Question

Efectuati pe DVA:

$( \frac{1}{X-1}) ^{-1} + \frac{X}{X+1}- \frac{X+2}{X^{2}-1 }$

Answer 1

=(x-1)+x/(x+1)-(x+2)/(x+1)(x-1)=aducem la acelasi numitor=
=[(x-1)(x²-1)+x(x-1)-x-2)]/(x²-1)=
=[(x³-x²-x+1+x²-x-x-2)/(x²-1)=
=(x³-3x-2)/(x²-1)
=[x(x²-1)-2(x+1)]/(x²-1)=[x(x+1)(x-1)-2(x+1)]/(x+1)(x-1)=
=(x+1)[(x(x-1)-2]/(x+1)(x-1)=simplif cu x+1=
=(x²-x-2)/(x-1)
cond:x≠1
x²-x-2=0
Δ=1+8=9
x₁=(1-3)/2=-1
x₂=(1+3)/2=2

Obs.
 x³-3x-2      x²-1
-x³+x            x
 =-2x-2