Descartes’rule

Descartes Rule of Signs The number of positive real roots of a polynomial equation with real coefficients either is equal to the number V of its variations in sign or is less than i by a positive even integer. The number of negative roots of P(x) = 0 either is equal to the number of variations of sign of P - ) or is less than that number by a positive even integer. 

From the Descartes rule of signs, since there is one change in the sign of the coefficients in (C), there is only one positive real root. (The same rule applied to - fA in (C) indicates that there may be two negative real roots for fA, but these are not allowable values.) Solution of (C) by trial or by means of the E-Z Solve software (file exl4-6.msp) gives... 

This equation may have three real, positive roots (as indicated by the Descartes rule of signs see Example 14-6). Three roots can be obtained by trial or by a suitable root-finding technique, for example, as provided by the E-Z Solve software (see note below) ... 

Equation 21.3-10 provides the value of the bed diameter D for a given allowable pressure drop, (-AP), a value of W calculated as described in Section 21.5, and known values of the other quantities. Since a, j8, and are all positive, from the Descartes rule of signs (Section 14.3.3), there is only one positive real root of equation 21.3-10. If the equations are solved for L instead of D, a cubic equation results. 

By Descartes rule (on alternation of signs), this general equation can have only one positive real root (5) which then must be the root of physical significance and must lie between 0 and Mtot- In application to a real system, use of the equation in this form requires a method for measuring Ma, and hence it is necessary to distinguish experimentally between M and Pa. Often such a measurement is not possible, and what can be measured is Mff°, the total number of monomer equivalents in the external phase. By definition... 

How many real roots does a given polynomial have Generally, in chemical problems, x represents a quantity that can only be positive, such as a concentration or an equilibrium constant. Descartes Rule of Signs states that for any polynomial F x) written in decreasing powers of x, the number of positive... 

This follows from Descartes rule of signs (Burnside and Panton, 1928), which is an extremely handy tool for doing linear stability analysis. Consider the polynomial equation... 

According to Descartes Rule of Signs [68], (6.51) has a positive solution which corresponds to a stable periodic solution of the original system. The schematic plots of the amplitude equation, (6.41), when trivial equilibrium point is imstable and when it is stable are given in Fig. 6.5b, c, respectively. 

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