structural:expansion, linguistic:voice_change
Division

The division of one polynomial by another is not typically a polynomial. Instead, such ratios are a more general family of objects, called rational fractions, rational expressions, or rational functions, depending on context. This is analogous to the fact that the ratio of two integers is a rational number, not necessarily an integer. For example, the fraction 1/(x + 1) is not a polynomial, and it cannot be written as a finite sum of powers of the variable x.

For polynomials in one variable, there is a notion of Euclidean division of polynomials, generalizing the Euclidean division of integers. This notion of the division a (x)/ b (x) results in two polynomials, a quotient q (x) and a remainder r (x), such that a = b q + r and degree(r) < degree(b). The quotient and remainder may be computed by any of several algorithms, including polynomial long division and synthetic division.

When the denominator b (x) is monic and linear, that is, b (x) = x − c for some constant c, then the polynomial remainder theorem asserts that the remainder of the division of a (x) by b (x) is the evaluation a (c). In this case, the quotient may be computed by Ruffini's rule, a special case of synthetic division.