(x-1)(x+1)(x+2)(x+4)-72 因数分解

2 min read Jun 17, 2024
(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

  • Group the first two factors and the last two factors:

    [(x-1)(x+1)][(x+2)(x+4)] - 72

  • Apply the difference of squares pattern:

    (x² - 1)(x² + 6x + 8) - 72

3. Expand and Simplify

  • Multiply the two quadratics:

    x⁴ + 6x³ + 8x² - x² - 6x - 8 - 72

  • Combine like terms:

    x⁴ + 6x³ + 7x² - 6x - 80

4. Factor the Resulting Polynomial

The resulting polynomial can be factored further:

  • Factor by grouping:

    (x⁴ + 6x³ + 7x²) - (6x + 80) x²(x² + 6x + 7) - 8(x + 10)

  • Factor the quadratic within the parentheses:

    x²(x + 1)(x + 7) - 8(x + 10)

  • 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.

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