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.