Constructing a position domain of the fourth order polynomials with the active coefficient roots

The paper considers a technique in constructing regions of position of the fourth-order algebraic polynomial roots with the real coefficients. The regions of stable and unstable polynomials are delimited. For the region of stable polynomials, equations of surfaces separating regions with different character of the roots are obtained. These surfaces are constructed in the space of the normalized polynomial coefficients directly related to the original coefficients. The proposed technique is of practical interest in studying roots of the characteristic polynomial in a system of linear differential equations of the fourth order, which makes it possible to determine the nature of its solution. The paper provides an example of implementing the proposed technique in the MATLAB environment.

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