https://file.aiquickdraw.com/mathbot/file/17265450251471htll4yi.jpg

The given expression is:

\[
\frac{x^3 + y^3 + z^3 - 3xyz}{(x - y)^2 + (y - z)^2 + (z - x)^2}
\]

### Step-by-Step Simplification:

1. **Numerator Simplification**:
   The numerator \( x^3 + y^3 + z^3 - 3xyz \) resembles the identity for the factorization of the sum of cubes:

   \[
   x^3 + y^3 + z^3 - 3xyz = (x + y + z) \left( (x - y)^2 + (y - z)^2 + (z - x)^2 \right)
   \]

2. **Denominator Simplification**:
   The denominator is given as \( (x - y)^2 + (y - z)^2 + (z - x)^2 \). Notice that this expression appears directly in the factorization of the numerator.

3. **Final Simplification**:
   Since the term \( (x - y)^2 + (y - z)^2 + (z - x)^2 \) is common in both the numerator and the denominator, they cancel out, and we are left with:

   \[
   \frac{(x + y + z) \left( (x - y)^2 + (y - z)^2 + (z - x)^2 \right)}{(x - y)^2 + (y - z)^2 + (z - x)^2} = x + y + z
   \]

### Simplified Result:
The simplified expression is:

\[
x + y + z
\]

---

Would you like further details on any steps or have additional questions? Here are five related questions for further exploration:

1. What is the geometric interpretation of this algebraic identity?
2. How can we apply the identity \(x^3 + y^3 + z^3 - 3xyz\) in real-world problems?
3. What would happen if the values of \(x\), \(y\), and \(z\) were complex numbers?
4. How does this factorization relate to symmetric polynomials?
5. What are some other common algebraic identities involving cubes?

**Tip**: When simplifying algebraic expressions, look for common factorization identities, especially in terms that seem familiar like sums of cubes or squares.

Simplify the expression: (x^3 + y^3 + z^3 - 3xyz) / ((x - y)^2 + (y - z)^2 + (z - x)^2)

Learn how to simplify the algebraic expression (x^3 + y^3 + z^3 - 3xyz) over ((x - y)^2 + (y - z)^2 + (z - x)^2) using the sum of cubes factorization identity. This step-by-step guide is suitable for students in Grades 9-12 and covers core algebraic concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation