Equal Roots (Tomato subjective 70)

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.
Now all of \({\displaystyle{\alpha}} \), \({\displaystyle{\beta}} \), \({\displaystyle{\gamma}} \) and \({\displaystyle{\lambda}} \) are positive so AM-GM inequality is applicable.
\({\displaystyle{\frac{\alpha + \beta + \gamma + \lambda}{4}}}{\ge}\) \({(\alpha\beta\lambda)^{\frac{1}{4}}}\)
\({\Rightarrow} \) \({\frac{4}{4}} {\ge}\) \({1^{\frac{1}{4}}}\)
\({\Rightarrow} \) 1 \({\ge} \) 1
Now we know equality in AM-GM occours if all the numbers are equal.So \({\displaystyle{\alpha}} \), \({\displaystyle{\beta}} \), \({\displaystyle{\gamma}} \) and \({\displaystyle{\lambda}} \) are all equal.