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受欢迎的代数 >

展开 (x+1/x)^{10}

解答

展开

(x + \frac{1}{x})^{10}

解答

x^{10} + 10 x^{8} + 45 x^{6} + 120 x^{4} + 210 x^{2} + 252 + \frac{210}{x ^{2}} + \frac{120}{x ^{4}} + \frac{45}{x ^{6}} + \frac{10}{x ^{8}} + \frac{1}{x ^{10}}

求解步骤

(x + \frac{1}{x})^{10}

使用二项式定理:

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

a = x, b = \frac{1}{x}

= i = 0 \sum 10 (i 10) x^{(10 - i)} (\frac{1}{x})^{i}

展开求和

= \frac{10 !}{0 ! ( 10 - 0 ) !} x^{10} (\frac{1}{x})^{0} + \frac{10 !}{1 ! ( 10 - 1 ) !} x^{9} (\frac{1}{x})^{1} + \frac{10 !}{2 ! ( 10 - 2 ) !} x^{8} (\frac{1}{x})^{2} + \frac{10 !}{3 ! ( 10 - 3 ) !} x^{7} (\frac{1}{x})^{3} + \frac{10 !}{4 ! ( 10 - 4 ) !} x^{6} (\frac{1}{x})^{4} + \frac{10 !}{5 ! ( 10 - 5 ) !} x^{5} (\frac{1}{x})^{5} + \frac{10 !}{6 ! ( 10 - 6 ) !} x^{4} (\frac{1}{x})^{6} + \frac{10 !}{7 ! ( 10 - 7 ) !} x^{3} (\frac{1}{x})^{7} + \frac{10 !}{8 ! ( 10 - 8 ) !} x^{2} (\frac{1}{x})^{8} + \frac{10 !}{9 ! ( 10 - 9 ) !} x^{1} (\frac{1}{x})^{9} + \frac{10 !}{10 ! ( 10 - 10 ) !} x^{0} (\frac{1}{x})^{10}

化简

\frac{10 !}{0 ! ( 10 - 0 ) !} x^{10} (\frac{1}{x})^{0} : x^{10}

化简

\frac{10 !}{1 ! ( 10 - 1 ) !} x^{9} (\frac{1}{x})^{1} : 10 x^{8}

化简

\frac{10 !}{2 ! ( 10 - 2 ) !} x^{8} (\frac{1}{x})^{2} : 45 x^{6}

化简

\frac{10 !}{3 ! ( 10 - 3 ) !} x^{7} (\frac{1}{x})^{3} : 120 x^{4}

化简

\frac{10 !}{4 ! ( 10 - 4 ) !} x^{6} (\frac{1}{x})^{4} : 210 x^{2}

化简

\frac{10 !}{5 ! ( 10 - 5 ) !} x^{5} (\frac{1}{x})^{5} : 252

化简

\frac{10 !}{6 ! ( 10 - 6 ) !} x^{4} (\frac{1}{x})^{6} : \frac{210}{x ^{2}}

化简

\frac{10 !}{7 ! ( 10 - 7 ) !} x^{3} (\frac{1}{x})^{7} : \frac{120}{x ^{4}}

化简

\frac{10 !}{8 ! ( 10 - 8 ) !} x^{2} (\frac{1}{x})^{8} : \frac{45}{x ^{6}}

化简

\frac{10 !}{9 ! ( 10 - 9 ) !} x^{1} (\frac{1}{x})^{9} : \frac{10}{x ^{8}}

化简

\frac{10 !}{10 ! ( 10 - 10 ) !} x^{0} (\frac{1}{x})^{10} : \frac{1}{x ^{10}}

= x^{10} + 10 x^{8} + 45 x^{6} + 120 x^{4} + 210 x^{2} + 252 + \frac{210}{x ^{2}} + \frac{120}{x ^{4}} + \frac{45}{x ^{6}} + \frac{10}{x ^{8}} + \frac{1}{x ^{10}}

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