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Popolare Algebra >

espandere (x^3-1/x)^{20}

Soluzione

espandere

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

Soluzione

x^{60} - 20 x^{56} + 190 x^{52} - 1140 x^{48} + 4845 x^{44} - 15504 x^{40} + 38760 x^{36} - 77520 x^{32} + 125970 x^{28} - 167960 x^{24} + 184756 x^{20} - 167960 x^{16} + 125970 x^{12} - 77520 x^{8} + 38760 x^{4} - 15504 + \frac{4845}{x ^{4}} - \frac{1140}{x ^{8}} + \frac{190}{x ^{12}} - \frac{20}{x ^{16}} + \frac{1}{x ^{20}}

Fasi della soluzione

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

Applicare il teorema binomiale:

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

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

= i = 0 \sum 20 (i 20) (x^{3})^{(20 - i)} (- \frac{1}{x})^{i}

Espandere la sommatoria

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

Semplificare

\frac{20 !}{0 ! ( 20 - 0 ) !} (x^{3})^{20} (- \frac{1}{x})^{0} : x^{60}

Semplificare

\frac{20 !}{1 ! ( 20 - 1 ) !} (x^{3})^{19} (- \frac{1}{x})^{1} : - 20 x^{56}

Semplificare

\frac{20 !}{2 ! ( 20 - 2 ) !} (x^{3})^{18} (- \frac{1}{x})^{2} : 190 x^{52}

Semplificare

\frac{20 !}{3 ! ( 20 - 3 ) !} (x^{3})^{17} (- \frac{1}{x})^{3} : - 1140 x^{48}

Semplificare

\frac{20 !}{4 ! ( 20 - 4 ) !} (x^{3})^{16} (- \frac{1}{x})^{4} : 4845 x^{44}

Semplificare

\frac{20 !}{5 ! ( 20 - 5 ) !} (x^{3})^{15} (- \frac{1}{x})^{5} : - 15504 x^{40}

Semplificare

\frac{20 !}{6 ! ( 20 - 6 ) !} (x^{3})^{14} (- \frac{1}{x})^{6} : 38760 x^{36}

Semplificare

\frac{20 !}{7 ! ( 20 - 7 ) !} (x^{3})^{13} (- \frac{1}{x})^{7} : - 77520 x^{32}

Semplificare

\frac{20 !}{8 ! ( 20 - 8 ) !} (x^{3})^{12} (- \frac{1}{x})^{8} : 125970 x^{28}

Semplificare

\frac{20 !}{9 ! ( 20 - 9 ) !} (x^{3})^{11} (- \frac{1}{x})^{9} : - 167960 x^{24}

Semplificare

\frac{20 !}{10 ! ( 20 - 10 ) !} (x^{3})^{10} (- \frac{1}{x})^{10} : 184756 x^{20}

Semplificare

\frac{20 !}{11 ! ( 20 - 11 ) !} (x^{3})^{9} (- \frac{1}{x})^{11} : - 167960 x^{16}

Semplificare

\frac{20 !}{12 ! ( 20 - 12 ) !} (x^{3})^{8} (- \frac{1}{x})^{12} : 125970 x^{12}

Semplificare

\frac{20 !}{13 ! ( 20 - 13 ) !} (x^{3})^{7} (- \frac{1}{x})^{13} : - 77520 x^{8}

Semplificare

\frac{20 !}{14 ! ( 20 - 14 ) !} (x^{3})^{6} (- \frac{1}{x})^{14} : 38760 x^{4}

Semplificare

\frac{20 !}{15 ! ( 20 - 15 ) !} (x^{3})^{5} (- \frac{1}{x})^{15} : - 15504

Semplificare

\frac{20 !}{16 ! ( 20 - 16 ) !} (x^{3})^{4} (- \frac{1}{x})^{16} : \frac{4845}{x ^{4}}

Semplificare

\frac{20 !}{17 ! ( 20 - 17 ) !} (x^{3})^{3} (- \frac{1}{x})^{17} : - \frac{1140}{x ^{8}}

Semplificare

\frac{20 !}{18 ! ( 20 - 18 ) !} (x^{3})^{2} (- \frac{1}{x})^{18} : \frac{190}{x ^{12}}

Semplificare

\frac{20 !}{19 ! ( 20 - 19 ) !} (x^{3})^{1} (- \frac{1}{x})^{19} : - \frac{20}{x ^{16}}

Semplificare

\frac{20 !}{20 ! ( 20 - 20 ) !} (x^{3})^{0} (- \frac{1}{x})^{20} : \frac{1}{x ^{20}}

= x^{60} - 20 x^{56} + 190 x^{52} - 1140 x^{48} + 4845 x^{44} - 15504 x^{40} + 38760 x^{36} - 77520 x^{32} + 125970 x^{28} - 167960 x^{24} + 184756 x^{20} - 167960 x^{16} + 125970 x^{12} - 77520 x^{8} + 38760 x^{4} - 15504 + \frac{4845}{x ^{4}} - \frac{1140}{x ^{8}} + \frac{190}{x ^{12}} - \frac{20}{x ^{16}} + \frac{1}{x ^{20}}

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