PolynomialRemainder

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`PolynomialRemainder`

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PolynomialRemainder[p,q,x]

gives the remainder from dividing p by q, treated as polynomials in x.

Details and Options

The degree of the result in x is guaranteed to be smaller than the degree of q.
Unlike PolynomialMod, PolynomialRemainder performs divisions in generating its results.
With the option Modulus->n, the remainder is computed modulo n.

Examples

open allclose all

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

Find the remainder after dividing one polynomial by another:

In[1]:=1

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https://wolfram.com/xid/0dc01xkv4ft2-wlj8df

Out[1]=1

The difference of the dividend and the remainder is a polynomial multiple of the divisor:

In[2]:=2

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https://wolfram.com/xid/0dc01xkv4ft2-qi5nrv

Out[2]=2

If the dividend is a multiple of the divisor, then the remainder is zero:

In[1]:=1

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https://wolfram.com/xid/0dc01xkv4ft2-c94jxg

Out[1]=1

Find the remainder of division for polynomials with symbolic coefficients:

In[1]:=1

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https://wolfram.com/xid/0dc01xkv4ft2-5kvosz

Out[1]=1

Coefficients of the quotient are rational functions of the input coefficients:

In[2]:=2

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https://wolfram.com/xid/0dc01xkv4ft2-jd58k0

Out[2]=2

Scope (4)Survey of the scope of standard use cases

The resulting polynomial will have coefficients that are rational expressions of input coefficients:

In[1]:=1

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https://wolfram.com/xid/0dc01xkv4ft2-h2awq

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0dc01xkv4ft2-5mye0w

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0dc01xkv4ft2-zhit9v

Out[3]=3

Polynomial remainder over the integers modulo :

In[1]:=1

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https://wolfram.com/xid/0dc01xkv4ft2-5fo91b

Out[1]=1

Polynomial remainder over a finite field:

In[1]:=1

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https://wolfram.com/xid/0dc01xkv4ft2-cjmue

In[2]:=2

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https://wolfram.com/xid/0dc01xkv4ft2-iw6si

Out[2]=2

PolynomialRemainder also works for rational functions:

In[1]:=1

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https://wolfram.com/xid/0dc01xkv4ft2-56t47m

Out[1]=1

The quotient and remainder of division of by are and , where :

In[2]:=2

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https://wolfram.com/xid/0dc01xkv4ft2-rh8sfu

Out[2]=2

and are uniquely determined by the condition that the degree of is less than the degree of :

In[3]:=3

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https://wolfram.com/xid/0dc01xkv4ft2-bod2z6

Out[3]=3

Options (1)Common values & functionality for each option

Modulus (1)

Use a prime modulus:

In[1]:=1

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https://wolfram.com/xid/0dc01xkv4ft2-cba8n

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0dc01xkv4ft2-ct6d1k

Out[2]=2

Applications (1)Sample problems that can be solved with this function

Euclid's algorithm for the greatest common divisor:

In[1]:=1

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https://wolfram.com/xid/0dc01xkv4ft2-d1x7ks

In[2]:=2

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https://wolfram.com/xid/0dc01xkv4ft2-hhdxkm

In[3]:=3

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https://wolfram.com/xid/0dc01xkv4ft2-d7mapl

In[4]:=4

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https://wolfram.com/xid/0dc01xkv4ft2-barvpf

Out[4]=4

Divide by the leading coefficient:

In[5]:=5

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https://wolfram.com/xid/0dc01xkv4ft2-cflfxk

Out[5]=5

In[6]:=6

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https://wolfram.com/xid/0dc01xkv4ft2-c8coa7

Out[6]=6

Properties & Relations (3)Properties of the function, and connections to other functions

For a polynomial , , where is given by PolynomialQuotient:

In[1]:=1

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https://wolfram.com/xid/0dc01xkv4ft2-dbkxd7

In[2]:=2

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https://wolfram.com/xid/0dc01xkv4ft2-lz5v1

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0dc01xkv4ft2-h1h08b

Out[3]=3

Use Expand to verify identity:

In[4]:=4

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https://wolfram.com/xid/0dc01xkv4ft2-hbesvd

Out[4]=4

To get both quotient and remainder use PolynomialQuotientRemainder:

In[5]:=5

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https://wolfram.com/xid/0dc01xkv4ft2-kbxtgh

Out[5]=5

PolynomialReduce generalizes PolynomialRemainder for multivariate polynomials:

In[1]:=1

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https://wolfram.com/xid/0dc01xkv4ft2-dlu8re

Out[1]=1

In[1]:=1

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https://wolfram.com/xid/0dc01xkv4ft2-d5vp5u

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0dc01xkv4ft2-p9vxdq

Out[2]=2

Possible Issues (1)Common pitfalls and unexpected behavior

The variable assumed for the polynomials matters:

In[1]:=1

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https://wolfram.com/xid/0dc01xkv4ft2-dgjffr

Out[1]=1

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