Find the value of 'k' if x+3, is a factor of the polynomial `x^(4)-x^(3)-11x^(2)-x+k`.

Options

A. `k=-13`
B. `k=-12`
C. `k=-14`
D. `k=-15`

Explanation

To find the value of 'k' such that \( x + 3 \) is a factor of the polynomial \( P(x) = x^4 - x^3 - 11x^2 - x + k \), we can use the Factor Theorem. According to the theorem, if \( x + 3 \) is a factor of \( P(x) \), then \( P(-3) = 0 \).

### Step-by-Step Solution:

1. **Substituting \( x = -3 \) into the polynomial:**
\[ P(-3) = (-3)^4 - (-3)^3 - 11(-3)^2 - (-3) + k \]

2. **Calculating each term:**
- Calculate \( (-3)^4 \):
\[ (-3)^4 = 81 \]
- Calculate \( -(-3)^3 \):
\[ -(-3)^3 = -(-27) = 27 \]
- Calculate \( -11(-3)^2 \):
\[ -11(-3)^2 = -11(9) = -99 \]
- Calculate \( -(-3) \):
\[ -(-3) = 3 \]

3. **Putting it all together:**
\[ P(-3) = 81 + 27 - 99 + 3 + k \]

4. **Simplifying the expression:**
\[ P(-3) = (81 + 27 + 3 - 99) + k \]
\[ = (111 - 99) + k \]
\[ = 12 + k \]

5. **Setting the polynomial equal to zero:**
Since \( P(-3) = 0 \):
\[ 12 + k = 0 \]

6. **Solving for \( k \):**
\[ k = -12 \]

### Final Answer:
The value of \( k \) is \( -12 \).

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