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Polynomial Roots Calculator

Find all roots of a real polynomial of degree 1–8. Closed form for linear and quadratic; Durand–Kerner for cubics and higher. Not the quadratic formula page. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 12/12 tests · Production surface contract 1/1 · v1.0.0
Input interpretation
Enter values to calculate.
Result
—
Model
All roots of a real polynomial of degree 1–8.
Scope
Real coefficients
Verification
Engine tested · 12/12 tests · Production surface contract 1/1 · Specification checked · v1.0.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
3 golden · 4 boundary · 5 property · Production surface contract 1/1 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (1 capability; 163 remain CURRENT) @ 2026-09-19T00:00:17.039Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

All n rootsc0 + c1 x + … + cn x^n = 0
Linearx = −c0 / c1
Quadraticx = (−b ± √(b² − 4ac)) / (2a)
Degree 3–8Durand–Kerner simultaneous iteration
iNot /calc/math/quadratic (discriminant, vertex, parabola). Not a single Newton/bisection root. Not symbolic factoring. Leading coefficient must be nonzero. Degree ≤ 8.

How to use

1

Enter coefficients constant-first

(x−1)(x−2)(x−3) is −6, 11, −6, 1. Trailing zeros are rejected; drop them instead.

2

Read every root

Real roots first by value; complex conjugates appear as a ± bi.

Example calculations

Common configurations with formula and result.

ϟ

Cubic with three real roots

(x−1)(x−2)(x−3) = 0

coeffs −6, 11, −6, 1
1, 2, 3
ϟ

x² + 1 = 0

coeffs 1, 0, 1

closed-form quadratic
± i

Polynomial Roots calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A degree-n polynomial c0 + c1 x + … + cn x^n = 0 has n roots in the complex plane (counting multiplicity). This page returns all of them. Coefficients are constant-first.
What it calculates
All roots of a real polynomial of degree 1–8.
Inputs
  • coeffs
  • method?
Outputs
  • roots
  • degree
  • solver
  • residual_max
Formula
closed form (deg 1–2) or Durand–Kerner (deg 3–8)
Assumptions
  • Real coefficients
  • Leading coefficient nonzero
  • Degree ≤ 8
  • Not a second quadratic page
  • Not CAS factoring
Units
  • dimensionless
Boundary conditions
  • degree 0 or one coefficient → INVALID_INPUT
  • leading coefficient 0 → INVALID_INPUT
  • degree > 8 → VALUE_OUT_OF_RANGE
  • method quadratic / cas / newton / eigen → INVALID_MODE
Example
coeffs=-6,11,-6,1 → 1, 2, 3
Validation cases

2 published on this page · 12/12 tests · Production surface contract 1/1 · View evidence

  • coeffs=-6,11,-6,1 → 1, 2, 3
  • method=quadratic coeffs=8,-6,1 → INVALID_MODE
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.0.0

Other calculators in this family: Linear Equation Calculator, Linear System Calculator, LU Factorization Calculator, Matrix Determinant Calculator, Matrix Eigenvalues Calculator, Matrix Inverse Calculator, Matrix Rank Calculator, QR Factorization Calculator . Explore all Algebra & Equations.

→ Linear Equation Calculator Solve ax + b = rhs for x. Identity and inconsistent cases when a = 0. Not the quadratic formula. Runs locally. › → Linear System Calculator Solve small linear systems Ax = b, plus det, inverse, rank, LU, QR, and eigenvalues. Same engine as the matrix workspace. Not a CAS. Runs locally. › → LU Factorization Calculator Doolittle LU factorization with partial pivoting for a small real matrix. Same engine as Linear System. Not a CAS. Runs locally. › → Matrix Determinant Calculator Determinant of a small real matrix. Same engine as Linear System and the matrix workspace. Not a CAS. Runs locally. › → Matrix Eigenvalues Calculator Eigenvalues and eigenvectors of a small real matrix via the characteristic polynomial. Same engine as Linear System. Not a CAS. Runs locally. › → Matrix Inverse Calculator Inverse of a small real matrix via Gauss–Jordan. Same engine as Linear System. Singular matrices fail. Not a CAS. Runs locally. › → Matrix Rank Calculator Rank of a small real matrix via Gaussian elimination. Same engine as Linear System. Not a CAS. Runs locally. › → QR Factorization Calculator Householder QR factorization of a small real matrix. Same engine as Linear System. Not a CAS. Runs locally. ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this the quadratic formula page?

No. /calc/math/quadratic is ax²+bx+c with discriminant, vertex, and a parabola. This page returns every root of a general degree-n polynomial.

How is this different from numerical root?

Numerical root finds one real root of a polynomial with Newton, bisection, or secant. This page returns all n roots, including complex conjugates.

Can I factor symbolically?

No. Symbolic CAS is out of scope. The solver is closed form for degree 1–2 and Durand–Kerner for degree 3–8.

Where does this run?

Locally in the browser by default. REST and MCP call the same polynomial-roots engine.