What is the value of ` (( a ^(2) + b^(2)) ( a - b ) - ( a - b) ^(3))/( a^(2) b - a b ^(2))`

Options

A. 0
B. 1
C. `-1`
D. 2

Answer

To solve the expression

\[ \frac{(a^2 + b^2)(a - b) - (a - b)^3}{a^2b - ab^2} \]

we will follow these steps:

### Step 1: Simplify the Numerator
The numerator is

\[ (a^2 + b^2)(a - b) - (a - b)^3. \]

We can factor out \( (a - b) \) from both terms:

\[ = (a - b) \left( (a^2 + b^2) - (a - b)^2 \right). \]

### Step 2: Expand \( (a - b)^2 \)
Now, we need to expand \( (a - b)^2 \):

\[ (a - b)^2 = a^2 - 2ab + b^2. \]

### Step 3: Substitute Back into the Numerator
Substituting this back into the expression gives us:

\[ = (a - b) \left( (a^2 + b^2) - (a^2 - 2ab + b^2) \right). \]

### Step 4: Simplify Inside the Parentheses
Now simplify the expression inside the parentheses:

\[ = (a - b) \left( a^2 + b^2 - a^2 + 2ab - b^2 \right) = (a - b)(2ab). \]

### Step 5: Simplify the Denominator
Now let's simplify the denominator:

\[ a^2b - ab^2 = ab(a - b). \]

### Step 6: Combine the Numerator and Denominator
Now we can substitute back into the original expression:

\[ \frac{(a - b)(2ab)}{ab(a - b)}. \]

### Step 7: Cancel Common Terms
Since \( a \neq b \) (to avoid division by zero), we can cancel \( (a - b) \):

\[ = \frac{2ab}{ab}. \]

### Step 8: Final Simplification
This simplifies to:

\[ = 2. \]

Thus, the value of the expression is

\[ \boxed{2}. \]

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