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

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

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

This article will guide you through the process of simplifying the given algebraic expression:

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

Step 1: Factor the Denominators

  • x^2 + 4x + 4: This is a perfect square trinomial, factoring to (x+2)^2
  • x^2 - 4: This is a difference of squares, factoring to (x+2)(x-2)

Step 2: Rewrite the Expression with Factored Denominators

Now the expression becomes:

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

Step 3: Find a Common Denominator for Each Parenthesis

  • Left Parenthesis: The common denominator is (x+2)^2.
    • Multiply the first term by (x+2)/(x+2)
  • Right Parenthesis: The common denominator is (x+2)(x-2).
    • Multiply the second term by (x+2)/(x+2)

This gives us:

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

Step 4: Simplify the Numerators

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

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

Step 5: Multiply the Fractions

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

Step 6: Simplify the Expression

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

Therefore, the simplified form of the expression is 2x(x + 4) / (x+2)^3 (x-2). This form is considered more compact and easier to work with in further algebraic manipulations.

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