If `z=re^(itheta)` ( r gt 0 & `0 le theta lt 2pi`) is a root of the equation `z^8-z^7+z^6-z^5+z^4-z^3+z^2 -z + 1=0` then number of value of `'theta'` is :

Options

A. 6
B. 7
C. 8
D. 9

Answer

To solve the problem, we need to find the number of values of \( \theta \) such that \( z = re^{i\theta} \) (where \( r > 0 \) and \( 0 \leq \theta < 2\pi \)) is a root of the polynomial equation:

\[ z^8 - z^7 + z^6 - z^5 + z^4 - z^3 + z^2 - z + 1 = 0 \]

### Step-by-step Solution:

1. **Identify the Degree of the Polynomial**:
The given polynomial is of degree 8, as the highest power of \( z \) is \( z^8 \). According to the Fundamental Theorem of Algebra, a polynomial of degree \( n \) has exactly \( n \) roots in the complex number system (counting multiplicities).

**Hint**: The degree of the polynomial gives you the total number of roots.

2. **Count the Roots**:
Since the polynomial is of degree 8, it will have exactly 8 roots. These roots can be represented in the form \( z_k = r_k e^{i\theta_k} \) for \( k = 1, 2, \ldots, 8 \).

**Hint**: Each root corresponds to a unique value of \( \theta \).

3. **Express the Roots**:
The roots can be expressed as:
\[ z_1, z_2, z_3, z_4, z_5, z_6, z_7, z_8 \]
Each \( z_k \) can be written as \( r_k e^{i\theta_k} \), where \( r_k > 0 \) and \( 0 \leq \theta_k < 2\pi \).

**Hint**: Each root has a unique angle \( \theta_k \) associated with it.

4. **Determine the Values of \( \theta \)**:
Since there are 8 distinct roots, there will be 8 corresponding values of \( \theta \):
\[ \theta_1, \theta_2, \theta_3, \theta_4, \theta_5, \theta_6, \theta_7, \theta_8 \]

**Hint**: The number of distinct angles \( \theta \) is equal to the number of roots.

5. **Conclusion**:
Therefore, the number of values of \( \theta \) is 8.

### Final Answer:
The number of values of \( \theta \) is **8**.