(This was supposed to be a comment, but it got too long.)
Robert raises a good question in the comments. Of course, general analytical expressions for roots of polynomials of high degree aren't really used much in practice, precisely because the expressions themselves are a bit unwieldy, and the special functions (theta functions, hypergeometric functions) that are involved in the closed-form expressions have to be numerically evaluated anyway, and for numerics, there's a bunch of more efficient numerical methods for getting a pile of roots than evaluating special functions.
Now, there are a number of reasons for the interest in solving polynomials numerically: for instance, the behavior of solutions to difference and differential equations can be easily analyzed by looking at the roots of a "characteristic polynomial". In signal processing and a bunch of other applications, one is often interested where in the complex plane the roots of a certain polynomial are, whether they are located within a disk, or to the left or right of a half plane (on the other hand, if one just wants to check for existence of roots in such regions, there are computationally less intensive methods). As lhf mentions, CAGD often relies on the solution of polynomials, one application being in finding intersections of shapes represented as piecewise polynomials.
Eigenvalues of matrices, pencils, and matrix polynomials have a wide variety of applications, which I won't go into detail here; just search around.
So yes, there'll always be a use for a better way to find roots of polynomials than current methods.