{
  "id": "partial-fraction",
  "version": "22742e792e29",
  "status": "published",
  "name": "Partial Fraction Decomposition Calculator",
  "question": "Partial fraction decomposition solver",
  "summary": "Splits a ratio of polynomials into partial fractions, checking their sum.",
  "category": "math",
  "subcategory": "algebra",
  "url": "https://www.acalculator.org/math/partial-fraction-calculator",
  "markdown": "https://www.acalculator.org/math/partial-fraction-calculator.md",
  "kind": "function",
  "method": "A computer algebra system finds the partial fractions; they show only when their sum equals the fraction at 20 points.",
  "assumptions": [
    "The denominator is factored over the rationals.",
    "An answer that fails its check is not shown."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "f": {
        "title": "Rational function",
        "description": "A ratio of two polynomials in one letter.",
        "type": "string",
        "maxLength": 200
      }
    }
  },
  "outputs": {
    "decomposition": {
      "label": "Partial fractions",
      "description": "The same function as a sum of simpler fractions.",
      "format": "math"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "f": "(3x + 2)/(x^3 - x^2 - 2x)"
    },
    "outputs": {
      "decomposition": "4/(3 (x - 2)) - 1/(3 (x + 1)) - 1/x"
    },
    "text": "The partial fraction decomposition of (3x + 2)/(x^3 - x^2 - 2x) is 4/(3 (x - 2)) - 1/(3 (x + 1)) - 1/x."
  },
  "examples": [
    {
      "given": {
        "f": "(3x + 2)/(x^3 - x^2 - 2x)"
      },
      "expect": {
        "decomposition": "4/(3 (x - 2)) - 1/(3 (x + 1)) - 1/x"
      },
      "source": "OpenStax Calculus Vol. 2, 3.4, Ex. 3.29. https://openstax.org/books/calculus-volume-2/pages/3-4-partial-fractions"
    },
    {
      "given": {
        "f": "(x^2 + 3x + 1)/(x^2 - 4)"
      },
      "expect": {
        "decomposition": "1 + 1/(4 (x + 2)) + 11/(4 (x - 2))"
      },
      "source": "OpenStax Calculus Vol. 2, 3.4, Ex. 3.30"
    },
    {
      "given": {
        "f": "(2x - 3)/(x (x^2 + 1))"
      },
      "expect": {
        "decomposition": "(3x + 2)/(x^2 + 1) - 3/x"
      },
      "source": "OpenStax Calculus Vol. 2, 3.4, Ex. 3.33"
    }
  ],
  "sources": [
    "OpenStax, Calculus Volume 2, section 3.4 Partial Fractions: https://openstax.org/books/calculus-volume-2/pages/3-4-partial-fractions",
    "OpenStax, College Algebra 2e, section 7.6 Partial Fractions: https://openstax.org/books/college-algebra-2e/pages/7-6-partial-fractions"
  ],
  "related": [
    "factoring",
    "integral",
    "simplify",
    "calculus"
  ],
  "changelog": []
}