N° 9 Espressioni Algebriche

$[(\frac{x + 5}{x^{2} - 5 x + 6} - \frac{x - 3}{x^{2} + 3 x - 10} + \frac{x - 2}{x^{2} + 2 x - 15}) \cdot (\frac{x}{x + 10} - \frac{1}{x + 2} + \frac{2 x}{x^{2} + 12 x + 20}) + 1] \cdot (3 - \frac{x}{x - 2}) =$

$= [(\frac{x + 5}{(x - 2) (x - 3)} - \frac{x - 3}{(x - 2) (x + 5)} + \frac{x - 2}{(x + 5) (x - 3)}) \cdot (\frac{x}{x + 10} - \frac{1}{x + 2} + \frac{2 x}{(x + 2) (x + 10)}) + 1] \cdot (\frac{3 x - 6 - x}{x - 2}) =$

$= [(\frac{{(x + 5)}^{2} - {(x - 3)}^{2} + {(x - 2)}^{2}}{(x - 2) (x + 5) (x - 3)}) \cdot (\frac{x (x + 2) - (x + 10) + 2 x}{(x + 2) (x + 10)}) + 1] \cdot \frac{2 x - 6}{x - 2} =$

$= [(\frac{x^{2} + 25 + 10 x - x^{2} + 6 x - 9 + x^{2} - 4 x + 4}{(x - 2) (x + 5) (x - 3)}) \cdot (\frac{x^{2} + 2 x - x - 10 + 2 x}{(x + 2) (x + 10)}) + 1] \cdot 2 \cdot \frac{x - 3}{x - 2} =$

$= [(\frac{x^{2} + (10 + 6 - 4) x + (25 - 9 + 4)}{(x - 2) (x + 5) (x - 3)}) \cdot (\frac{x^{2} + 3 x - 10}{(x + 2) (x + 10)}) + 1] \cdot 2 \cdot \frac{x - 3}{x - 2} =$

$= [(\frac{x^{2} + 12 x + 20}{(x - 2) (x + 5) (x - 3)}) \cdot (\frac{(x + 5) (x - 2)}{(x + 2) (x + 10)}) + 1] \cdot 2 \cdot \frac{x - 3}{x - 2} =$

$= (\frac{x^{2} + 12 x + 20}{(x - 3) (x + 10) (x + 2)} + 1) \cdot 2 \cdot \frac{x - 3}{x - 2} =$

$= (\frac{(x + 2) (x + 10)}{(x - 3) (x + 10) (x + 2)} + 1) \cdot 2 \cdot \frac{x - 3}{x - 2} =$

$= (\frac{1}{x - 3} + 1) \cdot 2 \cdot \frac{x - 3}{x - 2} =$

$= (\frac{1 + x - 3}{x - 3}) \cdot 2 \cdot \frac{x - 3}{x - 2} =$

$= \frac{x - 2}{x - 3} \cdot 2 \cdot \frac{x - 3}{x - 2} = 2$