How to Use Partial Fraction Decomposition Calculator
The Partial Fraction Decomposition Calculator breaks a rational function into a sum of simpler fractions. Enter the constant numerator and the roots of the denominator, then read the partial fraction coefficients directly.
- Enter the numerator — type the constant value P (e.g. 1 for 1/((x−1)(x−2))).
- Enter the denominator roots — type the roots separated by commas (e.g. 1, 2, 3). Each root rᵢ corresponds to the factor (x − rᵢ).
- Read the result — the Partial Fraction Decomposition Calculator shows the original expression, the factored denominator, the coefficient for each partial fraction term, and the complete expansion.
The calculator supports up to 5 distinct roots. The roots must all be different; if two roots are equal, the decomposition is undefined for distinct linear factors.
Formula & Theory - Partial Fraction Decomposition Calculator
The Partial Fraction Decomposition Calculator uses the cover-up (Heaviside) method for distinct linear factors:
P / [(x − r₁)(x − r₂)...(x − rₙ)] = A₁/(x − r₁) + A₂/(x − r₂) + ... + Aₙ/(x − rₙ)
Coefficient: Aᵢ = P / ∏_{j ≠ i} (rᵢ − rⱼ)| Symbol | Meaning |
|---|---|
| P | Constant numerator |
| rᵢ | Root of the i-th linear factor |
| Aᵢ | Coefficient of the i-th partial fraction |
The cover-up method works by multiplying both sides of the decomposition by (x − rᵢ) and evaluating at x = rᵢ, which causes all other terms to vanish. This is algebraically equivalent to the method of undetermined coefficients but is far faster for distinct linear factors.
Assumptions and Limits
- The numerator must be a constant (degree 0). If the degree of the numerator equals or exceeds the degree of the denominator, a polynomial long division step is required first; this calculator does not perform that step automatically.
- All denominator roots must be real and distinct.
- Coefficients are displayed with up to 4 decimal places for non-integer values.
Use Cases for Partial Fraction Decomposition Calculator
The Partial Fraction Decomposition Calculator is most commonly used in calculus and algebra. Typical use cases include:
- Integral calculus — partial fractions simplify integrals of rational functions into a sum of logarithms, making the Partial Fraction Decomposition Calculator an essential step-checker for calculus assignments.
- Inverse Laplace transforms — control systems and differential equations courses routinely decompose transfer functions into partial fractions before applying the inverse Laplace table.
- Algebra simplification — simplifying complex rational expressions for further algebraic manipulation.
- Homework verification — students can check manually computed coefficients against the Partial Fraction Decomposition Calculator to catch arithmetic mistakes before submission.
The transparent display of each coefficient and factor makes the Partial Fraction Decomposition Calculator equally useful for learning the technique and for verifying answers.