Cardinality of $R[x]/\langle f\rangle$ via canonical remainder reps. Suppose $R$ is a field and $f$ is a polynomial of degree $d$ in $R[x]$.  How do you show that each coset in $R[x]/\langle f\rangle$ may be represented by a unique polynomial of degree less than $d$?  Secondly, if $R$ is finite with $n$ elements, how do you show that $R[x]/\langle f\rangle$ has exactly $n^d$ cosets?

Hint $ $ Recall $\rm\ R[x]/(f)\:$ has a complete system of reps being the least degree elements in the cosets, i.e. the remainders mod $\rm\:f,\:$ which uniquely exist by the Polynomial Division Algorithm.
Therefore the cardinality of the quotient ring equals the number of such reps, i.e. the number of  polynomials $\rm\in R[x]\:$ with degree smaller than that of $\rm\:f.$
Remark $\ $ This is a generalization of the analogous argument for $\rm\:\Bbb Z/m.\:$ The argument generalizes to any ring with a Division (with Remainder) Algorithm, i.e. any Euclidean domain, as explained in the linked answer.
