If `a+b+c=0` then ` (a^3+b^3+c^3)` is

Options

A. 0
B. abc
C. 2abc
D. 3abc

Explanation

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

### Step-by-Step Solution:

1. **Start with the identity**:
We know the identity for the sum of cubes:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]

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

3. **Simplify the equation**:
The right side of the equation becomes zero:
\[ a^3 + b^3 + c^3 - 3abc = 0 \]
This simplifies to:
\[ a^3 + b^3 + c^3 = 3abc \]

4. **Final result**:
Therefore, we conclude that:
\[ a^3 + b^3 + c^3 = 3abc \]

### Conclusion:
Thus, if \( a + b + c = 0 \), then \( a^3 + b^3 + c^3 = 3abc \).