# Prove that the roots are equal

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.

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 $\alpha=\beta=\gamma=\delta$.
• Too fast - $\frac{{\alpha + \beta + \gamma + \delta = 4}}{4}$?? ...THAT'S BETTER! – user328032 Apr 22 '16 at 8:07