Rational Expressions

A rational expression is a type of algebraic expression that represents the ratio of two polynomials.

Definition

A rational expression is an expression of the form P/Q, where P and Q are polynomials and Q ≠ 0. The polynomials P and Q are respectively called the numerator and the denominator of the rational expression.

For example, (2x + 1)/(x^2 - 1) is a rational expression.

Properties

  1. Domain: The domain of a rational expression is all real numbers except those for which the denominator is 0 (since division by zero is undefined).

  2. Simplification: Rational expressions can often be simplified by factoring both the numerator and the denominator and cancelling common factors.

  3. Operations: Rational expressions can be added, subtracted, multiplied, and divided, much like fractions. However, when adding or subtracting rational expressions, a common denominator must be found.

Applications

Rational expressions are used extensively in algebra and calculus. They are used to solve equations, model relationships, and describe rates of change. Understanding rational expressions is crucial for further study in mathematics, physics, engineering, and many other fields.