= Vieta's formulas
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{title2=$e_k=(-1)^ka_{n-k}/a_n$}
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= Vieta formulas
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<Vieta's formulas> connect two descriptions of a <polynomial>: its <coefficients> and its <roots of a polynomial>. They let us read the <sum> and <product> of the <roots of a polynomial> without finding the individual <roots of a polynomial>, and turn a proposed <factorization of polynomials> into checkable <coefficient> identities. The key is that <coefficients> encode <elementary symmetric polynomials>, which are unchanged by a <permutation> of the <roots of a polynomial>.
For a <polynomial> of <polynomial degree> $n\geq1$, with nonzero <leading coefficient> $a_n$, write its <factorization of polynomials> in a <splitting field> as
$$
p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0
=a_n\prod_{i=1}^n(x-r_i).
$$
List every <root of a polynomial> according to its <multiplicity of a root>. Then
$$
\boxed{e_k(r_1,\ldots,r_n)=(-1)^k\frac{a_{n-k}}{a_n},\qquad 0\leq k\leq n.}
$$
In particular, $\sum_i r_i=-a_{n-1}/a_n$ and $\prod_i r_i=(-1)^na_0/a_n$. With <coefficients> in the <real numbers> or <complex numbers>, the <fundamental theorem of algebra> ensures that a list of $n$ <roots of a polynomial> in the <complex numbers> exists when <multiplicity of a root> is counted.<NIST Digital Library of Mathematical Functions/1 Algebraic and Analytic Methods>[NIST Digital Library of Mathematical Functions; §1.11(ii), root multiplicity paragraph and equation 1.11.10]{ref}
For a <monic polynomial>, this is the familiar convention
$$
X^n+c_1X^{n-1}+\cdots+c_n=\prod_{i=1}^n(X-r_i),
$$
where $(-1)^kc_k$ is the $k$th <elementary symmetric polynomial> in the <roots of a polynomial>; in particular, $\prod_i r_i=(-1)^nc_n$. The ratio by $a_n$ in the general statement matters: multiplying every <coefficient> by the same nonzero <scalar> preserves both the <roots of a polynomial> and these ratios.
A short derivation explains every alternating sign. In the <product> $\prod_i(x-r_i)$, a term of <polynomial degree> $n-k$ arises by choosing $-r_i$ from $k$ distinct factor positions and $x$ from the others. Summing the resulting <products> gives $(-1)^ke_k$. Values at different positions may coincide: the selection is of positions, not distinct root values. Matching <coefficients> gives the formula. This is the <distributive property> organized by the number of chosen root factors.
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