Solution 5.8.4.8 Wewish to factor the polynomial s 5 +71s 4 +1470s 3 +9400s 2 +8000s: Using root locus techniques. Wedonot need root locus analysis to see that one rootisats =0.Soall wereally need to nd our the roots of the fourth order equation s 4 +71s 3 +1470s 2 +9400s+8000=0: Wecan reformulate this as a root locus problem as follows. Divide both sides of the equation by s 4 +71s 3 +1470s 2 ;; to obtain s 4 +71s 3 +1470s 2 s 4 +71s 3 +1470s 2 + 9400s+8000 s 4 +71s 3 +1470s 2 =0;; or 1+ 9400(s+40=47) s 4 +71s 3 +1470s 2 =0: Wecaneasily factor s 4 +71s 3 +1470s 2 ,toobtain 1+ 9400(s+40=47) s 2 (s +35:5; j14:4827)(s+35:5+j14:4827) =0: Nowconsider the root locus problem 1+ K(s +40=47) s 2 (s +35:5; j14:4827)(s+35:5+j14:4827) =0: The root locus is shown in Figure 1. This locus was generated with MATLAB and shows one of MATLAB's de ciencies, namely that it sometimes confuses points on the root locus. However, the root locus is sucienttoshowusthat the system could have all real roots or all complex roots, depending on the gain. Since the real roots lie between s = ;40=47 and s ;33, webegin by searching along the real axis to the left of s = ;40=47 for the point where the gain is 9400. Figure 2 shows howwecalculate K for s = ;2. 1 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 -25 -20 -15 -10 -5 0 5 10 15 20 25 Real Axis I mag A x i s Figure 1: Root Locus Im(s) Re(s) -(40/47) 2 s + 40/47 s s +35.5 -j14.4827 s +35.5 +j14.4827 Figure 2: Gain Calculation 2 s -1 -2 -5 -8 -10 -15 -20 -30 -35 -40 K 9400 4637.3 6868.2 8648 9400 1001.8 9400 7410.2 7533 9400 Table 1: Searchforthe value K =9400 For s = ;2we get K = jsjjsjjs +35:5; j14:4827jjs+35:5+j14:4827j js +40=47j s=;2 = 4637:3 The value of K wehavefound is too small. This tells us that the root we are looking for is to the rightofs = ;2. So wenext guess s = ;1, for whichwe obtain: K = jsjjsjjs +35:5; j14:4827jjs+35:5+j14:4827j js +40=47j s=;1 = 9400: Thus, wehavefound one of the roots of the polynomial. Table 1 summarizes the searchofthe value 9400. In this case the polynomial has four real roots and the factorization is Thus s 4 +71s 3 +1470s 2 +9400s+8000 = (s +1)(s +10)(s +20)(s+40) 3