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In this paper, a novel approach for estimating the range of parameters, which exist as coefficients in the characteristic polynomial of a given Linear Time Invariant system, is presented. The approach involves the application of lower degree column polynomials obtained from the Routh table, which are termed as Pseudo Routh Column Polynomials. The approximate ranges of parameters obtained from the proposed approach are further sharpened to achieve marginal stability with the help of Routh's table formed for the original higher degree characteristic polynomial. The proposed approach is simple and straightforward in applications and is illustrated with suitable examples.