展開する (1+x+1/2 x^2)^8

解

展開する

(1 + x + \frac{1}{2} x^{2})^{8}

\frac{x ^{16}}{256} + \frac{x ^{15}}{16} + \frac{x ^{14}}{2} + \frac{21 x ^{13}}{8} + \frac{161 x ^{12}}{16} + \frac{119 x ^{11}}{4} + 70 x^{10} + \frac{267 x ^{9}}{2} + \frac{1667 x ^{8}}{8} + 267 x^{7} + 280 x^{6} + 238 x^{5} + 161 x^{4} + 84 x^{3} + 32 x^{2} + 8 x + 1

解答ステップ

(1 + x + \frac{1}{2} x^{2})^{8}

\frac{1}{2} x^{2} = \frac{x ^{2}}{2}

= (1 + x + \frac{x ^{2}}{2})^{8}

(1 + x + \frac{x ^{2}}{2})^{8} = (1 + x + \frac{x ^{2}}{2}) (1 + x + \frac{x ^{2}}{2}) (1 + x + \frac{x ^{2}}{2}) (1 + x + \frac{x ^{2}}{2}) (1 + x + \frac{x ^{2}}{2}) (1 + x + \frac{x ^{2}}{2}) (1 + x + \frac{x ^{2}}{2}) (1 + x + \frac{x ^{2}}{2})

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

拡張

(1 + x + \frac{x ^{2}}{2}) (1 + x + \frac{x ^{2}}{2}) (1 + x + \frac{x ^{2}}{2}) (1 + x + \frac{x ^{2}}{2}) (1 + x + \frac{x ^{2}}{2}) (1 + x + \frac{x ^{2}}{2}) (1 + x + \frac{x ^{2}}{2}) (1 + x + \frac{x ^{2}}{2}) : \frac{x ^{16} + 16 x ^{15} + 128 x ^{14} + 672 x ^{13} + 2576 x ^{12} + 7616 x ^{11} + 17920 x ^{10} + 34176 x ^{9} + 53344 x ^{8} + 68352 x ^{7} + 71680 x ^{6} + 60928 x ^{5} + 41216 x ^{4} + 21504 x ^{3} + 8192 x ^{2} + 2048 x}{256} + 1

= \frac{x ^{16} + 16 x ^{15} + 128 x ^{14} + 672 x ^{13} + 2576 x ^{12} + 7616 x ^{11} + 17920 x ^{10} + 34176 x ^{9} + 53344 x ^{8} + 68352 x ^{7} + 71680 x ^{6} + 60928 x ^{5} + 41216 x ^{4} + 21504 x ^{3} + 8192 x ^{2} + 2048 x}{256} + 1

分数の規則を適用する:

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

\frac{x ^{16} + 16 x ^{15} + 128 x ^{14} + 672 x ^{13} + 2576 x ^{12} + 7616 x ^{11} + 17920 x ^{10} + 34176 x ^{9} + 53344 x ^{8} + 68352 x ^{7} + 71680 x ^{6} + 60928 x ^{5} + 41216 x ^{4} + 21504 x ^{3} + 8192 x ^{2} + 2048 x}{256} = \frac{x ^{16}}{256} + \frac{16 x ^{15}}{256} + \frac{128 x ^{14}}{256} + \frac{672 x ^{13}}{256} + \frac{2576 x ^{12}}{256} + \frac{7616 x ^{11}}{256} + \frac{17920 x ^{10}}{256} + \frac{34176 x ^{9}}{256} + \frac{53344 x ^{8}}{256} + \frac{68352 x ^{7}}{256} + \frac{71680 x ^{6}}{256} + \frac{60928 x ^{5}}{256} + \frac{41216 x ^{4}}{256} + \frac{21504 x ^{3}}{256} + \frac{8192 x ^{2}}{256} + \frac{2048 x}{256}

= \frac{x ^{16}}{256} + \frac{16 x ^{15}}{256} + \frac{128 x ^{14}}{256} + \frac{672 x ^{13}}{256} + \frac{2576 x ^{12}}{256} + \frac{7616 x ^{11}}{256} + \frac{17920 x ^{10}}{256} + \frac{34176 x ^{9}}{256} + \frac{53344 x ^{8}}{256} + \frac{68352 x ^{7}}{256} + \frac{71680 x ^{6}}{256} + \frac{60928 x ^{5}}{256} + \frac{41216 x ^{4}}{256} + \frac{21504 x ^{3}}{256} + \frac{8192 x ^{2}}{256} + \frac{2048 x}{256} + 1

\frac{16 x ^{15}}{256} = \frac{x ^{15}}{16}

\frac{128 x ^{14}}{256} = \frac{x ^{14}}{2}

\frac{672 x ^{13}}{256} = \frac{21 x ^{13}}{8}

\frac{2576 x ^{12}}{256} = \frac{161 x ^{12}}{16}

\frac{7616 x ^{11}}{256} = \frac{119 x ^{11}}{4}

\frac{17920 x ^{10}}{256} = 70 x^{10}

\frac{34176 x ^{9}}{256} = \frac{267 x ^{9}}{2}

\frac{53344 x ^{8}}{256} = \frac{1667 x ^{8}}{8}

\frac{68352 x ^{7}}{256} = 267 x^{7}

\frac{71680 x ^{6}}{256} = 280 x^{6}

\frac{60928 x ^{5}}{256} = 238 x^{5}

\frac{41216 x ^{4}}{256} = 161 x^{4}

\frac{21504 x ^{3}}{256} = 84 x^{3}

\frac{8192 x ^{2}}{256} = 32 x^{2}

\frac{2048 x}{256} = 8 x

= \frac{x ^{16}}{256} + \frac{x ^{15}}{16} + \frac{x ^{14}}{2} + \frac{21 x ^{13}}{8} + \frac{161 x ^{12}}{16} + \frac{119 x ^{11}}{4} + 70 x^{10} + \frac{267 x ^{9}}{2} + \frac{1667 x ^{8}}{8} + 267 x^{7} + 280 x^{6} + 238 x^{5} + 161 x^{4} + 84 x^{3} + 32 x^{2} + 8 x + 1