Derivatives of x^n(x - 1)(x - a) with Rational Roots

Abstract

Let n >= 3 denote an integer and a != 0, 1 denote a rational number. For the family of polynomials f (x) = x^n(x - 1)(x - a) with ï¬xed value of n, we show that there exist inï¬nitely many values of a such that the ï¬rst two derivatives of f (x) have rational roots. We ï¬nd two examples of n and a for which the ï¬rst three derivatives of f (x) have rational roots.