# 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.

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 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 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.

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