Value of k for which `(x-2)` is a factor of `(x^2-kx+2)` is-

Options

A. 2
B. 3
C. 4
D. 5

Answer

To find the value of \( k \) for which \( (x - 2) \) is a factor of \( (x^2 - kx + 2) \), we can use the Factor Theorem. According to this theorem, if \( (x - 2) \) is a factor of the polynomial, then substituting \( x = 2 \) into the polynomial should yield zero.

### Step-by-Step Solution:

1. **Substitute \( x = 2 \) into the polynomial**:
\[ x^2 - kx + 2 \quad \text{becomes} \quad 2^2 - k(2) + 2 \]

2. **Calculate \( 2^2 \)**:
\[ 2^2 = 4 \]
So the expression now looks like:
\[ 4 - 2k + 2 \]

3. **Combine like terms**:
\[ 4 + 2 = 6 \]
Thus, we have:
\[ 6 - 2k \]

4. **Set the expression equal to zero** (since \( (x - 2) \) is a factor):
\[ 6 - 2k = 0 \]

5. **Solve for \( k \)**:
\[ 6 = 2k \]
Divide both sides by 2:
\[ k = \frac{6}{2} = 3 \]

### Conclusion:
The value of \( k \) for which \( (x - 2) \) is a factor of \( (x^2 - kx + 2) \) is \( k = 3 \).

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