PolynomialRemainder

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

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Basic Examples  (1)

Find the remainder after dividing one polynomial by another:

Scope  (2)

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

PolynomialRemainder also works for rational functions:

Options  (1)

Modulus  (1)

Use a prime modulus:

Applications  (1)

Euclid's algorithm for the greatest common divisor:

Divide by the leading coefficient:

Properties & Relations  (3)

For a polynomial , , where is given by PolynomialQuotient:

Use Expand to verify identity:

To get both quotient and remainder use PolynomialQuotientRemainder:

PolynomialReduce generalizes PolynomialRemainder for multivariate polynomials:

Possible Issues  (1)

The variable assumed for the polynomials matters:

Introduced in 1988
 (1.0)