Properties of Polynomial Roots - Polynomials With Real Roots

Polynomials With Real Roots

It is possible to determine the bounds of the roots of a polynomial using Samuelson's inequality. This method is due to a paper by Laguerre.

Let

be a polynomial with all real roots. The roots are located in the interval with endpoints

.

Example: The polynomial

has four real roots -3, -2, -1 and 1. The formula gives

,

its roots are contained in

I = .

If the polynomial f has real simple roots the Hessian H(f) evaluated on the interval is always ≥ 0. In symbols

H(f) = (n − 1)2 f' 2 − n(n − 1) f f' ≥ 0

where f' is the derivative of f with respect to x.

When n > 1 this simplifies to

f'(x) ≥ n f(x)

This relation applied to polynomials with complex roots is known as Bernstein's inequality.