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Populaire Algèbre >

développer (3x^4+2/(x^2))^{11}

Solution

développer

(3 x^{4} + \frac{2}{x ^{2}})^{11}

Solution

177147 x^{44} + 1299078 x^{38} + 4330260 x^{32} + 8660520 x^{26} + 11547360 x^{20} + 10777536 x^{14} + 7185024 x^{8} + 3421440 x^{2} + \frac{1140480}{x ^{4}} + \frac{253440}{x ^{10}} + \frac{33792}{x ^{16}} + \frac{2048}{x ^{22}}

étapes des solutions

(3 x^{4} + \frac{2}{x ^{2}})^{11}

Appliquer le théorème du binôme:

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

a = 3 x^{4}, b = \frac{2}{x ^{2}}

= i = 0 \sum 11 (i 11) (3 x^{4})^{(11 - i)} (\frac{2}{x ^{2}})^{i}

Développer la somme

= \frac{11 !}{0 ! ( 11 - 0 ) !} (3 x^{4})^{11} (\frac{2}{x ^{2}})^{0} + \frac{11 !}{1 ! ( 11 - 1 ) !} (3 x^{4})^{10} (\frac{2}{x ^{2}})^{1} + \frac{11 !}{2 ! ( 11 - 2 ) !} (3 x^{4})^{9} (\frac{2}{x ^{2}})^{2} + \frac{11 !}{3 ! ( 11 - 3 ) !} (3 x^{4})^{8} (\frac{2}{x ^{2}})^{3} + \frac{11 !}{4 ! ( 11 - 4 ) !} (3 x^{4})^{7} (\frac{2}{x ^{2}})^{4} + \frac{11 !}{5 ! ( 11 - 5 ) !} (3 x^{4})^{6} (\frac{2}{x ^{2}})^{5} + \frac{11 !}{6 ! ( 11 - 6 ) !} (3 x^{4})^{5} (\frac{2}{x ^{2}})^{6} + \frac{11 !}{7 ! ( 11 - 7 ) !} (3 x^{4})^{4} (\frac{2}{x ^{2}})^{7} + \frac{11 !}{8 ! ( 11 - 8 ) !} (3 x^{4})^{3} (\frac{2}{x ^{2}})^{8} + \frac{11 !}{9 ! ( 11 - 9 ) !} (3 x^{4})^{2} (\frac{2}{x ^{2}})^{9} + \frac{11 !}{10 ! ( 11 - 10 ) !} (3 x^{4})^{1} (\frac{2}{x ^{2}})^{10} + \frac{11 !}{11 ! ( 11 - 11 ) !} (3 x^{4})^{0} (\frac{2}{x ^{2}})^{11}

Simplifier

\frac{11 !}{0 ! ( 11 - 0 ) !} (3 x^{4})^{11} (\frac{2}{x ^{2}})^{0} : 177147 x^{44}

Simplifier

\frac{11 !}{1 ! ( 11 - 1 ) !} (3 x^{4})^{10} (\frac{2}{x ^{2}})^{1} : 1299078 x^{38}

Simplifier

\frac{11 !}{2 ! ( 11 - 2 ) !} (3 x^{4})^{9} (\frac{2}{x ^{2}})^{2} : 4330260 x^{32}

Simplifier

\frac{11 !}{3 ! ( 11 - 3 ) !} (3 x^{4})^{8} (\frac{2}{x ^{2}})^{3} : 8660520 x^{26}

Simplifier

\frac{11 !}{4 ! ( 11 - 4 ) !} (3 x^{4})^{7} (\frac{2}{x ^{2}})^{4} : 11547360 x^{20}

Simplifier

\frac{11 !}{5 ! ( 11 - 5 ) !} (3 x^{4})^{6} (\frac{2}{x ^{2}})^{5} : 10777536 x^{14}

Simplifier

\frac{11 !}{6 ! ( 11 - 6 ) !} (3 x^{4})^{5} (\frac{2}{x ^{2}})^{6} : 7185024 x^{8}

Simplifier

\frac{11 !}{7 ! ( 11 - 7 ) !} (3 x^{4})^{4} (\frac{2}{x ^{2}})^{7} : 3421440 x^{2}

Simplifier

\frac{11 !}{8 ! ( 11 - 8 ) !} (3 x^{4})^{3} (\frac{2}{x ^{2}})^{8} : \frac{1140480}{x ^{4}}

Simplifier

\frac{11 !}{9 ! ( 11 - 9 ) !} (3 x^{4})^{2} (\frac{2}{x ^{2}})^{9} : \frac{253440}{x ^{10}}

Simplifier

\frac{11 !}{10 ! ( 11 - 10 ) !} (3 x^{4})^{1} (\frac{2}{x ^{2}})^{10} : \frac{33792}{x ^{16}}

Simplifier

\frac{11 !}{11 ! ( 11 - 11 ) !} (3 x^{4})^{0} (\frac{2}{x ^{2}})^{11} : \frac{2048}{x ^{22}}

= 177147 x^{44} + 1299078 x^{38} + 4330260 x^{32} + 8660520 x^{26} + 11547360 x^{20} + 10777536 x^{14} + 7185024 x^{8} + 3421440 x^{2} + \frac{1140480}{x ^{4}} + \frac{253440}{x ^{10}} + \frac{33792}{x ^{16}} + \frac{2048}{x ^{22}}

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