展開する ((2x+1)^3(2x-3)^3)/(x^6)

解

展開する

\frac{( 2 x + 1 ) ^{3} ( 2 x - 3 ) ^{3}}{x ^{6}}

64 - \frac{192}{x} + \frac{48}{x ^{2}} + \frac{224}{x ^{3}} - \frac{36}{x ^{4}} - \frac{108}{x ^{5}} - \frac{27}{x ^{6}}

解答ステップ

\frac{( 2 x + 1 ) ^{3} ( 2 x - 3 ) ^{3}}{x ^{6}}

(2 x + 1)^{3} = 8 x^{3} + 12 x^{2} + 6 x + 1

= \frac{( 8 x ^{3} + 12 x ^{2} + 6 x + 1 ) ( 2 x - 3 ) ^{3}}{x ^{6}}

(2 x - 3)^{3} = 8 x^{3} - 36 x^{2} + 54 x - 27

= \frac{( 8 x ^{3} + 12 x ^{2} + 6 x + 1 ) ( 8 x ^{3} - 36 x ^{2} + 54 x - 27 )}{x ^{6}}

拡張

(8 x^{3} + 12 x^{2} + 6 x + 1) (8 x^{3} - 36 x^{2} + 54 x - 27) : 64 x^{6} - 192 x^{5} + 48 x^{4} + 224 x^{3} - 36 x^{2} - 108 x - 27

= \frac{64 x ^{6} - 192 x ^{5} + 48 x ^{4} + 224 x ^{3} - 36 x ^{2} - 108 x - 27}{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{64 x ^{6} - 192 x ^{5} + 48 x ^{4} + 224 x ^{3} - 36 x ^{2} - 108 x - 27}{x ^{6}} = \frac{64 x ^{6}}{x ^{6}} - \frac{192 x ^{5}}{x ^{6}} + \frac{48 x ^{4}}{x ^{6}} + \frac{224 x ^{3}}{x ^{6}} - \frac{36 x ^{2}}{x ^{6}} - \frac{108 x}{x ^{6}} - \frac{27}{x ^{6}}

= \frac{64 x ^{6}}{x ^{6}} - \frac{192 x ^{5}}{x ^{6}} + \frac{48 x ^{4}}{x ^{6}} + \frac{224 x ^{3}}{x ^{6}} - \frac{36 x ^{2}}{x ^{6}} - \frac{108 x}{x ^{6}} - \frac{27}{x ^{6}}

\frac{64 x ^{6}}{x ^{6}} = 64

\frac{192 x ^{5}}{x ^{6}} = \frac{192}{x}

\frac{48 x ^{4}}{x ^{6}} = \frac{48}{x ^{2}}

\frac{224 x ^{3}}{x ^{6}} = \frac{224}{x ^{3}}

\frac{36 x ^{2}}{x ^{6}} = \frac{36}{x ^{4}}

\frac{108 x}{x ^{6}} = \frac{108}{x ^{5}}

= 64 - \frac{192}{x} + \frac{48}{x ^{2}} + \frac{224}{x ^{3}} - \frac{36}{x ^{4}} - \frac{108}{x ^{5}} - \frac{27}{x ^{6}}