Analytic solutions exists for n up to 3 otherwise if the expression is not expressible in terms of lower polynomials we have to use field theory, I think.

If you are asking about polynomials in 1 variable, there are well-known solution expressions built from the coefficients using radicals (square and cube roots of expressions) for degrees up to 4 - no such expressions exist for n>=5 (this is a famous result of niels abel, and also follows from work of galois in the 1800's) - these expressions get increasingly complicated as n goes from 2 (the quadratic formula) up to 4

this does not mean that your question has no answer, however: for n >= 5, there may be expressions to find solutions in terms of the coefficients, but they are not based on simple or even common mathematical functions that are familiar to most - part of the difficulty is that even when the coefficients are simple rationals or even integers, the expressions have to account for the fact that the solutions need not be real numbers (there are many theorems that provide restrictions on the solutions based on the coefficients, but they are not sufficient to solve the equations - they do often help for finding good starting points to use with numerical methods, but that isn't what you asked)

if you were not asking about one variable polynomials only, then the answers are different - one polynomial in more than one variable usually has solutions in a complex space of dimension one less than the number of variables, with even more and stranger complications than the ones with one variable, and multiple polynomials have even more possibilities, with one variable or more

hope this helps; sorry for the bad news