On Newton's Rule of Signs

Analysing the cubic sectors of a real polynomial of degree n, a modification of the Newton Rule is Signs is proposed with which stricter upper bound on the number of real roots can be found. A new necessary condition for reality of the roots of a polynomial is also proposed. Relationship among the quadratic elements of the polynomial is established through its roots and those of its derivatives. Some aspects of polynomial discriminants are also discussed - the relationship between the discriminants of real polynomials, the discriminants of their derivatives, and the quadratic elements - following a discriminant of the discriminant approach.