I have a polynomial, and I want to get the conditions for the number of positive roots  

What are the different methods out there to determine these conditions?


this is the polynomial:


f(g)=A1*g^5 + A2*g^4 + A3*g^3 + A4*g^2 + A5\*g + A6  

and A1 to A6 are constants  

I will appreciate any help
Warm regards



finding positive roots for a polynomial
Abel-Ruffini Theorem postulates the existence of polynomials of every degree above $4$ that are not solvable by radicals (see also Galois' results on the same topic at the beginning of the XIX century, about the necessary and sufficient condition to let a polynomial equation be solvable by radicals).


A good (simple to be read) online source is "Galois Theory: Polynomials of Degree 5 and Up" by Tokuei Higashino.


P.S. Without solving $f(g)=0$, you can use Descartes' Rule of Signs, stating that *the possible number of the positive roots of a polynomial is equal to the number of sign changes in the coefficients of the terms **or** less than the sign changes by a multiple of $2$*.


