# Conditions to get 2 real negative roots for a 3rd order polynomial

What is the conditions required to make the 2 roots of the 3rd prder polynomial $ax^3+bx^2+cx+d$ real and negative, if $a<0$ and $d>0$.

Thanks

• Can you explain what you've tried on this problem? In what context did you encounter this? – T. Bongers Dec 10 '15 at 22:54
• Since $\frac da \lt 0$ is the product of the roots, you will have to have all three negative. Do you know what $\frac ba$ and $\frac ca$ are as functions of the roots? – Ross Millikan Dec 10 '15 at 22:57
• @RossMillikan but they will not be necessary real – Mohamed Abdelaal Dec 11 '15 at 6:54

## 1 Answer

This problem is motivated by getting a condition for the intersection of 2 ellipses. The conditions are : $b < 0$ and $\Delta > 0$ where $\Delta=18 a b c d-4 b^3 d+b^2 c^2-4 a c^3-27 a^2 d^2;$ is the discriminant. First condition is obtained via routh-criteria and guarantee that we have 2 negative roots, and the second guarantee that all the roots are real.