Enter Polynomial Data
Example Data Table
| Example Polynomial | Coefficient Input | Expected Roots | Notes |
|---|---|---|---|
| x3 - 6x2 + 11x - 6 | 1, -6, 11, -6 | 1, 2, 3 | Classic cubic with three distinct real roots. |
| x4 + 1 | 1, 0, 0, 0, 1 | Four complex roots | Shows symmetry on the complex plane. |
| x2 - 2x + 1 | 1, -2, 1 | 1, 1 | Repeated root, useful for multiplicity inspection. |
Formula Used
P(x) = anxn + an-1xn-1 + ... + a1x + a0
b0 = an, then bk = bk-1x + an-k. This evaluates the polynomial efficiently with fewer operations.
rk(m+1) = rk(m) - P(rk(m)) / Πj≠k(rk(m) - rj(m))
Residual = |P(r)|. Smaller residuals indicate more accurate roots.
How to Use This Calculator
- Enter polynomial coefficients in descending powers.
- Set tolerance for the stopping threshold.
- Choose maximum iterations for the numerical solver.
- Adjust graph limits or leave them blank.
- Pick how many sample points the graph should use.
- Press Find Roots to solve and draw plots.
- Review the roots table, residuals, and complex-plane map.
- Use the CSV or PDF buttons to export the computed results.
Frequently Asked Questions
1) What coefficients should I enter?
Enter them from the highest exponent to the constant term. For x3 - 6x2 + 11x - 6, type 1, -6, 11, -6.
2) Can this calculator return complex roots?
Yes. The solver handles both real and complex roots. Complex solutions appear with real and imaginary parts, and they are also plotted on the complex plane.
3) Why are some roots repeated?
Repeated roots occur when a factor appears more than once. For example, (x - 1)2 has root 1 with multiplicity two. The multiplicity guess helps you inspect that behavior.
4) What does the residual value mean?
The residual is |P(r)| after substitution of a computed root. A value close to zero means the root is numerically accurate for the entered tolerance.
5) What tolerance should I choose?
A smaller tolerance usually gives more precise roots but may need more iterations. For many cases, 1e-10 is a strong starting point.
6) Why does the graph look clipped?
High-degree polynomials can grow very quickly. The chart limits extreme values to keep the plot readable. Narrow the x-range for a more detailed view.
7) Does the method always converge?
It converges well for many practical polynomials, but difficult repeated-root cases may need tighter settings or more iterations. The status note helps you judge reliability.
8) Can I export my results?
Yes. After calculation, you can download the roots table as CSV or export a compact PDF summary containing the polynomial and computed solutions.