If `AB+BA=0`, then which of the following is equivalent to `A^(3)-B^(3)`

Options

A. `(A-B)(A^(2)+AB+B^(2))`
B. `(A-B)(A^(2)-AB-B^(2))`
C. `(A+B)(A^(2)-AB-B^(2))`
D. `(A+B)(A^(2)+AB-B^(2))`

Explanation

To solve the problem, we need to find which of the given options is equivalent to \( A^3 - B^3 \) given that \( AB + BA = 0 \). This implies that \( AB = -BA \).

### Step-by-Step Solution:

1. **Recall the identity for the difference of cubes**:
The difference of cubes can be factored using the formula:
\[ A^3 - B^3 = (A - B)(A^2 + AB + B^2) \]
However, we need to manipulate this expression based on the given condition \( AB + BA = 0 \).

2. **Use the given condition**:
From \( AB + BA = 0 \), we can express \( AB \) in terms of \( BA \):
\[ AB = -BA \]

3. **Substituting \( AB \) in the identity**:
We can rewrite \( A^2 + AB + B^2 \) using \( AB = -BA \):
\[ A^2 + AB + B^2 = A^2 - BA + B^2 \]

4. **Rearranging the expression**:
Now, we can factor out \( A - B \) from the expression:
\[ A^3 - B^3 = (A - B)(A^2 - BA + B^2) \]

5. **Testing the options**:
We need to check which of the options matches \( (A - B)(A^2 - BA + B^2) \).

- **Option 1**: \( A - B)(A^2 + AB + B^2) \)
- **Option 2**: \( (A - B)(A^2 - AB - B^2) \)
- **Option 3**: \( (A + B)(A^2 - AB - B^2) \)
- **Option 4**: \( (A + B)(A^2 + AB - B^2) \)

6. **Verifying Option 3**:
Let's check Option 3: \( (A + B)(A^2 - AB - B^2) \):
\[ (A + B)(A^2 - AB - B^2) = A^3 - A^2B - AB^2 + BA^2 - B^3 \]
Using \( AB = -BA \):
\[ = A^3 - A^2B + A^2B - B^3 = A^3 - B^3 \]

7. **Conclusion**:
Since we have shown that Option 3 simplifies to \( A^3 - B^3 \), we conclude that:
\[ A^3 - B^3 \text{ is equivalent to } (A + B)(A^2 - AB - B^2) \]

### Final Answer:
The correct option is **Option 3**: \( (A + B)(A^2 - AB - B^2) \).