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

### Step 1: Rearrange the expression
The given expression is:
\[ x^2 - 2xy + y^2 - x + y \]
We can group the first three terms together:
\[ (x^2 - 2xy + y^2) - x + y \]

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

### Step 3: Rewrite the expression
Now, let's rewrite the expression:
\[ (x - y)^2 + (y - x) \]
This can be simplified to:
\[ (x - y)^2 - (x - y) \]

### Step 4: Factor out the common term
Now we can factor out \( (x - y) \):
\[ (x - y)((x - y) - 1) \]

### Step 5: Identify the factors
The factors of the expression \( x^2 - 2xy + y^2 - x + y \) are:
1. \( x - y \)
2. \( (x - y) - 1 \) or \( x - y + 1 \)

Thus, one of the factors of the expression is:
\[ \boxed{x - y} \]