Built-in Mathematica Symbol

PolynomialQuotient

PolynomialQuotient[p, q, x]
gives the quotient of p and q, treated as polynomials in x, with any remainder dropped.

The quotient of

by

, with the remainder dropped:

The quotient of

by

, with the remainder dropped:

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

PolynomialQuotient also works for rational functions:

Use a prime modulus:

For a polynomial f, f

gq+r, where r is given by PolynomialRemainder:

Use Expand to verify identity:

To get both quotient and remainder use PolynomialQuotientRemainder:

PolynomialReduce generalizes PolynomialQuotient for multivariate polynomials:

Use PolynomialGCD to find a common divisor:

Use PolynomialQuotient to see the resulting factorization:

For rational functions common divisors are not automatically canceled:

Cancel effectively uses PolynomialQuotient to cancel common divisors:

The Cyclotomic polynomials are defined as quotients:

The result depends on what is assumed to be a variable:

The result from PolynomialQuotient depends on recognizing zeros:

This is a hidden zero:

The result is as if the hidden zero was not zero: