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受欢迎的代数 >

展开 (1+x/2)^9

解答

展开

(1 + \frac{x}{2})^{9}

解答

1 + \frac{9 x}{2} + 9 x^{2} + \frac{21 x ^{3}}{2} + \frac{63 x ^{4}}{8} + \frac{63 x ^{5}}{16} + \frac{21 x ^{6}}{16} + \frac{9 x ^{7}}{32} + \frac{9 x ^{8}}{256} + \frac{x ^{9}}{512}

求解步骤

(1 + \frac{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 = 1, b = \frac{x}{2}

= i = 0 \sum 9 (i 9) \cdot 1^{(9 - i)} (\frac{x}{2})^{i}

展开求和

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

化简

\frac{9 !}{0 ! ( 9 - 0 ) !} \cdot 1^{9} (\frac{x}{2})^{0} : 1

化简

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

化简

\frac{9 !}{2 ! ( 9 - 2 ) !} \cdot 1^{7} (\frac{x}{2})^{2} : 9 x^{2}

化简

\frac{9 !}{3 ! ( 9 - 3 ) !} \cdot 1^{6} (\frac{x}{2})^{3} : \frac{21 x ^{3}}{2}

化简

\frac{9 !}{4 ! ( 9 - 4 ) !} \cdot 1^{5} (\frac{x}{2})^{4} : \frac{63 x ^{4}}{8}

化简

\frac{9 !}{5 ! ( 9 - 5 ) !} \cdot 1^{4} (\frac{x}{2})^{5} : \frac{63 x ^{5}}{16}

化简

\frac{9 !}{6 ! ( 9 - 6 ) !} \cdot 1^{3} (\frac{x}{2})^{6} : \frac{21 x ^{6}}{16}

化简

\frac{9 !}{7 ! ( 9 - 7 ) !} \cdot 1^{2} (\frac{x}{2})^{7} : \frac{9 x ^{7}}{32}

化简

\frac{9 !}{8 ! ( 9 - 8 ) !} \cdot 1^{1} (\frac{x}{2})^{8} : \frac{9 x ^{8}}{256}

化简

\frac{9 !}{9 ! ( 9 - 9 ) !} \cdot 1^{0} (\frac{x}{2})^{9} : \frac{x ^{9}}{512}

= 1 + \frac{9 x}{2} + 9 x^{2} + \frac{21 x ^{3}}{2} + \frac{63 x ^{4}}{8} + \frac{63 x ^{5}}{16} + \frac{21 x ^{6}}{16} + \frac{9 x ^{7}}{32} + \frac{9 x ^{8}}{256} + \frac{x ^{9}}{512}

流行的例子

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e x p an d cos (4) (t + π)

展开 (9(x^2+49))/((16-x^2)^3)

e x p an d \frac{9 ( x ^{2} + 49 )}{( 16 - x ^{2} ) ^{3}}

展开 (2x^5+4x^3-3x^2+3x-1)/((x^3+1)^3)

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e x p an d \frac{1}{( x ^{2} - x + 1 ) ( x ^{2} - x + 2 )}

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e x p an d \frac{1}{( s + 10 ) ( s + 2 ) ^{2}}