= Polynomial long division
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To divide $q$ by a nonzero <polynomial> $Q$ over a <field>, subtract the multiple of $Q$ that cancels the leading term of the current remainder, and add that multiple to the quotient. Each step strictly decreases the remainder degree, so the process terminates with $q=Qs+r$ and $\deg r<\deg Q$. If two such pairs existed, $Q(s-s\prime)=r\prime-r$ would have degree both at least $\deg Q$ and less than $\deg Q$, unless both differences vanished.
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