`(x^(29)-x^(25)+x^(13)-1)` is divisible by :

Options

A. `(x+1)` but not by `(x-1)`
B. `(x-1)` but not by `(x+1)`
C. both `(x+1)` and `(x-1)`
D. neither `(x-1)` nor `(x+1)`

Answer

To determine if the polynomial \( P(x) = x^{29} - x^{25} + x^{13} - 1 \) is divisible by \( x + 1 \) or \( x - 1 \), we will evaluate \( P(x) \) at \( x = -1 \) and \( x = 1 \).

### Step-by-Step Solution:

1. **Evaluate \( P(-1) \)**:
\[ P(-1) = (-1)^{29} - (-1)^{25} + (-1)^{13} - 1 \]
- Calculate each term:
- \( (-1)^{29} = -1 \)
- \( (-1)^{25} = -1 \)
- \( (-1)^{13} = -1 \)
- Substitute these values into the polynomial:
\[ P(-1) = -1 - (-1) + (-1) - 1 \]
- Simplify:
\[ P(-1) = -1 + 1 - 1 - 1 = -2 \]

2. **Evaluate \( P(1) \)**:
\[ P(1) = 1^{29} - 1^{25} + 1^{13} - 1 \]
- Calculate each term:
- \( 1^{29} = 1 \)
- \( 1^{25} = 1 \)
- \( 1^{13} = 1 \)
- Substitute these values into the polynomial:
\[ P(1) = 1 - 1 + 1 - 1 \]
- Simplify:
\[ P(1) = 1 - 1 + 1 - 1 = 0 \]

3. **Conclusion**:
- Since \( P(-1) = -2 \), \( x + 1 \) is **not** a factor of \( P(x) \).
- Since \( P(1) = 0 \), \( x - 1 \) **is** a factor of \( P(x) \).
- Therefore, \( P(x) \) is divisible by \( x - 1 \) but not by \( x + 1 \).

### Final Answer:
The polynomial \( x^{29} - x^{25} + x^{13} - 1 \) is divisible by \( x - 1 \) but not by \( x + 1 \).

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