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

Jasonbradley

3 min read

·

Jun 16, 2024

29

Share

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

This article will guide you through the process of simplifying the complex expression: (1/x-2-2x/4-x^2+1/2+x)(2/x-1)

Step 1: Factor the Denominators

Begin by factoring the denominators of the fractions within the expression.

• 4-x²: This can be factored as (2+x)(2-x).

Step 2: Find a Common Denominator

The next step is to find a common denominator for all the fractions in the first set of parentheses. The common denominator is 2x(2+x)(2-x).

Step 3: Rewrite the Fractions with the Common Denominator

Rewrite each fraction in the first set of parentheses with the common denominator:

• 1/x-2: (2+x)(2-x)/2x(2+x)(2-x)
• -2x/4-x²: -4x²/2x(2+x)(2-x)
• 1/2+x: x(2-x)/2x(2+x)(2-x)
• x: 2x²(2+x)(2-x)/2x(2+x)(2-x)

Step 4: Combine the Fractions

Combine the numerators of the fractions over the common denominator:

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

Step 5: Simplify the Numerator

Expand and simplify the numerator:

[4 - x² - 4x² + 2x - x² + 4x⁴ + 8x³ + 4x²] / 2x(2+x)(2-x)

Combine like terms:

[4x⁴ + 8x³ - 3x² + 2x + 4] / 2x(2+x)(2-x)

Step 6: Multiply by the Second Fraction

Now, multiply the simplified expression by the second fraction (2/x-1):

[4x⁴ + 8x³ - 3x² + 2x + 4] / 2x(2+x)(2-x) * (2/x-1)

Step 7: Simplify the Expression

• Cancel common factors: The 2 in the numerator and denominator cancel.

• Multiply the remaining terms:

  (4x⁴ + 8x³ - 3x² + 2x + 4) / (x(2+x)(2-x)(x-1))

Final Simplified Expression:

The simplified expression is (4x⁴ + 8x³ - 3x² + 2x + 4) / (x(2+x)(2-x)(x-1)).

Important Note: The expression is undefined when x = 0, x = -2, x = 2, and x = 1 because these values would make the denominator equal to zero.