General Simplification of (1+x)/(1-x)

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Discussion Overview

The discussion revolves around the simplification of the expression (1-x)/(1+x). Participants explore various methods to derive the equivalence to (2/(x+1)) - 1, focusing on algebraic manipulations and techniques such as polynomial long division.

Discussion Character

  • Exploratory, Technical explanation, Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about the steps to simplify (1-x)/(1+x) to (2/(x+1)) - 1, as presented in their textbook.
  • Another participant suggests rewriting 1-x as -x-1+2, leading to a simplification that is a common trick for such fractions.
  • A different approach is proposed involving polynomial long division, where dividing -x + 1 by x + 1 results in -1 + 2/(x + 1).
  • Additionally, a participant mentions an alternative method of replacing 1 with (1+x)/(1+x) to achieve a common denominator, which simplifies to 1 - x.

Areas of Agreement / Disagreement

Participants present multiple methods for simplifying the expression, indicating that there is no single agreed-upon approach, and the discussion remains open to various techniques.

Contextual Notes

Some methods rely on specific algebraic tricks or techniques that may not be universally known, and the discussion does not resolve which method is preferable or most effective.

remorris
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I know that (1-x)/(1+x) is equivalent to (2/(x+1)) - 1 via 'hand waving' in my textbook but i cannot figure out the steps to arrive at this result. Suggestions??
 
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You can write 1-x as -x-1+2=-(x+1)+2 and simplify.
That is a usual trick for fractions like yours.
 
Neat Trick. Thanks!
 
remorris said:
I know that (1-x)/(1+x) is equivalent to (2/(x+1)) - 1 via 'hand waving' in my textbook but i cannot figure out the steps to arrive at this result. Suggestions??

mfb said:
You can write 1-x as -x-1+2=-(x+1)+2 and simplify.
That is a usual trick for fractions like yours.

An alternate approach is to divide - x + 1 by x + 1 using polynomial long division. If you don't know this technique, you can search Wikipedia using the search string "polynomial long division". Doing this, you get -1 + 2/(x + 1).
 
remorris said:
I know that (1-x)/(1+x) is equivalent to (2/(x+1)) - 1 via 'hand waving' in my textbook but i cannot figure out the steps to arrive at this result. Suggestions??

Yet another equivalent way to see this is to replace 1 by (1+x)/(1+x) and you now have a common denominator of 1 + x. Gathering terms in the numerator yields 1 - x.
 

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