Answer Happy

6. a. (2 pts) Give an example of a polynomial P(z) with all real coefficients, and four different non-real roots, one of which is 2i. b. (1 pt) Give an example of a polynomial P(z) with all real coefficients, where 2i is a repeated root. c. (3 pts) Suppose that r is a double root of a polynomial P(x). Show that r is a root of the derivative P'(z). d. (1 pt) Give an example of a non-constant polynomial in z and the conju- gate variable z so that P(2,7) is positive (never zero) for all z in C. e. (2 pts) Let n > 2 be a positive integer. Factor the degree n polynomial z" - 1 as a linear polynomial times a degree n-1 polynomial. (Hint: there is a real number that is a root for any n). f. (5 pts) Let f(z) be a rational function: f = PCE), for polynomials P, Q, with Q not identically 0. Show (using the Fundamental Theorem of Algebra) that, if f is non-constant, then for all but finitely many points we C, there exists a solution 2 the equation f(x) = w. Also give a specific example of a rational function f and a number w so that f(x) = w has no solution z.