(x-1)(x-3)(2x%2B3)%2B9.jpeg "(2x-1)(x-1)(x-3)(2x+3)+9")
Factoring and Simplifying the Expression (2x-1)(x-1)(x-3)(2x+3)+9
This article explores the process of factoring and simplifying the expression (2x-1)(x-1)(x-3)(2x+3)+9. We'll break down the steps to achieve a simplified form of the expression.
Step 1: Expanding the Expression
The first step is to expand the given expression by multiplying the factors together.
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Multiply the first two factors: (2x-1)(x-1) = 2x² - 3x + 1
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Multiply the next two factors: (x-3)(2x+3) = 2x² - 3x - 9
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Now multiply the results from above: (2x² - 3x + 1)(2x² - 3x - 9) = 4x⁴ - 12x³ - 13x² + 27x + 9
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Finally, add the constant term: 4x⁴ - 12x³ - 13x² + 27x + 9 + 9 = 4x⁴ - 12x³ - 13x² + 27x + 18
Step 2: Factoring the Simplified Expression
The simplified expression, 4x⁴ - 12x³ - 13x² + 27x + 18, can be factored further.
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Grouping: We can group the terms to find a common factor: (4x⁴ - 12x³) + (-13x² + 27x) + 18
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Factoring out common factors: 4x³(x - 3) - x(13x - 27) + 18
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Recognizing a pattern: Notice that the terms in parentheses are similar. We can rewrite the expression as: 4x³(x - 3) - (13x - 27)x + 18
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Factoring by grouping: Now we can factor out (x - 3) and (13x - 27): (x - 3)(4x³ - x) + 18
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Further factoring: x(x - 3)(4x² - 1) + 18
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Final factorization: (x - 3)(4x² - 1)(x) + 18
Final Result
The fully factored and simplified form of the expression (2x-1)(x-1)(x-3)(2x+3)+9 is (x - 3)(4x² - 1)(x) + 18.
This demonstrates the steps involved in factoring and simplifying complex expressions through multiplication, grouping, and recognizing common patterns.