erweitern (x-2/x)^4(x^2+2/x)^3

Lösung

erweitern

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

x^{10} + 6 x^{7} + 12 x^{4} + 8 x - \frac{8 ( x ^{9} + 6 x ^{6} + 12 x ^{3} + 8 )}{x} + \frac{24 ( x ^{9} + 6 x ^{6} + 12 x ^{3} + 8 )}{x ^{3}} - \frac{32 ( x ^{9} + 6 x ^{6} + 12 x ^{3} + 8 )}{x ^{5}} + \frac{16 ( x ^{9} + 6 x ^{6} + 12 x ^{3} + 8 )}{x ^{7}}

Schritte zur Lösung

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

Schreibe

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

um:

\frac{x ^{9} + 6 x ^{6} + 12 x ^{3} + 8}{x ^{3}}

= (x - \frac{2}{x})^{4} \frac{x ^{9} + 6 x ^{6} + 12 x ^{3} + 8}{x ^{3}}

= \frac{x ^{9} + 6 x ^{6} + 12 x ^{3} + 8}{x ^{3}} (x - \frac{2}{x})^{4}

Schreibe

(x - \frac{2}{x})^{4}

um:

x^{4} - 8 x^{2} + 24 - \frac{32}{x ^{2}} + \frac{16}{x ^{4}}

= \frac{x ^{9} + 6 x ^{6} + 12 x ^{3} + 8}{x ^{3}} (x^{4} - 8 x^{2} + 24 - \frac{32}{x ^{2}} + \frac{16}{x ^{4}})

Setze Klammern

= \frac{x ^{9} + 6 x ^{6} + 12 x ^{3} + 8}{x ^{3}} x^{4} + \frac{x ^{9} + 6 x ^{6} + 12 x ^{3} + 8}{x ^{3}} (- 8 x^{2}) + \frac{x ^{9} + 6 x ^{6} + 12 x ^{3} + 8}{x ^{3}} \cdot 24 + \frac{x ^{9} + 6 x ^{6} + 12 x ^{3} + 8}{x ^{3}} (- \frac{32}{x ^{2}}) + \frac{x ^{9} + 6 x ^{6} + 12 x ^{3} + 8}{x ^{3}} \cdot \frac{16}{x ^{4}}

Vereinfache

\frac{x ^{9} + 6 x ^{6} + 12 x ^{3} + 8}{x ^{3}} x^{4} + \frac{x ^{9} + 6 x ^{6} + 12 x ^{3} + 8}{x ^{3}} (- 8 x^{2}) + \frac{x ^{9} + 6 x ^{6} + 12 x ^{3} + 8}{x ^{3}} \cdot 24 + \frac{x ^{9} + 6 x ^{6} + 12 x ^{3} + 8}{x ^{3}} (- \frac{32}{x ^{2}}) + \frac{x ^{9} + 6 x ^{6} + 12 x ^{3} + 8}{x ^{3}} \cdot \frac{16}{x ^{4}} : x^{10} + 6 x^{7} + 12 x^{4} + 8 x - \frac{8 ( x ^{9} + 6 x ^{6} + 12 x ^{3} + 8 )}{x} + \frac{24 ( x ^{9} + 6 x ^{6} + 12 x ^{3} + 8 )}{x ^{3}} - \frac{32 ( x ^{9} + 6 x ^{6} + 12 x ^{3} + 8 )}{x ^{5}} + \frac{16 ( x ^{9} + 6 x ^{6} + 12 x ^{3} + 8 )}{x ^{7}}

= x^{10} + 6 x^{7} + 12 x^{4} + 8 x - \frac{8 ( x ^{9} + 6 x ^{6} + 12 x ^{3} + 8 )}{x} + \frac{24 ( x ^{9} + 6 x ^{6} + 12 x ^{3} + 8 )}{x ^{3}} - \frac{32 ( x ^{9} + 6 x ^{6} + 12 x ^{3} + 8 )}{x ^{5}} + \frac{16 ( x ^{9} + 6 x ^{6} + 12 x ^{3} + 8 )}{x ^{7}}

s im pl i f y (1 + \frac{1}{jk r}) (1 - \frac{1}{jk r})

e x p an d (x + ux) (u d x + x d u)

e x p an d \frac{m - 2}{( m - 3 ) ( m + 2 )}

e x p an d \frac{2 + x}{3 ( - x + 1 ) ( x - 2 )}

e x p an d \frac{12 ( x ^{2} + 12 )}{( x ^{2} - 36 ) ^{3}}