Factoring (a²  b²)³ + (b²  c²)³ + (c²  a²)³
This problem involves factoring a sum of cubes expression. Let's break down the process stepbystep.
Understanding the Pattern
The key to solving this problem is recognizing the following pattern:
x³ + y³ + z³  3xyz = (x + y + z)(x² + y² + z²  xy  xz  yz)
This pattern is a wellknown factorization formula.
Applying the Pattern

Identify x, y, and z: In our expression, we have:
 x = a²  b²
 y = b²  c²
 z = c²  a²

Calculate x + y + z: (a²  b²) + (b²  c²) + (c²  a²) = 0

Since (x + y + z) = 0, the entire expression simplifies to: (a²  b²)³ + (b²  c²)³ + (c²  a²)³ = 0
Conclusion
Therefore, the factored form of (a²  b²)³ + (b²  c²)³ + (c²  a²)³ is simply 0. This is because the sum of the three cubes is equal to zero due to the specific pattern of the terms.