Question

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

Answer 1

Given polynomial $f\left(x\right)=x^3+3x^2+3x+1$ and $g\left(x\right)=x+1$

If $x+1$ is factor then $f(-1)=0$

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

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

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

or,$f(-1)=-1+3-3+1$

or,$f(-1)=0$

So $g\left(x\right)=x+1$ is factor of $f\left(x\right)=x^3+3x^2+3x+1$