6. This problem deals with complex polynomials. Consider a first order polynomial Pi(s)s-So. Show that the phase change in L(jw - so) as w increases from-x to +x is (-1)", where k is 0 if the sign of the real part of so is negative and equals +1 if the sign of the real part of so is positive. Consider a nth degree complex polynomial, Pn (s) with r roots which have a positive real part and n r roots with a negative real part. What is the net phase change in LP(j) as w increases from-oo to +oo? . Let P(s) be a complex polynomial such that P(jw)(w + 4) (w+2)(w -1)(w - 3) (w- 5) +j w +3) (w - 2) (w- 4) Pr(w) P:(w) - Sketch Pr(w) as a function of w. Similarly, sketch P (u) as a function of w. For values of w--4,-3,-2, 1,2,3, 4,5, identify the corresponding complex numbers P(jw) in the com plex plane. What is the net phase change in P(j as varies from -oo to oo? - How many roots of P(s) have a negative real part? Suppose P.(s) and P(s) are real polynomials and P(s) P(o)-Po) P-(s) +jP(s) be a complex polynomial, Let P() -Suppose + jw is a root of P(s), what is a corresponding root for P. (s)? How would you use or appropriately modify the Routh-Hurwitz procedure to check if the polynomial P(s) is Hurwitz?