뒤로Rational Expressions: Operations and Simplification
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Rational Expressions
Definition and Domain
A rational expression is a quotient of two polynomials. The domain of a rational expression consists of all real numbers for which the expression is defined. Since division by zero is undefined, any value that makes the denominator zero must be excluded from the domain.
Rational Expression: , where and are polynomials and .
Domain Exclusion: Solve to find values to exclude.
Example: For , exclude from the domain.
Finding Excluded Values
To determine which values are excluded from the domain, set the denominator equal to zero and solve for the variable.
Example: : Exclude .
Example: : Exclude (since ).
Example: : Exclude (since ).
Simplifying Rational Expressions
Procedure for Simplification
A rational expression is simplified when its numerator and denominator have no common factors other than 1 or -1.
Step 1: Factor the numerator and denominator completely.
Step 2: Divide both numerator and denominator by any common factors.
Example: Simplify .
Solution: , . Cancel : .
Operations with Rational Expressions
Multiplying Rational Expressions
To multiply rational expressions:
Step 1: Factor all numerators and denominators completely.
Step 2: Divide numerators and denominators by common factors.
Step 3: Multiply remaining factors in numerators and denominators.
Example:
Dividing Rational Expressions
To divide rational expressions, multiply the first expression by the reciprocal of the second.
Reciprocal: Interchange numerator and denominator of the divisor.
Example:
Solution:
Adding and Subtracting Rational Expressions
With Same Denominator
When rational expressions have the same denominator:
Step 1: Add or subtract the numerators.
Step 2: Place the result over the common denominator.
Step 3: Simplify if possible.
Example:
With Different Denominators
When denominators differ, find the least common denominator (LCD).
Step 1: Factor each denominator completely.
Step 2: List all unique factors from both denominators.
Step 3: Multiply factors to form the LCD.
Step 4: Rewrite each expression with the LCD as denominator.
Step 5: Add or subtract numerators, place over LCD, and simplify.
Example:
Solution: Factor denominators: , . LCD is .
Complex Rational Expressions
Definition and Simplification
A complex rational expression (or complex fraction) has a numerator or denominator containing one or more rational expressions. To simplify, ensure neither the numerator nor denominator contains rational expressions.
Example:
Procedure: Find LCD for numerator and denominator, combine, and simplify.
Example:
Summary Table: Operations with Rational Expressions
Operation | Procedure | Example |
|---|---|---|
Multiplication | Factor, cancel common factors, multiply remaining | |
Division | Multiply by reciprocal of divisor | |
Add/Subtract (Same Denominator) | Add/subtract numerators, keep denominator | |
Add/Subtract (Different Denominator) | Find LCD, rewrite, add/subtract numerators | (if and are relatively prime) |
Complex Fractions | Combine numerator and denominator, simplify |
Key Concepts and Formulas
Excluded Values: Values that make the denominator zero.
LCD (Least Common Denominator): The smallest expression that is a common multiple of all denominators.
Reciprocal: For , the reciprocal is .
Complex Fraction Simplification:
Examples and Applications
Example 1: Simplify .
Factor: ,
Cancel :
Example 2: Add and .
LCD:
Rewrite:
Example 3: Simplify complex fraction .
Numerator:
Denominator:
Result:
Additional info: The notes cover all major operations with rational expressions, including domain restrictions, simplification, multiplication, division, addition, subtraction, and complex fractions. These are foundational skills for College Algebra and are directly relevant to polynomial and rational function chapters.
