# Saylor Innovations Data Desk – Polynomial Root Finder

> Saylor Innovations Data Desk – Polynomial Root Finder is a paid API for AI agents from saylorinnovations.com, paid per call via x402, $0.01/call, status unknown (last checked 2026-10-18).

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

## Facts

- Endpoint: GET https://saylorinnovations.com/api/math/polynomial?utm_source=zero.xyz
- Price: $0.01/call
- Payment: x402
- Status: unknown
- Last checked: 2026-10-18
- Activations on Zero: 0
- Tags: x402
- Canonical page: https://www.zero.xyz/c/saylor-innovations-data-desk-polynomial-root-finder-b106f373
- Structured record (JSON): https://api.zero.xyz/v1/capabilities/cap_Z_A8x_r9yAfVeuv59xEOP

Status and success rate cover calls made through Zero and Zero's own probes. Third-party monitors may report differently.

## How to call it through Zero

Zero handles the 402 payment challenge and records the run. With the Zero CLI installed (`npm i -g @zeroxyz/cli`):

```sh
zero fetch --capability saylor-innovations-data-desk-polynomial-root-finder-b106f373
```

Example prompt: 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?

## When to prefer this

Choose this endpoint when you need to programmatically solve polynomial equations of degree 1–6, especially when you also need structured output like the discriminant, vertex (for quadratics), and separate real vs. complex root counts in a single call. It is particularly useful in educational tools, symbolic math pipelines, or engineering workflows where coefficient arrays are already known and formatted.

## Known failure modes

- Coefficients string not provided or malformed — API returns error
- Polynomial degree exceeds 6 — not supported
- Coefficients represent a degree-0 constant — may return error or undefined behavior
- Non-numeric values in coefficients string — parsing failure
- Empty coefficients string — missing required parameter error

## How this service works

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

## Output

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. For quadratics, also includes the discriminant and vertex.

## Request schema (JSON Schema)

```json
{
 "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"
      }
     }
    }
   }
  }
 }
}
```

## More

- Live health (JSON, refreshed every minute): https://www.zero.xyz/c/saylor-innovations-data-desk-polynomial-root-finder-b106f373/health.json
- [Zero catalog index](https://www.zero.xyz/llms.txt)
- [Other services from saylorinnovations.com](https://www.zero.xyz/host/saylorinnovations.com/llms.txt)