(x+2y)(x-2y)(x^2+4y^2)

Jasonbradley

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Jun 16, 2024

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Simplifying the Expression (x+2y)(x-2y)(x^2+4y^2)

This expression involves a combination of multiplication and factoring, and it can be simplified using the difference of squares pattern. Here's how:

Understanding the Difference of Squares Pattern

The difference of squares pattern states: (a + b)(a - b) = a² - b²

Applying the Pattern

1. Focus on the first two factors: (x + 2y)(x - 2y)

   • This perfectly matches the difference of squares pattern, where:
     • a = x
     • b = 2y

2. Apply the pattern:

   • (x + 2y)(x - 2y) = x² - (2y)² = x² - 4y²

3. Multiply the result by the third factor: (x² - 4y²)(x² + 4y²)

   • Now we have another difference of squares pattern:
     • a = x²
     • b = 4y²

4. Apply the pattern again:

   • (x² - 4y²)(x² + 4y²) = (x²)² - (4y²)² = x⁴ - 16y⁴

Final Result

Therefore, the simplified form of the expression (x + 2y)(x - 2y)(x² + 4y²) is x⁴ - 16y⁴.