If `x - lambda` is a factor of `x^(4)-lambda x^(3)+2x+6`. Then the value of `lambda` is

Options

A. `2`
B. `-2`
C. `-3`
D. `3`

Explanation

To solve the problem, we need to determine the value of \( \lambda \) such that \( x - \lambda \) is a factor of the polynomial \( x^4 - \lambda x^3 + 2x + 6 \).

### Step-by-Step Solution:

1. **Identify the condition for a factor**: Since \( x - \lambda \) is a factor of the polynomial, it implies that when we substitute \( x = \lambda \) into the polynomial, the result should be zero. Therefore, we need to evaluate the polynomial at \( x = \lambda \).

2. **Substitute \( x = \lambda \)**: We substitute \( x \) with \( \lambda \) in the polynomial:
\[ \lambda^4 - \lambda(\lambda^3) + 2\lambda + 6 = 0 \]

3. **Simplify the equation**: The equation simplifies as follows:
\[ \lambda^4 - \lambda^4 + 2\lambda + 6 = 0 \]
Here, \( \lambda^4 - \lambda^4 \) cancels out.

4. **Rearranging the equation**: We are left with:
\[ 2\lambda + 6 = 0 \]

5. **Solve for \( \lambda \)**: To find \( \lambda \), we isolate it:
\[ 2\lambda = -6 \]
Dividing both sides by 2 gives:
\[ \lambda = -3 \]

6. **Conclusion**: The value of \( \lambda \) is \( -3 \).

### Final Answer:
\[ \lambda = -3 \]