%5E3-(3x%2B4y)%5E3.jpeg "(3x-4y)^3-(3x+4y)^3")
Factoring the Expression (3x-4y)³ - (3x+4y)³
This article will explore the process of factoring the expression (3x-4y)³ - (3x+4y)³. We will use the difference of cubes formula and other algebraic techniques to simplify the expression.
Understanding the Difference of Cubes Formula
The difference of cubes formula states: a³ - b³ = (a-b)(a² + ab + b²)
This formula is essential for factoring expressions that involve the difference of two cubes.
Applying the Formula to (3x-4y)³ - (3x+4y)³
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Identify a and b: In our expression, a = (3x-4y) and b = (3x+4y).
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Apply the difference of cubes formula: (3x-4y)³ - (3x+4y)³ = [(3x-4y) - (3x+4y)][(3x-4y)² + (3x-4y)(3x+4y) + (3x+4y)²]
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Simplify the expression:
- [(3x-4y) - (3x+4y)] = -8y
- [(3x-4y)² + (3x-4y)(3x+4y) + (3x+4y)²] = (9x² - 24xy + 16y²) + (9x² - 16y²) + (9x² + 24xy + 16y²) = 27x² + 16y²
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Combine the simplified terms: -8y * (27x² + 16y²) = -216x²y - 128y³
Final Factored Expression
Therefore, the factored form of (3x-4y)³ - (3x+4y)³ is -216x²y - 128y³.
Conclusion
By applying the difference of cubes formula, we successfully factored the given expression into a simpler form. This process demonstrates how understanding algebraic formulas and techniques can help simplify complex expressions and solve mathematical problems efficiently.