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  (3)

Compute the greatest common divisor of polynomials:

Compute the GCD of multivariate polynomials:

Show that polynomials are relatively prime:

Scope  (8)

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  (4)

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

Compute the GCD of polynomials over the integers modulo :

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

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:

Wolfram Research (1991), PolynomialGCD, Wolfram Language function, https://reference.wolfram.com/language/ref/PolynomialGCD.html (updated 1996).

Text

Wolfram Research (1991), PolynomialGCD, Wolfram Language function, https://reference.wolfram.com/language/ref/PolynomialGCD.html (updated 1996).

BibTeX

@misc{reference.wolfram_2020_polynomialgcd, author="Wolfram Research", title="{PolynomialGCD}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/PolynomialGCD.html}", note=[Accessed: 18-January-2021 ]}

BibLaTeX

@online{reference.wolfram_2020_polynomialgcd, organization={Wolfram Research}, title={PolynomialGCD}, year={1996}, url={https://reference.wolfram.com/language/ref/PolynomialGCD.html}, note=[Accessed: 18-January-2021 ]}

CMS

Wolfram Language. 1991. "PolynomialGCD." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 1996. 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