The expression `x^2-2xy+y^2-x+y` has one factor which is

Options

A. `x-y+1`
B. `x+y-1`
C. `x-y-1`
D. `x+y+1`

Answer

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

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

### Step 2: Recognize a perfect square
The first part of the expression \( x^2 - 2xy + y^2 \) can be recognized as a perfect square:
\[ x^2 - 2xy + y^2 = (x - y)^2 \]
So we can rewrite the expression as:
\[ (x - y)^2 - x + y \]

### Step 3: Factor out common terms
Now, we can factor out the common terms. Notice that we can rewrite \( -x + y \) as \( -(x - y) \):
\[ (x - y)^2 - (x - y) \]

### Step 4: Factor using the difference of squares
Now we can factor the expression:
\[ (x - y)((x - y) - 1) \]
This gives us:
\[ (x - y)(x - y - 1) \]

### Conclusion
Thus, one factor of the expression \( x^2 - 2xy + y^2 - x + y \) is:
\[ x - y - 1 \]