If ` x^(3) + 2x ^(2) - 5 x + k` is divisible by ` x + 1 ` then what is the value of k ?

Options

A. `- 6`
B. `-1`
C. 0
D. 6

Answer

To find the value of \( k \) such that the polynomial \( P(x) = x^3 + 2x^2 - 5x + k \) is divisible by \( x + 1 \), we can use the Factor Theorem. According to the Factor Theorem, if a polynomial \( P(x) \) is divisible by \( x + a \), then \( P(-a) = 0 \).

### Step-by-Step Solution:

1. **Identify the polynomial and the divisor**:
We have the polynomial \( P(x) = x^3 + 2x^2 - 5x + k \) and we want to check for divisibility by \( x + 1 \).

2. **Set up the equation using the Factor Theorem**:
Since we are checking for divisibility by \( x + 1 \), we set \( a = -1 \). Therefore, we need to evaluate \( P(-1) \) and set it equal to 0:
\[ P(-1) = (-1)^3 + 2(-1)^2 - 5(-1) + k = 0 \]

3. **Calculate \( P(-1) \)**:
- Calculate \( (-1)^3 = -1 \)
- Calculate \( 2(-1)^2 = 2 \times 1 = 2 \)
- Calculate \( -5(-1) = 5 \)

Now substitute these values back into the equation:
\[ P(-1) = -1 + 2 + 5 + k = 0 \]

4. **Combine the constants**:
\[ -1 + 2 + 5 = 6 \]
So, we have:
\[ 6 + k = 0 \]

5. **Solve for \( k \)**:
\[ k = -6 \]

Thus, the value of \( k \) is \( -6 \).

### Final Answer:
\[ k = -6 \]