Lesson 3.5
Finding the LCD
Before you can add or subtract rational expressions, you need a common denominator. The Least Common Denominator (LCD) is the smallest expression that all denominators divide into evenly.
Introduction
Just like adding requires finding the LCD of 12, adding rational expressions requires finding a common polynomial denominator.
Past Knowledge
Finding the LCM of numbers and factoring polynomials.
Today's Goal
Factor each denominator and build the LCD using the highest power of each factor.
Future Success
The LCD is essential for adding/subtracting rational expressions in Lessons 3.6 and 3.7.
Key Concepts
Building the LCD
Factor each denominator completely
List every unique factor that appears
For each factor, use the HIGHEST power it appears with
Multiply all those factors together → that's the LCD
Numeric Analogy
LCD of 12 and 18:
Take the highest power of each prime. Same logic for polynomials!
Worked Examples
Example 1: Simple Denominators
BasicFind the LCD of and .
Factors
Denominator 1: . Denominator 2: .
Highest power of is
LCD =
Example 2: Distinct Binomial Factors
IntermediateFind the LCD of and .
Both are already fully factored — no common factors
LCD =
Example 3: Overlapping Factors
AdvancedFind the LCD of and .
Factor both
Highest powers
appears as , and appears once.
LCD =
Common Pitfalls
Just Multiplying the Denominators
That always works, but it's not the least common denominator. You'll end up with unnecessarily large expressions.
Forgetting to Factor First
If you skip factoring, you might not see the shared factor and end up with a much larger LCD than necessary.
Real-Life Applications
In project scheduling, when tasks repeat at different intervals (e.g., every 4 days and every 6 days), the LCM tells you when they'll coincide. The LCD of polynomial denominators extends this same idea to algebraic settings.
Practice Quiz
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