In order to study the theory of FujiwaraHermite and RouthHurwitz criteria under the Bernstein polynomials basis，the authors use the algebraic method of the transformation relation between the classical Bezout matrix and Bernstein Bezout matrix to give some investigations on the polynomial inertia and stability theory in terms of the Bernstein Bezout matrix.The results obtained can be viewed as the generalizations of the corresponding classical FujiwaraHermite and RouthHurwitz criteria to the cases under the Bernstein polynomials basis.