Polynomial Roots Calculator

What does this polynomial roots calculator compute?

A degree-n polynomial c0 + c1 x + … + cn x^n = 0 has n roots in the complex plane (counting multiplicity). This page returns a complete validated set or an explicit numerical error. Coefficients are constant-first.

The published formula or method is closed form (deg 1–3 Cardano) or Durand–Kerner (deg 4–8). The following references support the stated mathematics or method:

• NIST DLMF §3.8 — Nonlinear Equations: Iterative root finding, convergence considerations, and polynomial-root methods.
• NumPy — numpy.polynomial.polynomial.polyval: Polynomial evaluation using Horner’s method; NumPy’s documented coefficients are in ascending degree order. CalculatorX’s input order must follow its own specification. Mathematical reference only; not validation or endorsement of this implementation.

Which inputs and limits should you check?

Follow the input domain, units, branch or solve direction, and rounding policy in the specification. Preserve exact outputs and any additional candidates or warnings.

The worked example uses (x−1)(x−2)(x−3) = 0 with coeffs −6, 11, −6, 1. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.polynomial.roots v1.2.2 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Numerical scope and interpretation

Coefficients are constant first, finite real binary64 values, with degree 1–8 and a nonzero leading coefficient. Decimal/scientific strings are limited to 512 characters per token and 4608 characters per coefficient list. Arrays contain 2–9 entries. A nonzero textual number that rounds to zero is rejected with NUMERIC_UNDERFLOW; invalid numbers, unknown fields, conflicting aliases, and inapplicable evaluation inputs are rejected. The cubic page locks Cardano; the evaluation page locks Horner.

The accepted binary64 coefficients define the polynomial. Exact rational square-free decomposition preserves its repeated-root multiplicities. Stable closed forms handle degree 1–3; higher degrees use scaled, bounded iteration with exact real-root isolation. Each square-free factor must pass disjoint exact rational Weierstrass/Gershgorin inclusion disks, a relative L1 inclusion-error budget of 2e-10, and normalized residual at most 2e-12. A small residual alone does not establish a complete root set. Overlapping disks or exhausted iteration reject NON_CONVERGENCE; difficult finite inputs need not succeed. Exact arithmetic is bounded to 16384 bits; iteration is limited to two attempts of 2048 steps and real isolation to 512 levels / 4096 steps.

Horner evaluation uses exact rational arithmetic on the accepted binary64 inputs and rounds once to binary64. Unrepresentable nonzero roots or values reject overflow/underflow. residual_max is the largest unscaled Euclidean |p(z)|, or null if its binary64 projection is outside range. normalized_residual uses (|Re p|+|Im p|) / Σ |cᵢ|(|Re z|+|Im z|)ⁱ. The exact depressed cubic discriminant determines cubic_case; delta can be null when projection loses range. Rounded zero residuals do not prove exact roots.

Independent CPython Fraction factor constructions and Decimal250/750 references check 67 complete result packs, root multiplicities and returned-root diagnostics. The fixed-vector root budget is 4e-10 relative L2 plus two subnormal quanta; exact zero roots remain zero and Horner values are checked within one ULP. These tests do not establish accuracy for every possible input or identical results across runtimes. Closely spaced roots can be highly sensitive to coefficient changes.

The real curve uses the same exact Horner evaluator. Complex roots are listed outside the real plane; repeated roots retain multiplicity in the result. The graph is omitted with a text explanation when finite coordinates or labels cannot be distinguished. Scalar results and native CSV/JSON exports retain the full result pack and raw inputs without snapping small nonzero roots to zero.

Typing computes locally. Share URLs and REST/MCP requests transmit inputs and may enter browser, server, or intermediary logs. Avoid confidential values in shared URLs. See NIST DLMF polynomial definitions and evaluation and Neumaier–Polzer polynomial zero enclosures for mathematical background; implementation checks are independent of these references.