If `x^3 + 5x^2+10k` leaves remainder -2x when divided by `x^2+2` then the value of k is:

Options

A. -2
B. -1
C. 1
D. 2

Answer

To solve the problem, we need to find the value of \( k \) such that the polynomial \( f(x) = x^3 + 5x^2 + 10k \) leaves a remainder of \( -2x \) when divided by \( x^2 + 2 \).

### Step-by-Step Solution:

1. **Understand the Division**:
When dividing a polynomial \( f(x) \) by \( g(x) \) (where \( g(x) = x^2 + 2 \)), the remainder will be a polynomial of degree less than the degree of \( g(x) \). Since \( g(x) \) is a quadratic polynomial, the remainder will be linear, of the form \( ax + b \).

2. **Set Up the Equation**:
According to the problem, the remainder when \( f(x) \) is divided by \( g(x) \) is given as \( -2x \). Therefore, we can write:
\[ f(x) = (x^2 + 2)Q(x) - 2x \]
for some polynomial \( Q(x) \).

3. **Substituting Values**:
We need to evaluate \( f(x) \) at specific values that satisfy \( x^2 + 2 = 0 \). The roots of \( x^2 + 2 = 0 \) are \( x = i\sqrt{2} \) and \( x = -i\sqrt{2} \). However, we can also evaluate \( f(x) \) directly using the expression for \( x^2 \).

4. **Substituting \( x^2 = -2 \)**:
Since \( x^2 = -2 \), we can express \( x^3 \) in terms of \( x^2 \):
\[ x^3 = x \cdot x^2 = x \cdot (-2) = -2x \]
Now substitute \( x^2 = -2 \) into \( f(x) \):
\[ f(x) = x^3 + 5x^2 + 10k = -2x + 5(-2) + 10k \]
Simplifying this gives:
\[ f(x) = -2x - 10 + 10k \]

5. **Setting the Equation Equal to the Remainder**:
Since we know that \( f(x) \) should equal \( -2x \) when substituting \( x^2 = -2 \):
\[ -2x - 10 + 10k = -2x \]

6. **Solving for \( k \)**:
To isolate \( k \), we can simplify the equation:
\[ -10 + 10k = 0 \]
Adding \( 10 \) to both sides:
\[ 10k = 10 \]
Dividing both sides by \( 10 \):
\[ k = 1 \]

### Final Answer:
The value of \( k \) is \( 1 \).