展開する (x-1/x)^4(x+1/x)^2

解

展開する

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

x^{6} - 2 x^{4} - 7 x^{2} + \frac{6 ( x ^{4} + 2 x ^{2} + 1 )}{x ^{2}} - \frac{4 ( x ^{4} + 2 x ^{2} + 1 )}{x ^{4}} + \frac{x ^{4} + 2 x ^{2} + 1}{x ^{6}} - 4

解答ステップ

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

拡張

(x + \frac{1}{x})^{2} : \frac{x ^{4} + 2 x ^{2} + 1}{x ^{2}}

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

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

拡張

(x - \frac{1}{x})^{4} : x^{4} - 4 x^{2} + 6 - \frac{4}{x ^{2}} + \frac{1}{x ^{4}}

= \frac{x ^{4} + 2 x ^{2} + 1}{x ^{2}} (x^{4} - 4 x^{2} + 6 - \frac{4}{x ^{2}} + \frac{1}{x ^{4}})

括弧を分配する

= \frac{x ^{4} + 2 x ^{2} + 1}{x ^{2}} x^{4} + \frac{x ^{4} + 2 x ^{2} + 1}{x ^{2}} (- 4 x^{2}) + \frac{x ^{4} + 2 x ^{2} + 1}{x ^{2}} \cdot 6 + \frac{x ^{4} + 2 x ^{2} + 1}{x ^{2}} (- \frac{4}{x ^{2}}) + \frac{x ^{4} + 2 x ^{2} + 1}{x ^{2}} \cdot \frac{1}{x ^{4}}

簡素化

\frac{x ^{4} + 2 x ^{2} + 1}{x ^{2}} x^{4} + \frac{x ^{4} + 2 x ^{2} + 1}{x ^{2}} (- 4 x^{2}) + \frac{x ^{4} + 2 x ^{2} + 1}{x ^{2}} \cdot 6 + \frac{x ^{4} + 2 x ^{2} + 1}{x ^{2}} (- \frac{4}{x ^{2}}) + \frac{x ^{4} + 2 x ^{2} + 1}{x ^{2}} \cdot \frac{1}{x ^{4}} : x^{6} - 2 x^{4} - 7 x^{2} + \frac{6 ( x ^{4} + 2 x ^{2} + 1 )}{x ^{2}} - \frac{4 ( x ^{4} + 2 x ^{2} + 1 )}{x ^{4}} + \frac{x ^{4} + 2 x ^{2} + 1}{x ^{6}} - 4

= x^{6} - 2 x^{4} - 7 x^{2} + \frac{6 ( x ^{4} + 2 x ^{2} + 1 )}{x ^{2}} - \frac{4 ( x ^{4} + 2 x ^{2} + 1 )}{x ^{4}} + \frac{x ^{4} + 2 x ^{2} + 1}{x ^{6}} - 4