MathDB

The set of polynomials

f_1, f_2, \ldots, f_n

with real coefficients is called special , if for any different

i,j,k \in \{ 1,2, \ldots, n\}

polynomial

\dfrac{2}{3}f_i + f_j + f_k

has no real roots, but for any different

p,q,r,s \in \{ 1,2, \ldots, n\}

of a polynomial

f_p + f_q + f_r + f_s

there is a real root. a) Give an example of a special set of four polynomials whose sum is not a zero polynomial. b) Is there a special set of five polynomials?

Problem Statement