`x^(3) + xy^(2) - x^(2) y - y^(3) = "_______"`

Options

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

Explanation

To factor the expression \( x^3 + xy^2 - x^2y - y^3 \), we will follow these steps:

### Step 1: Group the terms
We can group the terms in pairs to make factoring easier. The expression can be rewritten as:
\[ (x^3 - x^2y) + (xy^2 - y^3) \]

### Step 2: Factor out common terms from each group
Now, we will factor out the common terms from each group:
- From the first group \( x^3 - x^2y \), we can factor out \( x^2 \):
\[ x^2(x - y) \]
- From the second group \( xy^2 - y^3 \), we can factor out \( y^2 \):
\[ y^2(x - y) \]

### Step 3: Combine the factored terms
Now we can rewrite the expression using the factored terms:
\[ x^2(x - y) + y^2(x - y) \]

### Step 4: Factor out the common binomial
Notice that both terms have a common factor of \( (x - y) \):
\[ (x - y)(x^2 + y^2) \]

### Final Result
Thus, the factored form of the expression \( x^3 + xy^2 - x^2y - y^3 \) is:
\[ (x - y)(x^2 + y^2) \]