If `omega (ne 1)` is a cube root of unity and `(1 + omega)^(2017) = A + B omega`. Then A and B are respectively the numbers

Options

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

Answer

To solve the problem, we need to evaluate \((1 + \omega)^{2017}\) where \(\omega\) is a cube root of unity (and \(\omega \neq 1\)). The cube roots of unity satisfy the equation \(1 + \omega + \omega^2 = 0\).

### Step-by-step Solution:

1. **Identify the properties of cube roots of unity**:
The cube roots of unity are \(1\), \(\omega\), and \(\omega^2\) where:
\[ \omega^3 = 1 \quad \text{and} \quad 1 + \omega + \omega^2 = 0 \]
From this, we can express \(1 + \omega\) as:
\[ 1 + \omega = -\omega^2 \]

2. **Rewrite the expression**:
We can rewrite \((1 + \omega)^{2017}\) using the expression we found:
\[ (1 + \omega)^{2017} = (-\omega^2)^{2017} \]

3. **Simplify the power**:
We can simplify \((-1)^{2017}\) and \((\omega^2)^{2017}\):
\[ (-\omega^2)^{2017} = (-1)^{2017} \cdot (\omega^2)^{2017} = -(\omega^2)^{2017} \]
Since \(\omega^3 = 1\), we can reduce the exponent \(2017\) modulo \(3\):
\[ 2017 \mod 3 = 1 \quad \text{(since } 2017 = 3 \times 672 + 1\text{)} \]
Thus, \((\omega^2)^{2017} = \omega^{2 \cdot 1} = \omega^2\).

4. **Combine the results**:
Therefore, we have:
\[ (1 + \omega)^{2017} = -\omega^2 \]

5. **Express in terms of \(A + B\omega\)**:
We need to express \(-\omega^2\) in the form \(A + B\omega\). From the relation \(1 + \omega + \omega^2 = 0\), we can express \(\omega^2\) as:
\[ \omega^2 = -1 - \omega \]
Thus,
\[ -\omega^2 = 1 + \omega \]

6. **Identify coefficients**:
Now, we can identify \(A\) and \(B\):
\[ A + B\omega = 1 + 1\omega \]
Therefore, \(A = 1\) and \(B = 1\).

### Final Answer:
Thus, the values of \(A\) and \(B\) are:
\[ A = 1, \quad B = 1 \]