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  "body": "\n## Statement\n\nLet $R$ be a [[commutative ring|0GYGHKX834]], let $f \\in R[x]$ and let $g \\in R[x]$ be a [[monic polynomial|0BDX4MBKT4]].\nThen there are unique polynomials $q, r \\in R[x]$ such that $f = q g + r$ and $\\deg r \u003c \\deg g$.\n\n## Remarks\n\nThe polynomials $q$ and $r$ are the quotient and the remainder. Long division of polynomials computes them.\nOver a [[field|0QA8CHH8MG]] $K$, every nonzero $g$ is a unit times a monic polynomial. So the statement holds for every nonzero $g \\in K[x]$.\nFor example, $x^3 + 1 = (x^2 - x + 1)(x + 1) + 0$ and $x^2 + 1 = (x - 1)(x + 1) + 2$.\nThe statement fails for a divisor that is not monic: in $\\mathbb{Z}[x]$, no $q$ and $r$ with $\\deg r \u003c 1$ give $x = q \\cdot 2x + r$.\n\n## Explanations\n\n### L1 human\n\nThis is the polynomial version of division with remainder for integers.\nYou can divide by a monic polynomial and get a remainder of smaller degree.\nThe degree plays the role that the size of a number plays for integers.\n",
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