One of the factors of `a^(3) - b^(3) - a^(2)b + ab^(2) + a^(2) - b^(2)` is

Options

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

Answer

To find one of the factors of the expression \( a^3 - b^3 - a^2b + ab^2 + a^2 - b^2 \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Expression**:
We start with the expression:
\[ a^3 - b^3 - a^2b + ab^2 + a^2 - b^2 \]

2. **Group Terms**:
We can rearrange the expression to group similar terms:
\[ (a^3 - b^3) + (-a^2b + ab^2) + (a^2 - b^2) \]

3. **Factor the Cubic Term**:
The difference of cubes \( a^3 - b^3 \) can be factored using the formula:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
So we rewrite the expression:
\[ (a - b)(a^2 + ab + b^2) + (-a^2b + ab^2) + (a^2 - b^2) \]

4. **Factor the Quadratic Terms**:
The term \( -a^2b + ab^2 \) can be factored as:
\[ ab(b - a) \]
And \( a^2 - b^2 \) can be factored as:
\[ (a - b)(a + b) \]

5. **Combine the Factors**:
Now we can combine the factors:
\[ (a - b)(a^2 + ab + b^2) + ab(b - a) + (a - b)(a + b) \]
Notice that \( b - a = -(a - b) \), so we can rewrite:
\[ (a - b)(a^2 + ab + b^2 - ab + a + b) \]

6. **Simplify the Expression**:
Inside the parentheses, we simplify:
\[ a^2 + b^2 + a + b \]
Therefore, we have:
\[ (a - b)(a^2 + b^2 + a + b) \]

7. **Conclusion**:
Thus, one of the factors of the expression \( a^3 - b^3 - a^2b + ab^2 + a^2 - b^2 \) is:
\[ a - b \]

### Final Answer:
One of the factors is \( a - b \).