If a + b = 1, then 'a^(4) + b^(4) - a^(3) - b^(3) – 2a ^(2)b^(2)' + ab is equal to

Options

A. 1
B. 2
C. 4
D. 0

Answer

To solve the problem, we need to simplify the expression \( a^4 + b^4 - a^3 - b^3 - 2a^2b^2 + ab \) given that \( a + b = 1 \).

### Step-by-Step Solution:

1. **Start with the given expression:**
\[ a^4 + b^4 - a^3 - b^3 - 2a^2b^2 + ab \]

2. **Use the identity for \( a^4 + b^4 \):**
The identity states that:
\[ a^4 + b^4 = (a^2 + b^2)^2 - 2a^2b^2 \]
We can express \( a^2 + b^2 \) using \( (a + b)^2 \):
\[ a^2 + b^2 = (a + b)^2 - 2ab = 1^2 - 2ab = 1 - 2ab \]
So,
\[ a^4 + b^4 = (1 - 2ab)^2 - 2a^2b^2 \]

3. **Substitute \( a^4 + b^4 \) back into the expression:**
\[ (1 - 2ab)^2 - 2a^2b^2 - a^3 - b^3 + ab \]

4. **Use the identity for \( a^3 + b^3 \):**
The identity states that:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) = 1((1 - 2ab) - ab) = 1 - 3ab \]
Thus,
\[ a^3 + b^3 = 1 - 3ab \]

5. **Substituting \( a^3 + b^3 \) into the expression:**
\[ (1 - 2ab)^2 - 2a^2b^2 - (1 - 3ab) + ab \]

6. **Simplify the expression:**
\[ (1 - 2ab)^2 - 2a^2b^2 - 1 + 3ab + ab \]
\[ = (1 - 2ab)^2 - 2a^2b^2 + 4ab - 1 \]

7. **Expand \( (1 - 2ab)^2 \):**
\[ = 1 - 4ab + 4a^2b^2 - 2a^2b^2 + 4ab - 1 \]
\[ = 1 - 1 + (4a^2b^2 - 2a^2b^2) + 0 \]
\[ = 2a^2b^2 - 2a^2b^2 = 0 \]

8. **Final Result:**
\[ a^4 + b^4 - a^3 - b^3 - 2a^2b^2 + ab = 0 \]

### Conclusion:
The value of the expression is \( 0 \).