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  "body": "\n## Definition\n\nLet $K$ be a [[field|0QA8CHH8MG]] and let $f, g \\in K[x]$ be polynomials that are not both $0$.\nA greatest common divisor of $f$ and $g$ is a polynomial $d \\in K[x]$ such that:\n\n1. $d$ divides $f$ and $d$ divides $g$;\n2. every polynomial that divides both $f$ and $g$ divides $d$.\n\nThe greatest common divisor $\\gcd(f, g)$ is the unique [[monic|0BDX4MBKT4]] polynomial with these two properties.\n\n## Remarks\n\nTwo greatest common divisors differ by a nonzero constant factor, so exactly one of them is monic.\nExample: $\\gcd(x^2 - 1, x^2 - 3x + 2) = x - 1$.\nCheck: $x - 1$ divides both, and a common divisor divides $(x^2 - 1) - (x^2 - 3x + 2) = 3x - 3$, hence divides $x - 1$.\nNon-example: $2x - 2$ satisfies both clauses, but it is not monic, so it is a greatest common divisor and not $\\gcd(f, g)$.\nThe [[Euclidean algorithm|023VY493K0]] computes $\\gcd(f, g)$, and [[Bézout's identity|0VY3WPP9Y3]] gives $a, b \\in K[x]$ with $a f + b g = \\gcd(f, g)$.\nOver a [[unique factorisation domain|06YMXJMXYS]] such as $\\Z$, greatest common divisors in $\\Z[x]$ also exist, but only up to units, see [[greatest common divisor in an integral domain|0F7KNJ1ZYN]].\n\n## Explanations\n\n### L1 human\n\nThe greatest common divisor is the largest polynomial that goes into both.\nAsking for leading coefficient $1$ picks one answer among its constant multiples.\n",
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