You claim that you can express the root of a monic polynomial x^n +a_n-1x^(n-1) + +a_0=0 in terms of its coeficients (integer, rational, complex you do not say), but an equation of this kind can have all complex roots with rational coefficients. How can a complex number be expressed as a ratio rational numbers ? What is the GCF expansion of the largest root of x^2 +1 ? Where are the details of Ronan Larkin's work ? Are you a mathematician or an engineer ?