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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.

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    Products of monic polynomials will again be monic, as (therefore) will quotients of monic polynomials (in such cases that the quotient is a polynomial at all).2012-09-03

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The fact that the dividend is a power of $x$ does not make the division substantially simpler.

It is analogous to long division of $10^n$ by some arbitrary smaller number, say with 5 digits (to correspond to your 4th degree divisor); you cannot just write down the answer in that case either.