On Newton’s Rule of signs

Analysing the cubic sectors of a real polynomial of degree $n$, a modification of Newton’s Rule of 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 between 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.