Complex Numbers

Using veita's formula, you get to find that $u * v + v * w + w * u = a_1$ of the equation $a_{n}x^{n} + a_{n - 1}x^{n - 1} + ... + a_{2}x^{2} + a_{1}x + a_{0} = 0$ and looking at the equation $z^{3} - 1 = 0$ we can see that the value of $a_1$ is 0. Therefore, the answer is $\boxed{0}$ .

The roots are 1 and two conjugate complex numbers. Say u = 1. Then we have to calculate v + w + v.w . Since the sum of the roots is zero, we have 1 + v + w = 0 . Hence v + w = -1. We have to calculate -1 + v.w . Since the product of the roots is 1, we have 1.v.w = 1. So, u.v + v.w + w.u = -1 + v.w = -1 + 1 = 0