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Beliebt Algebra >

erweitern (x^2+1/x)^6

Lösung

erweitern

(x^{2} + \frac{1}{x})^{6}

Lösung

x^{12} + 6 x^{9} + 15 x^{6} + 20 x^{3} + 15 + \frac{6}{x ^{3}} + \frac{1}{x ^{6}}

Schritte zur Lösung

(x^{2} + \frac{1}{x})^{6}

Wende Binomialsetz an:

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

a = x^{2}, b = \frac{1}{x}

= i = 0 \sum 6 (i 6) (x^{2})^{(6 - i)} (\frac{1}{x})^{i}

Erweitere Summation

= \frac{6 !}{0 ! ( 6 - 0 ) !} (x^{2})^{6} (\frac{1}{x})^{0} + \frac{6 !}{1 ! ( 6 - 1 ) !} (x^{2})^{5} (\frac{1}{x})^{1} + \frac{6 !}{2 ! ( 6 - 2 ) !} (x^{2})^{4} (\frac{1}{x})^{2} + \frac{6 !}{3 ! ( 6 - 3 ) !} (x^{2})^{3} (\frac{1}{x})^{3} + \frac{6 !}{4 ! ( 6 - 4 ) !} (x^{2})^{2} (\frac{1}{x})^{4} + \frac{6 !}{5 ! ( 6 - 5 ) !} (x^{2})^{1} (\frac{1}{x})^{5} + \frac{6 !}{6 ! ( 6 - 6 ) !} (x^{2})^{0} (\frac{1}{x})^{6}

Vereinfache

\frac{6 !}{0 ! ( 6 - 0 ) !} (x^{2})^{6} (\frac{1}{x})^{0} : x^{12}

Vereinfache

\frac{6 !}{1 ! ( 6 - 1 ) !} (x^{2})^{5} (\frac{1}{x})^{1} : 6 x^{9}

Vereinfache

\frac{6 !}{2 ! ( 6 - 2 ) !} (x^{2})^{4} (\frac{1}{x})^{2} : 15 x^{6}

Vereinfache

\frac{6 !}{3 ! ( 6 - 3 ) !} (x^{2})^{3} (\frac{1}{x})^{3} : 20 x^{3}

Vereinfache

\frac{6 !}{4 ! ( 6 - 4 ) !} (x^{2})^{2} (\frac{1}{x})^{4} : 15

Vereinfache

\frac{6 !}{5 ! ( 6 - 5 ) !} (x^{2})^{1} (\frac{1}{x})^{5} : \frac{6}{x ^{3}}

Vereinfache

\frac{6 !}{6 ! ( 6 - 6 ) !} (x^{2})^{0} (\frac{1}{x})^{6} : \frac{1}{x ^{6}}

= x^{12} + 6 x^{9} + 15 x^{6} + 20 x^{3} + 15 + \frac{6}{x ^{3}} + \frac{1}{x ^{6}}

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