MATH 295 PROBLEMS 10

IGOR KRIZ

1. Calculate:

(3 + 4i)/(2 + 5i),
|\sqrt[7]{3 + 4i}|,
(1 + 5i)^{5}.

2. Using complex multiplication and the binomial theorem, find a formula for cos(nα), sin(nα). [Hint: it is probably better not to use the sum notation.]

3. Suppose that f, g are continuous functions of one variable defined on an interval J and such that f^{2} = g^{2}, and f(x) ≠ 0 for all x ∈ J. Prove that then either f(x) = g(x) for all x ∈ J, or f(x) = −g(x) for all x ∈ J.

4. Find an x ∈ ℂ such that x^{5} = 1, x ≠ 1. [Hint: because one knows one root, one can write this as a polynomial equation of 4th degree, and use the substitution z = x + \frac{1}{x}.]

5. Suppose f is a continuous function of one real variable defined on [a, b] such that f(a) = f(b) = 0, but there is an x_{0} ∈ (a, b) such that f(x_{0}) > 0. Prove that there exist c, d such that a ≤ c < x_{0} < d ≤ b such that f(c) = f(d) = 0, but f(x) > 0 for all x ∈ (c, d).