Value of `(x-y)^(3) +(y-z)^(3) +(z-x)^(3)` is

Options

A. `3xyz `
B. `3(x+y) (y+z) (z+x`
C. `3(x-y) (y-z) (z-x)`
D. 0

Answer

To find the value of \( (x-y)^3 + (y-z)^3 + (z-x)^3 \), we can use the property of cubes when the sum of the terms is zero.

### Step-by-Step Solution:

1. **Identify the Terms**:
Let \( A = x - y \), \( B = y - z \), and \( C = z - x \).
We want to find \( A^3 + B^3 + C^3 \).

2. **Sum of the Terms**:
Calculate \( A + B + C \):
\[ A + B + C = (x - y) + (y - z) + (z - x) \]
Simplifying this, we see:
\[ = x - y + y - z + z - x = 0 \]

3. **Using the Identity**:
Since \( A + B + C = 0 \), we can use the identity:
\[ A^3 + B^3 + C^3 = 3ABC \]
This holds true when the sum of the terms is zero.

4. **Calculate \( ABC \)**:
Now, we need to calculate \( ABC \):
\[ ABC = (x - y)(y - z)(z - x) \]

5. **Final Expression**:
Therefore, we have:
\[ A^3 + B^3 + C^3 = 3ABC = 3(x - y)(y - z)(z - x) \]

### Conclusion:
The value of \( (x-y)^3 + (y-z)^3 + (z-x)^3 \) is:
\[ 3(x - y)(y - z)(z - x) \]