Calculator inputs
Example data table
| Polynomial | Coefficient input | Expected structure |
|---|---|---|
| x³ − 6x² + 11x − 6 | 1, -6, 11, -6 | Three distinct real roots: 1, 2, 3 |
| x⁴ + 1 | 1, 0, 0, 0, 1 | No real roots, four complex roots |
| x³ − 3x + 2 | 1, 0, -3, 2 | Repeated root at 1 and simple root at −2 |
| x² − 4x + 4 | 1, -4, 4 | Repeated real root at 2 |
Formula used
Polynomial model: P(x) = anxn + an-1xn-1 + … + a1x + a0
Descartes’ rule of signs: sign changes in P(x) bound positive real roots, while sign changes in P(−x) bound negative real roots.
Derivative test: P′(x) identifies turning points and helps flag repeated roots when P(x) and P′(x) vanish together.
Cauchy bound: Every root r satisfies |r| ≤ 1 + max(|ai| / |an|).
Numerical roots: The page uses the Durand–Kerner method to approximate all roots simultaneously for higher-degree polynomials.
How to use this calculator
- Enter coefficients in descending powers, ending with the constant term.
- Adjust tolerance for stricter or looser numerical grouping.
- Set iteration count higher for harder or higher-degree polynomials.
- Press Analyze polynomial to show the result above the form.
- Review real roots, complex roots, estimated multiplicities, and turning points.
- Download the result as CSV for data work or PDF for reporting.
FAQs
1. What does root structure mean?
It describes how many roots are real or complex, whether any repeat, and how those roots are distributed across the number line or complex plane.
2. Why can a polynomial have complex roots?
When a real polynomial does not cross the x-axis enough times, the remaining roots occur as conjugate complex pairs, preserving the total degree count.
3. How are repeated roots detected here?
The calculator numerically groups nearly identical roots and also uses derivative behavior. Repeated roots often make the polynomial and derivative vanish together.
4. Are the roots exact?
The displayed roots are numerical approximations. For many practical cases they are highly accurate, but symbolic factorization may still be preferable for proofs.
5. What are rational root candidates?
For integer coefficients, the Rational Root Theorem lists possible fractional roots using factors of the constant term over factors of the leading coefficient.
6. Why are turning points useful?
Turning points reveal where the polynomial changes direction. They help explain how many real crossings can occur and where repeated roots may appear.
7. When should I raise iterations?
Increase iterations for higher-degree polynomials, tightly clustered roots, or cases where results seem unstable. More steps can improve convergence and grouping accuracy.