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受欢迎的代数 >

展开 (x^2-2/x)^5

解答

展开

(x^{2} - \frac{2}{x})^{5}

解答

x^{10} - 10 x^{7} + 40 x^{4} - 80 x + \frac{80}{x ^{2}} - \frac{32}{x ^{5}}

求解步骤

(x^{2} - \frac{2}{x})^{5}

使用二项式定理:

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

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

= i = 0 \sum 5 (i 5) (x^{2})^{(5 - i)} (- \frac{2}{x})^{i}

展开求和

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

化简

\frac{5 !}{0 ! ( 5 - 0 ) !} (x^{2})^{5} (- \frac{2}{x})^{0} : x^{10}

化简

\frac{5 !}{1 ! ( 5 - 1 ) !} (x^{2})^{4} (- \frac{2}{x})^{1} : - 10 x^{7}

化简

\frac{5 !}{2 ! ( 5 - 2 ) !} (x^{2})^{3} (- \frac{2}{x})^{2} : 40 x^{4}

化简

\frac{5 !}{3 ! ( 5 - 3 ) !} (x^{2})^{2} (- \frac{2}{x})^{3} : - 80 x

化简

\frac{5 !}{4 ! ( 5 - 4 ) !} (x^{2})^{1} (- \frac{2}{x})^{4} : \frac{80}{x ^{2}}

化简

\frac{5 !}{5 ! ( 5 - 5 ) !} (x^{2})^{0} (- \frac{2}{x})^{5} : - \frac{32}{x ^{5}}

= x^{10} - 10 x^{7} + 40 x^{4} - 80 x + \frac{80}{x ^{2}} - \frac{32}{x ^{5}}

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