If `2x^(2) +kx + 8` is divided by `(x+2)` leaves remainder 3k find k =?

Answer

To solve the problem, we need to find the value of \( k \) such that when the polynomial \( 2x^2 + kx + 8 \) is divided by \( x + 2 \), it leaves a remainder of \( 3k \).

### Step-by-Step Solution:

1. **Identify the Polynomial and the Divisor**:
We have the polynomial \( f(x) = 2x^2 + kx + 8 \) 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, we need to evaluate \( f(-2) \) since we are dividing by \( x + 2 \) (which is \( x - (-2) \)).

3. **Calculate \( f(-2) \)**:
Substitute \( x = -2 \) into the polynomial:
\[ f(-2) = 2(-2)^2 + k(-2) + 8 \]
Calculate \( (-2)^2 \):
\[ f(-2) = 2(4) - 2k + 8 \]
Simplify:
\[ f(-2) = 8 - 2k + 8 = 16 - 2k \]

4. **Set the Expression Equal to the Remainder**:
According to the problem, the remainder when dividing by \( x + 2 \) is \( 3k \). Therefore, we set up the equation:
\[ 16 - 2k = 3k \]

5. **Solve for \( k \)**:
Rearranging the equation:
\[ 16 = 3k + 2k \]
Combine like terms:
\[ 16 = 5k \]
Now, divide both sides by 5:
\[ k = \frac{16}{5} \]

6. **Convert to Decimal**:
To express \( k \) in decimal form:
\[ k = 3.2 \]

### Final Answer:
Thus, the value of \( k \) is \( \frac{16}{5} \) or \( 3.2 \).