If `x^(3) + 3x^(2)-kx + 4` is divided by (x-2), then remainder is k. Find the value of k.

Options

A. 4
B. 6
C. 8
D. 10

Answer

To solve the problem, we need to find the value of \( k \) such that when the polynomial \( f(x) = x^3 + 3x^2 - kx + 4 \) is divided by \( x - 2 \), the remainder is equal to \( k \).

### Step-by-Step Solution:

1. **Identify the polynomial and the divisor:**
The polynomial is \( f(x) = x^3 + 3x^2 - kx + 4 \) and we are dividing it by \( x - 2 \).

2. **Use the Remainder Theorem:**
According to the Remainder Theorem, the remainder of the division of a polynomial \( f(x) \) by \( x - a \) is \( f(a) \). Here, \( a = 2 \).

3. **Calculate \( f(2) \):**
We need to substitute \( x = 2 \) into the polynomial:
\[ f(2) = (2)^3 + 3(2)^2 - k(2) + 4 \]
Simplifying this gives:
\[ f(2) = 8 + 3(4) - 2k + 4 \]
\[ f(2) = 8 + 12 - 2k + 4 \]
\[ f(2) = 24 - 2k \]

4. **Set the remainder equal to \( k \):**
According to the problem, the remainder when \( f(x) \) is divided by \( x - 2 \) is \( k \). Therefore, we set up the equation:
\[ 24 - 2k = k \]

5. **Solve for \( k \):**
Rearranging the equation gives:
\[ 24 = k + 2k \]
\[ 24 = 3k \]
Now, divide both sides by 3:
\[ k = \frac{24}{3} = 8 \]

6. **Conclusion:**
The value of \( k \) is \( 8 \).

### Final Answer:
The value of \( k \) is \( 8 \).

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