展開する (x^2+1/x)^4

解

展開する

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

x^{8} + 4 x^{5} + 6 x^{2} + \frac{4}{x} + \frac{1}{x ^{4}}

解答ステップ

(x^{2} + \frac{1}{x})^{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 = x^{2}, b = \frac{1}{x}

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

総和を展開する

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

簡素化

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

簡素化

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

簡素化

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

簡素化

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

簡素化

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

= x^{8} + 4 x^{5} + 6 x^{2} + \frac{4}{x} + \frac{1}{x ^{4}}