# Rational expressions

A rational expression is made of two polynomial expressions: the numerator (or top) and the denominator (or bottom), where the denominator is not the constant polynomial 0. Think of a rational expression as a fraction: the numerator divided by the denominator. Every polynomial may be interpreted as a rational expression whose denominator is the constant polynomial 1.

To evaluate a rational expression at a particular value of each variable, evaluate the numerator and denominator and divide the results. The rational expression its undefined whenever the bottom evaluates to 0.

A rational expression is simplified if

• the top and bottom have only integer coefficients,
• the leading coefficient on the bottom is positive,
• the coefficients have no common factor greater than 1, and
• the top and bottom have no common factor that is not constant.
We can simplify rational expressions by factoring the top and bottom and cancelling and common factors; we may then leave the expression in factored form. (If the leading coefficient in the numerator is negative and we leave the expression in factored form, then we usually place the minus sign in front of the entire expression.)

We add, subtract, multiply, and divide rational expressions using the same techniques as for rational numbers.

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This web page was written in 2010 and 2011 by Toby Bartels, last edited on 2011 October 17. Toby reserves no legal rights to it.

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