Type a polynomial such as x^5 − 3x^3 + x − 4 or its coefficients, and get every root, real and complex, found numerically with the Durand-Kerner method and polished with Newton steps.
| # | Root | Type | Multiplicity | |p(root)| |
|---|
Runs entirely in your browser. Nothing is uploaded to any server.
Cubics, quartics and quintics up to degree 30.
Finds every root, not just the ones where the graph crosses zero.
Each root comes with the size of p(root) so you can trust it.
There is no general formula for polynomials of degree five or higher, so roots must be found numerically. This tool uses the Durand-Kerner (Weierstrass) method, which starts with guesses spread on a circle in the complex plane and improves all of them at the same time until they stop moving.
Each root is then polished with a few Newton steps on the original polynomial. Roots that are very close together are grouped and reported as one root with a multiplicity, because repeated roots can only be found to a few fewer decimal places.
Points where the graph crosses or touches the x-axis.
With real coefficients they always come in conjugate pairs a ± bi.
|p(root)| close to zero confirms the root is accurate.
A polynomial of degree n has exactly n complex roots when you count repeated roots by their multiplicity.
Repeated roots are numerically sensitive, so their last few digits are less accurate. The tool groups them and averages the values.
Yes. Type numbers separated by commas from the highest power down, such as 1, 0, -2, 1 for x³ − 2x + 1. Include zeros for missing powers.
No. Roots are numerical approximations. For exact quadratic roots, use the quadratic formula solver.
Real roots give factors (x − r), and each complex pair gives a real quadratic factor, so the product matches the polynomial up to rounding.
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