N° 11 Espressioni Algebriche

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

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

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

$= \frac{4 x (x + 1) (x^{2} - x + 1) + 4 x {(x + 1)}^{2} + 4 x^{2} (x - 1) - 8 x + {(x + 1)}^{2} (x^{3} - 2) - x^{3} {(x + 1)}^{2} + 2 {(x + 1)}^{2} + 4 x^{2} {(x + 1)}^{2}}{4 x^{2} {(x + 1)}^{2}} =$

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

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

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

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

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

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