erweitern (2x+1/x)^6

Lösung

erweitern

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

64 x^{6} + 192 x^{4} + 240 x^{2} + 160 + \frac{60}{x ^{2}} + \frac{12}{x ^{4}} + \frac{1}{x ^{6}}

Schritte zur Lösung

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

Wende Binomialsetz an:

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

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

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

Erweitere Summation

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

Vereinfache

\frac{6 !}{0 ! ( 6 - 0 ) !} (2 x)^{6} (\frac{1}{x})^{0} : 64 x^{6}

Vereinfache

\frac{6 !}{1 ! ( 6 - 1 ) !} (2 x)^{5} (\frac{1}{x})^{1} : 192 x^{4}

Vereinfache

\frac{6 !}{2 ! ( 6 - 2 ) !} (2 x)^{4} (\frac{1}{x})^{2} : 240 x^{2}

Vereinfache

\frac{6 !}{3 ! ( 6 - 3 ) !} (2 x)^{3} (\frac{1}{x})^{3} : 160

Vereinfache

\frac{6 !}{4 ! ( 6 - 4 ) !} (2 x)^{2} (\frac{1}{x})^{4} : \frac{60}{x ^{2}}

Vereinfache

\frac{6 !}{5 ! ( 6 - 5 ) !} (2 x)^{1} (\frac{1}{x})^{5} : \frac{12}{x ^{4}}

Vereinfache

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

= 64 x^{6} + 192 x^{4} + 240 x^{2} + 160 + \frac{60}{x ^{2}} + \frac{12}{x ^{4}} + \frac{1}{x ^{6}}

\frac{\partial}{\partial x} ((z) (2 x^{2} + y^{2}))

e x p an d A (- 4, - 7)

e x p an d (2 x + \frac{1}{x})^{5}

s im pl i f y (5 x - 10) (3 - x)

e x p an d \frac{20 ( s + 1 )}{s ( 5 s ^{2} + 6 s + 2 )}