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PolynomialLCM
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PolynomialLCM

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PolynomialLCM[poly1,poly2,…]

gives the least common multiple of the polynomials polyi.

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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

open allclose all

Basic Examples  (3)Summary of the most common use cases

Compute the least common multiple (LCM) of polynomials:

In[1]:=1
✖
https://wolfram.com/xid/0enyraham-ez0t51
Out[1]=1

Compute the least common multiple of several polynomials:

In[1]:=1
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https://wolfram.com/xid/0enyraham-e53zqt
Out[1]=1

Compute the least common multiple of multivariate polynomials:

In[1]:=1
✖
https://wolfram.com/xid/0enyraham-ye5dbn
Out[1]=1

Scope  (9)Survey of the scope of standard use cases

Basic Uses  (4)

The LCM of univariate polynomials:

In[1]:=1
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https://wolfram.com/xid/0enyraham-dmnnhd
Out[1]=1

The LCM of multivariate polynomials:

In[1]:=1
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https://wolfram.com/xid/0enyraham-w5552x
Out[1]=1

The LCM of more than two polynomials:

In[1]:=1
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https://wolfram.com/xid/0enyraham-46hd1a
Out[1]=1

The LCM of rational functions:

In[1]:=1
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https://wolfram.com/xid/0enyraham-l16cq1
Out[1]=1

Advanced Uses  (5)

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

In[1]:=1
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https://wolfram.com/xid/0enyraham-cjjv04
Out[1]=1

Compute the LCM over the integers modulo :

In[1]:=1
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https://wolfram.com/xid/0enyraham-eaapzc
Out[1]=1

Compute the LCM of polynomials over a finite field:

In[1]:=1
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https://wolfram.com/xid/0enyraham-cjmue
In[2]:=2
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https://wolfram.com/xid/0enyraham-iw6si
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0enyraham-okfr9n
Out[1]=1

The LCM of rational functions:

In[1]:=1
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https://wolfram.com/xid/0enyraham-mx889w
Out[1]=1

Options  (3)Common values & functionality for each option

Extension  (1)

By default, algebraic numbers are treated as independent variables:

In[1]:=1
✖
https://wolfram.com/xid/0enyraham-dtj4en
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0enyraham-iol7f
Out[2]=2

Modulus  (1)

Compute the LCM over the integers modulo 2:

In[1]:=1
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https://wolfram.com/xid/0enyraham-c5k1cw
Out[1]=1

Trig  (1)

By default, PolynomialLCM treats trigonometric functions as independent variables:

In[1]:=1
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https://wolfram.com/xid/0enyraham-ckmz7k
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0enyraham-dst8bi
Out[2]=2

Applications  (2)Sample problems that can be solved with this function

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

In[1]:=1
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https://wolfram.com/xid/0enyraham-cmgunl
In[3]:=3
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https://wolfram.com/xid/0enyraham-dp40tj
Out[3]=3

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

In[4]:=4
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https://wolfram.com/xid/0enyraham-ya5hyv
In[6]:=6
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https://wolfram.com/xid/0enyraham-faf1ts
Out[6]=6

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

In[7]:=7
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https://wolfram.com/xid/0enyraham-kqvo0y
In[9]:=9
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https://wolfram.com/xid/0enyraham-49sti4
Out[9]=9
In[10]:=10
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https://wolfram.com/xid/0enyraham-jvnehg
Out[10]=10

Use Together to prove the equality:

In[11]:=11
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https://wolfram.com/xid/0enyraham-oohqp9
Out[11]=11

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

In[1]:=1
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https://wolfram.com/xid/0enyraham-rtr5rw
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0enyraham-rw4r6p
Out[2]=2

The first cyclotomic polynomial is anti-palindromic:

In[3]:=3
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https://wolfram.com/xid/0enyraham-lu7bfi
Out[3]=3

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

In[4]:=4
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https://wolfram.com/xid/0enyraham-0nftw0
Out[4]=4

Properties & Relations  (1)Properties of the function, and connections to other functions

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

In[1]:=1
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https://wolfram.com/xid/0enyraham-gyrlio
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0enyraham-qr5oya
Out[4]=4

PolynomialGCD finds the greatest common divisor of polynomials:

In[5]:=5
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https://wolfram.com/xid/0enyraham-bi3zhw
Out[5]=5
In[6]:=6
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https://wolfram.com/xid/0enyraham-mwhwo
Out[6]=6
Wolfram Research (1991), PolynomialLCM, Wolfram Language function, https://reference.wolfram.com/language/ref/PolynomialLCM.html (updated 2023).
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Wolfram Research (1991), PolynomialLCM, Wolfram Language function, https://reference.wolfram.com/language/ref/PolynomialLCM.html (updated 2023).

Text

Wolfram Research (1991), PolynomialLCM, Wolfram Language function, https://reference.wolfram.com/language/ref/PolynomialLCM.html (updated 2023).

✖
Wolfram Research (1991), PolynomialLCM, Wolfram Language function, https://reference.wolfram.com/language/ref/PolynomialLCM.html (updated 2023).

CMS

Wolfram Language. 1991. "PolynomialLCM." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/PolynomialLCM.html.

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Wolfram Language. 1991. "PolynomialLCM." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/PolynomialLCM.html.

APA

Wolfram Language. (1991). PolynomialLCM. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PolynomialLCM.html

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Wolfram Language. (1991). PolynomialLCM. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PolynomialLCM.html

BibTeX

@misc{reference.wolfram_2025_polynomiallcm, author="Wolfram Research", title="{PolynomialLCM}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/PolynomialLCM.html}", note=[Accessed: 28-April-2025 ]}

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@misc{reference.wolfram_2025_polynomiallcm, author="Wolfram Research", title="{PolynomialLCM}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/PolynomialLCM.html}", note=[Accessed: 28-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_polynomiallcm, organization={Wolfram Research}, title={PolynomialLCM}, year={2023}, url={https://reference.wolfram.com/language/ref/PolynomialLCM.html}, note=[Accessed: 28-April-2025 ]}

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@online{reference.wolfram_2025_polynomiallcm, organization={Wolfram Research}, title={PolynomialLCM}, year={2023}, url={https://reference.wolfram.com/language/ref/PolynomialLCM.html}, note=[Accessed: 28-April-2025 ]}