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확장 (x^3-1/x)^{24}

해법

확장

(x^{3} - \frac{1}{x})^{24}

해법

x^{72} - 24 x^{68} + 276 x^{64} - 2024 x^{60} + 10626 x^{56} - 42504 x^{52} + 134596 x^{48} - 346104 x^{44} + 735471 x^{40} - 1307504 x^{36} + 1961256 x^{32} - 2496144 x^{28} + 2704156 x^{24} - 2496144 x^{20} + 1961256 x^{16} - 1307504 x^{12} + 735471 x^{8} - 346104 x^{4} + 134596 - \frac{42504}{x ^{4}} + \frac{10626}{x ^{8}} - \frac{2024}{x ^{12}} + \frac{276}{x ^{16}} - \frac{24}{x ^{20}} + \frac{1}{x ^{24}}

솔루션 단계

(x^{3} - \frac{1}{x})^{24}

이항 정리 적용:

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

a = x^{3}, b = - \frac{1}{x}

= i = 0 \sum 24 (i 24) (x^{3})^{(24 - i)} (- \frac{1}{x})^{i}

합계 확장

= \frac{24 !}{0 ! ( 24 - 0 ) !} (x^{3})^{24} (- \frac{1}{x})^{0} + \frac{24 !}{1 ! ( 24 - 1 ) !} (x^{3})^{23} (- \frac{1}{x})^{1} + \frac{24 !}{2 ! ( 24 - 2 ) !} (x^{3})^{22} (- \frac{1}{x})^{2} + \frac{24 !}{3 ! ( 24 - 3 ) !} (x^{3})^{21} (- \frac{1}{x})^{3} + \frac{24 !}{4 ! ( 24 - 4 ) !} (x^{3})^{20} (- \frac{1}{x})^{4} + \frac{24 !}{5 ! ( 24 - 5 ) !} (x^{3})^{19} (- \frac{1}{x})^{5} + \frac{24 !}{6 ! ( 24 - 6 ) !} (x^{3})^{18} (- \frac{1}{x})^{6} + \frac{24 !}{7 ! ( 24 - 7 ) !} (x^{3})^{17} (- \frac{1}{x})^{7} + \frac{24 !}{8 ! ( 24 - 8 ) !} (x^{3})^{16} (- \frac{1}{x})^{8} + \frac{24 !}{9 ! ( 24 - 9 ) !} (x^{3})^{15} (- \frac{1}{x})^{9} + \frac{24 !}{10 ! ( 24 - 10 ) !} (x^{3})^{14} (- \frac{1}{x})^{10} + \frac{24 !}{11 ! ( 24 - 11 ) !} (x^{3})^{13} (- \frac{1}{x})^{11} + \frac{24 !}{12 ! ( 24 - 12 ) !} (x^{3})^{12} (- \frac{1}{x})^{12} + \frac{24 !}{13 ! ( 24 - 13 ) !} (x^{3})^{11} (- \frac{1}{x})^{13} + \frac{24 !}{14 ! ( 24 - 14 ) !} (x^{3})^{10} (- \frac{1}{x})^{14} + \frac{24 !}{15 ! ( 24 - 15 ) !} (x^{3})^{9} (- \frac{1}{x})^{15} + \frac{24 !}{16 ! ( 24 - 16 ) !} (x^{3})^{8} (- \frac{1}{x})^{16} + \frac{24 !}{17 ! ( 24 - 17 ) !} (x^{3})^{7} (- \frac{1}{x})^{17} + \frac{24 !}{18 ! ( 24 - 18 ) !} (x^{3})^{6} (- \frac{1}{x})^{18} + \frac{24 !}{19 ! ( 24 - 19 ) !} (x^{3})^{5} (- \frac{1}{x})^{19} + \frac{24 !}{20 ! ( 24 - 20 ) !} (x^{3})^{4} (- \frac{1}{x})^{20} + \frac{24 !}{21 ! ( 24 - 21 ) !} (x^{3})^{3} (- \frac{1}{x})^{21} + \frac{24 !}{22 ! ( 24 - 22 ) !} (x^{3})^{2} (- \frac{1}{x})^{22} + \frac{24 !}{23 ! ( 24 - 23 ) !} (x^{3})^{1} (- \frac{1}{x})^{23} + \frac{24 !}{24 ! ( 24 - 24 ) !} (x^{3})^{0} (- \frac{1}{x})^{24}

\frac{24 !}{0 ! ( 24 - 0 ) !} (x^{3})^{24} (- \frac{1}{x})^{0}

간소화하다 :

x^{72}

\frac{24 !}{1 ! ( 24 - 1 ) !} (x^{3})^{23} (- \frac{1}{x})^{1}

간소화하다 :

- 24 x^{68}

\frac{24 !}{2 ! ( 24 - 2 ) !} (x^{3})^{22} (- \frac{1}{x})^{2}

간소화하다 :

276 x^{64}

\frac{24 !}{3 ! ( 24 - 3 ) !} (x^{3})^{21} (- \frac{1}{x})^{3}

간소화하다 :

- 2024 x^{60}

\frac{24 !}{4 ! ( 24 - 4 ) !} (x^{3})^{20} (- \frac{1}{x})^{4}

간소화하다 :

10626 x^{56}

\frac{24 !}{5 ! ( 24 - 5 ) !} (x^{3})^{19} (- \frac{1}{x})^{5}

간소화하다 :

- 42504 x^{52}

\frac{24 !}{6 ! ( 24 - 6 ) !} (x^{3})^{18} (- \frac{1}{x})^{6}

간소화하다 :

134596 x^{48}

\frac{24 !}{7 ! ( 24 - 7 ) !} (x^{3})^{17} (- \frac{1}{x})^{7}

간소화하다 :

- 346104 x^{44}

\frac{24 !}{8 ! ( 24 - 8 ) !} (x^{3})^{16} (- \frac{1}{x})^{8}

간소화하다 :

735471 x^{40}

\frac{24 !}{9 ! ( 24 - 9 ) !} (x^{3})^{15} (- \frac{1}{x})^{9}

간소화하다 :

- 1307504 x^{36}

\frac{24 !}{10 ! ( 24 - 10 ) !} (x^{3})^{14} (- \frac{1}{x})^{10}

간소화하다 :

1961256 x^{32}

\frac{24 !}{11 ! ( 24 - 11 ) !} (x^{3})^{13} (- \frac{1}{x})^{11}

간소화하다 :

- 2496144 x^{28}

\frac{24 !}{12 ! ( 24 - 12 ) !} (x^{3})^{12} (- \frac{1}{x})^{12}

간소화하다 :

2704156 x^{24}

\frac{24 !}{13 ! ( 24 - 13 ) !} (x^{3})^{11} (- \frac{1}{x})^{13}

간소화하다 :

- 2496144 x^{20}

\frac{24 !}{14 ! ( 24 - 14 ) !} (x^{3})^{10} (- \frac{1}{x})^{14}

간소화하다 :

1961256 x^{16}

\frac{24 !}{15 ! ( 24 - 15 ) !} (x^{3})^{9} (- \frac{1}{x})^{15}

간소화하다 :

- 1307504 x^{12}

\frac{24 !}{16 ! ( 24 - 16 ) !} (x^{3})^{8} (- \frac{1}{x})^{16}

간소화하다 :

735471 x^{8}

\frac{24 !}{17 ! ( 24 - 17 ) !} (x^{3})^{7} (- \frac{1}{x})^{17}

간소화하다 :

- 346104 x^{4}

\frac{24 !}{18 ! ( 24 - 18 ) !} (x^{3})^{6} (- \frac{1}{x})^{18}

간소화하다 :

134596

\frac{24 !}{19 ! ( 24 - 19 ) !} (x^{3})^{5} (- \frac{1}{x})^{19}

간소화하다 :

- \frac{42504}{x ^{4}}

\frac{24 !}{20 ! ( 24 - 20 ) !} (x^{3})^{4} (- \frac{1}{x})^{20}

간소화하다 :

\frac{10626}{x ^{8}}

\frac{24 !}{21 ! ( 24 - 21 ) !} (x^{3})^{3} (- \frac{1}{x})^{21}

간소화하다 :

- \frac{2024}{x ^{12}}

\frac{24 !}{22 ! ( 24 - 22 ) !} (x^{3})^{2} (- \frac{1}{x})^{22}

간소화하다 :

\frac{276}{x ^{16}}

\frac{24 !}{23 ! ( 24 - 23 ) !} (x^{3})^{1} (- \frac{1}{x})^{23}

간소화하다 :

- \frac{24}{x ^{20}}

\frac{24 !}{24 ! ( 24 - 24 ) !} (x^{3})^{0} (- \frac{1}{x})^{24}

간소화하다 :

\frac{1}{x ^{24}}

= x^{72} - 24 x^{68} + 276 x^{64} - 2024 x^{60} + 10626 x^{56} - 42504 x^{52} + 134596 x^{48} - 346104 x^{44} + 735471 x^{40} - 1307504 x^{36} + 1961256 x^{32} - 2496144 x^{28} + 2704156 x^{24} - 2496144 x^{20} + 1961256 x^{16} - 1307504 x^{12} + 735471 x^{8} - 346104 x^{4} + 134596 - \frac{42504}{x ^{4}} + \frac{10626}{x ^{8}} - \frac{2024}{x ^{12}} + \frac{276}{x ^{16}} - \frac{24}{x ^{20}} + \frac{1}{x ^{24}}

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