Find the value of the remainder, when ` x^(2) + (a + b) x + ab ` is divided by (x + a)

Answer

To find the value of the remainder when the polynomial \( f(x) = x^2 + (a + b)x + ab \) is divided by \( (x + a) \), we can use the Remainder Theorem. According to the Remainder Theorem, the remainder of the division of a polynomial \( f(x) \) by \( (x - c) \) is equal to \( f(c) \).

### Step-by-Step Solution:

1. **Identify the Polynomial and the Divisor**:
We have the polynomial \( f(x) = x^2 + (a + b)x + ab \) and we are dividing it by \( (x + a) \).

2. **Set the Divisor to Zero**:
To apply the Remainder Theorem, we set \( x + a = 0 \). This gives us:
\[ x = -a \]

3. **Substitute \( x = -a \) into the Polynomial**:
We need to evaluate \( f(-a) \):
\[ f(-a) = (-a)^2 + (a + b)(-a) + ab \]

4. **Calculate Each Term**:
- The first term:
\[ (-a)^2 = a^2 \]
- The second term:
\[ (a + b)(-a) = -a^2 - ab \]
- The third term:
\[ ab \]

5. **Combine the Terms**:
Now, we combine all the terms:
\[ f(-a) = a^2 - a^2 - ab + ab \]

6. **Simplify**:
Notice that \( a^2 \) and \(-a^2\) cancel out, and \(-ab\) and \(ab\) also cancel out:
\[ f(-a) = 0 \]

7. **Conclusion**:
Therefore, the remainder when \( f(x) \) is divided by \( (x + a) \) is:
\[ \text{Remainder} = 0 \]