As with integers, operations related to division are key to many computations with polynomials. Mathematica includes not only highly optimized univariate polynomial-division ...

An expression like x^2+2x-7==0 represents an equation in Mathematica. You will often need to solve equations like this, to find out for what values of x they are true. This ...

Although Diophantine equations provide classic examples of undecidability, Mathematica in practice succeeds in solving a remarkably wide range of such equations—automatically ...

MinimalPolynomial[s, x] gives the minimal polynomial in x for which the algebraic number s is a root.

SymmetricPolynomial[k, {x_1, ..., x_n}] gives the k\[Null]^th elementary symmetric polynomial in the variables x_1, ..., x_n.

The leading term of a polynomial can be chosen in many different ways. For multivariate polynomials, sorting by the total degree of the monomials is often useful. Different ...

PolynomialQuotient[p, q, x] gives the quotient of p and q, treated as polynomials in x, with any remainder dropped.

PolynomialMod[poly, m] gives the polynomial poly reduced modulo m. PolynomialMod[poly, {m_1, m_2, ...}] reduces modulo all of the m_i.

PolynomialQ[expr, var] yields True if expr is a polynomial in var, and yields False otherwise. PolynomialQ[expr, {var_1, ...}] tests whether expr is a polynomial in the var_i.

InterpolatingPolynomial[{f_1, f_2, ...}, x] constructs an interpolating polynomial in x which reproduces the function values f_i at successive integer values 1, 2, ... of x. ...