**problem**: Suppose that all roots of the polynomial equation

\({\displaystyle{x^4 – 4x^3 + ax^2 +bx + 1}} \) = 0 are positive real numbers.

Show that all the roots of the polynomial are equal.

**solution**: \({\displaystyle{x^4 – 4x^3 + ax^2 +bx + 1}} \) = 0

If the roots are \({\displaystyle{\alpha}} \), \({\displaystyle{\beta}} \), \({\displaystyle{\gamma}} \) and \({\displaystyle{\lambda}} \) .

then \({\displaystyle{\alpha}} \), \({\displaystyle{\beta}} \), \({\displaystyle{\gamma}} \) and \({\displaystyle{\lambda}} \) = 1

& \({\displaystyle{\alpha}} \) + \({\displaystyle{\beta}} \) + \({\displaystyle{\gamma}} \) + \({\displaystyle{\lambda}} \) = 4.

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