I just had a discussion with someone who said he thought about quotient rings of polynomials as simply adjoining an element that is a root of the polynomial defining the ideal. For example, consider a field, F, and a polynomial, x-a, in F[x]. If we let (x-a) denote the ideal generated by x-a, then we can form the quotient ring F[x]/(x-a). He was saying we can think about this ring as F[a] (i.e. adjoining a (which is a root of x-a) to the original field F). This seems like a very useful intuition to have, but I am still struggling to see how it is true. Up until now, I've been thinking of the quotient ring simply in terms of a bunch of equivalence classes of polynomials, which is difficult to have an intuition for.