Question

vă rog, e important!​

Answer 1

$a) \: x^2-10x+21=x^2-7x-3x+21=x(x-7)-3(x-7)=(x-7)(x-3)$

$\begin{aligned} F(x) &= \left(\frac{2x^2-7x-17}{x^2-10x+21} - \frac{x+1}{x-7}\right):\frac1{x^2-9}\\ &=\left(\frac{2x^2-7x-17}{(x-7)(x-3)}-\frac{x+1}{x-7}\right)\cdot(x^2-9)\\ & = \frac{2x^2-7x-17-(x+1)(x-3)}{(x-7)(x-3)}\cdot(x+3)(x-3)\\ & = \frac{2x^2-7x-17-x^2+2x+3}{x-7}\cdot(x+3)\\ & = \frac{x^2-5x-14}{x-7}\cdot(x+3)\\ & = \frac{x^2-7x+2x-14}{x-7}\cdot(x+3)\\ & = \frac{(x+2)(x-7)}{x-7}\cdot(x+3)\\ & = (x+2)(x+3), \forall x \in \mathbb{R}\backslash\{-3, 3, 7\} \end{aligned}$

$b)\:F(a)\overset{a)}{=}(a+2)(a+3)\\\textit{cazul I: } a\text{ par}\Rightarrow(a+2)\text{ par \c si }(a+3)\text{ impar}\Rightarrow F(a) = \text{par}\cdot\text{impar}=\text{par}\\\textit{cazul II: } a\text{ impar}\Rightarrow (a+2)\text{ impar \c si }(a+3)\text{ par}\Rightarrow F(a)=\text{impar}\cdot\text{par}=\text{par}$