Department of Mathematics --- College of Science --- University of Utah

Mathematics 1010 online

Rational Functions and Expressions

A rational expression is an algebraic expression that can be written as the ratio of two polynomial expressions. A rational function is a function whose value is given by a rational expression. Examples for rational functions (and associated expressions) include:

$r(x) = \frac{x^2+2x+3}{x-4}$ . This expression is obviously the ratio of two polynomials.
$r(x) = \frac{1}{1+\frac{1}{x}}$ . This expression is not in the standard form of a rational expression, but it can be converted to one by multiplying with in the numerator and denominator, giving

$\displaystyle \frac{1}{1+\frac{1}{x}} = \frac{x}{x+1}.$
Any polynomial expression is a rational expression, for example you can think of it as the polynomial divided by (the polynomial expression) .

Examples of algebraic expressions that are not rational can be constructed similarly as in the case of polynomials .

Another useful way of thinking about polynomial and rational expressions is that polynomial expressions are those algebraic expressions that can be evaluated using a finite number of additions and multiplications (and subtractions), whereas rational expressions may also require division for their evaluation. However, no operations beyond the basic four arithmetic operations are required to evaluate a polynomial or a rational expression.

We must not divide by zero, and hence a rational expression is undefined if the denominator is zero. For example, the rational expression

$\displaystyle \frac{1}{x^2-5x+6} = \frac{1}{(x-2)(x-3)}$

is undefined when

. On the other hand, the rational expression $\frac{1}{1+x^2}$ is defined for all (real) values of

since the denominator is never less than

and hence never zero.

The fundamental principle

There is just one fundamental principle governing the manipulation of rational expressions, and that is that they work exactly like fractions . Indeed, that is the main reason why fractions should still be emphasized in the elementary and secondary school curriculum. You may encounter the view that fractions are obsolete because calculators are ubiquitously available and it's more convenient to work with decimal numbers. However, not understanding fractions makes it impossible to understand algebra.

The basic operations

Understanding fractions, and appreciating the fact that rational expressions are like fractions is all you need to know to handle rational expressions successfully. However, for reference the remainder of this page lists the rules of fractional and rational algebra, and illustrates them with an example. So suppose that $\frac{r}{s}$ and $\frac{t}{u}$ are rational expressions where , , and are polynomial expressions. (This includes the special case that the polynomials are constants and the rational expressions are fractions.)

Then the following rules hold:

$\begin{displaymath} \begin{array}{rclr} \displaystyle\frac{ru}{su} &=& \displays... ...ru}{st} &\quad \hbox{division} \\ \\ \end{array}\qquad (*) \end{displaymath}$

It is worthwhile to think for a moment about why these rules are true. If the polynomial expressions are evaluated at specific values of the variable they turn into numbers and the rules turn into statements about fractions which hold as discussed on the fraction page . This is true for all possible values of the polynomials, and that's just what we mean by the equations in . We always reduce our problem to one that we have solved before .

Let's illustrate the rules with some very simple examples. Make sure you understand them! Suppose

$\displaystyle v=\frac{x-1}{x-2}\qquad\hbox{and}\qquad w = \frac{x+1}{x+2}.$

Then

$\begin{displaymath} \begin{array}{rcl} u+v &=& \frac{x-1}{x-2} + \frac{x+1}{x+2}... ...+2)}{(x-2)(x+1)} = \frac{x^2+x-2}{x^2-x-2}\\ \\ \end{array} \end{displaymath}$

In all of these cases, the final expression are in lowest terms, i.e., there are no factors that can be canceled in numerator and denominator.

The only possible pitfall is that we must avoid division by zero. That's easy to do when dealing with specific numbers but it's sometimes subtle when dealing with variables.