What Are the Sum and Product of the Roots of This Complex Polynomial?

[BlakeJA]

Question that I came across and that has stumped me for about a week hehe.
Let $$p(z)=z^n +i z^{n-1} - 10$$

if $$\omega_j$$ are the roots for j=1,2,...,ncompute: $$\sum_{j=1}^n \omega_j}$$

and

$$\prod_{j=1}^n \omega_j}$$

 

[Muzza]

Let's consider an easier example first. Let f(x) = x^2 + 3x + 5. If f has roots a and b, then

x^2 + 3x + 5 = f(x) = (x - a)(x - b) = x^2 + x(-a - b) + ab.

Hence a + b and ab equal what? Now generalize.