> ## Documentation Index > Fetch the complete documentation index at: https://docs.iotools.cloud/llms.txt > Use this file to discover all available pages before exploring further. # Descartes' Rule of Signs Calculator > Find the possible number of positive and negative real roots of a polynomial using Descartes' Rule of Signs. Counts sign changes in f(x) and f(-x), lists every possible root count, and builds the full positive/negative/complex root-combination table with a step-by-step derivation. [Try Descartes' Rule of Signs Calculator in your browser →](https://iotools.cloud/tool/descartes-rule-of-signs-calculator/) ## OpenAPI ````yaml https://api.iotools.cloud/v1/openapi post /v1/tool/descartes-rule-of-signs-calculator openapi: 3.1.0 info: title: iotools.cloud API version: 1.0.0 description: >- Run any iotools.cloud tool over HTTP. Authenticate with `Authorization: Bearer iot_live_…`. **Only `POST /v1/tool/{slug}` costs credits.** Every GET here — the catalog, a tool's schema, your balance — is free. A tool call is charged its own weight or your plan's per-call minimum, whichever is larger; `x-iotools-credit-cost` on each operation is quoted at the free-tier minimum, and `GET /v1/tools/list` returns the exact figure for your key. `GET /v1/me/credits` reports what you have left, and `GET /v1/me/usage` reports what it went on. Failures are RFC 9457 problem documents — branch on `code`. servers: - url: https://api.iotools.cloud description: Production security: - bearerAuth: [] tags: - name: Catalog description: Find a tool and read its contract. Free. - name: Converters description: >- Convert between formats, encodings, and units — Base64, CSV and JSON, timestamps, and more. Fast, free, and processed right in your browser. - name: Formatters description: >- Format, beautify, minify, and validate code and data — JSON, HTML, CSS, SQL, and regex. Clean up messy input in one click, with nothing to upload. - name: Generators description: >- Generate exactly what you need and on demand — passwords, UUIDs, QR codes, hashes, random numbers, and more. Secure, instant, and free. - name: Calculators description: >- Crunch the numbers fast — from everyday math to specialized conversions and unit work. Free online calculators that run entirely in your browser. - name: Editors description: >- Edit and transform text, code, and images with quick, focused editors that run entirely in your browser — nothing to install, and no sign-up needed. - name: Utilities description: >- Everyday developer and web utilities — DNS and IP lookups, redirect and certificate checkers, and other quick diagnostics. Free and privacy-friendly. - name: Account description: Your key's allowance, limits and usage history. paths: /v1/tool/descartes-rule-of-signs-calculator: post: tags: - Calculators summary: Descartes' Rule of Signs Calculator description: >- Find the possible number of positive and negative real roots of a polynomial using Descartes' Rule of Signs. Counts sign changes in f(x) and f(-x), lists every possible root count, and builds the full positive/negative/complex root-combination table with a step-by-step derivation. [Try Descartes' Rule of Signs Calculator in your browser →](https://iotools.cloud/tool/descartes-rule-of-signs-calculator/) operationId: run_descartes_rule_of_signs_calculator requestBody: required: true content: application/json: schema: type: object properties: coefficients: type: string description: Coefficients required: - coefficients examples: x_4_3x_3_2x_2_5x_1: summary: x^4 - 3x^3 + 2x^2 + 5x - 1 value: coefficients: 1,-3,2,5,-1 x_3_1_zero_coefficients_skipped: summary: x^3 - 1 (zero coefficients skipped) value: coefficients: 1,0,0,-1 x_2_1_no_real_roots: summary: x^2 + 1 (no real roots) value: coefficients: 1,0,1 responses: '200': description: Tool output content: application/json: examples: x_4_3x_3_2x_2_5x_1: summary: x^4 - 3x^3 + 2x^2 + 5x - 1 value: tool: descartes-rule-of-signs-calculator tool_version: 1.0.2 outputs: degree: '4' positiveSignChanges: '3' negativeSignChanges: '1' possiblePositiveRoots: 3, 1 possibleNegativeRoots: '1' rootCombinations: - positive: '3' negative: '1' complex: '0' - positive: '1' negative: '1' complex: '2' steps: >- f(x) = x^4 - 3x^3 + 2x^2 + 5x - 1 (degree 4) Coefficients of f(x), zeros skipped: 1, -3, 2, 5, -1 → 3 sign changes. f(-x) = x^4 + 3x^3 + 2x^2 - 5x - 1 Coefficients of f(-x), zeros skipped: 1, 3, 2, -5, -1 → 1 sign change. Possible positive real roots: 3, 1 (start at 3 sign changes, subtract 2 each step). Possible negative real roots: 1 (start at 1 sign changes, subtract 2 each step). Every (positive, negative) pair from those lists whose sum ≤ degree fills the rest with complex-conjugate pairs (degree − positive − negative): 2 valid combinations. credits_used: 5 credits_remaining: null x_3_1_zero_coefficients_skipped: summary: x^3 - 1 (zero coefficients skipped) value: tool: descartes-rule-of-signs-calculator tool_version: 1.0.2 outputs: degree: '3' positiveSignChanges: '1' negativeSignChanges: '0' possiblePositiveRoots: '1' possibleNegativeRoots: '0' rootCombinations: - positive: '1' negative: '0' complex: '2' steps: >- f(x) = x^3 - 1 (degree 3) Coefficients of f(x), zeros skipped: 1, -1 → 1 sign change. f(-x) = -x^3 - 1 Coefficients of f(-x), zeros skipped: -1, -1 → 0 sign changes. Possible positive real roots: 1 (start at 1 sign changes, subtract 2 each step). Possible negative real roots: 0 (start at 0 sign changes, subtract 2 each step). Every (positive, negative) pair from those lists whose sum ≤ degree fills the rest with complex-conjugate pairs (degree − positive − negative): 1 valid combination. credits_used: 5 credits_remaining: null x_2_1_no_real_roots: summary: x^2 + 1 (no real roots) value: tool: descartes-rule-of-signs-calculator tool_version: 1.0.2 outputs: degree: '2' positiveSignChanges: '0' negativeSignChanges: '0' possiblePositiveRoots: '0' possibleNegativeRoots: '0' rootCombinations: - positive: '0' negative: '0' complex: '2' steps: >- f(x) = x^2 + 1 (degree 2) Coefficients of f(x), zeros skipped: 1, 1 → 0 sign changes. f(-x) = x^2 + 1 Coefficients of f(-x), zeros skipped: 1, 1 → 0 sign changes. Possible positive real roots: 0 (start at 0 sign changes, subtract 2 each step). Possible negative real roots: 0 (start at 0 sign changes, subtract 2 each step). Every (positive, negative) pair from those lists whose sum ≤ degree fills the rest with complex-conjugate pairs (degree − positive − negative): 1 valid combination. credits_used: 5 credits_remaining: null schema: type: object required: - tool - tool_version - outputs - credits_used - credits_remaining properties: outputs: type: object properties: degree: type: string description: Degree positiveSignChanges: type: string description: Sign Changes in f(x) negativeSignChanges: type: string description: Sign Changes in f(-x) possiblePositiveRoots: type: string description: Possible Positive Real Roots possibleNegativeRoots: type: string description: Possible Negative Real Roots rootCombinations: type: array description: Root Combinations items: type: object properties: positive: type: string description: Positive negative: type: string description: Negative complex: type: string description: Complex x-iotools-columns: - positive - negative - complex steps: type: string description: Step-by-Step Derivation tool: type: string description: The tool's slug, echoing the {slug} in the request path. tool_version: type: string description: Output-contract version for this tool. credits_used: type: integer description: >- Credits this call consumed, after any settlement refund. 0 when metering is disabled. credits_remaining: type: - integer - 'null' description: >- Credits left in the current monthly allowance, or null when metering is disabled. request_id: type: string description: Correlation id, also sent as x-request-id. '400': $ref: '#/components/responses/ValidationError' '401': $ref: '#/components/responses/Unauthorized' '402': $ref: '#/components/responses/InsufficientCredits' '403': $ref: '#/components/responses/ToolNotAllowed' '404': $ref: '#/components/responses/ToolNotFound' '413': $ref: '#/components/responses/PayloadTooLarge' '422': $ref: '#/components/responses/UnprocessableInput' '429': $ref: '#/components/responses/RateLimited' '500': $ref: '#/components/responses/ToolFailed' '503': $ref: '#/components/responses/ToolDisabled' security: - bearerAuth: [] components: responses: ValidationError: description: Invalid request. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/validation_error title: type: string description: Short human-readable summary of the failure. examples: - Invalid request status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - validation_error detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. fields: type: - object - string additionalProperties: type: string description: >- What failed: a map of field name → message. Absent when the body itself is malformed; a single string when the failure isn't tied to one field. example: type: https://docs.iotools.cloud/errors/validation_error title: Invalid request status: 400 code: validation_error detail: One or more inputs are invalid — see `fields`. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 fields: inputString: Required Unauthorized: description: Missing or invalid API key. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/invalid_api_key title: type: string description: Short human-readable summary of the failure. examples: - Invalid API key status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - invalid_api_key detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. example: type: https://docs.iotools.cloud/errors/invalid_api_key title: Invalid API key status: 401 code: invalid_api_key detail: 'Provide ''Authorization: Bearer ''.' request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 InsufficientCredits: description: Monthly credit allowance exhausted. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/insufficient_credits - >- https://docs.iotools.cloud/errors/vendor_api_tier_limit_exceeded title: type: string description: Short human-readable summary of the failure. examples: - Insufficient credits - Free-tier API limit reached for this tool status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - insufficient_credits - vendor_api_tier_limit_exceeded detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. credits_used: type: integer description: Always 0 — a refused call charges nothing. credits_remaining: type: integer description: Credits left in the allowance — fewer than this call costs. example: type: https://docs.iotools.cloud/errors/insufficient_credits title: Insufficient credits status: 402 code: insufficient_credits detail: This call costs 1 credit and 0 remain in this month's allowance. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 credits_used: 0 credits_remaining: 0 ToolNotAllowed: description: Tool exists but has no API surface. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/tool_not_allowed title: type: string description: Short human-readable summary of the failure. examples: - Tool not available over the API status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - tool_not_allowed detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. example: type: https://docs.iotools.cloud/errors/tool_not_allowed title: Tool not available over the API status: 403 code: tool_not_allowed detail: >- "Background Remover" is available on iotools.cloud but has no API endpoint. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 ToolNotFound: description: No such tool. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/tool_not_found title: type: string description: Short human-readable summary of the failure. examples: - Tool not found status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - tool_not_found detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. example: type: https://docs.iotools.cloud/errors/tool_not_found title: Tool not found status: 404 code: tool_not_found detail: No tool with that slug. See GET /v1/tools/list. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 PayloadTooLarge: description: Body too large. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/payload_too_large title: type: string description: Short human-readable summary of the failure. examples: - Payload too large status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - payload_too_large detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. example: type: https://docs.iotools.cloud/errors/payload_too_large title: Payload too large status: 413 code: payload_too_large detail: Request body exceeds this tool's size limit. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 UnprocessableInput: description: Well-formed request, but there is no result to return. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/unprocessable_input title: type: string description: Short human-readable summary of the failure. examples: - Input cannot be processed status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - unprocessable_input detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. credits_used: type: integer description: >- Always 0 on `unprocessable_input` — a failed run is refunded, floor included. credits_remaining: type: integer description: >- Credits left after the refund. Absent when metering is disabled. example: type: https://docs.iotools.cloud/errors/unprocessable_input title: Input cannot be processed status: 422 code: unprocessable_input detail: >- This video has no captions or subtitles, so there is no transcript to extract. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 credits_used: 0 RateLimited: description: Per-minute rate limit exceeded. headers: Retry-After: description: Seconds to wait before retrying (RFC 9110 delta-seconds). schema: type: integer content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/rate_limited title: type: string description: Short human-readable summary of the failure. examples: - Rate limit exceeded status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - rate_limited detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. retry_after: type: integer description: >- Seconds until the window resets — the same value as the `Retry-After` header. example: type: https://docs.iotools.cloud/errors/rate_limited title: Rate limit exceeded status: 429 code: rate_limited detail: Too many requests. Retry in 30s. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 retry_after: 30 ToolFailed: description: Tool failed to run. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/processing_error - https://docs.iotools.cloud/errors/internal_error title: type: string description: Short human-readable summary of the failure. examples: - Tool failed to run - Internal error status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - processing_error - internal_error detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. credits_used: type: integer description: >- Always 0 on `processing_error` — a failed run is refunded, floor included. credits_remaining: type: integer description: >- Credits left after the refund. Absent when metering is disabled. example: type: https://docs.iotools.cloud/errors/processing_error title: Tool failed to run status: 500 code: processing_error detail: The tool failed to run. Please try again. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 credits_used: 0 ToolDisabled: description: Tool temporarily disabled. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/tool_disabled - https://docs.iotools.cloud/errors/api_unconfigured title: type: string description: Short human-readable summary of the failure. examples: - Tool temporarily disabled - API not configured status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - tool_disabled - api_unconfigured detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. example: type: https://docs.iotools.cloud/errors/tool_disabled title: Tool temporarily disabled status: 503 code: tool_disabled detail: This tool is temporarily unavailable. Try again shortly. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 securitySchemes: bearerAuth: type: http scheme: bearer bearerFormat: iot_live_… ````