Partial Fraction Decomposition Calculator
Step-by-Step Mathematical Solutions
Decompose rational functions into partial fractions with detailed step-by-step solutions and explanations.Used by 500K+ students • Supports all factor types • Educational explanations included
Partial Fraction Decomposition Calculator
Decompose rational functions with step-by-step solutions
Function Configuration
Example Functions
Ready to Decompose Your Function?
Enter your rational function to get step-by-step partial fraction decomposition
How to Use the Partial Fraction Decomposition Calculator
Decompose rational functions in 4 simple steps with detailed explanations
1. Enter Function
Input the numerator and denominator polynomials of your rational function
2. Select Type
Choose the decomposition type based on your denominator factors
3. Calculate
Click calculate to get the partial fraction decomposition with steps
4. Verify
Review the solution steps and verify the decomposition accuracy
Understanding Partial Fraction Decomposition
Master the key concepts and techniques for decomposing rational functions
Linear Factors
For factors like (x - a), use A/(x - a) form. Each distinct linear factor gets its own fraction with a constant numerator.
Quadratic Factors
For irreducible quadratic factors like (x² + bx + c), use (Ax + B)/(x² + bx + c) form with linear numerator.
Repeated Factors
For repeated factors like (x - a)ⁿ, include terms for each power: A₁/(x - a) + A₂/(x - a)² + ... + Aₙ/(x - a)ⁿ.
Coefficient Method
Clear denominators, expand, and equate coefficients of like terms to solve for unknown constants.
Verification
Always verify by adding the partial fractions back together to ensure they equal the original function.
Applications
Used in integration, Laplace transforms, differential equations, and signal processing applications.
Real-World Example: Sarah's Calculus Problem
See how partial fraction decomposition helps solve integration problems step by step
The Problem
Sarah needs to integrate: ∫ (3x + 2)/((x - 1)(x + 2)) dx
This rational function requires partial fraction decomposition before integration.
Solution Steps:
Set up: (3x + 2)/((x - 1)(x + 2)) = A/(x - 1) + B/(x + 2)
Clear denominators: 3x + 2 = A(x + 2) + B(x - 1)
Solve for coefficients: A = 8/3, B = 1/3
Result: (8/3)/(x - 1) + (1/3)/(x + 2)
The Benefits
Easy Integration
Each fraction integrates to simple logarithmic functions
Time Saving
Reduces complex integration to simple standard forms
Clear Method
Systematic approach with verifiable steps
Final Answer:
∫ (3x + 2)/((x - 1)(x + 2)) dx = (8/3)ln|x - 1| + (1/3)ln|x + 2| + C
Optimization Tips for Partial Fractions
Expert strategies to master partial fraction decomposition efficiently
Check Degree First
Always verify that the degree of the numerator is less than the degree of the denominator. If not, perform polynomial long division first.
Tip: Improper fractions need division before decomposition
Factor Completely
Factor the denominator into irreducible factors over the real numbers. Use techniques like grouping, quadratic formula, or synthetic division.
Tip: Look for common factors and special patterns first
Use Strategic Values
When solving for coefficients, substitute strategic values (like roots of factors) to simplify calculations and reduce system complexity.
Tip: Use x = root values to eliminate terms quickly
Handle Repeated Factors
For repeated factors (x - a)ⁿ, include all powers from 1 to n. Use the cover-up method for highest powers to find coefficients efficiently.
Tip: Start with highest power and work downward
Always Verify
Add your partial fractions back together and simplify to verify they equal the original function. This catches calculation errors early.
Tip: Verification prevents costly mistakes in integration
Use Technology Wisely
Use calculators for verification and complex arithmetic, but understand the underlying method. Practice by hand to build intuition.
Tip: Technology aids understanding, doesn't replace it
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Frequently Asked Questions
Common questions about partial fraction decomposition and mathematical problem solving
Ready to Master Partial Fraction Decomposition?
Use our calculator to practice and understand step-by-step solutions for any rational function