Elementary Symmetric Function/Examples/Monic Polynomial

Example of Elementary Symmetric Function: Monic Polynomial

Let $\set {x_1, x_2, \ldots, x_n}$ be a set of real or complex values, not required to be unique.

The expansion of the monic polynomial in variable $x$ with roots $\set {x_1, x_2, \ldots, x_n}$ has coefficients which are sign factors times an elementary symmetric function:

$\ds \prod_{j \mathop = 1}^n \paren {x - x_j} = x^n - \map {e_1} {\set {x_1, \ldots, x_n} } x^{n - 1} + \map {e_2} {\set {x_1, \ldots, x_n} } x^{n - 2} + \dotsb + \paren {-1}^n \map {e_n} {\set {x_1, \ldots, x_n} }$

Proof

Let:

$\map P x = \ds \prod_{j \mathop = 1}^n \paren {x - x_j}$

From Product of Sums: Corollary, $\map P x$ expands as a sum of powers $x^k$.

Induction on $n$ shows that the coefficient of $x^k$ is a sign factor times $\map {e_{n - k} } {\set {x_1, \ldots, x_n} }$, which is the sum of products for all subsets of $\set {x_1, \ldots, x_n}$ with $n - k$ elements.

$\blacksquare$

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