I have a cubic equation in $x$
$$x^3+bx^2+cx+d=0$$
where all the coefficients are positive.
I know that with Descartes' Rule, the equation has no positive real roots, it either has 3 negative real roots or 1 negative real root and 2 complex roots.
If I would somehow know for sure that all 3 roots are real and negative, then my problem would be solved. Is there a way of knowing if this is really the case? Or if it is not, how can I know if the complex roots have negative or positive real parts?
Best Answer
This is determined by the discriminant. You have three real roots if and only if $$b^2c^2 - 4c^3 - 4b^3d - 27d^2 + 18bcd \geq 0.$$