If `x^(3) +ax^(2) + 2x+3` is exactly divisible by `(x+1)`. Find the value of a

Answer

To find the value of \( a \) such that the polynomial \( f(x) = x^3 + ax^2 + 2x + 3 \) is exactly divisible by \( x + 1 \), we can follow these steps:

### Step 1: Understand the condition for divisibility
Since \( x + 1 \) is a factor of the polynomial, it means that when we substitute \( x = -1 \) into the polynomial, the result should be zero.

### Step 2: Substitute \( x = -1 \) into the polynomial
We substitute \( x = -1 \) into the polynomial:
\[ f(-1) = (-1)^3 + a(-1)^2 + 2(-1) + 3 \]

### Step 3: Simplify the expression
Calculating each term:
- \( (-1)^3 = -1 \)
- \( a(-1)^2 = a \)
- \( 2(-1) = -2 \)
- The constant term is \( 3 \)

So, we have:
\[ f(-1) = -1 + a - 2 + 3 \]

### Step 4: Combine like terms
Now, combine the constants:
\[ f(-1) = (-1 - 2 + 3) + a = 0 + a = a \]

### Step 5: Set the equation to zero
Since \( f(-1) \) must equal zero for \( x + 1 \) to be a factor:
\[ a = 0 \]

### Conclusion
Thus, the value of \( a \) is \( 0 \).

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