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Hints 203

16. Show that the roots of the cubic equation

64t3 - 192t2 - 60t - 1 = 0

are cos3(2r/7) sec(6?r/7), cos3(4a/7) sec(2?r/7), cos3(67r/7) sec(4n/7).

17. Solve for 2, y, t, w:

xfay+a2z+a3w=a4

x + by + b2z + b3w = b4

x + cy + C2% + c3w = c4

x+dy+d2z+d3w=d4

where a, b, c, d are all distinct.

18. Let x = U(V-w)‘, y = v(w-u)~, % = w(u-v)~ where U+V+W = 0.
    Eliminate U, V, w to obtain

x3 + y3 + z3 + a(x2y + x2% + y2x + y”.z + z2x + t2y) + bxyz = 0

for suitable a and 6.

19. Let U, v, w be distinct constants. Solve

A+Y+L,l

a+u b+u c+u

Z+Y+“,l

Q+v b+v c+v

L+Y+L=1.

a+w b+w c+w

Hints

Chapter 6

1.4. There are two approaches: either use symmetric functions, or else

make a substitution into t3 - 2t2 + 62 + 5 = 0 to determine the

equation that s = l/t must satisfy.

1.8. (a) Can a quadratic counterexample be found?

1.9. (a) Note that xyr = -c. For the latter part, express (x - Y)~ in terms

of symmetric functions of x and y, which are then e-xpressible in terms

of t.

(b) What does the negativity of 3%2 + 2az - (a2 - 4b) imply about

the relationship between % and the zeros of the quadratic?

(d) What is the sign of p(t) at the zeros of p’(t)?