not every root of a polynomial is of the form . while something akin to the discriminant for the quadratic exists for cubic equations, and quartics can be solved in terms of their resolvent cubics, for polynomials of degree > 4, there is no general formula at all.

wolframalpha.com and its "parent" program mathematica (and maple, too, i suspect, i don't have experience with that program much) will display roots in an exact form, if there is one. be prepared for some very ugly formulas.