展開する (2x-1/(x^2))^4

解

展開する

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

16 x^{4} - 32 x + \frac{24}{x ^{2}} - \frac{8}{x ^{5}} + \frac{1}{x ^{8}}

解答ステップ

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

2項定理を適用する:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = 2 x, b = - \frac{1}{x ^{2}}

= i = 0 \sum 4 (i 4) (2 x)^{(4 - i)} (- \frac{1}{x ^{2}})^{i}

総和を展開する

= \frac{4 !}{0 ! ( 4 - 0 ) !} (2 x)^{4} (- \frac{1}{x ^{2}})^{0} + \frac{4 !}{1 ! ( 4 - 1 ) !} (2 x)^{3} (- \frac{1}{x ^{2}})^{1} + \frac{4 !}{2 ! ( 4 - 2 ) !} (2 x)^{2} (- \frac{1}{x ^{2}})^{2} + \frac{4 !}{3 ! ( 4 - 3 ) !} (2 x)^{1} (- \frac{1}{x ^{2}})^{3} + \frac{4 !}{4 ! ( 4 - 4 ) !} (2 x)^{0} (- \frac{1}{x ^{2}})^{4}

簡素化

\frac{4 !}{0 ! ( 4 - 0 ) !} (2 x)^{4} (- \frac{1}{x ^{2}})^{0} : 16 x^{4}

簡素化

\frac{4 !}{1 ! ( 4 - 1 ) !} (2 x)^{3} (- \frac{1}{x ^{2}})^{1} : - 32 x

簡素化

\frac{4 !}{2 ! ( 4 - 2 ) !} (2 x)^{2} (- \frac{1}{x ^{2}})^{2} : \frac{24}{x ^{2}}

簡素化

\frac{4 !}{3 ! ( 4 - 3 ) !} (2 x)^{1} (- \frac{1}{x ^{2}})^{3} : - \frac{8}{x ^{5}}

簡素化

\frac{4 !}{4 ! ( 4 - 4 ) !} (2 x)^{0} (- \frac{1}{x ^{2}})^{4} : \frac{1}{x ^{8}}

= 16 x^{4} - 32 x + \frac{24}{x ^{2}} - \frac{8}{x ^{5}} + \frac{1}{x ^{8}}