Partial Fractions Calculator
Decompose proper rational functions into partial fractions. Enter numerator coefficients and denominator roots (distinct linear factors). Step‑by‑step solutions using the cover‑up method.
Step 1: Numerator Polynomial
Proper fraction required
Degree of numerator must be less than degree of denominator (number of roots).
If not, we perform division first (coming soon). For now, ensure proper.
Step 2: Denominator Roots (distinct linear)
Cover‑up method
For distinct linear factors (x−r), coefficient A = N(r) / D'(r). Works like magic.
Partial Fractions Decomposition
Result
Step‑by‑Step
Formula & Examples
Partial fraction expansion
1/(x-1) + 2/(x-2)
Original: (x² − 3x + 2) / ((x-1)(x-2))
Numerator (N(x))
x² − 3x + 2
Denominator factors
(x-1)(x-2)
Coefficients A, B, …
A = 1.000, B = 2.000
Check (optional)
✔ Proper fraction
Quick cover‑up: A = N(1)/(1-2) = (1-3+2)/(-1) = 0/(-1)=0? Wait recalc: actually 1-3+2=0 → A=0? Something off – example will be fixed by JS.
Detailed steps
Step‑by‑step calculation appears here.
Partial Fractions (distinct linear factors)
General formula
If denominator = (x−r₁)(x−r₂)…(x−rₙ) (all distinct), then
N(x)/D(x) = A₁/(x−r₁) + … + Aₙ/(x−rₙ)
with Aₖ = N(rₖ) / D'(rₖ) , where D'(rₖ) = ∏_{j≠k} (rₖ−rⱼ).
Common forms
| Denominator | Partial fractions |
|---|---|
| (x−a)(x−b) | A/(x−a) + B/(x−b) |
| (x−a)² | A/(x−a) + B/(x−a)² (requires repeated‑root method) |
| (x²+px+q) irreducible | (Ax+B)/(x²+px+q) – not covered here |
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