If `a + b = 1.` Find the value of `a ^(3) + b ^(3) - ab - (a ^(2) - b ^(2) ) ^(2)`

Options

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

Answer

To solve the problem, we need to find the value of the expression \( a^3 + b^3 - ab - (a^2 - b^2)^2 \) given that \( a + b = 1 \).

### Step 1: Use the identity for \( a^3 + b^3 \)
We can use the identity:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
Since \( a + b = 1 \), we can substitute:
\[ a^3 + b^3 = 1 \cdot (a^2 - ab + b^2) = a^2 - ab + b^2 \]

### Step 2: Express \( a^2 + b^2 \)
We can express \( a^2 + b^2 \) using the square of the sum:
\[ a^2 + b^2 = (a + b)^2 - 2ab = 1^2 - 2ab = 1 - 2ab \]

### Step 3: Substitute \( a^2 + b^2 \) into the expression for \( a^3 + b^3 \)
Now we substitute \( a^2 + b^2 \) into the equation from Step 1:
\[ a^3 + b^3 = (1 - 2ab) - ab = 1 - 3ab \]

### Step 4: Calculate \( (a^2 - b^2)^2 \)
We can use the identity:
\[ a^2 - b^2 = (a + b)(a - b) = 1(a - b) = a - b \]
Thus,
\[ (a^2 - b^2)^2 = (a - b)^2 \]

### Step 5: Express \( (a - b)^2 \)
Using \( a + b = 1 \):
\[ (a - b)^2 = (a + b)^2 - 4ab = 1^2 - 4ab = 1 - 4ab \]

### Step 6: Substitute back into the original expression
Now we substitute everything back into the original expression:
\[ a^3 + b^3 - ab - (a^2 - b^2)^2 = (1 - 3ab) - ab - (1 - 4ab) \]

### Step 7: Simplify the expression
Now we simplify:
\[ = 1 - 3ab - ab - 1 + 4ab \]
\[ = 1 - 1 + (-3ab - ab + 4ab) = 0 + 0ab = 0 \]

### Final Answer
Thus, the value of the expression \( a^3 + b^3 - ab - (a^2 - b^2)^2 \) is:
\[ \boxed{0} \]