The value of k for which `(x-1)` is a factor of the polynomial `x^(3)-kx^(2)+11x-6` is

Options

A. `-6`
B. `5`
C. `2`
D. `6`

Answer

To find the value of \( k \) for which \( (x - 1) \) is a factor of the polynomial \( P(x) = x^3 - kx^2 + 11x - 6 \), we can use the Factor Theorem. According to the Factor Theorem, if \( (x - 1) \) is a factor of \( P(x) \), then \( P(1) = 0 \).

### Step-by-Step Solution:

1. **Substitute \( x = 1 \) into the polynomial**:
\[ P(1) = (1)^3 - k(1)^2 + 11(1) - 6 \]

2. **Simplify the expression**:
\[ P(1) = 1 - k + 11 - 6 \]
\[ P(1) = 1 + 11 - 6 - k \]
\[ P(1) = 6 - k \]

3. **Set the polynomial equal to zero**:
Since \( (x - 1) \) is a factor, we have:
\[ 6 - k = 0 \]

4. **Solve for \( k \)**:
\[ k = 6 \]

Thus, the value of \( k \) for which \( (x - 1) \) is a factor of the polynomial \( x^3 - kx^2 + 11x - 6 \) is \( k = 6 \).