The polynomials `a^(2)-b+ab-a`, on factorization, reduces to ___________.

Options

A. `(a+b)(a^(2)+b^(2))`
B. `(a+b)(a+1)`
C. `(a-b)(a+b)`
D. `(a+b)(a-1)`

Answer

To factor the polynomial \( a^2 - b + ab - a \), we can follow these steps:

### Step 1: Rearrange the terms
We start by rearranging the polynomial:
\[ a^2 - a + ab - b \]

### Step 2: Group the terms
Next, we group the terms in pairs:
\[ (a^2 - a) + (ab - b) \]

### Step 3: Factor out common factors from each group
Now, we factor out the common factors from each group:
- From the first group \( a^2 - a \), we can factor out \( a \):
\[ a(a - 1) \]
- From the second group \( ab - b \), we can factor out \( b \):
\[ b(a - 1) \]

### Step 4: Combine the factored terms
Now we can combine the factored terms:
\[ a(a - 1) + b(a - 1) \]

### Step 5: Factor out the common binomial factor
We notice that \( (a - 1) \) is a common factor:
\[ (a - 1)(a + b) \]

### Final Answer
Thus, the polynomial \( a^2 - b + ab - a \) on factorization reduces to:
\[ (a - 1)(a + b) \]

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