# rational function

A real function $R(x)$ of a single variable $x$ is called if it can be written as a quotient

 $R(x)=\frac{P(x)}{Q(x)},$

where $P(x)$ and $Q(x)$ are polynomials    in $x$ with real coefficients. When one is only interested in algebraic  properties of $R(x)$ or $P(x)$ and $Q(x)$, it is convenient to forget that they define functions and simply treat them as algebraic expressions in $x$. In this case $R(x)$ is referred to as a rational expression.

In general, a rational function (expression) $R(x_{1},\ldots,x_{n})$ has the form

 $R(x_{1},\ldots,x_{n})=\frac{P(x_{1},\ldots,x_{n})}{Q(x_{1},\ldots,x_{n})},$

where $P(x_{1},\ldots,x_{n})$ and $Q(x_{1},\ldots,x_{n})$ are polynomials in the variables $(x_{1},\ldots,x_{n})$ with coefficients in some field or ring $S$.

In this sense, $R(x_{1},\ldots,x_{n})$ can be regarded as an element of the fraction field $S(x_{1},\ldots,x_{n})$ of the polynomial ring $S[x_{1},\ldots,x_{n}]$.

 Title rational function Canonical name RationalFunction Date of creation 2013-03-22 13:38:54 Last modified on 2013-03-22 13:38:54 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 6 Author CWoo (3771) Entry type Definition Classification msc 26C15 Synonym rational expression Related topic PolynomialRing Related topic FractionField Related topic RealFunction Related topic PropertiesOfEntireFunctions Related topic IntegrationOfFractionPowerExpressions