If `a + b + c = 0` , then the value of `a^3 + b^3 + c^3` is :

Options

A. 0
B. abc
C. 3abc
D. none of these

Answer

To solve the problem where \( a + b + c = 0 \) and we need to find the value of \( a^3 + b^3 + c^3 \), we can use the identity for the sum of cubes. The identity states:

\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]

### Step-by-step Solution:

1. **Identify the Given Information**:
We know that \( a + b + c = 0 \).

2. **Substitute into the Identity**:
Substitute \( a + b + c = 0 \) into the identity:
\[ a^3 + b^3 + c^3 - 3abc = (0)(a^2 + b^2 + c^2 - ab - ac - bc) \]

3. **Simplify the Equation**:
Since \( (0)(\text{anything}) = 0 \), we have:
\[ a^3 + b^3 + c^3 - 3abc = 0 \]

4. **Rearrange the Equation**:
Rearranging gives us:
\[ a^3 + b^3 + c^3 = 3abc \]

5. **Conclusion**:
Therefore, the value of \( a^3 + b^3 + c^3 \) is \( 3abc \).

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