Partial Fraction Decomposition (Basic)
Breaking apart complex fractions into a sum of simpler ones—the reverse of adding fractions.
Introduction
You know how to add fractions by finding a common denominator. Partial fraction decomposition is the reverse: we break apart a complicated fraction into a sum of simpler ones. This technique is essential in Calculus, where integrating is far easier than integrating the original combined form.
Prerequisite Connection
You can add fractions by finding a common denominator and factor polynomials.
Today's Increment
We reverse the process—splitting one fraction into a sum of simpler ones.
Why This Matters
In Calculus, this technique makes integration of rational functions possible.
Key Concepts
The Goal
Find the constants and
Rule for Distinct Linear Factors
The Cover-Up Method (Heaviside)
To find , "cover" and substitute
Evaluate the remaining fraction at that value
Repeat for each factor
Worked Examples
Example 1: Two Linear Factors (Basic)
Decompose:
Step 1: Set up the form
Step 2: Multiply both sides by LCD
Step 3: Use strategic substitution
:
:
Example 2: Three Linear Factors (Intermediate)
Decompose:
Setup
Cover-up method
:
:
:
Example 3: Improper Fraction (Advanced)
Decompose:
Step 1: Check degrees
Numerator degree (2) ≥ denominator degree (2) → IMPROPER. Must divide first!
Step 2: Long division
Step 3: Decompose the proper part
Common Pitfalls
Forgetting to check for improper fractions
If degree of numerator ≥ denominator, you MUST divide first.
Sign errors with negative roots
When substituting into , you get , not .
Not factoring the denominator completely
You must factor fully before setting up partial fractions.
Real-World Application
Integration in Calculus
In Calculus, you'll need to integrate rational functions. Partial fractions turn one hard integral into several easy ones:
Practice Quiz
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