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

Options

A. 0
B. 1
C. `-1`
D. 3

Answer

To solve the expression

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

we will simplify it step by step.

### Step 1: Expand the numerator

The numerator consists of two parts: \((a^2 + b^2)(a - b)\) and \(-(a^3 - b^3)\). We will first expand each part.

1. Expand \((a^2 + b^2)(a - b)\):
\[ (a^2 + b^2)(a - b) = a^3 - a^2b + b^2a - b^3. \]

2. The second part is \(-(a^3 - b^3)\), which can be rewritten using the formula for the difference of cubes:
\[ -(a^3 - b^3) = -((a - b)(a^2 + ab + b^2)). \]

So, the numerator becomes:
\[ a^3 - a^2b + b^2a - b^3 - (a^3 - b^3) = a^3 - a^2b + b^2a - b^3 - a^3 + b^3. \]

### Step 2: Combine like terms in the numerator

Now, combine the terms in the numerator:
\[ -a^2b + b^2a. \]

### Step 3: Factor the numerator

The expression can be factored:
\[ -a^2b + b^2a = b(a^2 - b^2) = b(a - b)(a + b). \]

### Step 4: Simplify the denominator

The denominator is:
\[ a^2b - ab^2 = ab(a - b). \]

### Step 5: Substitute back into the expression

Now substituting the factored forms back into the expression, we have:
\[ \frac{b(a - b)(a + b)}{ab(a - b)}. \]

### Step 6: Cancel common terms

We can cancel \((a - b)\) from the numerator and denominator (assuming \(a \neq b\)):
\[ \frac{b(a + b)}{ab}. \]

### Step 7: Simplify further

This simplifies to:
\[ \frac{a + b}{a}. \]

### Step 8: Final simplification

This can be further simplified to:
\[ 1 + \frac{b}{a}. \]

### Conclusion

Thus, the final simplified value of the expression is:
\[ 1 + \frac{b}{a}. \]