By the way, here's the "cubic equation" version that would be somewhat familiar to you. You end up substituting a, b, c, and d into dummy variables named q, r, s, and t, which are in turn substituted into your three roots named x1, x2, and x3.

The 3rd method I listed would probably be the most effective "by-hand" method for the equations you look to be dealing with, while the cubic equation one would probably be best for a computer. The Newton's Method one was what first came to mind, and would be effective, but the cubic equation is simply easier (and infinitesimally more accurate in some cases).

The big advantage of the Newton's Method idea is that it works for n-degree polynomials (at least for real roots, without some manipulation), and not just 3rd degree ones.

Good luck! If you'd like an explanation/example on one of them I'd be happy to write one up.