Overview
This lecture introduces rational expressions, covering how to simplify, multiply, divide, add, subtract, and combine more complex forms, with emphasis on using factoring and least common denominators.
What Are Rational Expressions?
- Rational expressions are fractions where the numerator and/or denominator contains variables.
- They appear frequently in algebra, especially as preparation for calculus.
Simplifying Rational Expressions
- Simplify rational expressions by factoring numerator and denominator and cancelling common factors.
- Only factors (not terms combined by addition or subtraction) can be cancelled.
- A rational expression can't be simplified if numerator and denominator cannot be factored.
Multiplying Rational Expressions
- Multiply fractions by multiplying numerators together and denominators together.
- Always factor first and cancel any common factors before carrying out multiplication.
Dividing Rational Expressions
- To divide by a fraction, multiply by its reciprocal ("copy-dot-flip").
- Write all components as fractions, factor, cancel common factors, and then multiply across.
Adding and Subtracting Rational Expressions
- Add or subtract only when denominators are the same; otherwise, find the least common denominator (LCD).
- To find the LCD, factor each denominator and combine all distinct factors.
- Multiply each fraction by the necessary form of 1 to convert denominators to the LCD.
- Add or subtract numerators and keep the common denominator; simplify if possible.
Simplifying Compound (Complex) Rational Expressions
- Compound rational expressions have other rational expressions in the numerator, denominator, or both.
- Combine the numerator and denominator into single fractions, then divide using the reciprocal trick.
- Simplify the resulting expression as usual.
Key Terms & Definitions
- Rational expression — A fraction where the numerator and/or denominator includes variables.
- Least Common Denominator (LCD) — The smallest shared multiple of the denominators of two or more rational expressions.
- Compound rational expression — A rational expression with other rational expressions in its numerator and/or denominator.
- Reciprocal — Flipping a fraction so the numerator and denominator exchange places.