展开 ((2+x)^{10})/(1024)

解答

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\frac{( 2 + x ) ^{10}}{1024}

\frac{x ^{10}}{1024} + \frac{5 x ^{9}}{256} + \frac{45 x ^{8}}{256} + \frac{15 x ^{7}}{16} + \frac{105 x ^{6}}{32} + \frac{63 x ^{5}}{8} + \frac{105 x ^{4}}{8} + 15 x^{3} + \frac{45 x ^{2}}{4} + 5 x + 1

求解步骤

\frac{( 2 + x ) ^{10}}{1024}

(2 + x)^{10} = 1024 + 5120 x + 11520 x^{2} + 15360 x^{3} + 13440 x^{4} + 8064 x^{5} + 3360 x^{6} + 960 x^{7} + 180 x^{8} + 20 x^{9} + x^{10}

= \frac{1024 + 5120 x + 11520 x ^{2} + 15360 x ^{3} + 13440 x ^{4} + 8064 x ^{5} + 3360 x ^{6} + 960 x ^{7} + 180 x ^{8} + 20 x ^{9} + x ^{10}}{1024}

使用分式法则:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{1024 + 5120 x + 11520 x ^{2} + 15360 x ^{3} + 13440 x ^{4} + 8064 x ^{5} + 3360 x ^{6} + 960 x ^{7} + 180 x ^{8} + 20 x ^{9} + x ^{10}}{1024} = \frac{1024}{1024} + \frac{5120 x}{1024} + \frac{11520 x ^{2}}{1024} + \frac{15360 x ^{3}}{1024} + \frac{13440 x ^{4}}{1024} + \frac{8064 x ^{5}}{1024} + \frac{3360 x ^{6}}{1024} + \frac{960 x ^{7}}{1024} + \frac{180 x ^{8}}{1024} + \frac{20 x ^{9}}{1024} + \frac{x ^{10}}{1024}

= \frac{1024}{1024} + \frac{5120 x}{1024} + \frac{11520 x ^{2}}{1024} + \frac{15360 x ^{3}}{1024} + \frac{13440 x ^{4}}{1024} + \frac{8064 x ^{5}}{1024} + \frac{3360 x ^{6}}{1024} + \frac{960 x ^{7}}{1024} + \frac{180 x ^{8}}{1024} + \frac{20 x ^{9}}{1024} + \frac{x ^{10}}{1024}

使用法则

\frac{a}{a} = 1

\frac{1024}{1024} = 1

= 1 + \frac{5120 x}{1024} + \frac{11520 x ^{2}}{1024} + \frac{15360 x ^{3}}{1024} + \frac{13440 x ^{4}}{1024} + \frac{8064 x ^{5}}{1024} + \frac{3360 x ^{6}}{1024} + \frac{960 x ^{7}}{1024} + \frac{180 x ^{8}}{1024} + \frac{20 x ^{9}}{1024} + \frac{x ^{10}}{1024}

\frac{5120 x}{1024} = 5 x

\frac{11520 x ^{2}}{1024} = \frac{45 x ^{2}}{4}

\frac{15360 x ^{3}}{1024} = 15 x^{3}

\frac{13440 x ^{4}}{1024} = \frac{105 x ^{4}}{8}

\frac{8064 x ^{5}}{1024} = \frac{63 x ^{5}}{8}

\frac{3360 x ^{6}}{1024} = \frac{105 x ^{6}}{32}

\frac{960 x ^{7}}{1024} = \frac{15 x ^{7}}{16}

\frac{180 x ^{8}}{1024} = \frac{45 x ^{8}}{256}

\frac{20 x ^{9}}{1024} = \frac{5 x ^{9}}{256}

= 1 + 5 x + \frac{45 x ^{2}}{4} + 15 x^{3} + \frac{105 x ^{4}}{8} + \frac{63 x ^{5}}{8} + \frac{105 x ^{6}}{32} + \frac{15 x ^{7}}{16} + \frac{45 x ^{8}}{256} + \frac{5 x ^{9}}{256} + \frac{x ^{10}}{1024}

改写为标准形式

= \frac{x ^{10}}{1024} + \frac{5 x ^{9}}{256} + \frac{45 x ^{8}}{256} + \frac{15 x ^{7}}{16} + \frac{105 x ^{6}}{32} + \frac{63 x ^{5}}{8} + \frac{105 x ^{4}}{8} + 15 x^{3} + \frac{45 x ^{2}}{4} + 5 x + 1

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