1: Express cos3θ in terms of cos θ.
2: The polynomial f(x) = x4 + ax3 + bx2 + cx + d has real coefficients and f(2i) = f(2 + i) = 0. What is a + b + c + d?
3: Let m and n be integers such that each can be expressed as a sum of 2 squares, say m = a2 + b2 and n = c2 + d2. Using complex numbers, show that mn can also be expressed as the sum of two squares. (For example, 17 = 42 + 1 and 13 = 22 + 32 and sure enough, 17 · 13 = 221 = 142 + 52.)
4: (a) Factor z5 + z + 1 into two polynomials with integer coefficients.
(b) Find all roots of z6 + z4 + z3 + z2 + 1 = 0.
5: A1, ..., An are vertices of a regular polygon inscribed in a circle of radius r and center O. P is a point on OA1, extended beyond A1. Show that:
k=1∏n P Ak = OPn - rn.
6: A pocket calculator has the ability to add, subtract, and take reciprocals, but cannot multiply. One way to perform multiplication is by successive addition but this is tedious and particular difficult with fractions. Show that one can use the operations on the calculator to mulitiply two numbers.