(x-3)(x%2B4)(x-4).jpeg "(x+3)(x-3)(x+4)(x-4)")
Expanding and Factoring the Expression (x+3)(x-3)(x+4)(x-4)
This expression involves a pattern that allows for efficient expansion and factorization:
The Difference of Squares Pattern:
The difference of squares pattern states that: (a + b)(a - b) = a² - b²
Applying the pattern to the expression:
Notice that the expression contains two pairs of factors that follow the difference of squares pattern:
- (x + 3)(x - 3)
- (x + 4)(x - 4)
Expanding the expression:
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First Pair: (x + 3)(x - 3) = x² - 3² = x² - 9
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Second Pair: (x + 4)(x - 4) = x² - 4² = x² - 16
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Combining the results: (x² - 9)(x² - 16)
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Expanding further: x⁴ - 16x² - 9x² + 144 = x⁴ - 25x² + 144
Factoring the expression back:
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Recognize the pattern: The expanded form (x⁴ - 25x² + 144) resembles a quadratic equation if we substitute y = x²: y² - 25y + 144
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Factor the quadratic: (y - 16)(y - 9)
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Substitute back: (x² - 16)(x² - 9)
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Apply the difference of squares pattern again: (x + 4)(x - 4)(x + 3)(x - 3)
Therefore, we have successfully expanded and factored the given expression. The final factored form is (x + 4)(x - 4)(x + 3)(x - 3).
Key takeaways:
- The difference of squares pattern is a powerful tool for simplifying and factoring expressions.
- Recognizing patterns in expressions helps streamline the expansion and factorization process.
- By substituting variables, we can simplify expressions and make them easier to work with.