The common roots of the equation `z^3+2z^2+2z+1=0` and `z^(1985)+z^(100)+1=0` are: (A) `1,omega` (B) `1,omega^2` (C) `omega,omega^2` (D) none of these

Options

A. `-1`
B. 1
C. 0
D. 1

Explanation

To find the common roots of the equations \( z^3 + 2z^2 + 2z + 1 = 0 \) and \( z^{1985} + z^{100} + 1 = 0 \), we will follow these steps:

### Step 1: Factor the first polynomial
We start with the polynomial \( z^3 + 2z^2 + 2z + 1 \). We can factor it by grouping or using synthetic division.

Notice that:
\[
z^3 + 2z^2 + 2z + 1 = (z + 1)(z^2 + z + 1)
...