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 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:


 * $ \displaystyle \prod_{j \mathop = 1}^n \paren { x - x_j } = x^n - e_1 \paren { \set {x_1,\ldots,x_n} } x^{n-1} + e_2 \paren { \set {x_1,\ldots,x_n}  } x^{n-2} + \cdots +(-1)^n e_n \paren { \set {x_1,\ldots,x_n}  }$

Proof
Let:


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

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

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

Also see
Viète's Formulas