[(x-y)^3 + (y-z)^3 + (z-x)^3]/9((x-y)(y-z)(z-x) = ?

We are given the expression:

\[
\frac{(x - y)^3 + (y - z)^3 + (z - x)^3}{9 (x - y)(y - z)(z - x)}
\]

We need to simplify it. A known identity for this type of expression is:

\[
(x - y)^3 + (y - z)^3 + (z - x)^3 = 3(x - y)(y - z)(z - x)
\]

This is a symmetric identity, and it greatly simplifies the numerator.

Substituting this into the original expression, we get:

\[
\frac{3(x - y)(y - z)(z - x)}{9 (x - y)(y - z)(z - x)}
\]

The terms \( (x - y)(y - z)(z - x) \) in the numerator and denominator cancel out, leaving:

\[
\frac{3}{9} = \frac{1}{3}
\]

Thus, the simplified expression is:

\[
\boxed{\frac{1}{3}}
\]

Would you like further details or explanations? Here are some related questions you might find interesting:

1. How can we derive the identity \((x - y)^3 + (y - z)^3 + (z - x)^3 = 3(x - y)(y - z)(z - x)\)?
2. What are some applications of symmetric polynomials in algebra?
3. How does factorization help in simplifying complex expressions like this one?
4. Can this method be applied to more complex cubic expressions?
5. What other known identities simplify polynomials of degree three?

**Tip:** When simplifying expressions, always check if any known identities apply—they can save a lot of time!

[(x - y)^3 + (y - z)^3 + (z - x)^3] / 9((x - y)(y - z)(z - x)) = ?

Simplify the symmetric cubic expression [(x - y)^3 + (y - z)^3 + (z - x)^3] / 9((x - y)(y - z)(z - x)) using a known identity for cubic expressions. This step-by-step guide covers algebraic techniques and simplifications. Suitable for high school students in Grades 10-12.

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