a 6. a. (2 pts) Give an example of a polynomial P(2) with all real coefficients, and four different non-real roots, one

[answerhappygod]

A 6 A 2 Pts Give An Example Of A Polynomial P 2 With All Real Coefficients And Four Different Non Real Roots One 1 (66.51 KiB) Viewed 25 times

a 6. a. (2 pts) Give an example of a polynomial P(2) 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(2) with all real coefficients, where 2i is a repeated root. c. (3 pts) Suppose that r is a double root of a polynomial P(). Show that r is a root of the derivative P'(). d. (1 pt) Give an example of a non-constant polynomial in z and the conju- gate variable z so that P(2, 2) is positive (never zero) for all z in C. e. (2 pts) Let n > 2 be a positive integer. Factor the degree n polynomial 2" - 1 as a linear polynomial times a degree n - 1 polynomial. (Hint: there is a real number that is a root for any n). PE f. (5 pts) Let f(2) be a rational function: f = 3), 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 w E C, there exists a solution z of the equation f(-) = w. Also give a specific example of a rational function f and a number w so that f(2)= w has no solution 2. =w.