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

open allclose all

Basic Examples  (3)

Compute the greatest common divisor of polynomials:

Compute the GCD of multivariate polynomials:

Show that polynomials are relatively prime:

Scope  (10)

Basic Uses  (4)

The GCD of univariate polynomials:

The GCD of multivariate polynomials:

The GCD of more than two polynomials:

The GCD of polynomials with complex coefficients:

Advanced Uses  (6)

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

Compute the GCD of polynomials over the integers modulo :

Compute the GCD of polynomials over a finite field:

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

The GCD of rational functions:

Compute the GCD of two polynomials of degree :

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:

Wolfram Research (1991), PolynomialGCD, Wolfram Language function, https://reference.wolfram.com/language/ref/PolynomialGCD.html (updated 2023).

Text

Wolfram Research (1991), PolynomialGCD, Wolfram Language function, https://reference.wolfram.com/language/ref/PolynomialGCD.html (updated 2023).

CMS

Wolfram Language. 1991. "PolynomialGCD." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/PolynomialGCD.html.

APA

Wolfram Language. (1991). PolynomialGCD. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PolynomialGCD.html

BibTeX

@misc{reference.wolfram_2024_polynomialgcd, author="Wolfram Research", title="{PolynomialGCD}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/PolynomialGCD.html}", note=[Accessed: 22-November-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_polynomialgcd, organization={Wolfram Research}, title={PolynomialGCD}, year={2023}, url={https://reference.wolfram.com/language/ref/PolynomialGCD.html}, note=[Accessed: 22-November-2024 ]}