Calculator Inputs
Enter the numerator coefficients in descending powers, then define the denominator with linear or irreducible quadratic blocks.
Example Data Table
| Example numerator | Block 1 | Block 2 | Block 3 | Interpretation |
|---|---|---|---|---|
| 2,5,11,7,9 | Linear: 1,-1, m=1 | Linear: 1,2, m=2 | Quadratic: 1,0,1, m=1 | Builds a mixed denominator with one distinct linear factor, one repeated linear factor, and one irreducible quadratic factor. |
Formula Used
General structure
If R(x) = P(x) / D(x) and D(x) is built from linear and irreducible quadratic factors, then:
R(x) = Q(x) + [proper remainder decomposition]
For each repeated linear factor (ax + b)m, the calculator uses constants over each power level:
A1/(ax+b) + A2/(ax+b)2 + ... + Am/(ax+b)m
For each repeated irreducible quadratic factor (ax2 + bx + c)m, the calculator uses linear numerators:
(A1x+B1)/q(x) + (A2x+B2)/q(x)2 + ... + (Amx+Bm)/q(x)m
After multiplying both sides by the full denominator, the page matches polynomial coefficients and solves the resulting linear system.
How to Use This Calculator
- Enter numerator coefficients in descending order.
- Add each denominator block using either a linear or quadratic type.
- Enter coefficients for every block in descending order.
- Set the multiplicity for repeated factors.
- Choose the x-range for the Plotly graph.
- Press Compute Decomposition.
- Review the quotient, proper remainder, partial fraction form, and validation table.
- Use the export buttons to download the current result as CSV or PDF.
FAQs
1. What does this calculator solve?
It decomposes rational functions into partial fractions when the denominator is supplied as linear and irreducible quadratic blocks, including repeated powers and improper cases.
2. Does it handle improper rational functions?
Yes. It first performs polynomial long division. The quotient is shown separately, and only the proper remainder is decomposed into partial fractions.
3. Why must quadratic blocks be irreducible?
Standard real partial fractions use linear numerators over irreducible quadratics. If a quadratic factors into real linear terms, it should be entered as linear blocks instead.
4. What coefficient order should I use?
Use descending powers. For example, x² + 3x + 2 becomes 1,3,2. A linear factor x − 1 becomes 1,-1.
5. Why are there repeated denominator powers?
A repeated factor of multiplicity m requires one term for each power from one through m. That ensures the decomposition has enough unknown coefficients.
6. What does the validation table mean?
It compares sampled function values using the original rational form and the reconstructed result. Very small error confirms the solved coefficients are consistent.
7. Can I export my result?
Yes. The page includes CSV export for structured summaries and PDF export for a neat, shareable snapshot of the current decomposition.
8. Why might the calculator show an error?
Errors usually come from invalid coefficient lists, missing denominator blocks, a reducible quadratic, or a singular setup caused by inconsistent factor inputs.