Enter polynomial coefficients
Coefficients define: anxn + ... + a1x + a0.
Use decimals or integers. Leading coefficient must be nonzero.
Example data table
Sample input for x³ − 6x² + 11x − 6 with zeros 1, 2, 3.
| Degree | a3 | a2 | a1 | a0 | Expected zeros |
|---|---|---|---|---|---|
| 3 | 1 | -6 | 11 | -6 | 1, 2, 3 |
Formula used
- P(x) = anxn + an−1xn−1 + ... + a0
- Cauchy bound: all zeros satisfy |z| ≤ 1 + max(|ai/an|) for i<n.
- Durand–Kerner update: zᵢ ← zᵢ − P(zᵢ) / ∏(zᵢ − zⱼ) for all j≠i.
- Residual check: report |P(z)|; smaller means a better zero.
- Real scan: find sign changes, then refine with bisection.
How to use this calculator
- Select the polynomial degree.
- Enter coefficients from highest power down to constant.
- Choose a method based on your needs.
- Press Find Zeros to compute roots and residuals.
- Download CSV or PDF to save the results.
FAQs
1) What does “zero” mean for a polynomial?
A zero is a value z where the polynomial evaluates to exactly zero. In practice, numerical methods return approximations with a small residual |P(z)|.
2) Why do I see complex roots?
Polynomials can have complex solutions even with real coefficients. Complex roots often appear as conjugate pairs, like a+bi and a−bi.
3) Which method should I choose?
Use Auto for degree one or two. Use All roots when you need every solution, including complex. Use Real zeros only when you only care about real solutions.
4) What is the bound shown in results?
It is a safe radius that contains every root. It helps set starting guesses and gives a quick sanity check if a computed root is unexpectedly large.
5) What does |P(z)| tell me?
It is the residual magnitude after plugging the root back in. Smaller values indicate the root is closer to a true zero, improving reliability.
6) My results look unstable. What can I do?
Try scaling coefficients so they are similar in size, or switch methods. Very high degrees or extreme coefficients can amplify rounding error.
7) Does the real scan miss some roots?
It can miss roots if the scan resolution is too coarse or if roots are repeated and do not create a sign change. Increase degree accuracy by using All roots.
8) Can I export the computed roots?
Yes. After calculation, use the CSV or PDF buttons. Exports include the polynomial, method label, and a table of roots with residuals.
Built for learning, checking, and quick sharing of results.