# Polynomial factorization check

`polynomial-factorization-check` · version 1.0.0 · Numbers & math · free, no key needed

Check a claimed factorization c * prod f_i^e_i of a rational polynomial: monic distinct factors, degrees, coefficients.

**Use when you need to: check a factorization of a polynomial · verify polynomial factors multiply back to the original · is this polynomial factored correctly over the rationals.**

## Decide before calling

Read the [versioned contract](/v1/tools/polynomial-factorization-check/versions/1.0.0) and the supported scope below. Reuse `polynomial-factorization-check@1.0.0` when your input, required output and limits match it. Choose another approach for an unsupported operation.

## Explain the choice

"I can use `polynomial-factorization-check@1.0.0` for check a factorization of a polynomial. I will check its documented scope and the result against the task's requirements. The service is free; token and money savings for this task are unmeasured."

## Supported

- check a factorization of a polynomial
- verify polynomial factors multiply back to the original
- is this polynomial factored correctly over the rationals
- check factors are monic and the leading constant is right
- validate repeated factors and multiplicities of a polynomial
- verify a quadratic factor is irreducible over q

## Not supported

- factoring a polynomial (supply the claimed factorization)
- proving irreducibility of factors of degree 3 or more (accepted, reported as irreducibility_verified false)
- factorization over finite fields, Z[x] content conventions, or multivariate polynomials
- floating-point coefficients or approximate roots

## Behavior

- Polynomials are canonical ascending coefficient arrays (entry k = coefficient of x^k, last entry nonzero) of exact rational strings (<= 40 characters), degree <= 64. Claim: f = leading_coefficient * prod factors[i].coefficients ^ multiplicity with distinct monic non-constant factors (Cohen 1993, ch. 3; reference_version cohen-gtm138-1993-ch3). f = [] is not_computable.
- Order: per factor i ascending: degree >= 1 (factor_degree_violation), monic (factor_not_monic), distinct from earlier factors (duplicate_factor); then leading_coefficient = lc(f) (leading_coefficient_mismatch); then sum of multiplicity * degree = deg f (degree_mismatch); then coefficients from degree 0 upward (coefficient_mismatch); then each quadratic factor must have a discriminant that is not a rational square (factor_reducible, rhs = the square-root witness).
- first_discrepancy is {check, factor_index, degree, lhs, relation, rhs}. degree = deg f is always returned; irreducibility_verified is true only when valid and every factor has degree <= 2 (degree 1 always irreducible, degree 2 by the exact discriminant test).

## Input

- `polynomial` (array of string, required): max items 65; each max length 40; each pattern `^-?(0|[1-9][0-9]*)(/[1-9][0-9]*|\.[0-9]+)?$`
- `leading_coefficient` (string, required): max length 40; pattern `^-?(0|[1-9][0-9]*)(/[1-9][0-9]*|\.[0-9]+)?$`
- `factors` (array of object, required): max items 16

## Output

- `valid` (boolean, required)
- `reason` (one of "factor_degree_violation", "factor_not_monic", "duplicate_factor", "leading_coefficient_mismatch", "degree_mismatch", "coefficient_mismatch", "factor_reducible", null, required)
- `first_discrepancy` (object or null, required)
- `degree` (integer, required): min 0
- `irreducibility_verified` (boolean, required)
- `reference_version` (constant "cohen-gtm138-1993-ch3", required)

## Limits

- max degree: 64
- max factors: 16
- max multiplicity: 64
- max number chars: 40

## Example

Request input:

```json
{
  "polynomial": [
    "-2",
    "0",
    "2"
  ],
  "leading_coefficient": "2",
  "factors": [
    {
      "coefficients": [
        "-1",
        "1"
      ],
      "multiplicity": 1
    },
    {
      "coefficients": [
        "1",
        "1"
      ],
      "multiplicity": 1
    }
  ]
}
```

Response:

```json
{
  "result": {
    "valid": true,
    "reason": null,
    "first_discrepancy": null,
    "degree": 2,
    "irreducibility_verified": true,
    "reference_version": "cohen-gtm138-1993-ch3"
  }
}
```

## How to call it

### MCP

Connect `https://computefirst.net/mcp` ([setup](/docs#connect)), then call `execute` with:

```json
{
  "id": "polynomial-factorization-check",
  "version": "1.0.0",
  "input": {
    "polynomial": [
      "-2",
      "0",
      "2"
    ],
    "leading_coefficient": "2",
    "factors": [
      {
        "coefficients": [
          "-1",
          "1"
        ],
        "multiplicity": 1
      },
      {
        "coefficients": [
          "1",
          "1"
        ],
        "multiplicity": 1
      }
    ]
  }
}
```

### HTTP (no key)

```sh
curl -X POST https://computefirst.net/v1/tools/polynomial-factorization-check/versions/1.0.0/execute \
  -H "Content-Type: application/json" \
  -d '{"polynomial":["-2","0","2"],"leading_coefficient":"2","factors":[{"coefficients":["-1","1"],"multiplicity":1},{"coefficients":["1","1"],"multiplicity":1}]}'
```

The machine-readable contract is at [/v1/tools/polynomial-factorization-check/versions/1.0.0](/v1/tools/polynomial-factorization-check/versions/1.0.0).

### CLI

```sh
node cli.mjs run polynomial-factorization-check 1.0.0 --input input.json --base-url https://computefirst.net
```

Get the client at [/clients/cli/](/clients/cli/).

## Related tools

- [Polynomial division check](/tools/polynomial-division-check): Check a claimed polynomial division A = BQ + R over exact rationals: remainder degree, quotient degree, coefficients.
- [Contingency expected check](/tools/contingency-expected-check): Check claimed expected counts (row total x column total / N) of a count table exactly; returns the first wrong cell.
- [Lp primal feasibility check](/tools/lp-primal-feasibility-check): Check that a claimed LP point satisfies x >= 0 and Ax <= b exactly; report the first violated bound or row.
- [Polynomial identity check](/tools/polynomial-identity-check): Check a polynomial identity over exact rationals by expanding both sides; report the lowest disagreeing coefficient.
- [Regression normal equations check](/tools/regression-normal-equations-check): Check that claimed coefficients and residuals satisfy X^T e = 0 exactly; rejects singular, ambiguous designs.
- [Chi square decomposition check](/tools/chi-square-decomposition-check): Check a claimed Pearson chi-square, its per-cell (O-E)^2/E terms and df exactly; returns the first wrong term.