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Suppose we have the following polynomial equations: $$a_nx^n+a_{n-1}x^{n-1}+...a_0=0,$$ $$b_nz^n+b_{n-1}z^{n-1}+...b_0=0.$$ I need to analytically determine the relationship between $x^*$ and $z^*$ in terms of $>,<$ or $=$ signs.$x^*$ and $z^*$ are roots of the corresponding polynomial equations. In order to determine the relationship, the main problem is that we have polynomials, which are $n$-degree, where $n>5$. About coefficients is known only that $a_i\neq b_i$, for $i=1,...,n.$ Therefore, I think we need to apply unconventional techniques to solve the problem. Please guide me.

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  • $\begingroup$ Is there any relationship between the $a_i$ and the $b_i$? Also, the second polynomial has $z$ as a variable but you use $y$ to denote the roots. Is that significant in any way? What have you tried? And what kind of "relationship" are you looking for? Presumably you know how to solve first and second degree polynomials, so maybe you can show is what you mean in those cases, and we can help you generalize? $\endgroup$ – Arthur Feb 5 '19 at 6:04
  • $\begingroup$ @Arthur Thank you for your comment. I've edited the question. Please have a look. $\endgroup$ – sane Feb 5 '19 at 7:30
  • $\begingroup$ Perhaps apply Descartes' rule to the difference of the polynomials? $\endgroup$ – lhf Feb 5 '19 at 11:12
  • $\begingroup$ @lhf Thank you for your comment. Could you little bit expand your comment? Do you mean apply Descartes' rule to difference of the polynomials in order to determine number of roots? $\endgroup$ – sane Feb 5 '19 at 11:16
  • $\begingroup$ @sane, yes, the number of positive and negative roots. $\endgroup$ – lhf Feb 5 '19 at 11:19
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Use a good computer with good software to find the numerical solutions.

Otherwise, there's no easy way; the information you've given is completely insufficient to do any kind of analysis, unless we have the specific values of $a_i,b_i$.

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