展開する (x-2/x)^4(x^2+2/x)^3

解

展開する

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

x^{10} + 6 x^{7} + 12 x^{4} + 8 x - \frac{8 ( x ^{9} + 6 x ^{6} + 12 x ^{3} + 8 )}{x} + \frac{24 ( x ^{9} + 6 x ^{6} + 12 x ^{3} + 8 )}{x ^{3}} - \frac{32 ( x ^{9} + 6 x ^{6} + 12 x ^{3} + 8 )}{x ^{5}} + \frac{16 ( x ^{9} + 6 x ^{6} + 12 x ^{3} + 8 )}{x ^{7}}

解答ステップ

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

拡張

(x^{2} + \frac{2}{x})^{3} : \frac{x ^{9} + 6 x ^{6} + 12 x ^{3} + 8}{x ^{3}}

= (x - \frac{2}{x})^{4} \frac{x ^{9} + 6 x ^{6} + 12 x ^{3} + 8}{x ^{3}}

= \frac{x ^{9} + 6 x ^{6} + 12 x ^{3} + 8}{x ^{3}} (x - \frac{2}{x})^{4}

拡張

(x - \frac{2}{x})^{4} : x^{4} - 8 x^{2} + 24 - \frac{32}{x ^{2}} + \frac{16}{x ^{4}}

= \frac{x ^{9} + 6 x ^{6} + 12 x ^{3} + 8}{x ^{3}} (x^{4} - 8 x^{2} + 24 - \frac{32}{x ^{2}} + \frac{16}{x ^{4}})

括弧を分配する

= \frac{x ^{9} + 6 x ^{6} + 12 x ^{3} + 8}{x ^{3}} x^{4} + \frac{x ^{9} + 6 x ^{6} + 12 x ^{3} + 8}{x ^{3}} (- 8 x^{2}) + \frac{x ^{9} + 6 x ^{6} + 12 x ^{3} + 8}{x ^{3}} \cdot 24 + \frac{x ^{9} + 6 x ^{6} + 12 x ^{3} + 8}{x ^{3}} (- \frac{32}{x ^{2}}) + \frac{x ^{9} + 6 x ^{6} + 12 x ^{3} + 8}{x ^{3}} \cdot \frac{16}{x ^{4}}

簡素化

\frac{x ^{9} + 6 x ^{6} + 12 x ^{3} + 8}{x ^{3}} x^{4} + \frac{x ^{9} + 6 x ^{6} + 12 x ^{3} + 8}{x ^{3}} (- 8 x^{2}) + \frac{x ^{9} + 6 x ^{6} + 12 x ^{3} + 8}{x ^{3}} \cdot 24 + \frac{x ^{9} + 6 x ^{6} + 12 x ^{3} + 8}{x ^{3}} (- \frac{32}{x ^{2}}) + \frac{x ^{9} + 6 x ^{6} + 12 x ^{3} + 8}{x ^{3}} \cdot \frac{16}{x ^{4}} : x^{10} + 6 x^{7} + 12 x^{4} + 8 x - \frac{8 ( x ^{9} + 6 x ^{6} + 12 x ^{3} + 8 )}{x} + \frac{24 ( x ^{9} + 6 x ^{6} + 12 x ^{3} + 8 )}{x ^{3}} - \frac{32 ( x ^{9} + 6 x ^{6} + 12 x ^{3} + 8 )}{x ^{5}} + \frac{16 ( x ^{9} + 6 x ^{6} + 12 x ^{3} + 8 )}{x ^{7}}

= x^{10} + 6 x^{7} + 12 x^{4} + 8 x - \frac{8 ( x ^{9} + 6 x ^{6} + 12 x ^{3} + 8 )}{x} + \frac{24 ( x ^{9} + 6 x ^{6} + 12 x ^{3} + 8 )}{x ^{3}} - \frac{32 ( x ^{9} + 6 x ^{6} + 12 x ^{3} + 8 )}{x ^{5}} + \frac{16 ( x ^{9} + 6 x ^{6} + 12 x ^{3} + 8 )}{x ^{7}}