Algebraic Equations : Reciprocal Equations

As provided by E. Galua theory the general algebraic equations for a polynomial of fourth order

ax^4 + bx^3 + cx^2 + dx + f=0 (*)

is the maximum order type of algebraic equations the solution to which one can write down in radical expressions.
Among all the equations of fourth order (not counting the degenerated cases like ax^4 +bx^3 = 0, or biquadratic equations ax^4 + bx^2 + c = 0) there exist so called reciprocal algebraic equations that can be solved by repeating procedure of the quadratic equation solution. The question is how to identify the reciprocal equation if we meet it in its canonical form (*)

To give the criteria for identification of the fourth order reciprocal algebraic equations.


Solution Summary

A relation involving reciprocal equations is proven