Conditions imposed on roots of a cubic equation by restrictions on the coefficients

Includes bibliographical references (leaf 34)The problem that has been suggested is to place or find the restrictions on the coefficients of a cubic equation so that the nature of the roots can be determined before solution of the equation is begun. The cubic that will be examined will be a rational and integral function of the variable in question and will have rational coefficients. Hence, the roots of the given cubic must be algebraic. Since the set of real numbers is a subset of the set of complex numbers, all of the roots of the given cubic could be considered complex. However, throughout the remainder of this paper, the term complex shall be used only when indicating a root of the form a + bi, where b = 0 and a = 0 or a = 0 as the case demands. All remaining roots, that is, those of the form a + bi, where b = 0, will be called real. The real roots will be divided into two classes, the rationals and the irrationals. As shown in any book dealing with the theory of equations, an equation of the nth degree must contain n roots, not necessarily all distinct. Thus, there are only three roots with which we need to concern ourselves.M.A. (Master of Arts