(y-z)^3 + (z-x)^3 + (x-y)^3 is equal to:

Options

A. a. 3(y-z)(z+x)(y-x)
B. b. (y+z)(x-z)(x-y)
C. c. 3(y-z)(z-x)(y-x)
D. d. none of these

Answer

To solve the expression \((y-z)^3 + (z-x)^3 + (x-y)^3\), we can use a specific algebraic identity.

### Step-by-Step Solution:

1. **Identify the Variables**: Let \(a = y - z\), \(b = z - x\), and \(c = x - y\).

2. **Check the Condition**: We need to check if \(a + b + c = 0\):
\[ a + b + c = (y - z) + (z - x) + (x - y) \]
Simplifying this:
\[ = y - z + z - x + x - y = 0 \]
Thus, the condition \(a + b + c = 0\) is satisfied.

3. **Apply the Identity**: According to the identity, if \(a + b + c = 0\), then:
\[ a^3 + b^3 + c^3 = 3abc \]
Therefore, we can write:
\[ (y - z)^3 + (z - x)^3 + (x - y)^3 = 3(y - z)(z - x)(x - y) \]

4. **Final Expression**: The expression simplifies to:
\[ 3(y - z)(z - x)(x - y) \]

### Conclusion:
Thus, the value of \((y-z)^3 + (z-x)^3 + (x-y)^3\) is equal to \(3(y - z)(z - x)(x - y)\).