Exercise 1. a) If you multiply two non-repeating decimals, can you ever get a repeating decimal?

b) If you multiply two repeating decimals, can you ever get a non-repeating decimal?

Exercise 2. a) If is a polynomial, show that is divisible by

 b) Deduce that is divisible by if
c) Deduce that if is a polynomial of degree and are distinct zeros of , then there is a constant such that
d) If has degree and vanishes at distinct values, show that
e) If and are polynomials of degree which agree for distinct values , show that

Exercise 3.  If, for each real number , the polynomials and agree on some neighborhood of , show that they must agree for all complex numbers .

Exercise 4. Find the sum and product of the zeros of a polynomial in terms of the coefficients, and then apply to the equation

Exercise 5. If are the zeros of a polynomial of degree and are the zeros of a polynomial , show that

Exercise 6.  Suppose is an entire function with for some positive number . Show that is constant.

Exercise 7.    Suppose is an entire function with for some positive numbers and and some non-negative integer . Show that is a polynomial of degree at most .

Exercise 8.  A real-valued function which is continuous in a region and satisfying there is called a subharmonic function. Show that is a subharmonic function on any region where is analytic.

Exercise 9.  Suppose is an entire function such that Re for some constant and all values . Show that is a constant function

Exercise 10.   Let be a holomorphic function in a domain .  If  in   (and being real constants not all zero), then is constant in