Rational Expressions

Introduction

Rational expressions are quotients of polynomials and are a fundamental concept in algebra and precalculus. Understanding their domains, how to simplify them, and how to perform arithmetic operations is essential for success in higher mathematics.

Domain of Rational Expressions

The domain of a rational expression is the set of all real numbers for which the expression is defined. Division by zero is undefined, so any value that makes the denominator zero must be excluded from the domain.

• Key Point: To find the domain, set the denominator equal to zero and solve for the variable. Exclude these values from the domain.

• Example: Find the domain of .

Solution:

• Set denominator to zero:

• Solve:

• Domain: All real numbers except

Simplifying, Multiplying, and Dividing Rational Expressions

To simplify rational expressions, factor numerators and denominators and cancel common factors. Multiplication and division follow the same rules as with numerical fractions.

• Multiplication: Multiply numerators together and denominators together, then simplify.

• Division: Multiply by the reciprocal of the divisor.

• Formula:

• Example: Simplify .

Solution:

• Factor numerator and denominator:

• Simplify:

Adding and Subtracting Rational Expressions

To add or subtract rational expressions, they must have a common denominator. If denominators differ, find the least common denominator (LCD) by factoring each denominator and using each factor the greatest number of times it occurs in any denominator.

• Key Point: After finding the LCD, rewrite each expression with the LCD as the denominator, then add or subtract the numerators.

• Example: Add

Solution:

• Factor denominators:

• LCD:

• Rewrite each fraction with the LCD:

• Combine numerators:

Complex Rational Expressions

A complex rational expression has rational expressions in its numerator, denominator, or both. Simplifying these requires special techniques.

• Method 1: Find the LCD of all denominators within the complex rational expression. Multiply numerator and denominator by this LCD to clear denominators.

• Method 2: Simplify the numerator and denominator separately, then divide by multiplying by the reciprocal of the denominator.

Example

Simplify

• Find LCD for numerator and denominator:

• Multiply numerator and denominator by :

Numerator: Denominator:

So,

Summary Table: Operations with Rational Expressions

Additional info: These notes expand on the examples and definitions provided in the slides, adding step-by-step solutions and a summary table for clarity.