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  "body": "\n## Statement\n\nLet $R$ be a [[commutative ring|0GYGHKX834]] and let $f = a_n x^n + a_{n-1} x^{n-1} + \\dots + a_0 \\in R[x]$.\nSuppose that $f = a_n (x - r_1)(x - r_2) \\cdots (x - r_n)$ with $r_1, \\dots, r_n \\in R$.\nThen for $1 \\le k \\le n$,\n$$a_n \\, e_k(r_1, \\dots, r_n) = (-1)^k a_{n-k},$$\nwhere $e_k$ is the $k$-th [[elementary symmetric polynomial|0ZMY8441GW]].\n\n## Remarks\n\nIf $a_n$ is a unit, this says $e_k(r_1, \\dots, r_n) = (-1)^k a_{n-k}/a_n$.\nFor $x^2 + b x + c = (x - r_1)(x - r_2)$, it gives $r_1 + r_2 = -b$ and $r_1 r_2 = c$.\nThe proof expands the product and compares coefficients.\nOver a field, the hypothesis says that $f$ [[splits|0SHMC00RJV]] with roots $r_1, \\dots, r_n$, counted with multiplicity.\n\n## History\n\nThe formulas are named after the French mathematician François Viète, who derived them for positive roots.\n\n## Explanations\n\n### L1 human\n\nVieta's formulas say that the sum of the roots, the sum of their products in pairs, and so on, are given by the coefficients.\nYou can know these sums without finding the roots.\n",
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