Value of k for which x + k is a factor of the polynomial `x^(3)+kx^(2)-2x+k+4` is

Options

A. `-(4)/(3)`
B. `(3)/(4)`
C. `-(3)/(4)`
D. `(4)/(3)`

Explanation

To find the value of \( k \) for which \( x + k \) is a factor of the polynomial \( f(x) = x^3 + kx^2 - 2x + k + 4 \), we can follow these steps:

### Step 1: Understand the Factor Condition
Since \( x + k \) is a factor of the polynomial, by the Factor Theorem, we know that \( f(-k) = 0 \).

### Step 2: Substitute \( -k \) into the Polynomial
We will substitute \( x = -k \) into the polynomial:
\[ f(-k) = (-k)^3 + k(-k)^2 - 2(-k) + k + 4 \]

### Step 3: Simplify the Expression
Now, we simplify each term:
- The first term: \( (-k)^3 = -k^3 \)
- The second term: \( k(-k)^2 = k(k^2) = k^3 \)
- The third term: \( -2(-k) = 2k \)
- The fourth term: \( k \)
- The fifth term: \( 4 \)

Putting it all together, we have:
\[ f(-k) = -k^3 + k^3 + 2k + k + 4 \]

### Step 4: Combine Like Terms
Now, combine the like terms:
\[ f(-k) = (-k^3 + k^3) + (2k + k) + 4 = 0 + 3k + 4 \]
Thus, we have:
\[ f(-k) = 3k + 4 \]

### Step 5: Set the Expression Equal to Zero
Since \( f(-k) = 0 \), we set the expression equal to zero:
\[ 3k + 4 = 0 \]

### Step 6: Solve for \( k \)
Now, we solve for \( k \):
\[ 3k = -4 \\ k = -\frac{4}{3} \]

### Conclusion
The value of \( k \) for which \( x + k \) is a factor of the polynomial \( x^3 + kx^2 - 2x + k + 4 \) is:
\[ \boxed{-\frac{4}{3}} \]