If `a+b+1=0` then what is the value of `(a^(3)+b^(3)+1-3ab)`?

Options

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

Answer

To solve the problem, we need to find the value of \( a^3 + b^3 + 1 - 3ab \) given that \( a + b + 1 = 0 \).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ a + b + 1 = 0 \]
From this, we can express \( a + b \):
\[ a + b = -1 \]

2. **Use the identity for \( a^3 + b^3 \)**:
The identity for the sum of cubes is:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
We can also express \( a^2 + b^2 \) using the square of the sum:
\[ a^2 + b^2 = (a + b)^2 - 2ab \]
Substituting \( a + b = -1 \):
\[ a^2 + b^2 = (-1)^2 - 2ab = 1 - 2ab \]

3. **Substituting into the identity**:
Now, substituting \( a + b \) and \( a^2 + b^2 \) into the identity for \( a^3 + b^3 \):
\[ a^3 + b^3 = (-1)((1 - 2ab) - ab) = - (1 - 2ab - ab) = - (1 - 3ab) \]
Thus, we have:
\[ a^3 + b^3 = -1 + 3ab \]

4. **Now substitute into the expression we need to evaluate**:
We need to find:
\[ a^3 + b^3 + 1 - 3ab \]
Substituting \( a^3 + b^3 \):
\[ = (-1 + 3ab) + 1 - 3ab \]

5. **Simplifying the expression**:
\[ = -1 + 3ab + 1 - 3ab = 0 \]

### Final Result:
Thus, the value of \( a^3 + b^3 + 1 - 3ab \) is:
\[ \boxed{0} \]