Răspuns :
Răspuns:
[tex]a) {x}^{2} - 5x + 6 = {x}^{2} - 2x - 3x + 6 = x(x -2) - 3(x - 2) = (x - 2)(x - 3)[/tex]
=(x-2)(x-3),x€R
[tex]b)e(x) = ( \frac{6(x - 2)}{ {(x - 2)}^{2} } - \frac{5(x - 3)}{ {(x - 3)}^{2} } ) + \frac{3}{(x - 2)(x - 3)} \\ e(x) =( \frac{6}{x - 2} - \frac{5}{x - 3} ) + \frac{3}{(x - 2)(x - 3)} \\ e(x) = ( \frac{6(x - 3)}{(x - 2)(x - 3)} - \frac{5(x - 2)}{(x - 2)(x - 3)} ) + \frac{3}{(x - 2)(x - 3)} \\ e(x) = \frac{6x - 18 - 5x + 10}{(x - 2)(x - 3)} + \frac{3}{(x - 2)(x - 3)} \\ e(x) = \frac{6x - 5x - 18 + 10 + 3}{(x - 2)(x - 3)} \\ e(x) = \frac{x - 5}{(x - 2)(x - 3)} [/tex]
x€R≠{2;3}
Răspuns:
a) x² - 5x + 6 = x² - 2x - 3x + 6 = x(x - 2) - 3(x - 2) = (x - 2)(x - 3), pentru orice x număr real.
b) Folosim relația demonstrată la punctul a, dăm factor comun la numărător:
[tex]E(x) = \dfrac{6(x - 2)}{(x - 2)^2} - \dfrac{5(x - 3)}{(x - 3)^2} + \dfrac{3}{(x - 2)(x - 3)} = \\[/tex]
- am simplificat cu (x - 2) și (x - 3)
[tex]= \dfrac{^{x-3)}6}{x - 2} - \dfrac{^{x-2)}5}{x - 3} + \dfrac{3}{(x - 2)(x - 3)}[/tex]
- am adus la numitor comun, prin amplificare
[tex]= \dfrac{6(x - 3)}{(x - 2)(x - 3)} - \dfrac{5(x - 2)}{(x - 2)(x - 3)} + \dfrac{3}{(x - 2)(x - 3)}\\[/tex]
- efectuăm calculele
[tex]= \dfrac{6(x - 3) - 5(x - 2) + 3}{(x - 2)(x - 3)} = \dfrac{6x - 18 - 5x + 10 + 3}{(x - 2)(x - 3)}\\[/tex]
De unde obținem:
[tex]\boldsymbol{E(x) = \dfrac{x - 5}{(x - 2)(x - 3)}}, \ \forall x \in \Bbb{R} \setminus \{2;3\}[/tex]