PolynomialGCD

PolynomialGCD[poly1,poly2,]

gives the greatest common divisor of the polynomials polyi.

PolynomialGCD[poly1,poly2,,Modulusp]

evaluates the GCD modulo the prime p.

Details and Options

  • In PolynomialGCD[poly1,poly2,], all symbolic parameters are treated as variables.
  • PolynomialGCD[poly1,poly2,] will by default treat algebraic numbers that appear in the polyi as independent variables.
  • PolynomialGCD[poly1,poly2,,Extension->Automatic] extends the coefficient field to include algebraic numbers that appear in the polyi.

Examples

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Basic Examples  (1)

The greatest common divisor of polynomials:

Scope  (3)

The GCD of univariate polynomials:

The GCD of multivariate polynomials:

The GCD of more than two polynomials:

Generalizations & Extensions  (1)

The GCD of rational functions:

Options  (3)

Extension  (1)

By default, algebraic numbers are treated as independent variables:

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

Modulus  (1)

Compute the GCD over the integers modulo 2:

Trig  (1)

By default, PolynomialGCD treats trigonometric functions as independent variables:

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

Applications  (2)

Find common roots of univariate polynomials:

Find multiple roots of univariate polynomials:

Properties & Relations  (3)

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:

Introduced in 1991
 (2.0)
 |
Updated in 1996
 (3.0)