A rational function is a quotient of two polynomials 
and 
, written
 |
where 
is nonzero. It is sometimes called a rational polynomial function. More generally,
if 
and 
are polynomials in multiple variables, their quotient
is called a (multivariate) rational function. The term "rational polynomial"
is sometimes used as a synonym for rational function. However, this usage is strongly
discouraged since by analogy with complex polynomial
and integer polynomial, rational
polynomial should properly refer to a polynomial
with rational coefficients.
As an element of a field of fractions, a rational function is unchanged by canceling a common polynomial factor. However, this can enlarge the domain of evaluation of its quotient expression. For example,
 |
where the left side is defined only for 
. The constant
expression on the right is defined everywhere. Assigning the value 1 at the four
excluded points removes the removable
singularities of the original expression (Trott 2006).
Over the complex numbers, a rational function is meromorphic on the extended complex plane. Conversely, every meromorphic function on the extended complex plane is a rational function (Knopp 1996, p. 137). In addition, a rational function has a partial fraction decomposition (Knopp 1996, p. 139).
