Question

E (x) = ( 1/x^2-x-2 + 1/x+1 - 1/2-x) : 2x/x^2-4

a) determinati valorile lui x pt care expresie NU este definita
b) aratati ca forma cea mai simpla a expresiei este E(x)= x+2/x+1

Answer 1

b)E (x) = [1/(x^2-x-2) + 1/(x+1) - 1/(2-x)] :[2x/(x^2-4)]=

=[1/(x+1)(x-2) +1/(x+1) +1/(x-2)]*[(x^2-4)/2x]=

=[1/(x+1)(x-2) +(x-2)/(x+1)(x-2) +(x+1)/(x+1)(x-2)]*[(x-2)(x+2)/2x]=

=[(1+x-2+x+1)/(x+1)(x-2)]*[(x-2)(x+2)/2x]=

=[2x/(x+1)(x-2)]*[(x-2)(x+2)/2x]=2x(x-2)(x+2)/2x(x+1)(x-2)=

=(x+2)/(x+1)

a)
Toti numitori care apar in expresie si de-a lungul demonstratiei tb. sa fie diferiti de zero:

''d''=diferit

x^2 -x-2 d 0, (x+1)(x-2) d 0, x d -1 si +2

x+1 d 0, x d -1

2-x d 0, x d +2

x^2 -4 d 0, (x-2)(x+2) d 0, x d -2 si +2

2x d 0, x d 0

Asadar x diferit de {-2,-1,0,2}