Polynomial division check

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

Check a claimed polynomial division A = BQ + R over exact rationals: remainder degree, quotient degree, coefficients.

Use when you need to: check a polynomial long division result · verify quotient and remainder of polynomial division · is this synthetic division correct.

Decide before calling

Read the versioned contract and the supported scope below. Reuse polynomial-division-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-division-check@1.0.0 for check a polynomial long division result. 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 polynomial long division result
• verify quotient and remainder of polynomial division
• is this synthetic division correct
• check that a polynomial divides another exactly
• validate remainder degree is less than divisor degree
• verify a = b times q plus r for polynomials

Not supported

• performing the division (supply the claimed quotient and remainder)
• division over finite fields or integers modulo m
• multivariate or pseudo-division conventions
• floating-point coefficients

Behavior

• Polynomials are canonical ascending coefficient arrays (entry k = coefficient of x^k, last entry nonzero, [] = zero) of exact rational strings (<= 40 characters), degree <= 64. Euclidean division over Q: A = BQ + R with R = 0 or deg R < deg B (Cohen 1993, s3.1.3; reference_version cohen-gtm138-1993-ch3).
• Divisor [] is not_computable. remainder may be omitted, which claims R = 0 (exact divisibility).
• Order: (1) R = 0 or deg R < deg B (remainder_degree_violation); (2) deg Q = deg A - deg B when deg A >= deg B, else Q = 0 (quotient_degree_violation, "-infinity" denotes the zero polynomial); (3) A = BQ + R compared from degree 0 upward (identity_mismatch, lowest disagreeing coefficient). first_discrepancy is {check, factor_index: null, degree, lhs, relation, rhs}.
• dividend_degree, divisor_degree, quotient_degree and remainder_degree are always returned (null for a zero or omitted polynomial).

Input

• dividend (array of string, required): max items 65; each max length 40; each pattern ^-?(0|[1-9][0-9]*)(/[1-9][0-9]*|\.[0-9]+)?$
• divisor (array of string, required): max items 65; each max length 40; each pattern ^-?(0|[1-9][0-9]*)(/[1-9][0-9]*|\.[0-9]+)?$
• quotient (array of string, required): max items 65; each max length 40; each pattern ^-?(0|[1-9][0-9]*)(/[1-9][0-9]*|\.[0-9]+)?$
• remainder (array of string, optional): max items 65; each max length 40; each pattern ^-?(0|[1-9][0-9]*)(/[1-9][0-9]*|\.[0-9]+)?$

Output

• valid (boolean, required)
• reason (one of "remainder_degree_violation", "quotient_degree_violation", "identity_mismatch", null, required)
• first_discrepancy (object or null, required)
• dividend_degree (integer or null, required): min 0
• divisor_degree (integer, required): min 0
• quotient_degree (integer or null, required): min 0
• remainder_degree (integer or null, required): min 0
• reference_version (constant "cohen-gtm138-1993-ch3", required)

Limits

• max degree: 64
• max number chars: 40

Example

Request input:

{
  "dividend": [
    "-4",
    "0",
    "-2",
    "1"
  ],
  "divisor": [
    "-3",
    "1"
  ],
  "quotient": [
    "3",
    "1",
    "1"
  ],
  "remainder": [
    "5"
  ]
}

Response:

{
  "result": {
    "valid": true,
    "reason": null,
    "first_discrepancy": null,
    "dividend_degree": 3,
    "divisor_degree": 1,
    "quotient_degree": 2,
    "remainder_degree": 0,
    "reference_version": "cohen-gtm138-1993-ch3"
  }
}

How to call it

MCP

Connect https://computefirst.net/mcp (setup), then call execute with:

{
  "id": "polynomial-division-check",
  "version": "1.0.0",
  "input": {
    "dividend": [
      "-4",
      "0",
      "-2",
      "1"
    ],
    "divisor": [
      "-3",
      "1"
    ],
    "quotient": [
      "3",
      "1",
      "1"
    ],
    "remainder": [
      "5"
    ]
  }
}

HTTP (no key)

curl -X POST https://computefirst.net/v1/tools/polynomial-division-check/versions/1.0.0/execute \
  -H "Content-Type: application/json" \
  -d '{"dividend":["-4","0","-2","1"],"divisor":["-3","1"],"quotient":["3","1","1"],"remainder":["5"]}'

The machine-readable contract is at /v1/tools/polynomial-division-check/versions/1.0.0.

CLI

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

Get the client at /clients/cli/.

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