Factorise:
`ab ^(2) - bc^(2) - ab + c^(2)`

Answer

To factorise the expression \( ab^2 - bc^2 - ab + c^2 \), we will follow these steps:

### Step 1: Group the terms
We can group the terms in pairs to make it easier to factor:
\[ (ab^2 - ab) + (-bc^2 + c^2) \]

### Step 2: Factor out the common factors from each group
In the first group \( ab^2 - ab \), we can factor out \( ab \):
\[ ab(b - 1) \]
In the second group \( -bc^2 + c^2 \), we can factor out \( -c^2 \):
\[ -c^2(b - 1) \]

### Step 3: Rewrite the expression with the factored groups
Now we can rewrite the expression using the factored groups:
\[ ab(b - 1) - c^2(b - 1) \]

### Step 4: Factor out the common binomial factor
Now we see that \( (b - 1) \) is a common factor:
\[ (b - 1)(ab - c^2) \]

### Final Answer
Thus, the factorised form of the expression \( ab^2 - bc^2 - ab + c^2 \) is:
\[ (b - 1)(ab - c^2) \]

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