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

Options

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

Answer

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 can use the algebraic identity:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]
This identity relates the sum of cubes to the sum of the variables and their products.

2. **Substitute the Given 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. **Rearranging the Equation**:
Rearranging the equation gives us:
\[ a^3 + b^3 + c^3 = 3abc \]

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

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

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