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jverzani
2026-06-03 14:34:02 -04:00
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quarto/precalc/polynomial_roots.qmd
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@@ -1004,7 +1004,7 @@ Now consider the case $p_0 > 0$. There are two possibilities either `pos(p)` is
So there is parity between `var(p)` and `pos(p)`: if $p$ is monic and $p_0 < 0$ then both `var(p)` and `pos(p)` are both odd; and if $p_0 > 0$ both `var(p)` and `pos(p)` are both even.
Descartes' rule of signs will be established if it can be shown that `var(p)` is at least as big as `pos(p)`. Supppose $r$ is a positive real root of $p$ with $p = (x-r)q$. We show that `var(p) > var(q)` which can be repeatedly applied to show that if $p=(x-r_1)\cdot(x-r_2)\cdot \cdots \cdot (x-r_l) q$, where the $r_i$s are the positive real roots, then `var(p) >= l + var(q) >= l = pos(p)`.
Descartes' rule of signs will be established if it can be shown that `var(p)` is at least as big as `pos(p)`. Suppose $r$ is a positive real root of $p$ with $p = (x-r)q$. We show that `var(p) > var(q)` which can be repeatedly applied to show that if $p=(x-r_1)\cdot(x-r_2)\cdot \cdots \cdot (x-r_l) q$, where the $r_i$s are the positive real roots, then `var(p) >= l + var(q) >= l = pos(p)`.
As $p = (x-c)q$ we must have the leading term is $p_nx^n = x \cdot q_{n-1} x^{n-1}$ so $q_{n-1}$ will also be `+` under our monic assumption. Looking at a possible pattern for the signs of $q$, we might see the following unfinished synthetic division table for a specific $q$: