If `z_1 + z_2 + z_3 = A, z_1 + z_(2)omega+ z_(3)omega^(2)= B, z_1 + z_(2) omega^(2) + z_(3) omega=C` where `1, omega, omega^2` are the cube roots of unity, then

Options

A. `A +B+C=3z_1`
B. `ABC=z_1 z_2 z_3`
C. `z_(1)^(3) + z_(2)^(3) +z_(3)^(3)=3z_1 z_2 z_3`
D. `|A|^2 + |B|^2 + |C|^2 = 3(|z_1|^2+ |z_2|^2 +|z_3|^2)`

Answer

To solve the problem, we need to find the value of \( A + B + C \) given the equations involving complex numbers \( z_1, z_2, z_3 \) and the cube roots of unity \( 1, \omega, \omega^2 \).

### Step 1: Write down the equations
We have the following equations:
1. \( z_1 + z_2 + z_3 = A \)
2. \( z_1 + z_2 \omega + z_3 \omega^2 = B \)
3. \( z_1 + z_2 \omega^2 + z_3 \omega = C \)

### Step 2: Add the equations
Now, we will add all three equations together:
\[ A + B + C = (z_1 + z_2 + z_3) + (z_1 + z_2 \omega + z_3 \omega^2) + (z_1 + z_2 \omega^2 + z_3 \omega) \]
This simplifies to:
\[ A + B + C = 3z_1 + z_2(1 + \omega + \omega^2) + z_3(1 + \omega^2 + \omega) \]

### Step 3: Use the property of cube roots of unity
We know that for the cube roots of unity:
\[ 1 + \omega + \omega^2 = 0 \]
Thus, both \( 1 + \omega + \omega^2 \) and \( 1 + \omega^2 + \omega \) equal \( 0 \).

### Step 4: Substitute the values
Substituting these values into our equation gives:
\[ A + B + C = 3z_1 + z_2(0) + z_3(0) = 3z_1 \]

### Step 5: Conclusion
Thus, we find that:
\[ A + B + C = 3z_1 \]

### Final Answer
The final result is:
\[ A + B + C = 3z_1 \]

---