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

Lösung

erweitern

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

\frac{x ^{6} + 6 x ^{5} - 50 x ^{3} - 45 x ^{2} + 108 x + 108}{x ^{5} + 4 x ^{4} + x ^{3} - 10 x ^{2} - 4 x + 8}

Schritte zur Lösung

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

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

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

(x + 2)^{3} = x^{3} + 6 x^{2} + 12 x + 8

= \frac{( x - 2 ) ^{2} ( x + 1 ) ( x + 3 ) ^{3}}{( x ^{2} - 2 x + 1 ) ( x ^{3} + 6 x ^{2} + 12 x + 8 )}

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

= \frac{( x ^{2} - 4 x + 4 ) ( x + 1 ) ( x + 3 ) ^{3}}{( x ^{2} - 2 x + 1 ) ( x ^{3} + 6 x ^{2} + 12 x + 8 )}

(x + 3)^{3} = x^{3} + 9 x^{2} + 27 x + 27

= \frac{( x ^{2} - 4 x + 4 ) ( x + 1 ) ( x ^{3} + 9 x ^{2} + 27 x + 27 )}{( x ^{2} - 2 x + 1 ) ( x ^{3} + 6 x ^{2} + 12 x + 8 )}

Multipliziere aus

(x^{2} - 2 x + 1) (x^{3} + 6 x^{2} + 12 x + 8) : x^{5} + 4 x^{4} + x^{3} - 10 x^{2} - 4 x + 8

= \frac{( x ^{2} - 4 x + 4 ) ( x + 1 ) ( x ^{3} + 9 x ^{2} + 27 x + 27 )}{x ^{5} + 4 x ^{4} + x ^{3} - 10 x ^{2} - 4 x + 8}

Multipliziere aus

(x^{2} - 4 x + 4) (x + 1) (x^{3} + 9 x^{2} + 27 x + 27) : x^{6} + 6 x^{5} - 50 x^{3} - 45 x^{2} + 108 x + 108

= \frac{x ^{6} + 6 x ^{5} - 50 x ^{3} - 45 x ^{2} + 108 x + 108}{x ^{5} + 4 x ^{4} + x ^{3} - 10 x ^{2} - 4 x + 8}

s im pl i f y 4 (105 + \frac{- 4 2 - 50 5}{15})

e x p an d \frac{π ( 3 ln ( 3 ) + 2 )}{2}

e x p an d \frac{π ( 3 ln ( 3 ) + 2 )}{4}

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

e x p an d (2 x + 3 y) d x + (3 x + 2 y) d y