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I'm familiar with the concept of modular arithmetic, but only with constants. I've never seen it with polynomials before. How would I reduce $q(x)$ modulo $p(x)$? Do polynomial long division and take the remainder?

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Exactly as you said, if $q(x)$ can be divided with remainder by $p(x)$, you can keep $r(x)$ instead of $q(x)$.

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It means to calculate the remainer when dividing $q(x)$ by $p(x)$ using long division.

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