For the polynomial `p(x)=x^(5)+4x^(3)-5x^(2)+x-1`, one of the factors is

Options

A. `(x+1)`
B. `(x-1)`
C. `x`
D. `(x+2)`

Answer

To determine one of the factors of the polynomial \( p(x) = x^5 + 4x^3 - 5x^2 + x - 1 \), we can use the factor theorem, which states that if \( p(a) = 0 \) for some value \( a \), then \( (x - a) \) is a factor of the polynomial.

### Step-by-step Solution:

1. **Identify the Polynomial**:
We have the polynomial:
\[ p(x) = x^5 + 4x^3 - 5x^2 + x - 1 \]

2. **Use the Factor Theorem**:
We will test some simple values (using the trial and error method) to see if any of them yield \( p(a) = 0 \). We will start with \( a = 1 \).

3. **Calculate \( p(1) \)**:
Substitute \( x = 1 \) into the polynomial:
\[ p(1) = 1^5 + 4(1^3) - 5(1^2) + 1 - 1 \]
Simplifying this:
\[ p(1) = 1 + 4 - 5 + 1 - 1 \]
\[ = 1 + 4 - 5 + 1 - 1 = 0 \]

4. **Conclusion**:
Since \( p(1) = 0 \), according to the factor theorem, \( (x - 1) \) is a factor of the polynomial \( p(x) \).

5. **Final Answer**:
Therefore, one of the factors of the polynomial \( p(x) \) is:
\[ x - 1 \]