Is there a standard way to understand contructions like $(?):=F\left[X\right]/\left\langle a_{n}X^{n}+\cdots+a_{1}X+a_{0}\right\rangle $, where $F$ is a field ? Because I constantly have to do exercises with contructions like these and I'm really tired of having to
a) try "manually'' to figure out how the set $(?)$ looks like
b) think every time of a homomorphism such that I may apply the homomorphism theorem for rings to get an isomorphism between the above contstruction
c) think of some other clever way which tells me what $(?)$ looks like
I'm thinking of some algorithm/grand theorem, which only given the $F,a_{n},\ldots,a_{1}$ tells me how the set $(?)$ looks like.