Rational Expression

Definition

A rational expression is a mathematical expression that consists of one or more polynomials divided by one or more polynomials. It represents a fraction where the numerator and denominator are both polynomials, and it can be used to model and solve a variety of mathematical problems.

5 Must Know Facts For Your Next Test

1. Rational expressions can be used to model and solve problems involving rates, ratios, and proportions.
2. The degree of a rational expression is determined by the highest degree of the polynomials in the numerator and denominator.
3. Rational expressions can be multiplied, divided, added, and subtracted, following specific rules and procedures.
4. Rational expressions can have restrictions on their domain, which are the values of the variable(s) for which the expression is defined.
5. Rational expressions can be used to represent and solve equations and inequalities, which are essential in solving real-world problems.

Review Questions

• Explain the process of dividing polynomials and how it relates to rational expressions.
  • Dividing polynomials is a fundamental operation in the context of rational expressions. When dividing one polynomial by another, the result is a quotient polynomial and a remainder polynomial. This division process can be used to simplify rational expressions by factoring the numerator and denominator and canceling common factors. The ability to divide polynomials is crucial for manipulating and simplifying rational expressions, which is a key skill in solving a variety of mathematical problems.
• Describe the steps involved in multiplying and dividing rational expressions.
  • To multiply rational expressions, you multiply the numerators together and the denominators together, then simplify the resulting expression by factoring and canceling common factors. To divide rational expressions, you invert the divisor (denominator of the second expression) and multiply it by the first expression. Again, the resulting expression should be simplified by factoring and canceling common factors. Mastering these procedures for multiplying and dividing rational expressions is essential for manipulating and solving problems involving fractions with polynomial numerators and denominators.
• Analyze how the properties of rational expressions, such as degree and restrictions on the domain, impact their behavior and applications.
  • The degree of a rational expression, determined by the highest degree of the polynomials in the numerator and denominator, affects the behavior of the expression. Higher-degree rational expressions can exhibit more complex characteristics, such as asymptotes, that must be considered when solving problems. Additionally, rational expressions may have restrictions on their domain, meaning there are certain values of the variable(s) for which the expression is not defined. Understanding these properties of rational expressions and how they impact the behavior and applications of the expressions is crucial for effectively solving a variety of mathematical problems.

Related terms

Polynomial: A polynomial is an algebraic expression consisting of variables and coefficients, where the variables are raised to non-negative integer powers.

Fraction: A fraction is a mathematical representation of a part of a whole, expressed as a numerator divided by a denominator.

Simplify: Simplifying a rational expression involves reducing the expression to its lowest terms by factoring the numerator and denominator and canceling common factors.