# 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(9)

### 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:

With , 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:

Compute the GCD of two polynomials of degree :

## Options(3)

### Extension(1)

By default, algebraic numbers are treated as independent variables:

With , 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 2022).

#### Text

Wolfram Research (1991), PolynomialGCD, Wolfram Language function, https://reference.wolfram.com/language/ref/PolynomialGCD.html (updated 2022).

#### CMS

Wolfram Language. 1991. "PolynomialGCD." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. 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_2022_polynomialgcd, author="Wolfram Research", title="{PolynomialGCD}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/PolynomialGCD.html}", note=[Accessed: 20-March-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_polynomialgcd, organization={Wolfram Research}, title={PolynomialGCD}, year={2022}, url={https://reference.wolfram.com/language/ref/PolynomialGCD.html}, note=[Accessed: 20-March-2023 ]}