Suppose I have the set of polynomials $P$ in one real variable.
$p(t) = t^3 + 2t^2 + 3t + 5, p ∈ P$.
And I have an equivalence relation $\sim$ on P defined as $a \sim b$ if $a = b + pq$ for some $q \in P$.
How can I show that each equivalence class has a unique representative of a degree less than $3$?
