Let `omega` be a cube root of unity not equal to 1. Then the maximum possible value of `| a + bw + cw^(2) |` where a, b, c `in` {+1, -1} is

Options

A. 0
B. 2
C. `sqrt(3) `
D. `1 + sqrt(3)`

Answer

To solve the problem, we need to find the maximum possible value of the expression \( | a + b\omega + c\omega^2 | \), where \( a, b, c \in \{+1, -1\} \) and \( \omega \) is a cube root of unity not equal to 1.

### Step-by-Step Solution:

1. **Understanding Cube Roots of Unity:**
The cube roots of unity are the solutions to the equation \( x^3 = 1 \). The roots are \( 1, \omega, \omega^2 \), where:
\[ \omega = e^{2\pi i / 3} = -\frac{1}{2} + i\frac{\sqrt{3}}{2} \]
\[ \omega^2 = e^{-2\pi i / 3} = -\frac{1}{2} - i\frac{\sqrt{3}}{2} \]

2. **Expressing the Given Expression:**
We need to evaluate:
\[ | a + b\omega + c\omega^2 | \]
where \( a, b, c \) can be either \( +1 \) or \( -1 \).

3. **Substituting Values:**
We will consider all combinations of \( a, b, c \). The possible combinations of \( (a, b, c) \) are \( (+1, +1, +1), (+1, +1, -1), (+1, -1, +1), (+1, -1, -1), (-1, +1, +1), (-1, +1, -1), (-1, -1, +1), (-1, -1, -1) \).

4. **Calculating Each Case:**
Let's compute the modulus for each case:
- For \( (1, 1, 1) \):
\[ |1 + \omega + \omega^2| = |1 + (-\frac{1}{2} + i\frac{\sqrt{3}}{2}) + (-\frac{1}{2} - i\frac{\sqrt{3}}{2})| = |1 - 1| = 0 \]
- For \( (1, 1, -1) \):
\[ |1 + \omega - \omega^2| = |1 + (-\frac{1}{2} + i\frac{\sqrt{3}}{2}) - (-\frac{1}{2} - i\frac{\sqrt{3}}{2})| = |1 + i\sqrt{3}| = 2 \]
- For \( (1, -1, 1) \):
\[ |1 - \omega + \omega^2| = |1 - (-\frac{1}{2} + i\frac{\sqrt{3}}{2}) + (-\frac{1}{2} - i\frac{\sqrt{3}}{2})| = |1 + 1| = 2 \]
- For \( (1, -1, -1) \):
\[ |1 - \omega - \omega^2| = |1 - (-\frac{1}{2} + i\frac{\sqrt{3}}{2}) - (-\frac{1}{2} - i\frac{\sqrt{3}}{2})| = |1| = 1 \]
- For \( (-1, 1, 1) \):
\[ |-1 + \omega + \omega^2| = |-1 + 0| = 1 \]
- For \( (-1, 1, -1) \):
\[ |-1 + \omega - \omega^2| = |-1 + i\sqrt{3}| = 2 \]
- For \( (-1, -1, 1) \):
\[ |-1 - \omega + \omega^2| = |-1 + 0| = 1 \]
- For \( (-1, -1, -1) \):
\[ |-1 - \omega - \omega^2| = |-1| = 1 \]

5. **Finding the Maximum Value:**
The maximum value from all computed cases is \( 2 \).

### Conclusion:
The maximum possible value of \( | a + b\omega + c\omega^2 | \) is \( \boxed{2} \).