MathDB
discriminant of cubic polynomial into matrix form

Source: 2017 South Korea USCM P4

August 15, 2020
algebrapolynomiallinear algebramatrixcollege contests

Problem Statement

For a real coefficient cubic polynomial f(x)=ax3+bx2+cx+df(x)=ax^3+bx^2+cx+d, denote three roots of the equation f(x)=0f(x)=0 by α,β,γ\alpha,\beta,\gamma. Prove that the three roots α,β,γ\alpha,\beta,\gamma are distinct real numbers iff the real symmetric matrix \begin{pmatrix} 3 & p_1 & p_2 \\ p_1 & p_2 & p_3 \\ p_2 & p_3 & p_4 \end{pmatrix},  p_i = \alpha^i + \beta^i + \gamma^i is positive definite.
Back to ProblemsView on AoPS