%5E2%2B4x%5E2y%5E2.jpeg "(x^2-y^2)^2+4x^2y^2")
Exploring the Expression (x² - y²)² + 4x²y²
This expression, (x² - y²)² + 4x²y², holds a fascinating mathematical structure and can be simplified and manipulated in various ways. Let's delve into its properties and explore its potential forms.
Recognizing a Familiar Pattern
At first glance, the expression might seem complex, but a closer look reveals a familiar pattern. Notice that the first term, (x² - y²)², is a perfect square. This pattern is reminiscent of the algebraic identity:
** (a - b)² = a² - 2ab + b²**
Applying this identity to our expression, we get:
(x² - y²)² = (x²)² - 2(x²)(y²) + (y²)² = x⁴ - 2x²y² + y⁴
Simplifying the Expression
Now, substituting this result back into the original expression:
(x² - y²)² + 4x²y² = (x⁴ - 2x²y² + y⁴) + 4x²y²
Combining like terms:
** (x² - y²)² + 4x²y² = x⁴ + 2x²y² + y⁴ **
Factoring the Simplified Expression
The simplified expression also presents a pattern. Observe that it resembles the expansion of another algebraic identity:
** (a + b)² = a² + 2ab + b²**
Applying this identity, we can factor our simplified expression as:
** x⁴ + 2x²y² + y⁴ = (x² + y²)² **
Final Thoughts
The expression (x² - y²)² + 4x²y² can be simplified to (x² + y²)². This simplification highlights the importance of recognizing familiar patterns and applying algebraic identities. By understanding the underlying structure, we can manipulate complex expressions and reveal simpler, more insightful forms. This process not only simplifies calculations but also deepens our understanding of mathematical relationships.