(2x – 5y)3 – (2x + 5y)3

Jasonbradley

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Jun 16, 2024

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Expanding and Simplifying (2x – 5y)3 – (2x + 5y)3

This problem involves expanding and simplifying a difference of cubes. Here's how we approach it:

Understanding the Difference of Cubes Formula

The difference of cubes formula states: a³ - b³ = (a - b)(a² + ab + b²)

Applying the Formula

1. Identify 'a' and 'b': In our problem, a = (2x - 5y) and b = (2x + 5y).

2. Substitute into the formula: (2x - 5y)³ - (2x + 5y)³ = [(2x - 5y) - (2x + 5y)][(2x - 5y)² + (2x - 5y)(2x + 5y) + (2x + 5y)²]

3. Simplify:

   • First bracket: (2x - 5y) - (2x + 5y) = -10y
   • Second bracket:
     • (2x - 5y)² = 4x² - 20xy + 25y²
     • (2x - 5y)(2x + 5y) = 4x² - 25y²
     • (2x + 5y)² = 4x² + 20xy + 25y²

4. Combine terms: -10y (4x² - 20xy + 25y² + 4x² - 25y² + 4x² + 20xy + 25y²) = -10y (12x² + 25y²)

5. Final Simplification: -10y (12x² + 25y²) = -120x²y - 250y³

Conclusion

Therefore, the simplified expression for (2x – 5y)³ – (2x + 5y)³ is -120x²y - 250y³.