erweitern (x^3+1)/((x-1)^3(x+2)^2(x^2+1)^2)

Lösung

erweitern

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

\frac{x ^{3} + 1}{x ^{9} + x ^{8} - 3 x ^{7} + x ^{6} - x ^{5} - 5 x ^{4} + 11 x ^{3} - 9 x ^{2} + 8 x - 4}

Schritte zur Lösung

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

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

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

(x + 2)^{2} = x^{2} + 4 x + 4

= \frac{x ^{3} + 1}{( x ^{3} - 3 x ^{2} + 3 x - 1 ) ( x ^{2} + 4 x + 4 ) ( x ^{2} + 1 ) ^{2}}

(x^{2} + 1)^{2} = x^{4} + 2 x^{2} + 1

= \frac{x ^{3} + 1}{( x ^{3} - 3 x ^{2} + 3 x - 1 ) ( x ^{2} + 4 x + 4 ) ( x ^{4} + 2 x ^{2} + 1 )}

Multipliziere aus

(x^{3} - 3 x^{2} + 3 x - 1) (x^{2} + 4 x + 4) (x^{4} + 2 x^{2} + 1) : x^{9} + x^{8} - 3 x^{7} + x^{6} - x^{5} - 5 x^{4} + 11 x^{3} - 9 x^{2} + 8 x - 4

= \frac{x ^{3} + 1}{x ^{9} + x ^{8} - 3 x ^{7} + x ^{6} - x ^{5} - 5 x ^{4} + 11 x ^{3} - 9 x ^{2} + 8 x - 4}

e x p an d (α_{0} + α_{1} t + α_{2} t^{2})^{2} e^{t^{2}}

e x p an d \frac{- 18 x ^{2} - 18 x - 15}{( 3 x ^{2} + 3 x - 1 ) ^{2}}

e x p an d \frac{10}{3 x ( x + 1 )}

e x p an d (1 + 2 x) (1 + \frac{x}{2})^{9}

s im pl i f y \frac{1}{2} (x + \frac{1}{6} sin (6 x))