What are the factors of `[(a-b)-a^(3)+b^(3)]`?

Options

A. `(a-b)(1-a^(2)-b^(2)+ab)`
B. `(a-b)(1-a^(2)+b^(2)-ab)`
C. `(a-b)(1-a^(2)-b^(2)-ab)`
D. None of these

Answer

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

### Step 1: Rewrite the expression
Start with the original expression:
\[ (a - b) - a^3 + b^3 \]
This can be rearranged to:
\[ (a - b) + (b^3 - a^3) \]

### Step 2: Factor the difference of cubes
Recall the formula for the difference of cubes:
\[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) \]
In our case, let \(x = b\) and \(y = a\):
\[ b^3 - a^3 = (b - a)(b^2 + ab + a^2) \]
Since \(b - a = -(a - b)\), we can rewrite it as:
\[ b^3 - a^3 = -(a - b)(b^2 + ab + a^2) \]

### Step 3: Substitute back into the expression
Now substitute this back into the expression:
\[ (a - b) + (-(a - b)(b^2 + ab + a^2)) \]
This simplifies to:
\[ (a - b)(1 - (b^2 + ab + a^2)) \]

### Step 4: Final simplification
Thus, the expression can be factored as:
\[ (a - b)(1 - b^2 - ab - a^2) \]

### Conclusion
The factors of the expression \((a - b) - a^3 + b^3\) are:
\[ (a - b)(1 - b^2 - ab - a^2) \]

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