Here’s a curious result I ran across the other day. Suppose you have a quintic equation of the form z x5 – x – 1 = 0. (It’s possible to reduce a general quintic equation to this form, known as Bring-Jerrard normal form.) There is no elementary formula for the roots of this equation, but the following infinite series does give a root as a function of the leading coefficient z:
One reason this is interesting is that the series above has a special form that makes it a hypergeometric function of z. You can read more about it here.
I could imagine situations where having such an expression for a root is useful, though I doubt the series would be much use if you just wanted to find the roots of a fifth degree polynomial numerically. Direct application of something like Newton’s method would be much simpler.