The vlue of k, for which the expression `x^(3)+4x^(2)+5x-k` is divisible by `x-2` is

Options

A. 16
B. 18
C. 24
D. 34

Answer

To find the value of \( k \) for which the expression \( x^3 + 4x^2 + 5x - k \) is divisible by \( x - 2 \), we can use the Remainder Theorem. According to the theorem, if a polynomial \( f(x) \) is divisible by \( x - c \), then \( f(c) = 0 \).

### Step-by-Step Solution:

1. **Identify the polynomial**:
The polynomial given is \( f(x) = x^3 + 4x^2 + 5x - k \).

2. **Set up the equation using the Remainder Theorem**:
Since we want \( f(2) = 0 \) (because \( x - 2 \) is a factor), we substitute \( x = 2 \) into the polynomial.
\[ f(2) = 2^3 + 4(2^2) + 5(2) - k \]

3. **Calculate \( f(2) \)**:
\[ f(2) = 2^3 + 4(2^2) + 5(2) - k = 8 + 4(4) + 10 - k \]
Simplifying further:
\[ f(2) = 8 + 16 + 10 - k = 34 - k \]

4. **Set the equation to zero**:
Since we want \( f(2) = 0 \):
\[ 34 - k = 0 \]

5. **Solve for \( k \)**:
Rearranging gives:
\[ k = 34 \]

### Final Answer:
The value of \( k \) for which the expression \( x^3 + 4x^2 + 5x - k \) is divisible by \( x - 2 \) is \( k = 34 \).