If `omega ( cancel(=)1)` is a cube root of unity and `( 1+ omega^(7))=A + B omega`, then A and B are respectively

Options

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

Answer

To solve the problem, we need to find the values of \( A \) and \( B \) in the equation \( 1 + \omega^7 = A + B\omega \), where \( \omega \) is a cube root of unity and \( \omega \neq 1 \).

### 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 \( \omega = e^{2\pi i / 3} \) and \( \omega^2 = e^{4\pi i / 3} \).

We know that:
\[ \omega^3 = 1 \quad \text{and} \quad 1 + \omega + \omega^2 = 0 \]

2. **Finding \( \omega^7 \)**:
Since \( \omega^3 = 1 \), we can reduce \( \omega^7 \):
\[ \omega^7 = \omega^{6} \cdot \omega = (\omega^3)^2 \cdot \omega = 1^2 \cdot \omega = \omega \]

3. **Substituting \( \omega^7 \) into the equation**:
Now substitute \( \omega^7 \) back into the equation:
\[ 1 + \omega^7 = 1 + \omega \]

4. **Setting up the equation**:
We have:
\[ 1 + \omega = A + B\omega \]

5. **Comparing coefficients**:
To find \( A \) and \( B \), we compare both sides of the equation:
- The constant term on the left side is \( 1 \).
- The coefficient of \( \omega \) on the left side is \( 1 \).

Therefore, we can equate:
\[ A = 1 \quad \text{and} \quad B = 1 \]

6. **Final values**:
Thus, the values of \( A \) and \( B \) are:
\[ A = 1, \quad B = 1 \]

### Conclusion:
The final answer is \( A = 1 \) and \( B = 1 \).