`xy^(2)- yz^(2)-xy +z^(2)` =_____

Options

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

Answer

To factor the expression \( xy^2 - yz^2 - xy + z^2 \), we can follow these steps:

### Step 1: Group the terms
We can rearrange the expression by grouping the terms in pairs:
\[ xy^2 - xy - yz^2 + z^2 \]

### Step 2: Factor out common terms from each group
Now, we can factor out common factors from each pair:
1. From the first group \( xy^2 - xy \), we can factor out \( xy \):
\[ xy(y - 1) \]
2. From the second group \( -yz^2 + z^2 \), we can factor out \( -z^2 \):
\[ -z^2(y - 1) \]

### Step 3: Combine the factored terms
Now we can combine the factored terms:
\[ xy(y - 1) - z^2(y - 1) \]

### Step 4: Factor out the common binomial factor
Notice that \( (y - 1) \) is common in both terms:
\[ (y - 1)(xy - z^2) \]

### Final Answer
Thus, the factored form of the expression \( xy^2 - yz^2 - xy + z^2 \) is:
\[ (y - 1)(xy - z^2) \]

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