erweitern (x^5(x-1)^2)/((x+2)^2(x^2+1)^4)

Lösung

erweitern

\frac{x ^{5} ( x - 1 ) ^{2}}{( x + 2 ) ^{2} ( x ^{2} + 1 ) ^{4}}

\frac{x ^{7} - 2 x ^{6} + x ^{5}}{x ^{10} + 4 x ^{9} + 8 x ^{8} + 16 x ^{7} + 22 x ^{6} + 24 x ^{5} + 28 x ^{4} + 16 x ^{3} + 17 x ^{2} + 4 x + 4}

Schritte zur Lösung

\frac{x ^{5} ( x - 1 ) ^{2}}{( x + 2 ) ^{2} ( x ^{2} + 1 ) ^{4}}

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

= \frac{x ^{5} ( x - 1 ) ^{2}}{( x ^{2} + 4 x + 4 ) ( x ^{2} + 1 ) ^{4}}

(x^{2} + 1)^{4} = x^{8} + 4 x^{6} + 6 x^{4} + 4 x^{2} + 1

= \frac{x ^{5} ( x - 1 ) ^{2}}{( x ^{2} + 4 x + 4 ) ( x ^{8} + 4 x ^{6} + 6 x ^{4} + 4 x ^{2} + 1 )}

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

= \frac{x ^{5} ( x ^{2} - 2 x + 1 )}{( x ^{2} + 4 x + 4 ) ( x ^{8} + 4 x ^{6} + 6 x ^{4} + 4 x ^{2} + 1 )}

Multipliziere aus

(x^{2} + 4 x + 4) (x^{8} + 4 x^{6} + 6 x^{4} + 4 x^{2} + 1) : x^{10} + 4 x^{9} + 8 x^{8} + 16 x^{7} + 22 x^{6} + 24 x^{5} + 28 x^{4} + 16 x^{3} + 17 x^{2} + 4 x + 4

= \frac{x ^{5} ( x ^{2} - 2 x + 1 )}{x ^{10} + 4 x ^{9} + 8 x ^{8} + 16 x ^{7} + 22 x ^{6} + 24 x ^{5} + 28 x ^{4} + 16 x ^{3} + 17 x ^{2} + 4 x + 4}

Multipliziere aus

x^{5} (x^{2} - 2 x + 1) : x^{7} - 2 x^{6} + x^{5}

= \frac{x ^{7} - 2 x ^{6} + x ^{5}}{x ^{10} + 4 x ^{9} + 8 x ^{8} + 16 x ^{7} + 22 x ^{6} + 24 x ^{5} + 28 x ^{4} + 16 x ^{3} + 17 x ^{2} + 4 x + 4}

s im pl i f y (x - 6 - 3) (x - 6 + 3)

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

e x p an d e^{t - 2} u (t - 2)

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

e x p an d lo g_{10} (4 \frac{a}{b})