What are the components of `(X^(6 ) - X^(4) - X^(5) - X^(4) + X^(2) - 4)`
A. (X - 1)
B. (X + 1)
C. Both (X - 1) & (X + 1)
D. Neither (x - 1) or (X + 1)

Options

A. B
B. A
C. D
D. C

Answer

To determine the components of the polynomial \( X^6 - X^5 - 2X^4 + X^2 - 4 \), we will check if \( (X - 1) \) or \( (X + 1) \) are factors of the polynomial. We can do this by substituting \( X = 1 \) and \( X = -1 \) into the polynomial and checking if the result is zero.

### Step-by-step solution:

1. **Write the polynomial**:
\[ P(X) = X^6 - X^5 - 2X^4 + X^2 - 4 \]

2. **Check if \( X - 1 \) is a factor**:
Substitute \( X = 1 \) into the polynomial:
\[ P(1) = 1^6 - 1^5 - 2 \cdot 1^4 + 1^2 - 4 \]
Simplifying this:
\[ P(1) = 1 - 1 - 2 + 1 - 4 = -5 \]
Since \( P(1) \neq 0 \), \( X - 1 \) is not a factor.

3. **Check if \( X + 1 \) is a factor**:
Substitute \( X = -1 \) into the polynomial:
\[ P(-1) = (-1)^6 - (-1)^5 - 2 \cdot (-1)^4 + (-1)^2 - 4 \]
Simplifying this:
\[ P(-1) = 1 + 1 - 2 + 1 - 4 = -3 \]
Since \( P(-1) \neq 0 \), \( X + 1 \) is not a factor.

4. **Conclusion**:
Since neither \( X - 1 \) nor \( X + 1 \) are factors of the polynomial, the correct answer is:
\[ \text{D. Neither } (X - 1) \text{ nor } (X + 1) \]