MathDB

F(x) = f(x) + h f'(x) + h^2 f''(x) +... + h^n f^{(n)}(x) has no roots

Source: 1968-69 Germany R4 12.6

October 13, 2024

algebrapolynomialanalysiscalculus

Problem Statement

Let

n

be a positive integer,

h

a real number and

f(x)

a polynomial (whole rational function) with real coefficients of degree n, which has no real zeros. Prove that then also the polynomial

F(x) = f(x) + h f'(x) + h^2 f''(x) +... + h^n f^{(n)}(x)

has no real zeros.