0
$\begingroup$

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.

$\endgroup$
6
$\begingroup$

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 $\alpha=\beta=\gamma=\delta$.

$\endgroup$
  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.