Enter Polynomial Data
Use descending coefficients. Example: 1, -10, 35, -50, 24
Example Data Table
| Example | Polynomial | Initial r | Initial s | Expected Roots | Why It Helps |
|---|---|---|---|---|---|
| 1 | 1, -10, 35, -50, 24 | 0.5 | -0.5 | 1, 2, 3, 4 | Shows two clean quadratic deflations. |
| 2 | 1, 0, 0, 0, 1 | 0.5 | -0.5 | ±0.7071 ± 0.7071i | Tests complex conjugate root handling. |
| 3 | 1, -5, 8, -4, 0 | 1 | -1 | 0, 1, 2, 2 | Demonstrates repeated and zero roots. |
Formula Used
Quadratic factor assumption: P(x) = (x² - rx - s)Q(x) + (b₁x + b₀)
Synthetic sequence: bn = an, bn-1 = an-1 + rbn
bi = ai + rbi+1 + sbi+2
Correction sequence: cn = bn, cn-1 = bn-1 + rcn
ci = bi + rci+1 + sci+2
Updates: Solve for Δr and Δs, then use r = r + Δr and s = s + Δs until the error falls below tolerance.
Once a quadratic factor converges, the calculator deflates the polynomial and repeats the process until only a linear or quadratic tail remains.
How to Use This Calculator
- Enter coefficients in descending powers, including zeros for missing terms.
- Provide starting guesses for r and s.
- Set tolerance for convergence and choose a practical iteration cap.
- Click Solve Polynomial to generate roots and iteration details.
- Review the factor breakdown and log table.
- Use the export buttons to save results as CSV or PDF.
Frequently Asked Questions
1. What does Bairstow’s method solve?
It estimates real and complex roots of higher degree polynomials by iteratively locating quadratic factors, then deflating the original polynomial step by step.
2. Why do I need initial guesses for r and s?
Bairstow’s method is iterative. Better starting values usually improve stability, reduce iteration count, and lower the chance of slow or failed convergence.
3. Can this calculator return complex roots?
Yes. When the quadratic factor has a negative discriminant, the solver returns conjugate complex roots in a + bi and a - bi form.
4. What should I do if convergence fails?
Try different starting guesses, allow more iterations, or relax the tolerance slightly. Repeated roots and poor guesses can slow convergence noticeably.
5. Why must coefficients be in descending order?
The recurrence relations for the synthetic division arrays follow descending powers. Missing powers should be entered as zero coefficients to preserve structure.
6. Does the solver work for repeated roots?
Yes, but repeated roots can reduce numerical stability. Good initial guesses and suitable tolerance settings help the iteration finish more reliably.
7. What does the iteration table show?
It records each quadratic-factor search, including updated r and s values, corrections, remainder terms, and the current convergence error.
8. What is exported in CSV and PDF files?
The exported data includes summary details, computed roots, and the full iteration log so you can document or review the numerical process later.