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Beliebt Algebra >

erweitern (2(2iz^2-15z^2+15iz+16z-14i))/((z-i)^5)

Lösung

erweitern

\frac{2 ( 2 i z ^{2} - 15 z ^{2} + 15 i z + 16 z - 14 i )}{( z - i ) ^{5}}

Lösung

\frac{28 - 124 z ^{2} + 12 z ^{6} - 140 z ^{4} + 150 z ^{3} + 150 z ^{5} - 30 z - 30 z ^{7}}{1 + z ^{10} + 10 z ^{4} + 10 z ^{6} + 5 z ^{2} + 5 z ^{8}} + i \frac{- 108 z + 120 z ^{2} - 120 z ^{6} - 20 z ^{3} + 4 z ^{7} + 92 z ^{5}}{1 + z ^{10} + 10 z ^{4} + 10 z ^{6} + 5 z ^{2} + 5 z ^{8}}

Schritte zur Lösung

\frac{2 ( 2 i z ^{2} - 15 z ^{2} + 15 i z + 16 z - 14 i )}{( z - i ) ^{5}}

(z - i)^{5} = z^{5} - 5 i z^{4} - 10 z^{3} + 10 i z^{2} + 5 z - i

= \frac{2 ( 2 i z ^{2} - 15 z ^{2} + 15 i z + 16 z - 14 i )}{z ^{5} - 5 i z ^{4} - 10 z ^{3} + 10 i z ^{2} + 5 z - i}

Multipliziere aus

2 (2 i z^{2} - 15 z^{2} + 15 i z + 16 z - 14 i) : 4 i z^{2} - 30 z^{2} + 30 i z + 32 z - 28 i

= \frac{4 i z ^{2} - 30 z ^{2} + 30 i z + 32 z - 28 i}{z ^{5} - 5 i z ^{4} - 10 z ^{3} + 10 i z ^{2} + 5 z - i}

Rationalisiere

\frac{4 i z ^{2} - 30 z ^{2} + 30 i z + 32 z - 28 i}{z ^{5} - 5 i z ^{4} - 10 z ^{3} + 10 i z ^{2} + 5 z - i} : \frac{- 30 z ^{7} + 12 z ^{6} + 150 z ^{5} - 140 z ^{4} + 150 z ^{3} - 124 z ^{2} - 30 z + i ( 4 z ^{7} - 120 z ^{6} + 92 z ^{5} - 20 z ^{3} + 120 z ^{2} - 108 z ) + 28}{z ^{10} + 5 z ^{8} + 10 z ^{6} + 10 z ^{4} + 5 z ^{2} + 1}

= \frac{- 30 z ^{7} + 12 z ^{6} + 150 z ^{5} - 140 z ^{4} + 150 z ^{3} - 124 z ^{2} - 30 z + 28 + i ( 4 z ^{7} - 120 z ^{6} + 92 z ^{5} - 20 z ^{3} + 120 z ^{2} - 108 z )}{z ^{10} + 5 z ^{8} + 10 z ^{6} + 10 z ^{4} + 5 z ^{2} + 1}

Schreibe

\frac{- 30 z ^{7} + 12 z ^{6} + 150 z ^{5} - 140 z ^{4} + 150 z ^{3} - 124 z ^{2} - 30 z + 28 + i ( 4 z ^{7} - 120 z ^{6} + 92 z ^{5} - 20 z ^{3} + 120 z ^{2} - 108 z )}{z ^{10} + 5 z ^{8} + 10 z ^{6} + 10 z ^{4} + 5 z ^{2} + 1}

in der Standard komplexen Form um:

\frac{- 30 z ^{7} + 12 z ^{6} + 150 z ^{5} - 140 z ^{4} + 150 z ^{3} - 124 z ^{2} - 30 z + 28}{z ^{10} + 5 z ^{8} + 10 z ^{6} + 10 z ^{4} + 5 z ^{2} + 1} + \frac{4 z ^{7} - 120 z ^{6} + 92 z ^{5} - 20 z ^{3} + 120 z ^{2} - 108 z}{z ^{10} + 5 z ^{8} + 10 z ^{6} + 10 z ^{4} + 5 z ^{2} + 1} i

= \frac{- 30 z ^{7} + 12 z ^{6} + 150 z ^{5} - 140 z ^{4} + 150 z ^{3} - 124 z ^{2} - 30 z + 28}{z ^{10} + 5 z ^{8} + 10 z ^{6} + 10 z ^{4} + 5 z ^{2} + 1} + \frac{4 z ^{7} - 120 z ^{6} + 92 z ^{5} - 20 z ^{3} + 120 z ^{2} - 108 z}{z ^{10} + 5 z ^{8} + 10 z ^{6} + 10 z ^{4} + 5 z ^{2} + 1} i

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