# 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 is undefined whenever the bottom evaluates to 0.

## Arithmetic and simplifying

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 integer factor greater than 1, and
• the top and bottom have no common polynomial factor that is not constant.
We can simplify rational expressions by factoring the top and bottom and cancelling any 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 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. But systematic factoring is now more important, since less can be done by trial and error. When multiplying or dividing, you should factor all of the numerators and denominators, and there is no need to get common denominators; when adding or subtracting, you only need to factor the denominators at first, and you need to make all of these denominators the same. (Even when adding or subtracting, you'll still need to factor the numerator of the final answer, but that's only to simplify the result.)

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This web page was written between 2010 and 2017 by Toby Bartels, last edited on 2017 August 28. Toby reserves no legal rights to it.

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