Polynomial roots

Enter an expression such as x^3 - 6x^2 + 11x - 6 or real coefficients from highest to lowest power: 1 -6 11 -6. Use a decimal point, optional scientific notation (1e-20), x or X and integer powers 0–32. A multiplication sign before x is optional. Separate coefficient lists with spaces, semicolons or commas; decimal commas are not accepted.

Calculate explicitly; editing or cancelling removes the previous result. Copy exports up to 17 significant digits.

Leading coefficients are removed only when exactly zero. Trailing zeros give exact roots at zero with their multiplicity. Constant zero vanishes everywhere; other constants have no roots. The table retains repeated roots and the plane shows their positions, so coincident points overlap.

The coefficients and variable are scaled before calculation. Degrees 1–3 use division or closed formulas; higher degrees use companion-matrix QR with a Durand–Kerner fallback. A complete set is checked by reconstructing coefficients as well as evaluating each root. Calculations run in a cancellable worker; the activity bar does not claim an estimated percentage.

η = |p(z)| / Σ |aₖ| |z|ᵏ is a scale-independent residual, not an error bound on z. Near or repeated roots can move considerably under rounding, even with tiny η; approximate multiple roots may have small imaginary parts. Severe scale disparities, coefficient underflow or failure to converge cause an error rather than a partial result. No arbitrary-precision or symbolic guarantee.

Netlib · Roots of a Polynomial, §D

Raw input, selected example and expanded panels are saved locally. Previously calculated roots are restored without rerunning the solver; unfinished input and cancelled calculations remain uncalculated. Save JSON backs up the session; Load JSON checks the root set and recomputes residuals before restoring it. This is a numerical consistency check, not a proof of accurate roots for ill-conditioned polynomials. PNG exports the displayed root plane with its input.