Partial Fractions Calculator
Decompose rational functions into partial fractions with step-by-step solutions.
Input Parameters
Enter the numerator and denominator of the rational function.
What is Partial Fraction Decomposition?
Partial fraction decomposition is a technique used in algebra to express a rational function as a sum of simpler fractions. This is particularly useful for integrating rational functions and solving differential equations.
The general form is: P(x)/Q(x) = A₁/(x - r₁) + A₂/(x - r₂) + ... + (Bx + C)/(x² + px + q) + ...
Where Q(x) is factored into linear and irreducible quadratic factors.
Properties of Partial Fractions
- Linear Factors: For distinct roots, use A/(x - r)
- Repeated Linear Factors: For repeated roots, use A₁/(x - r) + A₂/(x - r)² + ...
- Quadratic Factors: For irreducible quadratics, use (Bx + C)/(x² + px + q)
- Proper Fractions: Degree of numerator must be less than denominator
- Improper Fractions: Perform polynomial division first
Formula
General Partial Fraction Decomposition:
P(x)/Q(x) = ∑ [Aᵢ / (x - rᵢ)] + ∑ [(Bx + C) / (x² + px + q)]
Where Q(x) = ∏ (x - rᵢ) × ∏ (x² + pⱼx + qⱼ)
Examples
Example 1: Distinct Linear Factors
(x+1)/(x²-1) = 1/2/(x-1) + 1/2/(x+1)
Example 2: Repeated Linear Factor
(x²)/(x-1)³ = 1/(x-1) + 2/(x-1)² + 1/(x-1)³
Example 3: Quadratic Factor
(x)/(x²+1) = (1/2)x/(x²+1) + 1/2/(x²+1)
Applications
- Integration: Simplifies integration of rational functions
- Differential Equations: Used in solving linear DEs
- Signal Processing: Partial fraction expansion in Laplace transforms
- Control Systems: Transfer function analysis
- Physics: Wave function decomposition
Frequently Asked Questions
When do I need partial fraction decomposition?
It's needed when integrating rational functions or solving differential equations involving rational expressions.
What if the fraction is improper?
Perform polynomial division first to separate the improper part, then decompose the remaining proper fraction.
How do I find the coefficients?
Multiply both sides by the denominator and equate coefficients, or use the method of undetermined coefficients.
Can complex numbers appear?
Yes, when quadratic factors don't factor over reals, but typically we work with real coefficients.
Academic & Scientific References
For further understanding and validation of the formulas used above, we recommend exploring these authoritative resources: