If (x-6) is the HCF of `x^2- 2x - 24` and `x^2-kx-6`, then what is the value of k is

Options

A. 3
B. 5
C. 6
D. 8

Answer

To solve the problem, we need to find the value of \( k \) given that \( (x-6) \) is the highest common factor (HCF) of the polynomials \( x^2 - 2x - 24 \) and \( x^2 - kx - 6 \).

### Step-by-step Solution:

1. **Factor the first polynomial \( x^2 - 2x - 24 \)**:
- We need to factor \( x^2 - 2x - 24 \).
- We look for two numbers that multiply to \(-24\) (the constant term) and add to \(-2\) (the coefficient of \( x \)).
- The numbers \(-6\) and \(4\) satisfy this condition.
- Thus, we can factor it as:
\[ x^2 - 2x - 24 = (x - 6)(x + 4) \]

2. **Since \( (x-6) \) is a factor, substitute \( x = 6 \) into the second polynomial**:
- We substitute \( x = 6 \) into the second polynomial \( x^2 - kx - 6 \):
\[ f(6) = 6^2 - k(6) - 6 \]
- This simplifies to:
\[ f(6) = 36 - 6k - 6 = 30 - 6k \]

3. **Set the result equal to zero**:
- Since \( (x-6) \) is a factor of \( x^2 - kx - 6 \), we set \( f(6) = 0 \):
\[ 30 - 6k = 0 \]

4. **Solve for \( k \)**:
- Rearranging the equation gives:
\[ 6k = 30 \]
- Dividing both sides by \( 6 \):
\[ k = 5 \]

### Final Answer:
The value of \( k \) is \( 5 \).