# 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(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 , 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)