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

Options

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

Answer

To solve the expression \( a^4 + b^4 - a^3 - b^3 - 2a^2b^2 + ab \) given that \( a + b = 1 \), we can follow these steps:

### Step 1: Substitute values for \( a \) and \( b \)
Since we know \( a + b = 1 \), we can choose specific values for \( a \) and \( b \) to simplify our calculations. A convenient choice is to let \( a = 1 \) and \( b = 0 \).

### Step 2: Calculate each term in the expression
Now we substitute \( a = 1 \) and \( b = 0 \) into the expression:

\[ a^4 + b^4 - a^3 - b^3 - 2a^2b^2 + ab \]

Substituting the values:

\[ 1^4 + 0^4 - 1^3 - 0^3 - 2(1^2)(0^2) + (1)(0) \]

Calculating each term:

- \( 1^4 = 1 \)
- \( 0^4 = 0 \)
- \( 1^3 = 1 \)
- \( 0^3 = 0 \)
- \( 2(1^2)(0^2) = 2(1)(0) = 0 \)
- \( (1)(0) = 0 \)

### Step 3: Combine the results
Now, we combine the results:

\[ 1 + 0 - 1 - 0 - 0 + 0 = 1 - 1 = 0 \]

### Conclusion
Thus, the value of the expression \( a^4 + b^4 - a^3 - b^3 - 2a^2b^2 + ab \) is \( 0 \).

### Final Answer
The answer is \( 0 \).

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