Factoring a quadratic polynomial in one variable is straightforward. But the Wolfram Language routinely factors degree-100 polynomials in 3 variables—by making use of a tower of sophisticated algorithms, carefully tuned at Wolfram Research over the course of two decades.

Factor — fully factor any polynomial

FactorSquareFree — pull out only factors that occur as squares or higher powers

FactorTerms — pull out only overall numerical factors

FactorList, FactorSquareFreeList, FactorTermsList — give lists of factors

IrreduciblePolynomialQ  ▪  SquareFreeQ

Modulus — specify a modulus for a finite field

Extension — specify a general algebraic extension field

GaussianIntegers — factor over Gaussian integers

Expand — expand out polynomials

MonomialList — the list of monomials in a specified ordering

Decompose — decompose into a composition of polynomials

HornerForm — rearrange a polynomial into Horner form

SymmetricReduction — reduce a polynomial into a symmetric and nonsymmetric part

InterpolatingPolynomial — construct a polynomial that goes through a list of points

Rational Functions

Apart  ▪  Together  ▪  ApartSquareFree