If `omega (ne 1)` is a cube root of unity and `(1 + omega^(2))^(11) = a + b omega + c omega^(2)`, then (a, b, c) equals

Options

A. (1, 1, 0)
B. (0, 1, 1)
C. (1, 0, 1)
D. (1, 1, 1)

Answer

To solve the problem, we need to evaluate \((1 + \omega^2)^{11}\) where \(\omega\) is a cube root of unity (and \(\omega \neq 1\)). The cube roots of unity are \(1, \omega, \omega^2\) where \(\omega^3 = 1\) and \(1 + \omega + \omega^2 = 0\).

### Step-by-Step Solution:

1. **Identify the value of \(1 + \omega^2\)**:
Since \(1 + \omega + \omega^2 = 0\), we can rearrange this to find:
\[ 1 + \omega^2 = -\omega \]

2. **Substitute into the expression**:
We substitute \(1 + \omega^2\) into the expression we need to evaluate:
\[ (1 + \omega^2)^{11} = (-\omega)^{11} \]

3. **Simplify \((- \omega)^{11}\)**:
We can separate the negative sign and the power of \(\omega\):
\[ (-\omega)^{11} = -(\omega^{11}) \]

4. **Reduce \(\omega^{11}\)**:
Since \(\omega^3 = 1\), we can reduce the exponent \(11\) modulo \(3\):
\[ 11 \mod 3 = 2 \quad \text{(because \(11 = 3 \times 3 + 2\))} \]
Therefore, \(\omega^{11} = \omega^2\).

5. **Combine the results**:
Now we can substitute back:
\[ -(\omega^{11}) = -\omega^2 \]

6. **Express \(-\omega^2\) in terms of \(a\), \(b\), and \(c\)**:
We need to express \(-\omega^2\) in the form \(a + b\omega + c\omega^2\):
\[ -\omega^2 = 0 + 0\omega - 1\omega^2 \]
This gives us:
\[ a = 0, \quad b = 0, \quad c = -1 \]

### Final Result:
Thus, the values of \(a\), \(b\), and \(c\) are:
\[ (a, b, c) = (0, 0, -1) \]