Question 109
Given a polynomial $p(x) = a_{2n}x^{2n} + a_{2n-1}x^{2n-1} + \dots + a_1x + a_0$ of even degree with positive coefficients $a_0, a_1, \dots, a_{2n}$.
a) Prove that there exists a permutation of these coefficients such that the polynomial obtained has no real roots.
b) Does the statement of a) remain true if some coefficients of $p(x)$ are non-positive?
a) Prove that there exists a permutation of these coefficients such that the polynomial obtained has no real roots.
b) Does the statement of a) remain true if some coefficients of $p(x)$ are non-positive?