When P (x) = `2x^(3) - 6x^(2) + Kx` is divided by x + 2 , the remainder is `-10` . Then K =

Options

A. `-30`
B. `-15`
C. `-6`
D. `-1`

Explanation

To solve the problem, we will use the Remainder Theorem, which states that when a polynomial \( P(x) \) is divided by \( x - a \), the remainder is \( P(a) \).

### Step-by-Step Solution:

1. **Identify the Polynomial and the Divisor**:
The polynomial is given as:
\[ P(x) = 2x^3 - 6x^2 + Kx \]
The divisor is \( x + 2 \), which can be rewritten as \( x - (-2) \). Thus, \( a = -2 \).

2. **Apply the Remainder Theorem**:
According to the Remainder Theorem, the remainder when \( P(x) \) is divided by \( x + 2 \) is \( P(-2) \). We know from the problem that this remainder is \( -10 \):
\[ P(-2) = -10 \]

3. **Substitute \(-2\) into the Polynomial**:
Now we substitute \(-2\) into the polynomial:
\[ P(-2) = 2(-2)^3 - 6(-2)^2 + K(-2) \]

4. **Calculate Each Term**:
- Calculate \( 2(-2)^3 \):
\[ 2(-2)^3 = 2 \times (-8) = -16 \]
- Calculate \( -6(-2)^2 \):
\[ -6(-2)^2 = -6 \times 4 = -24 \]
- Calculate \( K(-2) \):
\[ K(-2) = -2K \]

5. **Combine the Results**:
Now, combine these results:
\[ P(-2) = -16 - 24 - 2K \]
This simplifies to:
\[ P(-2) = -40 - 2K \]

6. **Set Up the Equation**:
We know from the problem that \( P(-2) = -10 \):
\[ -40 - 2K = -10 \]

7. **Solve for \( K \)**:
Rearranging the equation gives:
\[ -2K = -10 + 40 \]
\[ -2K = 30 \]
Dividing both sides by -2:
\[ K = -15 \]

### Final Answer:
Thus, the value of \( K \) is:
\[ \boxed{-15} \]