Suppose that all roots of the polynomial equation $x^4 - 4x^3 + ax^2 +bx + 1 = 0$ are positive real numbers. Show that all roots of the polynomial are equal.

I am not getting any idea as to how to solve it. Please help.


Suppose the roots are $\alpha,\beta,\gamma,\delta$. Then we have $\frac{\alpha+\beta+\gamma+\delta}{4}=\alpha\beta\gamma\delta=1$. But the arithmetic and geometric means of $\alpha,\beta,\gamma,\delta$ can only be equal if $\alpha=\beta=\gamma=\delta$.

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    $\begingroup$ That was fast +1! $\endgroup$ – Macavity Apr 22 '16 at 8:04
  • $\begingroup$ thank you very much...stupid of me for not thinking like that....thnx again.. $\endgroup$ – Abhijit A J Apr 22 '16 at 8:07
  • $\begingroup$ Too fast - $\frac{{\alpha + \beta + \gamma + \delta = 4}}{4}$?? ...THAT'S BETTER! $\endgroup$ – user328032 Apr 22 '16 at 8:07
  • $\begingroup$ @Benedict Many thanks, fixed! $\endgroup$ – almagest Apr 22 '16 at 8:08

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