{
    "name": "Polynomial Long Division Calculator",
    "url": "https://calculators.geekmidst.com/calculators/polynomial-long-division-calculator",
    "markdown": "https://calculators.geekmidst.com/calculators/polynomial-long-division-calculator.md",
    "category": "Algebra, geometry & advanced math",
    "description": "Divide one polynomial by another and get the exact quotient and remainder, with each long-division step and fractional coefficients kept exact.",
    "summary": "Enter the dividend and the divisor. Each step divides the leading terms, multiplies back and subtracts, and fractions stay exact (1/2, not 0.5).",
    "also_called": [
        "Polynomial division calculator",
        "Algebraic long division calculator"
    ],
    "applies_to": "Worldwide",
    "last_updated": "2026-10-29",
    "computation": "Results are computed in the visitor’s browser on the HTML page. This API describes the calculator; it does not compute results.",
    "inputs": [
        {
            "name": "dividend",
            "label": "Dividend P(x)",
            "type": "text",
            "default": "x^3 - 2x^2 - 4"
        },
        {
            "name": "divisor",
            "label": "Divisor D(x)",
            "type": "text",
            "default": "x - 3"
        }
    ],
    "outputs": [
        {
            "key": "result",
            "label": "P(x) ÷ D(x)",
            "format": "text"
        },
        {
            "key": "quotient",
            "label": "Quotient Q(x)",
            "format": "text"
        },
        {
            "key": "remainder",
            "label": "Remainder R(x)",
            "format": "text"
        },
        {
            "key": "divides",
            "label": "Does D(x) divide P(x)?",
            "format": "text"
        }
    ],
    "formula": {
        "text": "P(x) = D(x)·Q(x) + R(x), with deg R < deg D;  each step: next quotient term = (leading term of remainder) ÷ (leading term of D)",
        "variables": {
            "P": "the dividend",
            "D": "the divisor",
            "Q": "the quotient",
            "R": "the remainder"
        }
    },
    "worked_example": {
        "inputs": {
            "dividend": "x^3 - 2x^2 - 4",
            "divisor": "x - 3"
        },
        "steps": [
            "x³ ÷ x = x²; subtract x²(x − 3) = x³ − 3x² to leave x² − 4.",
            "x² ÷ x = x; subtract x(x − 3) = x² − 3x to leave 3x − 4.",
            "3x ÷ x = 3; subtract 3(x − 3) = 3x − 9 to leave 5, whose degree is below the divisor’s."
        ],
        "result": "(x³ − 2x² − 4) ÷ (x − 3) = x² + x + 3 with remainder 5."
    },
    "interpretation": "A zero remainder means the divisor is a factor of the dividend. The remainder always has a lower degree than the divisor.",
    "assumptions": [
        "Coefficients are rational numbers (decimals and fractions are kept exact)."
    ],
    "limitations": [
        "One variable (x) only; degree up to 50."
    ],
    "rules": null,
    "faqs": [
        {
            "question": "When should I use synthetic division instead?",
            "answer": "Synthetic division is a shortcut that only works for divisors of the form x − r. Long division works for any divisor, such as x² + 1 or 2x + 3."
        },
        {
            "question": "Do I need to include missing powers?",
            "answer": "Not when you type an expression; missing powers are treated as zero. When you list coefficients, include a 0 for each missing power."
        }
    ],
    "sources": [],
    "related": [
        {
            "name": "Synthetic Division Calculator",
            "url": "https://calculators.geekmidst.com/calculators/polynomial-synthetic-division-calculator"
        },
        {
            "name": "Polynomial Roots Calculator",
            "url": "https://calculators.geekmidst.com/calculators/polynomial-roots-calculator"
        },
        {
            "name": "Long Division Calculator",
            "url": "https://calculators.geekmidst.com/calculators/long-division-calculator"
        }
    ],
    "disclaimer": null,
    "publisher": "GM¹X CalcVerse"
}