Factoring the Difference of Squares: (3x² + 5y³) (3x² - 5y³)
This expression represents a classic example of the difference of squares factorization pattern. Here's how it works:
Understanding the Pattern
The difference of squares pattern states: a² - b² = (a + b)(a - b)
Applying the Pattern
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Identify the squares:
- In our expression, (3x²) is the square of (3x).
- (5y³) is the square of (5y).
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Apply the pattern:
- (3x² + 5y³) (3x² - 5y³) = [(3x)² + (5y)³][(3x)² - (5y)³]
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Simplify:
- This directly follows the difference of squares pattern, resulting in:
- (3x)² - (5y)³ = 9x⁴ - 25y⁶
Therefore, the factored form of (3x² + 5y³) (3x² - 5y³) is 9x⁴ - 25y⁶.
Key Takeaways
- The difference of squares pattern is a powerful tool for simplifying algebraic expressions.
- Recognizing the pattern can save time and effort when factoring.
- This pattern is applicable in various mathematical contexts, including solving equations and simplifying expressions.