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3.8. Other Problems 115

21. Let f, g, h be polynomials over R or C. Suppose that f = g”‘h,
    where g does not divide h.

(a) Show that gm-l divides f’, the derivative of f.

(b) The example g = (t - 1)3, f = (t - 1)3(t3 - 1)2 shows that it is

possible for gm to divide f’. Is it possible for gm+’ to divide f’?

22. Show that, for every positive integer n, there is a polynomial f(x) of
    degree n and a related polynomial g(x) for which

[f(x)l” - 1 = (x2 - W412.

23. Put {m}! = (xm - 1)(x”-1 - 1).. .(x - 1) for m^2 1 and {O}! = 1.
    Show that

is a polynomial with rational coefficients.

24. If x + y + .z = xyz, show that

2x 2Y 2% 2x

-+-

1 - x2 l-y2+ m= 1 - x2

2y 2%

jqyy2’.

25. Suppose that

22x3 + y2y3 = 23x1 + y3yl = 21x2 + yly2 = 1

and

dl = x2y3 - x3y2

d2 = X3Yl - XlY3

4 = 21~2 - 22~1.

Show that dl + d2 + d3 = dld2d3.

26. (a) For which integers a, b does the quadratic t2 - at + b have a zero
    which is a root of unity?
    (b) Show that, if t2 - (a” - 2b)t + b2 has a zero which is a root of
    unity, then so does t2 - at + b (where a, b E Z).


27. Suppose that a, b, c are nonzero integers and u, v are roots of unity for
    which u2 # 1, v2 # 1, and au + bv + c = 0. Show that Ial = lb] = ICI.


28. Show that x = sin(a/l4) is a root of the equation

8x3 - 4x2 - 4x + 1 = 0.