展開する ((x^2+1)(x^4+1)(x^6+1))/(x^6)

解

展開する

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

x^{6} + 2 + x^{2} + x^{4} + \frac{1}{x ^{2}} + \frac{1}{x ^{4}} + \frac{1}{x ^{6}}

解答ステップ

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

拡張

(x^{2} + 1) (x^{4} + 1) (x^{6} + 1) : x^{12} + 2 x^{6} + x^{8} + x^{10} + x^{4} + x^{2} + 1

= \frac{x ^{12} + 2 x ^{6} + x ^{8} + x ^{10} + x ^{4} + x ^{2} + 1}{x ^{6}}

分数の規則を適用する:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{x ^{12} + 2 x ^{6} + x ^{8} + x ^{10} + x ^{4} + x ^{2} + 1}{x ^{6}} = \frac{x ^{12}}{x ^{6}} + \frac{2 x ^{6}}{x ^{6}} + \frac{x ^{8}}{x ^{6}} + \frac{x ^{10}}{x ^{6}} + \frac{x ^{4}}{x ^{6}} + \frac{x ^{2}}{x ^{6}} + \frac{1}{x ^{6}}

= \frac{x ^{12}}{x ^{6}} + \frac{2 x ^{6}}{x ^{6}} + \frac{x ^{8}}{x ^{6}} + \frac{x ^{10}}{x ^{6}} + \frac{x ^{4}}{x ^{6}} + \frac{x ^{2}}{x ^{6}} + \frac{1}{x ^{6}}

\frac{x ^{12}}{x ^{6}} = x^{6}

\frac{2 x ^{6}}{x ^{6}} = 2

\frac{x ^{8}}{x ^{6}} = x^{2}

\frac{x ^{10}}{x ^{6}} = x^{4}

\frac{x ^{4}}{x ^{6}} = \frac{1}{x ^{2}}

\frac{x ^{2}}{x ^{6}} = \frac{1}{x ^{4}}

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