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受欢迎的代数 >

展开 (2x+1/x)^4

解答

展开

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

解答

16 x^{4} + 32 x^{2} + 24 + \frac{8}{x ^{2}} + \frac{1}{x ^{4}}

求解步骤

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

使用二项式定理:

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

a = 2 x, b = \frac{1}{x}

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

展开求和

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

化简

\frac{4 !}{0 ! ( 4 - 0 ) !} (2 x)^{4} (\frac{1}{x})^{0} : 16 x^{4}

化简

\frac{4 !}{1 ! ( 4 - 1 ) !} (2 x)^{3} (\frac{1}{x})^{1} : 32 x^{2}

化简

\frac{4 !}{2 ! ( 4 - 2 ) !} (2 x)^{2} (\frac{1}{x})^{2} : 24

化简

\frac{4 !}{3 ! ( 4 - 3 ) !} (2 x)^{1} (\frac{1}{x})^{3} : \frac{8}{x ^{2}}

化简

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

= 16 x^{4} + 32 x^{2} + 24 + \frac{8}{x ^{2}} + \frac{1}{x ^{4}}

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