Understanding the position in the complex plane of the roots of a polynomial of degree three

Question

I am working with the polynomial $x^3+ax^2+bx+c$, with $a>0, b>0, c>0$. I would like to understand under which conditions, on the coefficients, this polynomial has complex roots with positive real part. Does someone knows this conditions or someone has some literature to suggest?

Answer 1

I guess that you are looking for a condition such that the complex roots of the polynomial $x^3+ax^2+bx+c$ have all NEGATIVE real part (otherwise $a>0$ implies that at least a root has a negative real part). Then use Routh–Hurwitz stability criterion: together with $a,b,c>0$, impose that $ab>c$.

Answer 2

The sum of the roots is $-a<0$, so they never all have positive real part.