Built-in Mathematica Symbol

PolynomialGCD

PolynomialGCD[poly₁, poly₂, ...]
gives the greatest common divisor of the polynomials poly_i.

PolynomialGCD[poly₁, poly₂, ..., Modulus->p]
evaluates the GCD modulo the prime p.

In PolynomialGCD[poly₁, poly₂, ...], all symbolic parameters are treated as variables.

PolynomialGCD[poly₁, poly₂, ...] will by default treat algebraic numbers that appear in the poly_i as independent variables.

PolynomialGCD[poly₁, poly₂, ..., Extension->Automatic] extends the coefficient field to include algebraic numbers that appear in the poly_i.

The greatest common divisor of polynomials:

The GCD of univariate polynomials:

The GCD of multivariate polynomials:

The GCD of more than two polynomials:

The GCD of rational functions:

By default, algebraic numbers are treated as independent variables:

With Extension->Automatic, PolynomialGCD detects algebraically dependent coefficients:

Compute the GCD over the integers modulo 2:

By default, PolynomialGCD treats trigonometric functions as independent variables:

With Trig->True, PolynomialGCD recognizes dependencies between trigonometric functions:

Find common roots of univariate polynomials:

Find multiple roots of univariate polynomials:

The GCD of polynomials divides the polynomials; use PolynomialMod to prove it:

Cancel divides the numerator and the denominator of a rational function by their GCD:

PolynomialLCM finds the least common multiple of polynomials:

Resultant of two polynomials is zero if and only if their GCD has a nonzero degree:

Discriminant of a polynomial f is zero if and only if the degree of GCD(f, f') is nonzero:

Discriminant of a polynomial f is zero if and only if the polynomial has multiple roots: