= Elementary symmetric polynomial
{title2=$e_k(x_1,\ldots,x_n)$}
{wiki}
= Elementary symmetric function
{synonym}
The $k$th elementary symmetric polynomial adds every product obtained by choosing $k$ distinct positions from $x_1,\ldots,x_n$:
$$
e_k(x_1,\ldots,x_n)=\sum_{1\leq i_1<\cdots<i_k\leq n}x_{i_1}\cdots x_{i_k}.
$$
The conventions are $e_0=1$ and $e_k=0$ for $k>n$. The positions must be distinct, but their values may coincide. Each <permutation> of the variables merely rearranges the summands, so this is a <symmetric polynomial>.
For $x_1=1$, $x_2=2$, $x_3=3$, we obtain $e_1=1+2+3=6$, $e_2=1\cdot2+1\cdot3+2\cdot3=11$, and $e_3=1\cdot2\cdot3=6$. Expanding the product gives
$$
\boxed{(z-1)(z-2)(z-3)=z^3-6z^2+11z-6}.
$$
In general, $\prod_{i=1}^n(z-x_i)=z^n-e_1z^{n-1}+e_2z^{n-2}-\cdots+(-1)^ne_n$. Thus the <coefficients> of a <monic polynomial> collect <elementary symmetric polynomials> of its <roots of a polynomial>, with alternating signs. These are the <Vieta formulas>.<NIST Digital Library of Mathematical Functions/1 Algebraic and Analytic Methods>[NIST Digital Library of Mathematical Functions; §1.11(ii), equation 1.11.10]{ref}
\Image[/elementary-symmetric-root-products.png]
{title=Choosing one, two, or three root positions gives the coefficients of a cubic}
{description=For the stipulated roots $1,2,3$, node values add to $e_1=6$, edge products add to $e_2=11$, and the product of all three positions is $e_3=6$. These <elementary symmetric polynomials> enter the <monic polynomial> with alternating signs. The graph is an original illustration of the exact finite calculation.}
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