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

Options

A. `3a(a+b) (b+c) `
B. `3a (a+b) (c +a)`
C. `3a (b+c) (c+a)`
D. `3(a+b) ( b+c) (c+a)`

Answer

To find the value of \( a^3 + b^3 + c^3 \) given that \( a + b + c = 0 \), we can use the algebraic identity for the sum of cubes.

### Step-by-step Solution:

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

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

3. **Rearrange the equation**:
To isolate \( a^3 + b^3 + c^3 \), we can add \( 3abc \) to both sides:
\[ a^3 + b^3 + c^3 = 3abc \]

4. **Conclusion**:
Thus, the value of \( a^3 + b^3 + c^3 \) is:
\[ \boxed{3abc} \]