Question

Se da expresia: E(x)=(x-6 supra x patrat-25 - x supra 5-x - 2 supra x+5) : (2x patrat +x-6 toata paranteza supra x patrat - 25 ,unde x apartine R -{-5;-2;3/2;5}
A. aratati ca (x+2)(2x-3)=2x patrat +x-6,oricare x apartine Z.

B. Aratati ca E(x)=x+2 supra 2x-3 oricare x apartine Z\{-5;-2;3/2;5}

C. Aflati valorile intregi ale lui a pentru care E(a) apartine Z.

:3 Mulțumesc

Answer 1

A) (x+2)(2x-3)=2x²-3x+4x-6=2x²+x-6. B)aducem la acelsi numitor(x²-25). Prima fractie o amplifucam cu 1, a doua cu x+5, a treia cu x-5⇒E(x)=(x-6+x²+5x-2x+10)/(x²-25)*(x²-25)/(x+2)(2x-3). Simplifivam x²-25 ⇒E(x)=x²+4x+4/(x+2)(2x-3)=(x+2)²/(x+2)(2x-3). Simplificam si obtinemE(x)=(x+2)/2x-3. C)x+2/2x-3 si 2x-3/2x-3⇒2x+4/2x-3 si 2x-3/2x-3 scadem ⇒7/2x-3⇒2x-3∈D7. D7={+1;-1;-7;+7}=>2x-3=1⇒x=2; 2x-3=-1=>x=1; 2x-3=7=>x=5; 2x-3=-7=>x=-2