Newton’s Inequality, Maclaurin’s Inequality

Abstract

Let a ₁,a ₂,…,a _n be arbitrary real numbers.

Consider the polynomial

$$P(x) = (x + a_{1})(x + a_{2}) \cdots(x + a_{n}) = c_{0}x^{n} +c_{1}x^{n - 1} + \cdots + c_{n - 1}x + c_{n}.$$

Then the coefficients c ₀,c ₁,…,c _n can be expressed as functions of a ₁,a ₂,…,a _n , i.e. we have

For each k=1,2,…,n we define $p_{k} = \frac{c_{k}}{\binom{n}{k}} = \frac{k!(n - k)!}{n!}c_{k}$.