Factorise :
`a-b - a^(3) + b^(3)`

Answer

To factorise the expression \( a - b - a^3 + b^3 \), we can follow these steps:

### Step 1: Rearrange the expression
We start with the expression:
\[ a - b - a^3 + b^3 \]
We can rearrange it as:
\[ a - b + b^3 - a^3 \]

### Step 2: Group the terms
Next, we can group the terms:
\[ (a - b) + (b^3 - a^3) \]

### Step 3: Factor the difference of cubes
We know that \( b^3 - a^3 \) can be factored using the difference of cubes formula:
\[ b^3 - a^3 = (b - a)(b^2 + ba + a^2) \]
Since \( b - a = -(a - b) \), we can rewrite it as:
\[ b^3 - a^3 = -(a - b)(b^2 + ba + a^2) \]

### Step 4: Substitute back into the expression
Now we substitute this back into our grouped expression:
\[ (a - b) - (a - b)(b^2 + ba + a^2) \]

### Step 5: Factor out \( a - b \)
We can factor out \( (a - b) \):
\[ (a - b)(1 - (b^2 + ba + a^2)) \]

### Step 6: Simplify the expression
This simplifies to:
\[ (a - b)(1 - b^2 - ab - a^2) \]

### Final Answer
Thus, the factorised form of the expression \( a - b - a^3 + b^3 \) is:
\[ (a - b)(1 - b^2 - ab - a^2) \]

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