factorize : `x^2-(a+b)x+ab`

Answer

To factorize the expression \( x^2 - (a + b)x + ab \), we can follow these steps:

### Step 1: Identify the coefficients
The expression is in the standard quadratic form \( ax^2 + bx + c \). Here, we have:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = -(a + b) \) (coefficient of \( x \))
- \( c = ab \) (constant term)

### Step 2: Find two numbers that multiply and add
We need to find two numbers that multiply to \( c \) (which is \( ab \)) and add to \( b \) (which is \( -(a + b) \)). The two numbers we are looking for are \( -a \) and \( -b \) because:
- \( -a \times -b = ab \)
- \( -a + (-b) = -(a + b) \)

### Step 3: Rewrite the middle term
Using the two numbers found, we can rewrite the expression:
\[ x^2 - (a + b)x + ab = x^2 - ax - bx + ab \]

### Step 4: Group the terms
Now, we can group the terms:
\[ = (x^2 - ax) + (-bx + ab) \]

### Step 5: Factor by grouping
Next, we will factor out the common terms from each group:
- From the first group \( (x^2 - ax) \), we can factor out \( x \):
\[ x(x - a) \]
- From the second group \( (-bx + ab) \), we can factor out \( -b \):
\[ -b(x - b) \]

### Step 6: Combine the factors
Now, we can combine the factored terms:
\[ = x(x - a) - b(x - a) \]
Now, we can factor out \( (x - a) \):
\[ = (x - a)(x - b) \]

### Final Result
Thus, the factorization of the expression \( x^2 - (a + b)x + ab \) is:
\[ (x - a)(x - b) \]