I need to be solving a function of the form : $\frac{x}{f(a,b,c,d)}+ \sqrt{g(x,a,b,c,d)} + \frac{x}{\sqrt[3]{h(x,a,b,c,d)}} =0 $ for real root(s) of $x$ in terms of the parameters $a,b,c,d$.
Due to the complexity introduced by the square and cubic roots (especially in the denominator), I am unable to solve this expression and obtain the roots.
Now, I want to communicate this complexity in an (business/economics) academic paper without reproducing the expression in the paper (which is very long and ugly). I want to communicate this complexity as the reason why I resort to numerical analysis.
What would be a good way to do this? If I were to check the degree of the polynomial itself, it turns out to be just 1. $f(),g()$ and $h()$ are expressions with only $ x$ and not of higher degree ($x^2$ etc.).
Any help would be appreciated.
That's not a degree-1 polynomial - it's not actually a polynomial at all. Polynomials may not have $x$ in the denominator or inside a radical. In order to make it a polynomial, you would have to multiply both sides of the expression by $\sqrt[3]{h(x,a,b,c,d)}$ (to clear denominators) and then raise both sides to the sixth power (to get rid of the square root and the cube root). The result would be not just a polynomial, but a polynomial of sixth degree.
So I would say something along these lines: "The expression we wish to solve is equivalent to a sixth-degree polynomial; it is well-known that there is no closed-form expression for the roots of such a polynomial, so rather than attempt an exact solution we resort to numerical analysis."