is divisible by ?
`x^29-x^26-x^23+1`

Options

A. (x - 1), but not (x + 1)
B. (x + 1), but not (x - 1)
C. Both (x + 1) and (x - 1)
D. Neither (x + 1) nor (x - 1)

Answer

To determine whether the expression \( x^{29} - x^{26} - x^{23} + 1 \) is divisible by \( x - 1 \) and \( x + 1 \), we will evaluate the expression at \( x = 1 \) and \( x = -1 \) respectively.

### Step 1: Check divisibility by \( x - 1 \)

1. Substitute \( x = 1 \) into the expression:
\[ x^{29} - x^{26} - x^{23} + 1 = 1^{29} - 1^{26} - 1^{23} + 1 \]
Simplifying this gives:
\[ 1 - 1 - 1 + 1 = 0 \]

2. Since the result is \( 0 \), this means that \( x^{29} - x^{26} - x^{23} + 1 \) is divisible by \( x - 1 \).

### Step 2: Check divisibility by \( x + 1 \)

1. Substitute \( x = -1 \) into the expression:
\[ x^{29} - x^{26} - x^{23} + 1 = (-1)^{29} - (-1)^{26} - (-1)^{23} + 1 \]
Simplifying this gives:
\[ -1 - 1 + 1 + 1 = 0 \]

2. Since the result is \( 0 \), this means that \( x^{29} - x^{26} - x^{23} + 1 \) is also divisible by \( x + 1 \).

### Conclusion

Since the expression is divisible by both \( x - 1 \) and \( x + 1 \), we conclude that the correct answer is that the expression is divisible by both.

### Final Answer
The expression \( x^{29} - x^{26} - x^{23} + 1 \) is divisible by \( x - 1 \) and \( x + 1 \).

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