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

Options

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

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

2. **Substitute the given condition**:
We know from the problem that \( a + b + c = 0 \). Substituting this into the identity gives:
\[ a^3 + b^3 + c^3 - 3abc = 0 \cdot (a^2 + b^2 + c^2 - ab - ac - bc) \]

3. **Simplify the equation**:
Since \( 0 \) multiplied by anything is \( 0 \), we have:
\[ a^3 + b^3 + c^3 - 3abc = 0 \]
This simplifies to:
\[ a^3 + b^3 + c^3 = 3abc \]

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

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