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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]] and let $f, g \\in R[x]$, where $g$ has [[degree|0DPF6WRZW6]] $m$ and its [[leading coefficient|0P05RJC5XA]] $u$ is a unit of $R$.\nPolynomial long division of $f$ by $g$ is the following procedure.\n\n1. Start with $q = 0$ and $r = f$.\n2. While $r \\ne 0$ and $\\deg r \\ge m$, let $c$ be the leading coefficient of $r$ and $d = \\deg r$, and replace $q$ by $q + u^{-1} c x^{d - m}$ and $r$ by $r - u^{-1} c x^{d - m} g$.\n3. When $r = 0$ or $\\deg r \u003c m$, stop and return the quotient $q$ and the remainder $r$.\n\n## Remarks\n\nEach pass of step 2 removes the leading term of $r$, so $\\deg r$ drops and the procedure stops.\nThe equation $f = q g + r$ holds at every step, so the output is the unique pair of the [[division algorithm for polynomials|0MES3KS0ZC]].\nExample: divide $f = x^3 - 2x^2 - 4$ by $g = x - 3$ in $\\Z[x]$.\nThe steps add $x^2$, then $x$, then $3$ to the quotient, and the remainder is $5$.\nCheck: $(x - 3)(x^2 + x + 3) + 5 = x^3 - 2x^2 - 9 + 5 = x^3 - 2x^2 - 4$.\nNon-example: in $\\Z[x]$ the procedure does not apply to $g = 2x$, because the leading coefficient $2$ is not a unit, and the first step for $f = x$ would need the coefficient $\\frac{1}{2}$.\nDivision by a monic $g = x - a$ is often written in a compact table, called synthetic division; it is [[Horner's method|FYBQ3Z52Z7]].\n\n## Explanations\n\n### L1 human\n\nLong division of polynomials repeats one move: kill the highest term of what is left.\nIt stops when what is left has smaller degree than the divisor.\n",
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