0

Let's say we have a third-order polynomial given as follows:

$P(u)=-u^{3}+a_{1}u^{2}+a_{2}u-a_{3}^{2}$,

where $a_{1},a_{2},a_{3}$ are all constants. In the specific problem that I am dealing with it is also supposed that $u=k^{2}.$ So, I need to understand the behavior of the roots of the $P(u)$ i.e. are they all real or not?

What I see first is that $P(0)=-a_{3}^{2}\leq 0.$ Secondly, I also know that $\lim_{u\rightarrow \infty }=-\infty $ and $\lim_{u\rightarrow -\infty }=\infty.$ It gives me one real root such that $\alpha_{1}\leq 0$. However, I believe these observations do not enough to determine whether other roots say $\alpha_{2}$, $\alpha_{3}$ are real or not. I feel that I need also consider the fact that $u=k^{2}$ for somewhere but I do not know how to apply this fact to the problem and take advantage of it.

Thanks in advance.

ruudvaan
  • 65
  • 1
    $u=k^2$ tells you nothing about $u$ unless you know something about $k$ or unless $u$ is somehow constrained. If your coefficients are real, then if you have no better idea, you can use calculus to determine the locations of the local maximum and minimum of the cubic, and if the value at the maximum is greater than zero and the value at the minimum is less than zero you will have three real roots. There will be some degenerate cases where roots coincide which you can detect. – Mark Bennet May 23 '22 at 15:53
  • Thank you for changing my perspective. I guess I can also choose to find such u>0 such that it results in P(u)>0. Then the problem might be solved I believe. – ruudvaan May 23 '22 at 17:10

0 Answers0