## Factoring the Expression (a²-b²)³ + (b²-c²)³ + (c²-a²)³

This article explores the factorization of the expression **(a²-b²)³ + (b²-c²)³ + (c²-a²)³**. We will employ a combination of algebraic manipulation and the application of a helpful algebraic identity.

### Understanding the Problem

The given expression consists of three cubes, each involving the difference of squares. Our goal is to find a factored form for this expression, ideally in terms of simpler factors.

### Applying the Sum of Cubes Identity

We can utilize the following identity:

**x³ + y³ = (x + y)(x² - xy + y²)**

To apply this identity, we need to identify 'x' and 'y' in our expression. Let's set:

**x = (a² - b²)****y = (b² - c²)**

Substituting these values into the identity, we get:

**(a² - b²)³ + (b² - c²)³ = [(a² - b²) + (b² - c²)][(a² - b²)² - (a² - b²)(b² - c²) + (b² - c²)²]**

Simplifying the first factor:

**(a² - b²) + (b² - c²) = a² - c²**

Now, let's focus on the second factor. Expanding the squares and simplifying:

**(a² - b²)² - (a² - b²)(b² - c²) + (b² - c²)² = (a⁴ - 2a²b² + b⁴) - (a²b² - a²c² - b⁴ + b²c²) + (b⁴ - 2b²c² + c⁴)**

Combining like terms:

**(a⁴ - 2a²b² + b⁴) - (a²b² - a²c² - b⁴ + b²c²) + (b⁴ - 2b²c² + c⁴) = a⁴ + a²c² + 2b⁴ - 3a²b² - 3b²c² + c⁴**

Now, we have:

**(a² - b²)³ + (b² - c²)³ = (a² - c²)(a⁴ + a²c² + 2b⁴ - 3a²b² - 3b²c² + c⁴)**

### Dealing with the Remaining Cube

We still need to incorporate the third cube, **(c² - a²)³**. Notice that we can rewrite it as:

**(c² - a²)³ = - (a² - c²)³**

Now, we can substitute this back into our original expression:

**(a² - b²)³ + (b² - c²)³ + (c² - a²)³ = (a² - c²)(a⁴ + a²c² + 2b⁴ - 3a²b² - 3b²c² + c⁴) - (a² - c²)³**

We can factor out **(a² - c²)**:

**(a² - c²) [(a⁴ + a²c² + 2b⁴ - 3a²b² - 3b²c² + c⁴) - (a² - c²)²]**

Finally, expanding and simplifying the expression inside the brackets:

**(a² - c²) [(a⁴ + a²c² + 2b⁴ - 3a²b² - 3b²c² + c⁴) - (a⁴ - 2a²c² + c⁴)]**

**(a² - c²) [3a²c² + 2b⁴ - 3a²b² - 3b²c²]**

Therefore, the completely factored form of the expression is:

**(a² - b²)³ + (b² - c²)³ + (c² - a²)³ = (a² - c²)(3a²c² + 2b⁴ - 3a²b² - 3b²c²) **

### Conclusion

By applying the sum of cubes identity and carefully manipulating algebraic expressions, we successfully factored the given expression. The final result is a product of two factors, revealing the underlying structure of the original expression.