Problem 86 - College Algebra Essentials

Question

What is a rational expression?

Step-by-Step Solution

Answer

A rational expression is an algebraic expression that can be defined as the ratio of two polynomials, with the denominator not equal to zero.

Step 1: Understanding the concept of an Rational Expression

A rational expression, often referred to as a rational function, is any expression or function which includes a polynomial in its numerator and denominator. In short, it can be simply described as the ratio of two polynomials.

Step 2: Recognizing a Rational Expression

A rational expression is typically presented in the form \(\frac{p(x)}{q(x)}\) where p(x) and q(x) are polynomials and q(x) is not equal to zero (as division by zero is undefined in mathematics). An example would be \(\frac{x^2 - 2x + 1}{x^2 + 3}\).

Step 3: The Importance of Non-zero Denominators

One crucial point to note in a rational expression is that the denominator polynomial, q(x), should not result in zero. If it does, the rational expression is undefined. For example, in the rational expression \(\frac{2}{x-2}\), when x equals 2, the denominator becomes zero, making the entire expression undefined.