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受欢迎的代数 >

展开 (x-2/(x^2))^9

解答

展开

(x - \frac{2}{x ^{2}})^{9}

解答

x^{9} - 18 x^{6} + 144 x^{3} - 672 + \frac{2016}{x ^{3}} - \frac{4032}{x ^{6}} + \frac{5376}{x ^{9}} - \frac{4608}{x ^{12}} + \frac{2304}{x ^{15}} - \frac{512}{x ^{18}}

求解步骤

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

= i = 0 \sum 9 (i 9) x^{(9 - i)} (- \frac{2}{x ^{2}})^{i}

展开求和

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

化简

\frac{9 !}{0 ! ( 9 - 0 ) !} x^{9} (- \frac{2}{x ^{2}})^{0} : x^{9}

化简

\frac{9 !}{1 ! ( 9 - 1 ) !} x^{8} (- \frac{2}{x ^{2}})^{1} : - 18 x^{6}

化简

\frac{9 !}{2 ! ( 9 - 2 ) !} x^{7} (- \frac{2}{x ^{2}})^{2} : 144 x^{3}

化简

\frac{9 !}{3 ! ( 9 - 3 ) !} x^{6} (- \frac{2}{x ^{2}})^{3} : - 672

化简

\frac{9 !}{4 ! ( 9 - 4 ) !} x^{5} (- \frac{2}{x ^{2}})^{4} : \frac{2016}{x ^{3}}

化简

\frac{9 !}{5 ! ( 9 - 5 ) !} x^{4} (- \frac{2}{x ^{2}})^{5} : - \frac{4032}{x ^{6}}

化简

\frac{9 !}{6 ! ( 9 - 6 ) !} x^{3} (- \frac{2}{x ^{2}})^{6} : \frac{5376}{x ^{9}}

化简

\frac{9 !}{7 ! ( 9 - 7 ) !} x^{2} (- \frac{2}{x ^{2}})^{7} : - \frac{4608}{x ^{12}}

化简

\frac{9 !}{8 ! ( 9 - 8 ) !} x^{1} (- \frac{2}{x ^{2}})^{8} : \frac{2304}{x ^{15}}

化简

\frac{9 !}{9 ! ( 9 - 9 ) !} x^{0} (- \frac{2}{x ^{2}})^{9} : - \frac{512}{x ^{18}}

= x^{9} - 18 x^{6} + 144 x^{3} - 672 + \frac{2016}{x ^{3}} - \frac{4032}{x ^{6}} + \frac{5376}{x ^{9}} - \frac{4608}{x ^{12}} + \frac{2304}{x ^{15}} - \frac{512}{x ^{18}}

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