trying to simplify (1+x)/(1-x)?

Question

I think I remember a trick to make this very simple, but I can't remember. The best I've gotten so far is 2*x/(1+x^2) but I thought there was a simpler one.

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

This is the simplest I've gotten so far. What I really want is to get rid of the denominator.

sorry about my type-o earlier.

well, I'm having sing error trouble today, scratch that algebra above.

still looking for ideas on how to get rid of the denominator. if not, (1+x)^2/(1-x) is as good as I'll get.

Answer 1

Multiply numerator and denominator by (1+x) to get

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

To get rid of the denominator, try polynomial division
......_____
1-x| 1+x

It comes out 1 + 2x + 2x^2 + 2x^3 + ....

It's probably not what you're after, but here is how it's done:
http://img250.imageshack.us/img250/5976/tempbo9.png