PolynomialExtendedGCD

PolynomialExtendedGCD[poly1,poly2,x]

gives the extended GCD of poly1 and poly2 treated as univariate polynomials in x.

PolynomialExtendedGCD[poly1,poly2,x,Modulusp]

gives the extended GCD over the integers mod prime p.

Examples

open allclose all

Basic Examples  (2)

Compute the extended GCD:

The second part gives coefficients of a linear combination of polynomials that yields the GCD:

Compute the extended GCD of polynomials with coefficients involving symbolic parameters:

Scope  (5)

Polynomials with numeric coefficients:

Polynomials with symbolic coefficients:

Relatively prime polynomials:

Polynomials with complex coefficients:

Compute the extended GCD of polynomials over the integers modulo 3:

Options  (2)

Modulus  (2)

Extended GCD over the integers:

Extended GCD over the integers modulo 2:

Applications  (1)

Given polynomials , , and , find polynomials and such that :

A solution exists if and only if is divisible by :

Properties & Relations  (1)

The extended GCD of and is {d,{r,s}}, such that and :

d is equal to PolynomialGCD[f,g] up to a factor not containing x:

r and s are uniquely determined by the following Exponent conditions:

Use Cancel or PolynomialRemainder to prove that d divides f and g:

Wolfram Research (2007), PolynomialExtendedGCD, Wolfram Language function, https://reference.wolfram.com/language/ref/PolynomialExtendedGCD.html.

Text

Wolfram Research (2007), PolynomialExtendedGCD, Wolfram Language function, https://reference.wolfram.com/language/ref/PolynomialExtendedGCD.html.

BibTeX

@misc{reference.wolfram_2021_polynomialextendedgcd, author="Wolfram Research", title="{PolynomialExtendedGCD}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/PolynomialExtendedGCD.html}", note=[Accessed: 21-October-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_polynomialextendedgcd, organization={Wolfram Research}, title={PolynomialExtendedGCD}, year={2007}, url={https://reference.wolfram.com/language/ref/PolynomialExtendedGCD.html}, note=[Accessed: 21-October-2021 ]}

CMS

Wolfram Language. 2007. "PolynomialExtendedGCD." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PolynomialExtendedGCD.html.

APA

Wolfram Language. (2007). PolynomialExtendedGCD. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PolynomialExtendedGCD.html