Real roots

Upper Bound for the Real Roots Any number that exceeds all the roots is called an upper bound to the real roots. If the coefficients of a polynomial equation are all of hke sign, there is no positive root. Such equations are excluded here since zero is the upper oound to the real roots. If the coefficient of the highest power of P x) = 0 is negative, replace the equation by —P x) = 0. 

If in a polynomial P x) = Cqx + Cix -t- + c iX + c = 0, with Cq > 0, the first negative coefficient is preceded by k coefficients which are positive or zero, and if G denotes the greatest of the numerical values of the negative coefficients, then each real root is less than... 

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. 

Distinct Real Roots If the roots of the characteristic equation... 

A lower bound to the real roots may be found by applying the criterion to the equation P - ). 

One last result is helpful in getting an estimate of how many positive and negative real roots there are. 

In the event that three real roots obtain for these equations, only the largest 2 (smallest p) appropriate for the vapor phase has physical significance, because the viriaf equations are suitable only for vapors and gases. 

This equation is cubic in hquid depth. Below a minimum value of Ejp there are no real positive roots above the minimum value there are two positive real roots. At this minimum value of Ejp the flow is critical that is, Fr = 1, V= V, and Ejp = (3/2)h. Near critical flow conditions, wave motion ana sudden depth changes called hydraulic jumps are hkely. Chow (Open Channel Hydraulics, McGraw-Hill, New York, 1959), discusses the numerous surface profile shapes which may exist in nommiform open channel flows. 

When the damping eoeffieient C of a seeond-order system has its eritieal value Q, the system, when disturbed, will reaeh its steady-state value in the minimum time without overshoot. As indieated in Table 3.4, this is when the roots of the Charaeteristie Equation have equal negative real roots. 

From equations (5.1)-(5.4), it ean be seen that the stability of a dynamie system depends upon the sign of the exponential index in the time response funetion, whieh is in faet a real root of the eharaeteristie equation as explained in seetion 5.1.1. 

These results agree with the root loeus diagram in Figure 5.9, where A" = 4 produees two real roots of. v = —2,. v = —2 (i.e. eritieal damping). 

The solution of Eq. (4) gives for a third-power polynomial of Kc, its real root being computed with the Cardan formula14 resulting in the expression ... 

Here is a positive number and C equal to zero for a point located on the spheroid surface, but in accordance with Equation (2.307), inside it is negative. For this reason, the difference B —AC is positive and greater than B. Thus, this equation has two real roots, one of which is positive. The latter represents the distance ri from the point p to any point q( i, rji, Ci) on the spheroid surface, but the negative root has to be discarded. Respectively, we have... 

M. McCully, How do real roots work Some new views of root structure. Plant Physiol. 709 1 (1995). 

The secular equation (9.14) is satisfied only for certain values of S. Since this equation is of degree N in S, there are N real roots... 

For any particular distillation problem equation 11.28 will have only one real root k between 0 and 1... 

The next question is how to find the partial fractions in Eq. (2-25). One of the techniques is the so-called Heaviside expansion, a fairly straightforward algebraic method. We will illustrate three important cases with respect to the roots of the polynomial in the denominator (1) distinct real roots, (2) complex conjugate roots, and (3) multiple (or repeated) roots. In a given problem, we can have a combination of any of the above. Yes, we need to know how to do them all. 

Distinct Real Roots If the roots of the characteristic equation are distinct real roots, ri and r2, say, the solution is y = AeryX + Ber, where A and B are arbitrary constants. 

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... 

For the polynomial equation f(x) = 0 with real coefficients, the number of positive real roots of x is either equal to file number of changes in sign of the coefficients or less than that number by a positive even integer the number of negative real roots is similarly given, if x is replaced by - x. 

This equation has one positive real root, /A1 = 0.69, which can be obtained by trial or by means of the E-Z Solve software (file exl4-ll.msp). 

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. 

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