erweitern (x^2(x+1)^3)/((x-2)^2(x-4)^4)

Lösung

erweitern

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

\frac{x ^{5} + 3 x ^{4} + 3 x ^{3} + x ^{2}}{x ^{6} - 20 x ^{5} + 164 x ^{4} - 704 x ^{3} + 1664 x ^{2} - 2048 x + 1024}

Schritte zur Lösung

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

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

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

(x - 4)^{4} = x^{4} - 16 x^{3} + 96 x^{2} - 256 x + 256

= \frac{x ^{2} ( x + 1 ) ^{3}}{( x ^{2} - 4 x + 4 ) ( x ^{4} - 16 x ^{3} + 96 x ^{2} - 256 x + 256 )}

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

= \frac{x ^{2} ( x ^{3} + 3 x ^{2} + 3 x + 1 )}{( x ^{2} - 4 x + 4 ) ( x ^{4} - 16 x ^{3} + 96 x ^{2} - 256 x + 256 )}

Multipliziere aus

(x^{2} - 4 x + 4) (x^{4} - 16 x^{3} + 96 x^{2} - 256 x + 256) : x^{6} - 20 x^{5} + 164 x^{4} - 704 x^{3} + 1664 x^{2} - 2048 x + 1024

= \frac{x ^{2} ( x ^{3} + 3 x ^{2} + 3 x + 1 )}{x ^{6} - 20 x ^{5} + 164 x ^{4} - 704 x ^{3} + 1664 x ^{2} - 2048 x + 1024}

Multipliziere aus

x^{2} (x^{3} + 3 x^{2} + 3 x + 1) : x^{5} + 3 x^{4} + 3 x^{3} + x^{2}

= \frac{x ^{5} + 3 x ^{4} + 3 x ^{3} + x ^{2}}{x ^{6} - 20 x ^{5} + 164 x ^{4} - 704 x ^{3} + 1664 x ^{2} - 2048 x + 1024}

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

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

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

e x p an d 8 cos^{2} (2 t) + (t - 5)^{2} - 7 e^{4 t}

s im pl i f y δ (x - 3)