Unit 1: Advanced Algebra
Section C: Rational Expressions and Equations
Welcome
Welcome to Section C: Rational Expressions and Equations! In this section, you’ll explore rational expressions, which are fractions that contain polynomials in the numerator and denominator, and learn how to simplify, manipulate, and solve them.
Imagine
Imagine you’re analyzing the rate at which a chemical reaction occurs, where the rate can be modeled by a rational expression. Understanding how to work with these expressions is crucial in accurately predicting and optimizing the reaction.
Context
Previously, you’ve studied polynomials and their properties. Now, we’ll extend those ideas to rational expressions and equations, which involve ratios of polynomials and introduce new challenges in simplification and solution.
Overview
This section covers simplifying rational expressions, multiplying and dividing them, adding and subtracting rational expressions with like and unlike denominators, solving rational equations, and exploring their applications in real-world contexts.
Objectives
- Understand the properties and behavior of rational expressions and equations.
- Simplify rational expressions by factoring and canceling common factors.
- Perform operations on rational expressions, including multiplication, division, addition, and subtraction.
- Solve rational equations and apply them to real-world problems.
- Explore the use of rational expressions in fields like chemistry, physics, and engineering.
Preparatory Guidance
Definitions and Pronunciations
- Rational Expression: A fraction that contains polynomials in the numerator and denominator.
- Undefined Expression: A rational expression is undefined when its denominator is equal to zero.
- Extraneous Solution: A solution that arises from the process of solving an equation but does not satisfy the original equation.
Verbal Reading of Equations
- For
, say “P of x over Q of x.” - For
, say “a over b plus c over d equals a d plus b c over b d.”
Problem-Solving Strategies
- Simplify rational expressions by factoring the numerator and denominator, then canceling common factors.
- Perform operations on rational expressions, including multiplication, division, addition, and subtraction, by finding common denominators.
- Solve rational equations by clearing the denominators and solving the resulting polynomial equation.
- Identify and discard extraneous solutions that do not satisfy the original equation.
- Explore real-world applications of rational expressions in fields like chemistry, physics, and engineering.
Considerations
How do rational expressions model real-world phenomena? Why is it important to understand the behavior of rational expressions and equations? In what ways can you apply rational expressions in your daily life or future career?
