If `a+b+c= 0` find `a^(3) + b^(3) + c^(3) + 3abc`

Answer

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

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ a + b + c = 0 \]

2. **Use the identity for the sum of cubes**:
The identity states that:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]

3. **Substitute the value of \( a + b + c \)**:
Since \( a + b + c = 0 \), we can substitute this into the identity:
\[ a^3 + b^3 + c^3 - 3abc = 0 \cdot (a^2 + b^2 + c^2 - ab - ac - bc) \]

4. **Simplify the equation**:
The right-hand side becomes zero:
\[ a^3 + b^3 + c^3 - 3abc = 0 \]
Therefore:
\[ a^3 + b^3 + c^3 = 3abc \]

5. **Now, find \( a^3 + b^3 + c^3 + 3abc \)**:
Substitute the result from the previous step:
\[ a^3 + b^3 + c^3 + 3abc = 3abc + 3abc \]
This simplifies to:
\[ a^3 + b^3 + c^3 + 3abc = 6abc \]

### Final Answer:
\[ a^3 + b^3 + c^3 + 3abc = 6abc \]