Let r and z be real and complex numbers respectively, such that
(a) 0 < r < 1
(b) z^6  z^5  z + 1 = 0
(c) z^2  rz + 1 = 0
Find the value of r.
z^6  z^5  z + 1 = 0
==> (z  1)(z^5  1) = 0
==> z = 1 or z^5 = 1.
==> z = cos(72k) + i sin(72k)
for k = 0, +1, or +2
z^2  rz + 1 = 0
==> r = z + 1/z = z + conjugate(z)
= 2 Real(z) = 2 cos(72k)
0 < r < 1
==> r = 2 cos(72) ~= 0.618034

Posted by Bractals
on 20070123 15:32:50 