{"uid":"cap_Z_A8x_r9yAfVeuv59xEOP","slug":"saylor-innovations-data-desk-polynomial-root-finder-b106f373","name":"Saylor Innovations Data Desk – Polynomial Root Finder","description":"Find the real and complex roots of a polynomial up to degree 6, with discriminant and vertex for quadratics and a readable form of the equation","url":"https://saylorinnovations.com/api/math/polynomial?utm_source=zero.xyz","method":"GET","headers":{},"bodySchema":{"type":"object","$schema":"https://json-schema.org/draft/2020-12/schema","required":["input"],"properties":{"input":{"type":"object","required":["type","method"],"properties":{"type":{"type":"string","const":"http"},"method":{"enum":["GET"],"type":"string"},"queryParams":{"type":"object","required":["coefficients"],"properties":{"coefficients":{"type":"string","description":"Comma-separated coefficients from highest power down, degree 1-6, e.g. 1,-3,2 for x^2 - 3x + 2"}}}},"additionalProperties":false},"output":{"type":"object","required":["type"],"properties":{"type":{"type":"string","const":"json"},"example":{"type":"object","required":["degree","roots"],"properties":{"roots":{"type":"array","items":{"type":"object","properties":{"real":{"type":["number","null"],"description":"Real part"},"text":{"description":"Text content"},"is_real":{"description":"True when this is real"},"imaginary":{"type":["number","null"],"description":"Imaginary part"}}},"description":"Roots with real part, imaginary part and a text form"},"degree":{"type":"number","description":"Polynomial degree"},"source":{"type":"string","description":"Data source the values come from"},"polynomial":{"type":"string","description":"Polynomial written as text"},"real_roots":{"type":"number","description":"Real roots"},"coefficients":{"type":"array","description":"Coefficients from the highest power down"},"complex_roots":{"type":"number","description":"Complex roots"}}}}}}},"responseSchema":null,"example":null,"exampleRequest":null,"tags":["x402"],"displayCostAmount":"0.01","displayCostAsset":"USDC","priceDynamic":false,"priceHint":null,"priceStatus":"priced","priceSource":"probe","requiresHandshake":false,"reviewCount":0,"rating":{"score":"0.00","successRate":"0.00","reviews":0,"stars":null,"state":"unrated"},"availabilityStatus":"unknown","priceObserved":null,"sessionDeposit":null,"pricing":{"kind":"static","summary":"$0.01/call","primary":{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.01","per":"call","confidence":"exact"},"accepted":[{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.01","per":"call","confidence":"exact"}]},"paymentMethods":[{"uid":"pm_cL-yyoJTULLPHgk_craEo","protocol":"x402","methodType":"crypto","chain":"base","mode":"charge","costAmount":"0.01","costPer":"request","priority":0,"asset":"0x833589fCD6eDb6E08f4c7C32D4f71b54bdA02913","unit":"request","depositMicros":null,"planRef":null}],"brandName":null,"brandSlug":null,"brandBaseUrl":null,"brandDocsUrl":null,"whatItDoes":"Computes real and complex roots of a polynomial (degree 1–6) from its coefficients, with discriminant and vertex for quadratics and a human-readable equation string","exampleAgentPrompt":"Can you find all the roots of the polynomial x^2 - 3x + 2 — I need both the real and any complex roots, plus the vertex and discriminant if it's a quadratic?","exampleUseCases":[{"title":"Solve a quadratic for classroom use","prompt":"I need the roots, discriminant, and vertex of the quadratic 2x^2 + 4x - 6 — can you solve it and show me the equation in readable form?"},{"title":"Find complex roots of a cubic","prompt":"What are all the roots, including any complex ones, of the cubic polynomial x^3 - 6x^2 + 11x - 6?"},{"title":"Factor a sixth-degree polynomial","prompt":"Can you find the real and complex zeros of the degree-6 polynomial with coefficients 1, -2, 0, 3, -1, 0, 5 — highest power first?"}],"resultDescription":"Returns the polynomial degree, a readable text form of the equation, the input coefficients array, all roots as objects with real part, imaginary part, an is_real flag and a text label, plus counts of real and complex roots. 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It is particularly useful in educational tools, symbolic math pipelines, or engineering workflows where coefficient arrays are already known and formatted.","instructions":null,"reviewSummary":null,"reviewSummaryHighlights":null,"reviewSummaryConcerns":null,"reviewSummaryGeneratedAt":null,"activationCount":0,"lastUsedAt":null,"lastSuccessfullyRanAt":null,"lastHealthCheckAt":"2026-10-18T12:56:37.534Z","isFirstParty":false,"canonicalSlug":"saylor-innovations-data-desk-polynomial-root-finder-b106f373"}