erweitern (x(x+20)^3)/((ex+1)^3*(-x+1))

Lösung

erweitern

\frac{x \cdot ( x + 20 ) ^{3}}{( e \cdot x + 1 ) ^{3} \cdot ( - x + 1 )}

\frac{x ^{4} + 60 x ^{3} + 1200 x ^{2} + 8000 x}{- e ^{3} x ^{4} + e ^{3} x ^{3} - 3 e ^{2} x ^{3} + 3 e ^{2} x ^{2} - 3 e x ^{2} + 3 e x - x + 1}

Schritte zur Lösung

\frac{x ( x + 20 ) ^{3}}{( e x + 1 ) ^{3} ( - x + 1 )}

(e x + 1)^{3} = e^{3} x^{3} + 3 e^{2} x^{2} + 3 e x + 1

= \frac{x ( x + 20 ) ^{3}}{( e ^{3} x ^{3} + 3 e ^{2} x ^{2} + 3 e x + 1 ) ( - x + 1 )}

(x + 20)^{3} = x^{3} + 60 x^{2} + 1200 x + 8000

= \frac{x ( x ^{3} + 60 x ^{2} + 1200 x + 8000 )}{( e ^{3} x ^{3} + 3 e ^{2} x ^{2} + 3 e x + 1 ) ( - x + 1 )}

Multipliziere aus

(e^{3} x^{3} + 3 e^{2} x^{2} + 3 e x + 1) (- x + 1) : - e^{3} x^{4} + e^{3} x^{3} - 3 e^{2} x^{3} + 3 e^{2} x^{2} - 3 e x^{2} + 3 e x - x + 1

= \frac{x ( x ^{3} + 60 x ^{2} + 1200 x + 8000 )}{- e ^{3} x ^{4} + e ^{3} x ^{3} - 3 e ^{2} x ^{3} + 3 e ^{2} x ^{2} - 3 e x ^{2} + 3 e x - x + 1}

Multipliziere aus

x (x^{3} + 60 x^{2} + 1200 x + 8000) : x^{4} + 60 x^{3} + 1200 x^{2} + 8000 x

= \frac{x ^{4} + 60 x ^{3} + 1200 x ^{2} + 8000 x}{- e ^{3} x ^{4} + e ^{3} x ^{3} - 3 e ^{2} x ^{3} + 3 e ^{2} x ^{2} - 3 e x ^{2} + 3 e x - x + 1}

e x p an d \frac{[ ( 1 - x ^{2} ) ]}{[ x ( 1 - 2 x ) ]}

e x p an d \frac{d}{d x} ((- x^{2} + 5 x - 6) \cdot 4^{x}) + 3

e x p an d \frac{4}{x ^{2} ( x + 2 )}

e x p an d \frac{16 ( 2 x ^{3} - 1 )}{8 x ^{3} + 1}

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