# 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.