For what value (s) of k is `x^(2) - kx + k` divisible by x - k ?

Options

A. only 0
B. only 0 or `- (1)/(2)`
C. only 1
D. any value of k

Explanation

To determine the value(s) of \( k \) for which the polynomial \( x^2 - kx + k \) is divisible by \( x - k \), we can use the Factor Theorem. According to this theorem, a polynomial \( f(x) \) is divisible by \( x - k \) if and only if \( f(k) = 0 \).

### Step-by-Step Solution:

1. **Define the Polynomial**:
We have the polynomial \( f(x) = x^2 - kx + k \).

2. **Apply the Factor Theorem**:
According to the Factor Theorem, for \( f(x) \) to be divisible by \( x - k \), we need to evaluate \( f(k) \) and set it equal to zero:
\[ f(k) = k^2 - k \cdot k + k \]

3. **Simplify the Expression**:
Substitute \( k \) into the polynomial:
\[ f(k) = k^2 - k^2 + k \]
This simplifies to:
\[ f(k) = k \]

4. **Set the Expression to Zero**:
For \( f(k) \) to be zero, we set:
\[ k = 0 \]

5. **Conclusion**:
The value of \( k \) for which the polynomial \( x^2 - kx + k \) is divisible by \( x - k \) is:
\[ k = 0 \]

### Final Answer:
The value of \( k \) is \( 0 \).

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