(x-2)(x%2B3)(x-3).jpeg "(x+2)(x-2)(x+3)(x-3)")
Factoring and Expanding (x+2)(x-2)(x+3)(x-3)
This expression represents the product of four binomials: (x+2), (x-2), (x+3), and (x-3). Let's explore how to factor and expand this expression.
Understanding the Pattern
Notice that each binomial is in the form of (x + a)(x - a), which is a difference of squares pattern.
Key point: The difference of squares pattern states that (x + a)(x - a) = x² - a².
Factoring the Expression
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Factor the first two binomials: (x + 2)(x - 2) = x² - 2² = x² - 4
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Factor the last two binomials: (x + 3)(x - 3) = x² - 3² = x² - 9
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Combine the factored expressions: (x² - 4)(x² - 9)
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Apply the difference of squares pattern again: (x² - 4)(x² - 9) = (x² - 2²)(x² - 3²) = (x + 2)(x - 2)(x + 3)(x - 3)
Expanding the Expression
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Expand the first two factors: (x² - 4)(x² - 9) = x⁴ - 9x² - 4x² + 36
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Combine like terms: x⁴ - 9x² - 4x² + 36 = x⁴ - 13x² + 36
Therefore, the expanded form of (x+2)(x-2)(x+3)(x-3) is x⁴ - 13x² + 36.
Conclusion
By recognizing and applying the difference of squares pattern, we effectively factored and expanded the given expression. This highlights the importance of identifying patterns in mathematics for simplifying complex expressions.