Question

Fie expresia : E(x)=  ($\frac{x+1}{x-1} +\frac{3}{ x^{2} -1} - \frac{x+3}{x-1} ) * \frac{ x^{2} -4x+3}{7}$ , x apartine R-{1,-1}

Aduceti expresia la forma cea mai simpla si determinati numerele naturale n pentru care E(n) apartine Z.

Answer 1

E(x) = [(x+1)² + 3- (x+3)(x-1)]/(x² - 1) ·(x² - 4x +3)/7 =
= (x²+2x +1 + 3 - x² -2x +3)/[(x-1)(x+1)] · (x-3)(x-1)/7 =
=7/(x+1) · (x-3)/7 = (x-3)/(x+1)
E(n) ∈ Z  ⇔ (x+1) ∈ D(x - 3)
daca  (n+1) | (n-3)      (1)
         (n+1) | (n+1)      (2)  ⇒ (n+1) | [(2) - (1)] = 4 ⇒(n+1) ∈ D4
(n+1) ∈ {-4,-2,-1, 1,2,4}      n ∈ {-5,-3,-2,0,1,3}