erweitern (1+x+(x^2)/2+(x^3)/3)^3 1/6

Lösung

erweitern

(1 + x + \frac{x ^{2}}{2} + \frac{x ^{3}}{3})^{3} \frac{1}{6}

\frac{x ^{9}}{162} + \frac{x ^{8}}{36} + \frac{7 x ^{7}}{72} + \frac{35 x ^{6}}{144} + \frac{11 x ^{5}}{24} + \frac{17 x ^{4}}{24} + \frac{5 x ^{3}}{6} + \frac{3 x ^{2}}{4} + \frac{x}{2} + \frac{1}{6}

Schritte zur Lösung

(1 + x + \frac{x ^{2}}{2} + \frac{x ^{3}}{3})^{3} \frac{1}{6}

(1 + x + \frac{x ^{2}}{2} + \frac{x ^{3}}{3})^{3} = \frac{8 x ^{9} + 36 x ^{8} + 126 x ^{7} + 315 x ^{6} + 594 x ^{5} + 918 x ^{4} + 1080 x ^{3} + 972 x ^{2} + 648 x}{216} + 1

= \frac{1}{6} (\frac{8 x ^{9} + 36 x ^{8} + 126 x ^{7} + 315 x ^{6} + 594 x ^{5} + 918 x ^{4} + 1080 x ^{3} + 972 x ^{2} + 648 x}{216} + 1)

= \frac{1}{6} (\frac{8 x ^{9} + 36 x ^{8} + 126 x ^{7} + 315 x ^{6} + 594 x ^{5} + 918 x ^{4} + 1080 x ^{3} + 972 x ^{2} + 648 x}{216} + 1)

Wende das Distributivgesetz an:

a (b + c) = ab + a c

a = \frac{1}{6}, b = \frac{8 x ^{9} + 36 x ^{8} + 126 x ^{7} + 315 x ^{6} + 594 x ^{5} + 918 x ^{4} + 1080 x ^{3} + 972 x ^{2} + 648 x}{216}, c = 1

= \frac{1}{6} \cdot \frac{8 x ^{9} + 36 x ^{8} + 126 x ^{7} + 315 x ^{6} + 594 x ^{5} + 918 x ^{4} + 1080 x ^{3} + 972 x ^{2} + 648 x}{216} + \frac{1}{6} \cdot 1

= \frac{1}{6} \cdot \frac{8 x ^{9} + 36 x ^{8} + 126 x ^{7} + 315 x ^{6} + 594 x ^{5} + 918 x ^{4} + 1080 x ^{3} + 972 x ^{2} + 648 x}{216} + 1 \cdot \frac{1}{6}

Vereinfache

\frac{1}{6} \cdot \frac{8 x ^{9} + 36 x ^{8} + 126 x ^{7} + 315 x ^{6} + 594 x ^{5} + 918 x ^{4} + 1080 x ^{3} + 972 x ^{2} + 648 x}{216} + 1 \cdot \frac{1}{6} : \frac{8 x ^{9} + 36 x ^{8} + 126 x ^{7} + 315 x ^{6} + 594 x ^{5} + 918 x ^{4} + 1080 x ^{3} + 972 x ^{2} + 648 x}{1296} + \frac{1}{6}

= \frac{8 x ^{9} + 36 x ^{8} + 126 x ^{7} + 315 x ^{6} + 594 x ^{5} + 918 x ^{4} + 1080 x ^{3} + 972 x ^{2} + 648 x}{1296} + \frac{1}{6}

Wende Bruchregel an:

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

\frac{8 x ^{9} + 36 x ^{8} + 126 x ^{7} + 315 x ^{6} + 594 x ^{5} + 918 x ^{4} + 1080 x ^{3} + 972 x ^{2} + 648 x}{1296} = \frac{8 x ^{9}}{1296} + \frac{36 x ^{8}}{1296} + \frac{126 x ^{7}}{1296} + \frac{315 x ^{6}}{1296} + \frac{594 x ^{5}}{1296} + \frac{918 x ^{4}}{1296} + \frac{1080 x ^{3}}{1296} + \frac{972 x ^{2}}{1296} + \frac{648 x}{1296}

= \frac{8 x ^{9}}{1296} + \frac{36 x ^{8}}{1296} + \frac{126 x ^{7}}{1296} + \frac{315 x ^{6}}{1296} + \frac{594 x ^{5}}{1296} + \frac{918 x ^{4}}{1296} + \frac{1080 x ^{3}}{1296} + \frac{972 x ^{2}}{1296} + \frac{648 x}{1296} + \frac{1}{6}

\frac{8 x ^{9}}{1296} = \frac{x ^{9}}{162}

\frac{36 x ^{8}}{1296} = \frac{x ^{8}}{36}

\frac{126 x ^{7}}{1296} = \frac{7 x ^{7}}{72}

\frac{315 x ^{6}}{1296} = \frac{35 x ^{6}}{144}

\frac{594 x ^{5}}{1296} = \frac{11 x ^{5}}{24}

\frac{918 x ^{4}}{1296} = \frac{17 x ^{4}}{24}

\frac{1080 x ^{3}}{1296} = \frac{5 x ^{3}}{6}

\frac{972 x ^{2}}{1296} = \frac{3 x ^{2}}{4}

\frac{648 x}{1296} = \frac{x}{2}

= \frac{x ^{9}}{162} + \frac{x ^{8}}{36} + \frac{7 x ^{7}}{72} + \frac{35 x ^{6}}{144} + \frac{11 x ^{5}}{24} + \frac{17 x ^{4}}{24} + \frac{5 x ^{3}}{6} + \frac{3 x ^{2}}{4} + \frac{x}{2} + \frac{1}{6}

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

e x p an d 15 (\frac{2 x}{3} + \frac{1}{5})^{2}

s im pl i f y e^{(- a - i w) \cdot 2}

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

e x p an d (ln (y - 2) + \frac{x - 3}{y - 2} = C) y - 2