# What does it mean to reduce one polynomial “modulo” another polynomial?

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?

Exactly as you said, if $q(x)$ can be divided with remainder by $p(x)$, you can keep $r(x)$ instead of $q(x)$.
It means to calculate the remainer when dividing $q(x)$ by $p(x)$ using long division.