Prove that the roots are equal

Question

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.

Answer 1

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