Hints 203
- 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).
- 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.
- 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.
- 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)?