### Rational Functions and its geometric Notions

A function is called a rational function if and only if it can be written in the form

where and are polynomials in and is not the zero polynomial. The domain of is the set of all points for which the denominator is not zero, assuming and have no common factors.

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. 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[X]. Any rational expression can be written as the quotient of two polynomials P/Q with Q ≠ 0, although this representation isn't unique. P/Q is equivalent to R/S, for polynomials P, Q, R, and S, when PS = QR. However since F[X] is a unique factorization domain, there is a unique representation for any rational expression P/Q with P and Q polynomials of lowest degree and Q chosen to be monic. This is similar …

where and are polynomials in and is not the zero polynomial. The domain of is the set of all points for which the denominator is not zero, assuming and have no common factors.

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. 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[X]. Any rational expression can be written as the quotient of two polynomials P/Q with Q ≠ 0, although this representation isn't unique. P/Q is equivalent to R/S, for polynomials P, Q, R, and S, when PS = QR. However since F[X] is a unique factorization domain, there is a unique representation for any rational expression P/Q with P and Q polynomials of lowest degree and Q chosen to be monic. This is similar …