2011/003) If 1 , w , w^2 are cube root of unity

If 1 , w , w^2 are cube root of unity show that (a+bw+cw^2)/(b+cw+aw^2) + (a+bw+cw^2)/(c+aw+bw^2) = -1

Proof:

As 1 w and w^2 are cube root of 1 we have

1+w+w^2 = 0 … 1

w^3 = 1…2


now
(a+bw+cw^2) = aw^3 + bw+ cw^2 = bw+cw^2 + aw^3= w( b+ cw + aw^2)

Hence (a+bw+cw^2)/(b+cw^2+aw^2) = w

Further
As (a+bw+cw^2) = a + bw + c/w (as w^2 = 1/w) = 1/w(aw+bw^2+c)

So (a+bw+cw^2)/ (c+ aw+bw^2 ) = 1/w = w^2

Hence (a+bw+cw^2)/(b+cw^2+aw^2) + (a+bw+cw^2)/ (c+ aw+bw^2 ) = w + w^2 = - 1 (from 1)