Counting zeros of generalised polynomials: Descartes’ rule of signs and Laguerre’s extensions

A slightly different question is how many positive zeros a polynomial has. Here the basic result is known as “Descartes’ rule of signs”. It says that the number of positive zeros is no more than the number of sign changes in the sequence of coefficients. Descartes included it in his treatise La Geometrie, which appeared in 1637. It can be proved by a method based on factorization, but, again, just as easily by deduction from Rolle’s theorem.