Symbolic computation with roots of polynomials is both possible
and useful, by use of the
RootOf construct. It is not yet
perfected in Maple, as it does not yet allow useful distinctions
to be made between the roots, but even so it is useful.
Some applications of polynomials require only knowledge of the stability of the polynomials and not the exact location of the roots. Classical algorithms exist for this problem, and one has been implemented in Maple here.