When `x^3 + 5x^2 +10k` is divided by `(x^2+2) ` the remainder obtained is `- 2x `, then the value of `k` is

Options

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

Answer

To find the value of \( k \) in the expression \( x^3 + 5x^2 + 10k \) when it is divided by \( x^2 + 2 \) and the remainder is \( -2x \), we can follow these steps:

### Step 1: Set up the equation
We know that when \( x^3 + 5x^2 + 10k \) is divided by \( x^2 + 2 \), the remainder is \( -2x \). Therefore, we can express this as:
\[ x^3 + 5x^2 + 10k = (x^2 + 2)Q(x) - 2x \]
where \( Q(x) \) is the quotient.

### Step 2: Rearrange the equation
Rearranging the equation gives us:
\[ x^3 + 5x^2 + 10k + 2x = (x^2 + 2)Q(x) \]

### Step 3: Simplify the left-hand side
Now, we can combine the terms on the left:
\[ x^3 + 5x^2 + 2x + 10k = (x^2 + 2)Q(x) \]

### Step 4: Divide \( x^3 + 5x^2 + 2x + 10k \) by \( x^2 + 2 \)
We will now perform polynomial long division of \( x^3 + 5x^2 + 2x + 10k \) by \( x^2 + 2 \).

1. Divide the leading term \( x^3 \) by \( x^2 \) to get \( x \).
2. Multiply \( x \) by \( x^2 + 2 \) to get \( x^3 + 2x \).
3. Subtract this from \( x^3 + 5x^2 + 2x + 10k \):
\[ (x^3 + 5x^2 + 2x + 10k) - (x^3 + 2x) = 5x^2 + 10k \]

### Step 5: Continue the division
Next, we take \( 5x^2 \):
1. Divide \( 5x^2 \) by \( x^2 \) to get \( 5 \).
2. Multiply \( 5 \) by \( x^2 + 2 \) to get \( 5x^2 + 10 \).
3. Subtract this from \( 5x^2 + 10k \):
\[ (5x^2 + 10k) - (5x^2 + 10) = 10k - 10 \]

### Step 6: Set the remainder to zero
Since the remainder must be zero (as we are dividing completely), we set:
\[ 10k - 10 = 0 \]

### Step 7: Solve for \( k \)
Now, we solve for \( k \):
\[ 10k = 10 \implies k = 1 \]

### Final Answer
Thus, the value of \( k \) is:
\[ \boxed{1} \]