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  "body": "\n## Definition\n\nA function $f \\colon \\R \\to \\R$ is a *polynomial function* if there are an integer $n \\ge 0$ and real numbers $a_0, a_1, \\dots, a_n$ such that\n$$f(x) = a_0 + a_1 x + \\dots + a_n x^n \\quad \\text{for all } x \\in \\R.$$\n\n## Remarks\n\nThe coefficients are unique: if two such formulas give the same function, their difference is a polynomial with infinitely many roots, so all its coefficients are $0$.\nSo each polynomial function comes from exactly one element of the [[polynomial ring|0Q9EZS90GY]] $\\R[x]$, and its [[degree|0DPF6WRZW6]] is well defined.\nThe same definition works on an interval of $\\R$, or with $\\R$ replaced by $\\C$.\nEvery polynomial function is [[continuous|0P27W9RW0G]] and differentiable, and its derivative is the polynomial function $a_1 + 2 a_2 x + \\dots + n a_n x^{n-1}$.\n\nExample: $f(x) = (x - 1)^2$ is a polynomial function, since $(x - 1)^2 = 1 - 2x + x^2$ for all $x$, with $n = 2$, $a_0 = 1$, $a_1 = -2$, $a_2 = 1$.\n\nNon-example: $f(x) = \\abs{x}$ is not a polynomial function.\nIf it were equal to a polynomial $p$, then $p(x) - x$ would vanish for all $x \u003e 0$, so $p(x) = x$; but then $p(-1) = -1 \\ne \\abs{-1}$.\n",
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