Relations between roots and coefficients of cubic equations with one root negative the reciprocal of another
Under predetermined conditions on the roots and coefficients, necessary and sufficient conditions relating the coefficients of a given cubic equation x3 + ax2 + bx + c = 0 can be established so that the roots possess desired properties. In this note, the condition for one root of a cubic equation to be the negative reciprocal of another one is obtained. Given that the coefficients a, b, c of the cubic equation are in arithmetical or geometrical progression, further conditions are deived for one root to be the negative reciprocal of another. These results provide useful means for checking calculated roots of cubic equations and could serve the needs of teachers and students of Mathematical Sciences in tertiary institutions when the solution of cubic equations are first studied.