# PolynomialLCM PolynomialLCM[poly1,poly2,]

gives the least common multiple of the polynomials polyi.

PolynomialLCM[poly1,poly2,,Modulusp]

evaluates the LCM modulo the prime p.

# Details and Options • PolynomialLCM[poly1,poly2,] will by default treat algebraic numbers that appear in the polyi as independent variables.
• PolynomialLCM[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 least common multiple (LCM) of polynomials:

Compute the least common multiple of several polynomials:

Compute the least common multiple of multivariate polynomials:

## Scope(9)

### Basic Uses(4)

The LCM of univariate polynomials:

The LCM of multivariate polynomials:

The LCM of more than two polynomials:

The LCM of rational functions:

With , PolynomialLCM detects algebraically dependent coefficients:

Compute the LCM over the integers modulo :

Compute the LCM of polynomials over a finite field:

With Trig->True, PolynomialLCM recognizes identities between trigonometric functions:

The LCM of rational functions:

## Options(3)

### Extension(1)

By default, algebraic numbers are treated as independent variables:

With , PolynomialLCM detects algebraically dependent coefficients:

### Modulus(1)

Compute the LCM over the integers modulo 2:

### Trig(1)

By default, PolynomialLCM treats trigonometric functions as independent variables:

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

## Applications(2)

If divides , then their least common multiple is equal to :

If and are relatively prime, then their least common multiple is equal to :

In general, the least common multiple of and is divided by the greatest common divisor of and :

Use Together to prove the equality:

Compute the LCM of the first five cyclotomic polynomials. Notice the coefficients are anti-palindromic:

This results from the fact that every cyclotomic polynomial is palindromic except the first:

The first cyclotomic polynomial is anti-palindromic:

Thus when taking the product of palindromic polynomials with one anti-palindromic polynomial, we will always obtain an anti-palindromic polynomial:

## Properties & Relations(1)

The LCM of polynomials is divisible by the polynomials; use PolynomialMod to prove it:

PolynomialGCD finds the greatest common divisor of polynomials: