erweitern ((x-3)^4(x+4)^3(2x+1)^2(2x-1)^2)/(16)

Lösung

erweitern

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

x^{11} - \frac{85 x ^{9}}{2} + 28 x^{8} + \frac{10081 x ^{7}}{16} - 770 x^{6} - \frac{26649 x ^{5}}{8} + \frac{22255 x ^{4}}{4} + \frac{24801 x ^{3}}{16} - \frac{10557 x ^{2}}{4} - 189 x + 324

Schritte zur Lösung

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

(x - 3)^{4} = x^{4} - 12 x^{3} + 54 x^{2} - 108 x + 81

= \frac{( x ^{4} - 12 x ^{3} + 54 x ^{2} - 108 x + 81 ) ( x + 4 ) ^{3} ( 2 x + 1 ) ^{2} ( 2 x - 1 ) ^{2}}{16}

(x + 4)^{3} = x^{3} + 12 x^{2} + 48 x + 64

= \frac{( x ^{4} - 12 x ^{3} + 54 x ^{2} - 108 x + 81 ) ( x ^{3} + 12 x ^{2} + 48 x + 64 ) ( 2 x + 1 ) ^{2} ( 2 x - 1 ) ^{2}}{16}

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

= \frac{( x ^{4} - 12 x ^{3} + 54 x ^{2} - 108 x + 81 ) ( x ^{3} + 12 x ^{2} + 48 x + 64 ) ( 4 x ^{2} + 4 x + 1 ) ( 2 x - 1 ) ^{2}}{16}

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

= \frac{( x ^{4} - 12 x ^{3} + 54 x ^{2} - 108 x + 81 ) ( x ^{3} + 12 x ^{2} + 48 x + 64 ) ( 4 x ^{2} + 4 x + 1 ) ( 4 x ^{2} - 4 x + 1 )}{16}

Multipliziere aus

(x^{4} - 12 x^{3} + 54 x^{2} - 108 x + 81) (x^{3} + 12 x^{2} + 48 x + 64) (4 x^{2} + 4 x + 1) (4 x^{2} - 4 x + 1) : 16 x^{11} - 680 x^{9} + 448 x^{8} + 10081 x^{7} - 12320 x^{6} - 53298 x^{5} + 89020 x^{4} + 24801 x^{3} - 42228 x^{2} - 3024 x + 5184

= \frac{16 x ^{11} - 680 x ^{9} + 448 x ^{8} + 10081 x ^{7} - 12320 x ^{6} - 53298 x ^{5} + 89020 x ^{4} + 24801 x ^{3} - 42228 x ^{2} - 3024 x + 5184}{16}

Wende Bruchregel an:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{16 x ^{11} - 680 x ^{9} + 448 x ^{8} + 10081 x ^{7} - 12320 x ^{6} - 53298 x ^{5} + 89020 x ^{4} + 24801 x ^{3} - 42228 x ^{2} - 3024 x + 5184}{16} = \frac{16 x ^{11}}{16} - \frac{680 x ^{9}}{16} + \frac{448 x ^{8}}{16} + \frac{10081 x ^{7}}{16} - \frac{12320 x ^{6}}{16} - \frac{53298 x ^{5}}{16} + \frac{89020 x ^{4}}{16} + \frac{24801 x ^{3}}{16} - \frac{42228 x ^{2}}{16} - \frac{3024 x}{16} + \frac{5184}{16}

= \frac{16 x ^{11}}{16} - \frac{680 x ^{9}}{16} + \frac{448 x ^{8}}{16} + \frac{10081 x ^{7}}{16} - \frac{12320 x ^{6}}{16} - \frac{53298 x ^{5}}{16} + \frac{89020 x ^{4}}{16} + \frac{24801 x ^{3}}{16} - \frac{42228 x ^{2}}{16} - \frac{3024 x}{16} + \frac{5184}{16}

\frac{16 x ^{11}}{16} = x^{11}

\frac{680 x ^{9}}{16} = \frac{85 x ^{9}}{2}

\frac{448 x ^{8}}{16} = 28 x^{8}

\frac{12320 x ^{6}}{16} = 770 x^{6}

\frac{53298 x ^{5}}{16} = \frac{26649 x ^{5}}{8}

\frac{89020 x ^{4}}{16} = \frac{22255 x ^{4}}{4}

\frac{42228 x ^{2}}{16} = \frac{10557 x ^{2}}{4}

\frac{3024 x}{16} = 189 x

\frac{5184}{16} = 324

= x^{11} - \frac{85 x ^{9}}{2} + 28 x^{8} + \frac{10081 x ^{7}}{16} - 770 x^{6} - \frac{26649 x ^{5}}{8} + \frac{22255 x ^{4}}{4} + \frac{24801 x ^{3}}{16} - \frac{10557 x ^{2}}{4} - 189 x + 324

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

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

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

(3 y^{3} + 4 y x + 1 + 2 x^{2} + 9 y^{2} x)

e x p an d 3 x + 5 + 3 x + 5