This is how our lecture on equivalence classes (abstract algebra) went:
Let's consider an example, first. For $4$ modulo, the equivalence classes are $\bar{0},\bar{1},\bar{2},\bar{3}$. To find which equivalence class $7$ belongs to, write it as $4k+r$ (where $ 0\leq r\leq 3$).The $r$'s form an equivalence class. We can extend this concept to polynomials. Suppose we have to find the equivalence class to which $2x^2+4x+5$ under modulo $x^2+1$, we write it in the form $2(x^2+1)+(4x+3)$. That's is when divided by $(x^2+1)$, we get an equivalence class of remainders of the form $ax+b$. $ax+b$ and $a'x+b$ belong to the same class if $(a-a')$ and $(b-b')$ are divisible by $(x^2+1)$.
I lost it at the last line. What does the professor mean by if $(a-a')$ and $(b-b')$ are divisible by $(x^2+1)$ (how does division of numbers by polynomial make sense)? Or did I mishear/misinterpret something? Anyhow, I'd like to know the actual condition for $ax+b$ and $a'x+b'$ to be considered to belong to the equivalence classes.
If $a,b,a',b'$ are numbers, then it is correct. In fact numbers are divisible by $x^2+1$ iff they are zero.
What your teacher is trying to tell you is that $ax+b$ represent all equivalence classes, since every polynomial can be reduced to one of them, and they all belong to different classes of equivalence.