{
  "id": "synthetic-division",
  "version": "ae209fa70b1c",
  "status": "published",
  "name": "Synthetic Division Calculator",
  "question": "How do I do synthetic division?",
  "summary": "Divides a polynomial by a linear divisor such as x − 3 or 2x + 1 with synthetic division, in exact fractions: the quotient, the remainder, p(k), and every bring-down, multiply and add step.",
  "category": "math",
  "subcategory": "algebra",
  "url": "https://www.acalculator.org/math/synthetic-division-calculator",
  "markdown": "https://www.acalculator.org/math/synthetic-division-calculator.md",
  "kind": "function",
  "method": "For a divisor ax + b, k = −b ÷ a. Bring down the first coefficient; then multiply by k and add to the next coefficient, to the end. The last number is the remainder; the others, divided by a, are the quotient.",
  "assumptions": [
    "Coefficients are exact fractions: 0.5 is 1/2.",
    "The divisor has degree 1; the dividend has degree 1 to 30, in the same letter."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "p": {
        "title": "Dividend",
        "description": "The polynomial to divide, in one letter.",
        "type": "string",
        "maxLength": 200
      },
      "d": {
        "title": "Divisor",
        "description": "A divisor of degree 1, such as x - 3, x + 2 or 2x - 1, in the same letter.",
        "type": "string",
        "maxLength": 60
      }
    }
  },
  "outputs": {
    "quotient": {
      "label": "Quotient",
      "description": "The dividend ÷ the divisor, without the remainder.",
      "format": "math"
    },
    "remainder": {
      "label": "Remainder",
      "description": "What is left over: p(k), a number.",
      "format": "math"
    },
    "k": {
      "label": "k",
      "description": "The number synthetic division uses: the zero of the divisor.",
      "format": "text"
    },
    "value": {
      "label": "Remainder theorem",
      "description": "The dividend’s value at k, which equals the remainder.",
      "format": "text"
    },
    "row": {
      "label": "Bottom row",
      "description": "The numbers under the line, the last one the remainder.",
      "format": "text"
    },
    "steps": {
      "label": "Steps",
      "description": "Every step of the synthetic division.",
      "format": "text"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "p": "5x^2 - 3x - 36",
      "d": "x - 3"
    },
    "outputs": {
      "quotient": "5x + 12",
      "remainder": "0",
      "k": "3",
      "value": "p(3) = 0",
      "row": "5, 12 | 0",
      "steps": "The divisor x - 3 is 0 at x = 3, so k = 3; Write the coefficients of the dividend, highest power first, with 0 for a missing power: 5, -3, -36; Bring down 5; Multiply 3 × 5 = 15, then add -3 + 15 = 12; Multiply 3 × 12 = 36, then add -36 + 36 = 0; The last number, 0, is the remainder. The quotient is 5x + 12"
    },
    "text": "(5x^2 - 3x - 36) ÷ (x - 3) = 5x + 12, remainder 0."
  },
  "examples": [
    {
      "given": {
        "p": "5x^2 - 3x - 36",
        "d": "x - 3"
      },
      "expect": {
        "quotient": "5x + 12",
        "remainder": "0",
        "k": "3",
        "row": "5, 12 | 0",
        "value": "p(3) = 0"
      },
      "source": "OpenStax, Algebra and Trigonometry 2e, §5.4 Dividing Polynomials, https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-4-dividing-polynomials, Example 3 Using Synthetic Division to Divide a Second-Degree Polynomial: 5x + 12, remainder 0"
    },
    {
      "given": {
        "p": "4x^3 + 10x^2 - 6x - 20",
        "d": "x + 2"
      },
      "expect": {
        "quotient": "4x^2 + 2x - 10",
        "remainder": "0",
        "k": "-2"
      },
      "source": "OpenStax, Algebra and Trigonometry 2e, §5.4 Dividing Polynomials, https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-4-dividing-polynomials, Example 4 Using Synthetic Division to Divide a Third-Degree Polynomial: 4x^2 + 2x - 10"
    },
    {
      "given": {
        "p": "-9x^4 + 10x^3 + 7x^2 - 6",
        "d": "x - 1"
      },
      "expect": {
        "quotient": "-9x^3 + x^2 + 8x + 8",
        "remainder": "2",
        "row": "-9, 1, 8, 8 | 2"
      },
      "source": "OpenStax, Algebra and Trigonometry 2e, §5.4 Dividing Polynomials, https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-4-dividing-polynomials, Example 5 Using Synthetic Division to Divide a Fourth-Degree Polynomial: −9x^3 + x^2 + 8x + 8, remainder 2"
    },
    {
      "given": {
        "p": "6x^3 + 11x^2 - 31x + 15",
        "d": "3x - 2"
      },
      "expect": {
        "quotient": "2x^2 + 5x - 7",
        "remainder": "1",
        "k": "2/3",
        "row": "6, 15, -21 | 1"
      },
      "source": "OpenStax, Algebra and Trigonometry 2e, §5.4 Dividing Polynomials, https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-4-dividing-polynomials, Example 2 (by long division): 2x^2 + 5x - 7, remainder 1; hand calculation in content.mdx"
    }
  ],
  "sources": [
    "OpenStax, Algebra and Trigonometry 2e, section 5.4 Dividing Polynomials: synthetic division, Examples 3, 4 and 5 (5x² − 3x − 36 ÷ (x − 3) = 5x + 12; 4x³ + 10x² − 6x − 20 ÷ (x + 2) = 4x² + 2x − 10; −9x⁴ + 10x³ + 7x² − 6 ÷ (x − 1) = −9x³ + x² + 8x + 8, remainder 2) and Example 2. CC BY 4.0, retrieved 2026-10-02. https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-4-dividing-polynomials"
  ],
  "related": [
    "polynomial",
    "factoring",
    "equation-solver"
  ],
  "changelog": []
}