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(x-1)(x-2)(x-3)(x-4)-120 因数分解

3 min read Jun 17, 2024
(x-1)(x-2)(x-3)(x-4)-120 因数分解

Factoring (x-1)(x-2)(x-3)(x-4)-120

This problem involves factoring a polynomial expression. Let's break down the steps to factor the expression (x-1)(x-2)(x-3)(x-4)-120.

Understanding the Problem

The given expression is a product of four linear factors (x-1), (x-2), (x-3), (x-4) minus a constant term 120. Our goal is to rewrite this expression as a product of simpler factors.

Solution Strategy

  1. Recognize a Pattern: Notice that the constant term (-120) is the product of -1, -2, -3, and -4. This suggests a potential connection to the linear factors.

  2. Manipulate the Expression: Let's try to rearrange the expression to make this pattern more explicit. We can expand the first four terms and manipulate the expression as follows:

    (x-1)(x-2)(x-3)(x-4) - 120 
= (x^2 - 3x + 2)(x^2 - 7x + 12) - 120 
= (x^2 - 3x + 2)(x^2 - 7x + 12) - (-1)(-2)(-3)(-4) 
= (x^2 - 3x + 2)(x^2 - 7x + 12) + (-1)(x-1)(x-2)(x-3)(x-4)

  3. Factor by Grouping: Now we can factor out a common factor of (x-1)(x-2)(x-3)(x-4) from the last two terms:

    (x^2 - 3x + 2)(x^2 - 7x + 12) + (-1)(x-1)(x-2)(x-3)(x-4)
= (x-1)(x-2)(x-3)(x-4) [(x^2 - 3x + 2) - 1]
= (x-1)(x-2)(x-3)(x-4) (x^2 - 3x + 1)

Final Factored Form

The completely factored form of the expression is:

(x-1)(x-2)(x-3)(x-4)(x^2 - 3x + 1)

Important Note: The quadratic factor (x^2 - 3x + 1) cannot be factored further using real numbers.

Conclusion

By strategically manipulating the expression and recognizing patterns, we were able to factor the polynomial into a product of simpler factors. This demonstrates the power of algebraic techniques in simplifying complex expressions.

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