Verified calculation capability

Polynomial roots Calculation Capability

All roots of a real polynomial of degree 1–8. Closed form for linear and quadratic; Durand–Kerner otherwise. Not the quadratic formula page, not a single Newton root, not CAS. · Math · Polynomial

Capability ID math.polynomial.roots · version 1.0.0

Open calculator REST API MCP Evidence Evidence JSON Review policy

Model

All n roots

All n roots: c0 + c1 x + … + cn x^n = 0
Linear: x = −c0 / c1
Quadratic: x = (−b ± √(b² − 4ac)) / (2a)
Degree 3–8: Durand–Kerner simultaneous iteration

Inputs / Outputs

Canonical symbols for this capability. Agent schemas remain the contract.

Symbol	Quantity	Unit	Role
`coeffs`	c0,c1,…,cn constant first	—	input
`method`	—	—	input
`formula`	—	—	output
`api_version`	—	—	output
`api_contract_version`	—	—	output
`spec_version`	—	—	output
`page`	Canonical calculator page URL	—	output
`links`	—	—	output
`degree`	—	—	output
`n`	amount of substance	—	output
`solver`	—	—	output
`roots`	—	—	output
`residual_max`	—	—	output

Assumptions / Validity

Declared scope for this capability. Do not use it outside this domain.

Assumptions

Real coefficients
Leading coefficient nonzero
Degree ≤ 8
Not a second quadratic page
Not CAS factoring

Example

One numerical check. The calculator page holds worked steps and FAQ.

coeffs=-6,11,-6,1 → 1, 2, 3

Open this example in the calculator

Trust

Review-policy: engine tested — automated tests, sources, version lock, and evidence published. Supplemental to CVP; not Domain Expert Review. See CVP and Review policy.

Tier: Engine tested — catalog token engine_tested
Calculation version: 1.0.0
Evidence tests: 12/12 calculation tests · golden 3/3 · boundary 4/4 · property 5/5
Engine tested: Yes
Source checked: Yes
Supplemental domain review: Not performed
Named expert review: Not performed
Version lock: Supported (409 VERSION_MISMATCH)
Artifact consistency: Build verified · 1.0.0 (workspace artifacts agree)

Production attestation: Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (1 capability; 163 remain CURRENT)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
Evidence attested at: 2026-09-18T21:41:55.460Z
Last production check: 2026-09-18T21:41:55.460Z
Public / cache: —
Origin: —

Call this capability

Human page, REST, and MCP share the same engine and calculation_version. Build verified hashes workspace artifacts only. After deploy, attest:production GETs canonical URLs twice — the public/cached view an Agent actually hits, and origin — including every /capabilities/{id} HTML page. Public Capability HTML version must match catalog before production attestation passes. Live Trust rows above fetch /.well-known/calculatorx-production-attestation.json and the output schema const. Build hashes: release integrity. That proves currently published page / engine / schema / evidence / MCP agree — not that a historical version is independently immutable. Production surface contract 1/1.

Tool id: polynomial-roots
Engine: polynomial-roots
REST: POST /api/v1/calc/polynomial-roots
MCP: execute_calculation with capability_id: "math.polynomial.roots"
Agent input schema: Canonical (MCP / Agent)
Output schema: JSON Schema (latest)
Versioned input schema: Immutable pin (1.0.0)
Versioned output schema: Immutable pin (1.0.0)

References

Human-readable sources plus CVP reference evidence classes.

Sources

ISO 80000-2:2019 Quantities and units — Mathematics

Try with an agent

Inspect this capability via same-origin MCP, then call execute_calculation.

curl -sS -X POST 'https://www.calculatorx.com/mcp' \
  -H 'Content-Type: application/json' \
  -H 'MCP-Protocol-Version: 2026-07-28' \
  -H 'Mcp-Method: tools/call' \
  -H 'Mcp-Name: inspect_capability' \
  -d '{"jsonrpc":"2.0","id":1,"method":"tools/call","params":{"name":"inspect_capability","arguments":{"capability_id":"math.polynomial.roots"}}}'

Inspect the trust tier and required assurance level before execution. Pin calculation_version (1.0.0) when reproducibility matters. Full protocol notes: MCP docs.