{
  "id": "polynomial-division",
  "version": "59862e594dd4",
  "status": "published",
  "name": "Polynomial Division Calculator",
  "question": "What do I get dividing polynomials?",
  "summary": "Divides one polynomial by another by long division in exact fractions, showing every step, the quotient and the remainder, and synthetic division when the divisor is x − c.",
  "category": "math",
  "subcategory": "algebra",
  "url": "https://www.acalculator.org/math/polynomial-division-calculator",
  "markdown": "https://www.acalculator.org/math/polynomial-division-calculator.md",
  "kind": "function",
  "method": "Long division: divide the leading terms, multiply the divisor by that term, subtract, and repeat until the remainder’s degree is below the divisor’s.",
  "assumptions": [
    "Coefficients are exact fractions: 0.1 is 1/10.",
    "One letter, whole powers up to 50."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "p": {
        "title": "Dividend",
        "description": "The polynomial to divide, in one letter, for example 6x^3 + 11x^2 - 31x + 15.",
        "type": "string",
        "maxLength": 200
      },
      "q": {
        "title": "Divisor",
        "description": "The polynomial to divide by, in the same letter, not 0.",
        "type": "string",
        "maxLength": 200
      }
    }
  },
  "outputs": {
    "result": {
      "label": "Quotient",
      "description": "The quotient q(x) of the division.",
      "format": "math"
    },
    "remainder": {
      "label": "Remainder",
      "description": "What is left: 0, or a polynomial of lower degree than the divisor.",
      "format": "math"
    },
    "check": {
      "label": "Division algorithm",
      "description": "The dividend written as divisor × quotient + remainder.",
      "format": "text"
    },
    "synthetic": {
      "label": "Synthetic division",
      "description": "For a divisor x − c: c, then the bottom row of synthetic division (quotient coefficients, remainder).",
      "format": "text"
    },
    "steps": {
      "label": "Steps",
      "description": "Each step of the long division.",
      "format": "text"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "p": "6x^3 + 11x^2 - 31x + 15",
      "q": "3x - 2"
    },
    "outputs": {
      "result": "2x^2 + 5x - 7",
      "remainder": "1",
      "check": "6x^3 + 11x^2 - 31x + 15 = (3x - 2) × (2x^2 + 5x - 7) + 1",
      "steps": "Divide 6x^3 by 3x: 2x^2. Multiply: 2x^2 × (3x - 2) = 6x^3 - 4x^2. Subtract: 15x^2 - 31x + 15; Divide 15x^2 by 3x: 5x. Multiply: 5x × (3x - 2) = 15x^2 - 10x. Subtract: -21x + 15; Divide -21x by 3x: -7. Multiply: -7 × (3x - 2) = -21x + 14. Subtract: 1"
    },
    "text": "The quotient is 2x^2 + 5x - 7 with remainder 1."
  },
  "examples": [
    {
      "given": {
        "p": "6x^3 + 11x^2 - 31x + 15",
        "q": "3x - 2"
      },
      "expect": {
        "result": "2x^2 + 5x - 7",
        "remainder": "1",
        "check": "6x^3 + 11x^2 - 31x + 15 = (3x - 2) × (2x^2 + 5x - 7) + 1"
      },
      "source": "OpenStax, Algebra and Trigonometry 2e, §5.4 Dividing Polynomials: the Division Algorithm f(x) = d(x)q(x) + r(x) (https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-4-dividing-polynomials, retrieved 2026-10-02): (6x³ + 11x² − 31x + 15) ÷ (3x − 2) = 2x² + 5x − 7 remainder 1"
    },
    {
      "given": {
        "p": "5x^2 + 3x - 2",
        "q": "x + 1"
      },
      "expect": {
        "result": "5x - 2",
        "remainder": "0",
        "synthetic": "c = -1; bottom row: 5, -2, 0"
      },
      "source": "OpenStax, Algebra and Trigonometry 2e, §5.4 Dividing Polynomials: the Division Algorithm f(x) = d(x)q(x) + r(x) (https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-4-dividing-polynomials, retrieved 2026-10-02): (5x² + 3x − 2) ÷ (x + 1) = 5x − 2, remainder 0"
    },
    {
      "given": {
        "p": "x^3 - 1",
        "q": "2x^2 + 1"
      },
      "expect": {
        "result": "x/2",
        "remainder": "-x/2 - 1",
        "check": "x^3 - 1 = (2x^2 + 1) × x/2 - x/2 - 1"
      },
      "source": "hand calculation in content.mdx: x³ ÷ 2x² = x/2; x³ − 1 − (x³ + x/2) = −x/2 − 1; OpenStax, Algebra and Trigonometry 2e, §5.4 Dividing Polynomials: the Division Algorithm f(x) = d(x)q(x) + r(x) (https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-4-dividing-polynomials, retrieved 2026-10-02)"
    }
  ],
  "sources": [
    "OpenStax, Algebra and Trigonometry 2e, §5.4 Dividing Polynomials (the Division Algorithm; (5x² + 3x − 2) ÷ (x + 1) = 5x − 2; (6x³ + 11x² − 31x + 15) ÷ (3x − 2) = 2x² + 5x − 7 remainder 1), CC BY 4.0, retrieved 2026-10-02. https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-4-dividing-polynomials"
  ],
  "related": [
    "polynomial",
    "factoring",
    "partial-fraction",
    "remainder"
  ],
  "changelog": []
}