Olimpiadi di Matematica

Solution:

The answer is (B). Let $x_{0}$, $x_{1}$ and $-x_{1}$ be the roots of the polynomial; it must hold, for every $x$, that $(x - x_{0})(x - x_{1})(x + x_{1}) = x^{3} + a x^{2} + b x + c$. Expanding the products and equating the coefficients of the terms of the same degree (principle of identity of polynomials), we get $-x_{0} = a$, $-x_{1}^{2} = b$, $x_{1}^{2} x_{0} = c$, from which $a b = c$.