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  "body": "\n## Statement\n\nLet $f = a_n x^n + \\cdots + a_1 x + a_0$ be a nonzero polynomial with real coefficients.\nLet $V$ be the number of sign changes in the sequence of nonzero coefficients of $f$, taken in order of degree.\nLet $P$ be the number of positive [[roots|00EHX6FWC0]] of $f$, counted with [[multiplicity|03XVP8Q4GY]].\nThen $P \\le V$, and $V - P$ is even.\n\n## Remarks\n\nThe parity part follows by comparing the sign of the leading coefficient with the sign of the lowest nonzero coefficient.\nBoth $V$ and $P$ are even exactly when these signs agree.\nThe bound $P \\le V$ can be proved by induction on the degree, using [[Rolle's theorem|0194DY35BG]] to compare the positive roots of $f$ with those of $f'$.\nExample: $f = x^3 - x^2 - x + 1$ has signs $+, -, -, +$, so $V = 2$; indeed $f = (x - 1)^2 (x + 1)$ has the double positive root $1$, so $P = 2$.\nThe number of negative roots is bounded in the same way by the sign changes of $f(-x)$.\nThe bound need not be reached: $x^2 - x + 1$ has $V = 2$ and no real root, so $P = 0$.\n\n## Explanations\n\n### L1 human\n\nReading the signs of the coefficients of a polynomial gives an upper bound for the number of its positive roots.\nEach change from plus to minus, or from minus to plus, allows at most one positive root.\nThe count can only be too high by an even number.\n\n### L2 human\n\nThe rule is a quick first test before one locates roots more precisely.\nIf all roots of $f$ are real, then the bound is exact: $P = V$.\n[[Budan's theorem|7THS519D4A]] extends the rule from the half-line $(0, \\infty)$ to any interval, and [[Sturm's theorem|R44CV0KD9T]] gives the exact number of distinct real roots in an interval.\n",
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