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

3 min read Jun 16, 2024
(1/x^2+4x+4-1/x^2-4x+4) (1/x+2+1/x-2)

Simplifying the Expression: (1/x^2+4x+4 - 1/x^2-4x+4) (1/x+2 + 1/x-2)

This expression looks complex, but we can simplify it by using algebraic manipulation and factoring. Let's break it down step by step:

Step 1: Factoring the Denominators

  • x² + 4x + 4 is a perfect square trinomial and can be factored as (x + 2)²
  • x² - 4x + 4 is also a perfect square trinomial and can be factored as (x - 2)²

Now our expression becomes:

(1/(x + 2)² - 1/(x - 2)²) (1/(x + 2) + 1/(x - 2))

Step 2: Finding a Common Denominator

  • For the first part:
    • The least common denominator for (x + 2)² and (x - 2)² is (x + 2)²(x - 2)².
    • We need to multiply the first term by (x - 2)² / (x - 2)² and the second term by (x + 2)² / (x + 2)²:

( (x - 2)² / (x + 2)²(x - 2)² - (x + 2)² / (x + 2)²(x - 2)²) (1/(x + 2) + 1/(x - 2))

  • For the second part:
    • The least common denominator for (x + 2) and (x - 2) is (x + 2)(x - 2).
    • We need to multiply the first term by (x - 2) / (x - 2) and the second term by (x + 2) / (x + 2):

( (x - 2)² / (x + 2)²(x - 2)² - (x + 2)² / (x + 2)²(x - 2)²) ((x - 2) / (x + 2)(x - 2) + (x + 2) / (x + 2)(x - 2))

Step 3: Simplifying the Expression

  • Combine the terms in the first part:

( (x² - 4x + 4 - x² - 4x - 4) / (x + 2)²(x - 2)²) ((x - 2) / (x + 2)(x - 2) + (x + 2) / (x + 2)(x - 2))

  • Simplify further:

( -8x / (x + 2)²(x - 2)²) ((x - 2) / (x + 2)(x - 2) + (x + 2) / (x + 2)(x - 2))

  • Combine the terms in the second part:

( -8x / (x + 2)²(x - 2)²) ((x - 2 + x + 2) / (x + 2)(x - 2))

  • Simplify:

( -8x / (x + 2)²(x - 2)²) (2x / (x + 2)(x - 2))

  • Multiply the numerators and denominators:

( -16x² / (x + 2)³(x - 2)³ )

Conclusion

The simplified form of the expression is -16x² / (x + 2)³(x - 2)³. This is the most compact representation of the original expression.

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