# How to Distinguish Proper and Improper Rational Expressions

Integration by partial fractions works only with *proper rational expressions,* but not with *improper rational expressions**.* Telling a proper fraction from an improper one is easy: A fraction a/b is proper if the numerator (disregarding sign) is *less* than the denominator, and improper otherwise.

With rational expressions, the idea is similar, but instead of comparing the value of the numerator and denominator, you compare their *degrees.* The degree of a polynomial is its highest power of *x*.

A rational expression is proper if the degree of the numerator is less than the degree of the denominator, and improper otherwise.

For example, look at these three rational expressions:

In the first example, the numerator is a second-degree polynomial and the denominator is a third-degree polynomial, so the rational is *proper.* In the second example, the numerator is a fifth-degree polynomial and the denominator is a second-degree polynomial, so the expression is *improper.* In the third example, the numerator and denominator are both fourth-degree polynomials, so the rational function is *improper.*