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

My work:
I assume the contraposition that all the roots are not equal.
Assume that the roots are $\alpha,\beta,\gamma,\delta$
So,$\alpha+\beta+\gamma+\delta=4$
and,$\alpha\beta\gamma\delta=1$
Here, by observation I can see that this holds for all the roots to be equal to 1, but I cannot prove it. Please help!

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Hint : you are then in the equality case of the AM-GM inequality

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  • $\begingroup$ yes right! How did I miss that! Thank you... $\endgroup$ – Hawk Jan 12 '14 at 16:01
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Let $x_1$, $x_2$, $x_3$, $x_4$ be the roots of the equation Sum of roots $x_1+x_2+x_3+x_4 = 4$ Product of roots $x_1.x_2.x_3.x_4 = 1$ We know AM > GM unless the quantities are equal Hence $(x_1+x_2+x_3+x_4 )/4 = 1 > (x_1.x_2.x_3.x_4)^1/4$. Therefore $x_1 = x_2 = x_3 = x_4 = 1$

Thus $x^4-4x^3+6x^2-4x+1 = 0$

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