N° 5 Espressioni Algebriche

$\frac{2 x + 3}{x^{3} + x^{2} + x + 1} + \frac{2 x (x + 2) + 4}{x^{4} - 1} - \frac{3 x + 2}{x^{3} - x^{2} + x - 1} - \frac{1}{x^{2} + 1} =$

$= \frac{2 x + 3}{(x + 1) (x^{2} + 1)} + \frac{2 x^{2} + 4 x + 4}{(x^{2} + 1) (x^{2} - 1)} - \frac{3 x + 2}{(x - 1) (x^{2} + 1)} - \frac{1}{x^{2} + 1} =$

$= \frac{(x - 1) (2 x + 3) + 2 x^{2} + 4 x + 4 - (x + 1) (3 x + 2) - (x^{2} - 1)}{(x + 1) (x - 1) (x^{2} + 1)} =$

$= \frac{2 x^{2} + 3 x - 2 x - 3 + 2 x^{2} + 4 x + 4 - 3 x^{2} - 2 x - 3 x - 2 - x^{2} + 1}{(x + 1) (x - 1) (x^{2} + 1)} =$

$= \frac{(2 + 2 - 3 - 1) x^{2} + (3 - 2 + 4 - 2 - 3) x + (- 3 + 4 - 2 + 1)}{(x + 1) (x - 1) (x^{2} + 1)} =$

$= \frac{0}{(x + 1) (x - 1) (x^{2} + 1)} = 0$