![]() P/ Q is equivalent to R/ S, for polynomials P, Q, R, and S, when PS = QR. Any rational expression can be written as the quotient of two polynomials P/ Q with Q ≠ 0, although this representation isn't unique. In this setting given a field F and some indeterminate X, a rational expression is any element of the field of fractions of the polynomial ring F. In abstract algebra the concept of a polynomial is extended to include formal expressions in which the coefficients of the polynomial can be taken from any field. This is useful in solving such recurrences, since by using partial fraction decomposition we can write any proper rational function as a sum of factors of the form 1 / ( ax + b) and expand these as geometric series, giving an explicit formula for the Taylor coefficients this is the method of generating functions.Ībstract algebra and geometric notion ![]() The set of rational functions over a field K is a field, the field of fractions of the ring of the polynomial functions over K.Ī function f ( x ) Ĭonversely, any sequence that satisfies a linear recurrence determines a rational function when used as the coefficients of a Taylor series. Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is L. The values of the variables may be taken in any field L containing K. In this case, one speaks of a rational function and a rational fraction over K. The coefficients of the polynomials need not be rational numbers they may be taken in any field K. ![]() In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. ( September 2015) ( Learn how and when to remove this template message) Please help to improve this article by introducing more precise citations. This article includes a list of general references, but it lacks sufficient corresponding inline citations. ![]()
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