For what value(s) of k is `x^(2)+3x+k` divisible by x+k ?

Options

A. only 0
B. only 0 or 2
C. only 0 or -4
D. no value of k

Explanation

To determine the value(s) of \( k \) for which the polynomial \( x^2 + 3x + k \) is divisible by \( x + k \), we can follow these steps:

### Step 1: Set up the problem
We know that if \( x + k \) divides \( x^2 + 3x + k \), then substituting \( x = -k \) into the polynomial \( x^2 + 3x + k \) should yield zero.

### Step 2: Substitute \( x = -k \)
Substituting \( x = -k \) into the polynomial:
\[ (-k)^2 + 3(-k) + k = 0 \]

### Step 3: Simplify the equation
Now simplify the expression:
\[ k^2 - 3k + k = 0 \]
This simplifies to:
\[ k^2 - 2k = 0 \]

### Step 4: Factor the equation
We can factor the equation:
\[ k(k - 2) = 0 \]

### Step 5: Solve for \( k \)
Setting each factor equal to zero gives us the possible solutions:
1. \( k = 0 \)
2. \( k - 2 = 0 \) which implies \( k = 2 \)

### Conclusion
The values of \( k \) for which \( x^2 + 3x + k \) is divisible by \( x + k \) are:
\[ k = 0 \quad \text{or} \quad k = 2 \]