Higher degree equations

To solve higher degree equations is somewhat (to a lot) more difficult. My preferred method is to use Mathematica, which knows the cubic formula and knows how to factor quartics. Any system will fail on some quintics, however, since it is a consequence of field extension theory that there are quintic equations which are not solvable using radicals. Your best bet is to chip away at the problem by looking for possible rational roots and then using long division to reduce the degree of the equation.