MathDB

6

Part of 1969 German National Olympiad

Problems(1)

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

10/13/2024
Let nn be a positive integer, hh a real number and f(x)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)+hf′(x)+h2f′′(x)+...+hnf(n)(x)F(x) = f(x) + h f'(x) + h^2 f''(x) +... + h^n f^{(n)}(x) has no real zeros.
algebrapolynomialanalysiscalculus