Rational Expressions

Rational expressions are algebraic fractions in which both the numerator and the denominator are polynomials. These expressions follow the arithmetic rules of numerical fractions but require extra care due to the presence of variables. A fundamental part of working with rational expressions is identifying values that make the expression undefined, typically those that result in division by zero or undefined radicals.

Determining the Domain

The domain of a rational expression includes all real numbers except those that make the denominator zero or violate constraints from square roots. Consider the expression:

Factoring the denominator:

The domain excludes x=3 and x=-3, so the domain is all real numbers except these values.

Simplifying and Multiplying

Simplifying involves factoring both numerator and denominator and canceling common terms. For example:

For multiplication:

Dividing and Adding Rational Expressions

To divide, invert the second expression:

For addition:

Applications and Structure

Rational expressions are used in modeling rates, proportions, and other variable relationships. Operations involving these expressions appear in problems related to motion, growth, resistance in circuits, and optimization. Understanding how to simplify, multiply, divide, and combine rational expressions is foundational for problem-solving in algebra and applied sciences.