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(1/x-1-x/1-x^3.x^2+x+1/x+1) 2x+1/x^2+2x+1

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

Simplifying the Expression: (1/x-1 - x/1-x^3) * (x^2+x+1)/(x+1) * (2x+1)/(x^2+2x+1)

This expression looks complex, but we can simplify it using basic algebraic manipulations. Let's break it down step-by-step:

1. Factoring and Simplifying the First Term:

  • Factor out a -1 from the denominator of the first term: (1/x-1 - x/1-x^3) = (1/x-1 + x/(x^3-1))
  • Factor the difference of cubes in the denominator: (1/x-1 + x/(x^3-1)) = (1/x-1 + x/((x-1)(x^2+x+1)))
  • Find a common denominator: (1/x-1 + x/((x-1)(x^2+x+1))) = (x^2+x+1 + x^2)/((x-1)(x^2+x+1))
  • Simplify: (x^2+x+1 + x^2)/((x-1)(x^2+x+1)) = (2x^2+x+1)/((x-1)(x^2+x+1))

2. Simplifying the Second Term:

  • Factor the denominator: (x^2+x+1)/(x+1) = ((x+1)(x+1))/(x+1)
  • Simplify by canceling out the common factor: ((x+1)(x+1))/(x+1) = x+1

3. Simplifying the Third Term:

  • Factor the denominator: (2x+1)/(x^2+2x+1) = (2x+1)/((x+1)(x+1))

4. Combining the Simplified Terms:

Now we have:

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

5. Cancelling Common Factors:

  • Cancel out the (x+1) terms: (2x^2+x+1)/((x-1)(x^2+x+1)) * (2x+1)/(x+1)

6. Final Simplified Expression:

The simplified expression is:

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

Important Note: This expression is defined for all values of x except for x = 1 and x = -1, as these values would make the denominator equal to zero.

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