If `(x-2)` is a factor of `x^(3)-3x^(2)+k` , then k =

Answer

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

### Step-by-Step Solution:

1. **Identify the polynomial and the factor**:
The polynomial is \( P(x) = x^3 - 3x^2 + k \) and the factor is \( (x - 2) \).

2. **Set up the equation using the Factor Theorem**:
Since \( (x - 2) \) is a factor, we have:
\[ P(2) = 0 \]

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

4. **Calculate \( P(2) \)**:
\[ P(2) = 8 - 3 \cdot 4 + k \]
\[ P(2) = 8 - 12 + k \]
\[ P(2) = -4 + k \]

5. **Set the equation to zero**:
Since \( P(2) = 0 \), we have:
\[ -4 + k = 0 \]

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

### Final Answer:
Thus, the value of \( k \) is \( 4 \).

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