curl --request POST \
  --url https://api.iotools.cloud/v1/tool/partial-fraction-decomposition-calculator \
  --header 'Authorization: Bearer <token>' \
  --header 'Content-Type: application/json' \
  --data '
{
  "numerator": "3x + 5",
  "denominator": "(x - 1)(x + 2)"
}
'

{
  "tool": "partial-fraction-decomposition-calculator",
  "tool_version": "1.0.2",
  "outputs": {
    "decomposition": "(1/3)/(x + 2) + (8/3)/(x - 1)",
    "polynomialPart": "None — this is already a proper fraction (degree of the numerator is less than the denominator's).",
    "terms": [
      {
        "denominator-factor": "(x + 2)",
        "numerator": "1/3",
        "term": "(1/3)/(x + 2)"
      },
      {
        "denominator-factor": "(x - 1)",
        "numerator": "8/3",
        "term": "(8/3)/(x - 1)"
      }
    ],
    "steps": "Numerator: P(x) = 3x + 5\nDenominator (as entered): Q(x) = (x - 1)(x + 2)\nExpanded: Q(x) = x^2 + x - 2\ndeg(P) = 1 < deg(Q) = 2, so P(x)/Q(x) is already a proper fraction — no polynomial part.\nPartial fraction form: R(x)/Q(x) = A1/(x + 2) + A2/(x - 1)\nSolving the resulting linear system (comparing coefficients of each power of x):\n  A1 = 1/3\n  A2 = 8/3\nCover-up (Heaviside) cross-check — cover each (x - r) and evaluate the rest at x = r:\n  A1 = R(-2) / [Q(x)/(x + 2)](-2) = -1 / -3 = 1/3\n  A2 = R(1) / [Q(x)/(x - 1)](1) = 8 / 3 = 8/3\nResult: P(x)/Q(x) = (1/3)/(x + 2) + (8/3)/(x - 1)"
  },
  "credits_used": 5,
  "credits_remaining": null
}