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

Options

A. `(x^(2) + y^(2))(x+y)`
B. `(x^(2) + y^(2))(x-1)`
C. `(x-y)(x+y)^(2)`
D. `(x+y)(x-y)^(2)`

Answer

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

### Step 1: Group the terms
We can rearrange the expression to group similar terms together:
\[ x^3 - x^2 + xy^2 - y^2 \]

### Step 2: Factor by grouping
Now we can factor by grouping. We will take the first two terms and the last two terms separately:
\[ (x^3 - x^2) + (xy^2 - y^2) \]

### Step 3: Factor out the common factors
From the first group \( (x^3 - x^2) \), we can factor out \( x^2 \):
\[ x^2(x - 1) \]
From the second group \( (xy^2 - y^2) \), we can factor out \( y^2 \):
\[ y^2(x - 1) \]

### Step 4: Combine the factored terms
Now we have:
\[ x^2(x - 1) + y^2(x - 1) \]
We can see that \( (x - 1) \) is a common factor.

### Step 5: Factor out the common factor
Factoring out \( (x - 1) \) gives us:
\[ (x - 1)(x^2 + y^2) \]

### Final Answer
Thus, the factorized form of the expression \( x^3 + xy^2 - x^2 - y^2 \) is:
\[ (x - 1)(x^2 + y^2) \]

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