If `x + y + z= 0`, then `x^3 + y^3 + z^3` is equal to :

Options

A. 0
B. 3xyz
C. `(xy+yz+zx)/(xyz)`
D. `xyz(xy + yz + zx)`

Answer

To solve the problem, we need to find the value of \( x^3 + y^3 + z^3 \) given that \( x + y + z = 0 \).

### Step-by-Step Solution:

1. **Use the Identity for Cubes**:
We can use the identity for the sum of cubes:
\[ x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx) \]
This identity relates the sum of cubes to the sum of the variables and their products.

2. **Substitute the Given Condition**:
Since we are given that \( x + y + z = 0 \), we can substitute this into the identity:
\[ x^3 + y^3 + z^3 - 3xyz = 0 \cdot (x^2 + y^2 + z^2 - xy - yz - zx) \]
This simplifies to:
\[ x^3 + y^3 + z^3 - 3xyz = 0 \]

3. **Rearrange the Equation**:
Rearranging the equation gives us:
\[ x^3 + y^3 + z^3 = 3xyz \]

4. **Conclusion**:
Thus, we conclude that:
\[ x^3 + y^3 + z^3 = 3xyz \]

### Final Answer:
The value of \( x^3 + y^3 + z^3 \) is equal to \( 3xyz \).