Question

.Question 5 (10 points) Determine the following statements are true or false. If it is true, explain briefly. If it is false, give a counterexample. (a) A non-constant polynomial must have a zero in the complex plane. (b) A non-constant entire function must have a zero in the complex plane. (c) A non-constant entire function must be unbounded.

Answer 1

A) False

B) True

C) False

Explanation:

(a) False. Not all non-constant polynomials have a solution in the complex plane. For example, the polynomial 2 does not have any solutions.

(b) True. Every non-constant polynomial has at least one solution in the complex plane. So, a non-constant function that is defined everywhere on the complex plane must have at least one solution.

(c) False. Non-constant functions that are defined everywhere in the complex plane can be bounded. There are functions that are not constant but still have a limited range and do not go to infinity.