Step By Step Calculus » 2.8 - Rational Functions

Synopsis
A ratio of polynomials is called a rational function. This is a natural analogue of rational numbers. Polynomials have many similarities to the integers: there is a zero polynomial, they factor, there are prime polynomials, and the algebra is similar to that of the integers. So it is natural to call a fraction or ratio of polynomials a rational function. The only exception is that the denominator in a rational function can not be a zero polynomial. For example, \dfrac{x^2+3}{3}, \dfrac{x^3-x+2}{x-2}\dfrac{x^2+3}{3}, \dfrac{x^3-x+2}{x-2} are polynomial fractions but \dfrac{x^3+8}{0}\dfrac{x^3+8}{0} is not.
Multiplying Rational Functions If \dfrac{n(x)}{d(x)}\dfrac{n(x)}{d(x)} and \dfrac{a(x)}{b(x)}\dfrac{a(x)}{b(x)} are two rational functions, then
\displaystyle \frac{n(x)}{d(x)}\cdot\frac{a(x)}{b(x)}=\frac{n(x)a(x)}{d(x)b(x)}.
\displaystyle \frac{n(x)}{d(x)}\cdot\frac{a(x)}{b(x)}=\frac{n(x)a(x)}{d(x)b(x)}.
Adding Rational Functions If \dfrac{n(x)}{d(x)}\dfrac{n(x)}{d(x)} and \dfrac{a(x)}{b(x)}\dfrac{a(x)}{b(x)} are two rational functions, then
\displaystyle \frac{n(x)}{d(x)}+\frac{a(x)}{b(x)}=\frac{n(x)b(x)+a(x)d(x)}{d(x)b(x)}.
\displaystyle \frac{n(x)}{d(x)}+\frac{a(x)}{b(x)}=\frac{n(x)b(x)+a(x)d(x)}{d(x)b(x)}.
Proper Rational Function A rational function \displaystyle \frac{n(x)}{d(x)}\displaystyle \frac{n(x)}{d(x)} is proper if the degree of n(x)n(x) is strictly less than the degree of d(x)d(x).
Polynomial Long Division It is a method for expressing an improper rational function \displaystyle \frac{n(x)}{d(x)}\displaystyle \frac{n(x)}{d(x)} in the form \displaystyle q(x)+\frac{r(x)}{d(x)}\displaystyle q(x)+\frac{r(x)}{d(x)} where q(x)q(x) is called the quotient and r(x)r(x) is called the remainder, and the ratio \dfrac{r(x)}{d(x)}\dfrac{r(x)}{d(x)} is proper.
Remainder Theorem If a polynomial f(x)f(x) is divided by a linear polynomial x-cx-c, then the remainder rr is equal to f(c)f(c). Thus, \displaystyle \frac{f(x)}{x-c}=q(x)+\frac{f(c)}{x-c}\displaystyle \frac{f(x)}{x-c}=q(x)+\frac{f(c)}{x-c}.