What are the components of `(x^(29) -x^(24) +x^(13) -1)`?

Options

A. A) `(x-1)` only
B. B) `(x+1)` only
C. C) both `(x-1)` and `(x+1)`
D. D) Neither `(x-1)` nor `(x+1)`

Answer

To find the components of the expression \( x^{29} - x^{24} + x^{13} - 1 \), we will check if \( x - 1 \) and \( x + 1 \) are factors of the expression.

### Step 1: Check if \( x - 1 \) is a component
To check if \( x - 1 \) is a component, we substitute \( x = 1 \) into the expression.

\[ x^{29} - x^{24} + x^{13} - 1 = 1^{29} - 1^{24} + 1^{13} - 1 \]

Calculating this gives:

\[ 1 - 1 + 1 - 1 = 0 \]

Since the result is 0, \( x - 1 \) is indeed a component of the expression.

### Step 2: Check if \( x + 1 \) is a component
Next, we check if \( x + 1 \) is a component by substituting \( x = -1 \) into the expression.

\[ x^{29} - x^{24} + x^{13} - 1 = (-1)^{29} - (-1)^{24} + (-1)^{13} - 1 \]

Calculating this gives:

\[ -1 - 1 - 1 - 1 = -4 \]

Since the result is not 0, \( x + 1 \) is not a component of the expression.

### Conclusion
The only component of the expression \( x^{29} - x^{24} + x^{13} - 1 \) is \( x - 1 \).