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 $d>0$.


  • $\begingroup$ Can you explain what you've tried on this problem? In what context did you encounter this? $\endgroup$ – T. Bongers Dec 10 '15 at 22:54
  • $\begingroup$ 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? $\endgroup$ – Ross Millikan Dec 10 '15 at 22:57
  • $\begingroup$ @RossMillikan but they will not be necessary real $\endgroup$ – Mohamed Abdelaal Dec 11 '15 at 6:54

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.


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