1$

$\Rightarrow f(-1)=-5+1+5-1$

$\Rightarrow f(-1)=0$

So $f(-1)$ is zero by $g\left(x\right)=x+1$ then $g\left(x\right)=x+1$ is factor of $f\left(x\right)=5x^3+x^2-5x-1$