Hi there,
Believe it or not, there is an analytical solution to a fourth degree polynomial, but that solution isnt that simple anyway. It would take a few lines of text here to display a solution for just *one* of the four roots.
For any degree, check out Lin-Bairstow or even just Bairstow on Wikipedia which has a worked out example using Bairstow's Method and some good links too.
You should also keep in mind that regular floating point may not work for degrees above 8 or so. One way to test is to evaluate the polynomial to see if it really reaches a true zero with the supposed solution found.