Packed into functions like Solve and Reduce are a wealth of sophisticated algorithms, many created specifically for Mathematica. Routinely handling both dense and sparse polynomials with thousands of terms, Mathematica can represent results in terms of numerical approximations, exact radicals or its unique symbolic Root object constructs.
Solve — find generic solutions
Roots — roots of a univariate polynomial
Reduce — reduce a general polynomial system
CountRoots — count roots in an interval