A rational expression is a ratio of polynomials that can be evaluated to real (non-imaginary) values. Rational Expressions can contain any number of terms in the numerator and denominator (even a regular fraction like ½ is technically a rational expression). In most cases, when talking about a rational expression we are referring to a ratio of polynomials that contains variables in the numerator, denominator, or both.

Whenever the denominator of a fraction contains a variable, you should be alert to the possibility that certain values of the variable may result in the denominator equaling zero (if the denominator is 0, you are essentially dividing by 0, which is undefined). This means that when dealing with rational expressions, we can't always say that the variable represents “any real number” - certain values may have to be excluded. For example, look at the following expression:



If x = 0, then the denominator would equal 0 because 3(0) = 0. We cannot allow the variable x to equal 0, so we would add the comment (x 0). Let’s take a look at another example:



For this example, if x = 3 then the denominator would equal 0 (3-3 = 0), so we would write (x 3). Let’s take a look at one more:



Both x = 1 and x = –1 are not allowed, since either choice would make the denominator equal zero. Thus, (x 1, -1).

What happens if the numerator equals 0? This situation is perfectly fine. If the numerator equals 0, then the whole rational expression equals 0 (assuming that the denominator does not equal 0). Recall that 0/2 = 0, but 2/0 is undefined.

This is important information to keep in mind as you work with rational expressions because it is possible to get an answer that would be considered undefined. When this happens, you must discard the solution.