(x%2B1)(x%2B2)(x%2B4)-72-%E5%9B%A0%E6%95%B0%E5%88%86%E8%A7%A3.jpeg "(x-1)(x+1)(x+2)(x+4)-72 因数分解")
Factoring (x-1)(x+1)(x+2)(x+4) - 72
This problem involves factoring a polynomial expression. Here's a step-by-step breakdown of the process:
1. Recognize the Pattern
Notice that the first part of the expression, (x-1)(x+1)(x+2)(x+4), is a product of four factors. This suggests we might be able to manipulate it to create a difference of squares pattern.
2. Group and Apply Difference of Squares
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Group the first two factors and the last two factors:
[(x-1)(x+1)][(x+2)(x+4)] - 72
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Apply the difference of squares pattern:
(x² - 1)(x² + 6x + 8) - 72
3. Expand and Simplify
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Multiply the two quadratics:
x⁴ + 6x³ + 8x² - x² - 6x - 8 - 72
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Combine like terms:
x⁴ + 6x³ + 7x² - 6x - 80
4. Factor the Resulting Polynomial
The resulting polynomial can be factored further:
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Factor by grouping:
(x⁴ + 6x³ + 7x²) - (6x + 80) x²(x² + 6x + 7) - 8(x + 10)
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Factor the quadratic within the parentheses:
x²(x + 1)(x + 7) - 8(x + 10)
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Find a common factor:
(x + 10)[x²(x + 7) - 8]
5. Final Factored Form
The fully factored expression is:
(x + 10)(x³ + 7x² - 8)
This expression can be factored further, but it is generally considered "sufficiently" factored at this point.
Important Note: If the problem requires the complete factorization of the cubic polynomial (x³ + 7x² - 8), you would need to apply additional factoring techniques, possibly involving the rational root theorem.