The value of a for which `(x+a)` is a factor of the polynomial `x^(3)+ax^(2)-2x+a+6` is

Options

A. `4`
B. `2`
C. `-4`
D. `-2`

Answer

To find the value of \( a \) for which \( (x + a) \) is a factor of the polynomial \( P(x) = x^3 + ax^2 - 2x + a + 6 \), we can follow these steps:

### Step 1: Understand the Factor Condition
Since \( (x + a) \) is a factor of the polynomial, it implies that if we substitute \( x = -a \) into the polynomial, the result should be zero. This is based on the Factor Theorem.

### Step 2: Substitute \( x = -a \) into the Polynomial
We will substitute \( x = -a \) into the polynomial \( P(x) \):
\[ P(-a) = (-a)^3 + a(-a)^2 - 2(-a) + a + 6 \]

### Step 3: Simplify the Expression
Now, we simplify \( P(-a) \):
\[ P(-a) = -a^3 + a(a^2) + 2a + a + 6 \]
\[ = -a^3 + a^3 + 2a + a + 6 \]
\[ = 0 + 3a + 6 \]
\[ = 3a + 6 \]

### Step 4: Set the Expression to Zero
Since \( P(-a) \) must equal zero for \( (x + a) \) to be a factor, we set the expression to zero:
\[ 3a + 6 = 0 \]

### Step 5: Solve for \( a \)
Now, we solve for \( a \):
\[ 3a = -6 \]
\[ a = -2 \]

### Conclusion
Thus, the value of \( a \) for which \( (x + a) \) is a factor of the polynomial is:
\[ \boxed{-2} \]