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受欢迎的代数 >

展开 (x+3/x)^9

解答

展开

(x + \frac{3}{x})^{9}

解答

x^{9} + 27 x^{7} + 324 x^{5} + 2268 x^{3} + 10206 x + \frac{30618}{x} + \frac{61236}{x ^{3}} + \frac{78732}{x ^{5}} + \frac{59049}{x ^{7}} + \frac{19683}{x ^{9}}

求解步骤

(x + \frac{3}{x})^{9}

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{3}{x}

= i = 0 \sum 9 (i 9) x^{(9 - i)} (\frac{3}{x})^{i}

展开求和

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

化简

\frac{9 !}{0 ! ( 9 - 0 ) !} x^{9} (\frac{3}{x})^{0} : x^{9}

化简

\frac{9 !}{1 ! ( 9 - 1 ) !} x^{8} (\frac{3}{x})^{1} : 27 x^{7}

化简

\frac{9 !}{2 ! ( 9 - 2 ) !} x^{7} (\frac{3}{x})^{2} : 324 x^{5}

化简

\frac{9 !}{3 ! ( 9 - 3 ) !} x^{6} (\frac{3}{x})^{3} : 2268 x^{3}

化简

\frac{9 !}{4 ! ( 9 - 4 ) !} x^{5} (\frac{3}{x})^{4} : 10206 x

化简

\frac{9 !}{5 ! ( 9 - 5 ) !} x^{4} (\frac{3}{x})^{5} : \frac{30618}{x}

化简

\frac{9 !}{6 ! ( 9 - 6 ) !} x^{3} (\frac{3}{x})^{6} : \frac{61236}{x ^{3}}

化简

\frac{9 !}{7 ! ( 9 - 7 ) !} x^{2} (\frac{3}{x})^{7} : \frac{78732}{x ^{5}}

化简

\frac{9 !}{8 ! ( 9 - 8 ) !} x^{1} (\frac{3}{x})^{8} : \frac{59049}{x ^{7}}

化简

\frac{9 !}{9 ! ( 9 - 9 ) !} x^{0} (\frac{3}{x})^{9} : \frac{19683}{x ^{9}}

= x^{9} + 27 x^{7} + 324 x^{5} + 2268 x^{3} + 10206 x + \frac{30618}{x} + \frac{61236}{x ^{3}} + \frac{78732}{x ^{5}} + \frac{59049}{x ^{7}} + \frac{19683}{x ^{9}}

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