展開する (5(x-1)^4(x^2+x+1)^4(x^3+2))/(x^{11)}

解

展開する

\frac{5 ( x - 1 ) ^{4} ( x ^{2} + x + 1 ) ^{4} ( x ^{3} + 2 )}{x ^{11}}

5 x^{4} - 10 x - \frac{10}{x ^{2}} + \frac{40}{x ^{5}} - \frac{35}{x ^{8}} + \frac{10}{x ^{11}}

解答ステップ

\frac{5 ( x - 1 ) ^{4} ( x ^{2} + x + 1 ) ^{4} ( x ^{3} + 2 )}{x ^{11}}

(x - 1)^{4} = x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1

= \frac{5 ( x ^{4} - 4 x ^{3} + 6 x ^{2} - 4 x + 1 ) ( x ^{2} + x + 1 ) ^{4} ( x ^{3} + 2 )}{x ^{11}}

(x^{2} + x + 1)^{4} = x^{8} + 4 x^{7} + 10 x^{6} + 16 x^{5} + 19 x^{4} + 16 x^{3} + 10 x^{2} + 4 x + 1

= \frac{5 ( x ^{4} - 4 x ^{3} + 6 x ^{2} - 4 x + 1 ) ( x ^{8} + 4 x ^{7} + 10 x ^{6} + 16 x ^{5} + 19 x ^{4} + 16 x ^{3} + 10 x ^{2} + 4 x + 1 ) ( x ^{3} + 2 )}{x ^{11}}

拡張

5 (x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1) (x^{8} + 4 x^{7} + 10 x^{6} + 16 x^{5} + 19 x^{4} + 16 x^{3} + 10 x^{2} + 4 x + 1) (x^{3} + 2) : 5 x^{15} - 10 x^{12} - 10 x^{9} + 40 x^{6} - 35 x^{3} + 10

= \frac{5 x ^{15} - 10 x ^{12} - 10 x ^{9} + 40 x ^{6} - 35 x ^{3} + 10}{x ^{11}}

分数の規則を適用する:

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

\frac{5 x ^{15} - 10 x ^{12} - 10 x ^{9} + 40 x ^{6} - 35 x ^{3} + 10}{x ^{11}} = \frac{5 x ^{15}}{x ^{11}} - \frac{10 x ^{12}}{x ^{11}} - \frac{10 x ^{9}}{x ^{11}} + \frac{40 x ^{6}}{x ^{11}} - \frac{35 x ^{3}}{x ^{11}} + \frac{10}{x ^{11}}

= \frac{5 x ^{15}}{x ^{11}} - \frac{10 x ^{12}}{x ^{11}} - \frac{10 x ^{9}}{x ^{11}} + \frac{40 x ^{6}}{x ^{11}} - \frac{35 x ^{3}}{x ^{11}} + \frac{10}{x ^{11}}

\frac{5 x ^{15}}{x ^{11}} = 5 x^{4}

\frac{10 x ^{12}}{x ^{11}} = 10 x

\frac{10 x ^{9}}{x ^{11}} = \frac{10}{x ^{2}}

\frac{40 x ^{6}}{x ^{11}} = \frac{40}{x ^{5}}

\frac{35 x ^{3}}{x ^{11}} = \frac{35}{x ^{8}}

= 5 x^{4} - 10 x - \frac{10}{x ^{2}} + \frac{40}{x ^{5}} - \frac{35}{x ^{8}} + \frac{10}{x ^{11}}