N° 8 Prodotti Notevoli

${((x^{0.25} - 1) (x^{0.25} + 1) (x + 1) + 2)}^{2} - x (x - 1) ({(x - 1)}^{2} + 3 x) - (2 x - 1) (x + 1)$

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

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

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

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

$= x^{3} - 2 x^{\frac{5}{2}} + 3 x^{2} - x + 2 x^{\frac{1}{2}} + 1 - x^{4} - 2 x^{2} + 1 =$

$= - x^{4} + x^{3} - 2 x^{\frac{5}{2}} + x^{2} - x + 2 x^{\frac{1}{2}} + 2$