The exact proability that the roots of quadratic, cubic, and quartic equations are all real if the equation coefficients are random

Abstract

Suppose that the coefficients of a polynomial equation are Independent random variables defined on subsets of real numbers, The purpose of this paper is to find the exact probability that all roots of a random polynomial equation are real. Since a polynomial equation of degree higher than four with arbitrary coefficients cannot be solved algrebraically, this paper will consider quadratic, cubic and quartic equations only. The general results are obtained in each case, Also, a number of special cases are furnished.