erweitern ((1-x^3)/(1-x))^5

Lösung

erweitern

(\frac{1 - x ^{3}}{1 - x})^{5}

x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 51 x^{5} + 45 x^{4} + 30 x^{3} + 15 x^{2} + 5 x + 1

Schritte zur Lösung

(\frac{1 - x ^{3}}{1 - x})^{5}

\frac{1 - x ^{3}}{1 - x} = x^{2} + x + 1

= (x^{2} + x + 1)^{5}

(x^{2} + x + 1)^{5} = (x^{2} + x + 1) (x^{2} + x + 1) (x^{2} + x + 1) (x^{2} + x + 1) (x^{2} + x + 1)

= (x^{2} + x + 1) (x^{2} + x + 1) (x^{2} + x + 1) (x^{2} + x + 1) (x^{2} + x + 1)

Multipliziere aus

(x^{2} + x + 1) (x^{2} + x + 1) (x^{2} + x + 1) (x^{2} + x + 1) (x^{2} + x + 1) : x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 51 x^{5} + 45 x^{4} + 30 x^{3} + 15 x^{2} + 5 x + 1

= x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 51 x^{5} + 45 x^{4} + 30 x^{3} + 15 x^{2} + 5 x + 1

e x p an d (x + x^{3} + x^{5})^{7}

e x p an d 3 e^{x} (e^{x} + 1)

e x p an d \frac{- 2 x - 6}{( x + 1 ) ( x - 1 )}

e x p an d 2 e^{- x} x - e^{- x} (x^{2} + 1)

e x p an d \frac{x ^{3} + x ^{2} + 1}{x ( x - 1 ) ^{2} ( x ^{2} + 5 ) ( x ^{2} + x + 1 )}