In $\mathbb{Z}_3$
$ x^9 : x^4+x^3+x^2+2x+1 = x^5+2x^4+2x^2+2x$
with remainder of $x$.
In $\mathbb{Z}_7$
$x^7 : x^4+5x^3+x+5 = x^3+2x^2+4x$
with remainder of $x$.
Is this random? Or is there some kind of trick which I do not see to avoid polynomial long-divison by hand when computing the remainder of polynomials in finite fields of this form. Both calculations occured in example excerises for the calculation of distinct-degree factorization for polynomials in finite fields.