PolynomialReduce[poly, {poly_1, poly_2, ...}, {x_1, x_2, ...}] yields a list representing a reduction of poly in terms of the poly_i. The list has the form {{a_1, a_2, ...}, ...

Mathematica includes functionality to factor polynomials symbolically.

The main equations that Solve and related Mathematica functions deal with are polynomial equations. It is easy to solve a linear equation in  x. One can also solve quadratic ...

PolynomialRemainder[p, q, x] gives the remainder from dividing p by q, treated as polynomials in x.

PolynomialQuotientRemainder[p, q, x] gives a list of the quotient and remainder of p and q, treated as polynomials in x.

CharacteristicPolynomial[m, x] gives the characteristic polynomial for the matrix m. CharacteristicPolynomial[{m, a}, x] gives the generalized characteristic polynomial with ...

DiscreteLyapunovSolve[a, c] finds the numeric solution x of the discrete matrix equation a.x.a\[ConjugateTranspose] - x == c.DiscreteLyapunovSolve[a, b, c] solves a.x.b - x ...

LyapunovSolve[a, c] finds a solution x of the matrix Lyapunov equation a.x + x.a\[ConjugateTranspose] == c.LyapunovSolve[a, b, c] solves a.x + x.b == c.LyapunovSolve[{a, d}, ...

AlgebraicNumberPolynomial[a, x] gives the polynomial in x corresponding to the AlgebraicNumber object a.

You can give Solve a list of simultaneous equations to solve. Solve can find explicit solutions for a large class of simultaneous polynomial equations. Here is a simple ...