Question

Dividing second polynomial by first polynomial and test whether first polynomial is a factor of second polynomial.
$g\left(x\right)=x^3-3x+1,f\left(x\right)=x^5-4x^3+x^2+3x+1$

Answer 1

$f\left(x\right)=x^5-4x^3+x^2+3x+1$
$g\left(x\right)=x^3-3x+1$
If dividing $f\left(x\right)$ by $g\left(x\right)$ we get remainder $0$ then $g\left(x\right)$ will be a factor of $f\left(x\right)$.
Dividing $f\left(x\right)$ by $g\left(x\right)$,
$x^2-1x^3-3x+1)\overline{x^5-4x^3+x^2+3x+1}{x}^5-3x^3+x^2-+-\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_-x^3+3x+1-4x^3+3x-1+-+\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_2$
Remainder $r\left(x\right)=2$
Quotient $q\left(x\right)=x^2-1$
Remainder is not zero.
Therefore $g\left(x\right)$ is not a factor of $f\left(x\right)$.