Partial Fraction Decomposition (Advanced)
Handling repeated factors and irreducible quadratics in the denominator.
Introduction
What happens when the denominator has repeated factors like orirreducible quadratics like ? The basic rules don't apply directly. We need additional terms for each power and linear numerators for quadratics—patterns that handle any rational function you'll encounter.
Prerequisite Connection
You can decompose fractions with distinct linear factors.
Today's Increment
We handle repeated factors and quadratics .
Why This Matters
Repeated factors appear in differential equations; quadratics in oscillations and complex analysis.
Key Concepts
Rule for Repeated Linear Factors
One term for each power from 1 to
Rule for Irreducible Quadratic Factors
Numerator is a LINEAR expression (one degree less than denominator)
For Repeated Quadratics
Worked Examples
Example 1: Repeated Linear Factor (Basic)
Decompose:
Step 1: Set up form (need BOTH powers)
Step 2: Multiply by
Step 3: Strategic substitution
:
Compare coefficients:
Example 2: Quadratic Factor (Intermediate)
Decompose:
Setup (quadratic gets )
Multiply by LCD
Solve system
:
Compare :
Compare constants:
Example 3: Mixed Types (Advanced)
Decompose:
Setup (linear + repeated linear)
Strategic values
:
:
Compare coefficients
Common Pitfalls
Missing terms for repeated factors
needs BOTH AND
Using just A for quadratics
Quadratic factors need in the numerator, not just a constant.
Forgetting to compare coefficients
Strategic substitution may not give all constants. Compare , , etc.
Real-World Application
Inverse Laplace Transforms in Engineering
Engineers use partial fractions to invert Laplace transforms when solving differential equations for circuits and control systems. Repeated roots model systems with resonance; irreducible quadratics model oscillations.
The transfer function requires decomposition with repeated factors to find the time-domain response.
Practice Quiz
Loading...