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.
Read more about this topic: Properties Of Polynomial Roots
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