%5E3%2B(b%5E2-c%5E2)%5E3%2B(c%5E2-a%5E2)%5E3-factorise.jpeg "(a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3 Factorise")
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 step-by-step.
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 well-known factorization formula.
Applying the Pattern
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Identify x, y, and z: In our expression, we have:
- x = a² - b²
- y = b² - c²
- z = c² - a²
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Calculate x + y + z: (a² - b²) + (b² - c²) + (c² - a²) = 0
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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.