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. **Use the Identity for Cubes**:
We start with the identity:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]

2. **Substitute the Given Condition**:
Since we know that \( a + b + c = 0 \), we can substitute this into our 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 because anything multiplied by zero is zero:
\[ a^3 + b^3 + c^3 - 3abc = 0 \]

4. **Isolate \( a^3 + b^3 + c^3 \)**:
Now, we can rearrange the equation to solve for \( a^3 + b^3 + c^3 \):
\[ a^3 + b^3 + c^3 = 3abc \]

5. **Final Result**:
Thus, we conclude that:
\[ a^3 + b^3 + c^3 = 3abc \]

### Conclusion:
The value of \( a^3 + b^3 + c^3 \) is \( 3abc \).