What is expand ((z^6+1))/((z-i)) ?

The solution to expand ((z^6+1))/((z-i)) is (z^5-z^3+z)+i(1-z^2+z^4)

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expand ((z^6+1))/((z-i))

expand

\frac{( z ^{6} + 1 )}{( z - i )}

(z^{5} - z^{3} + z) + i (1 - z^{2} + z^{4})

Solution steps

\frac{z ^{6} + 1}{z - i}

Multiply by the conjugate

\frac{z + i}{z + i}

= \frac{( z ^{6} + 1 ) ( z + i )}{( z - i ) ( z + i )}

(z - i) (z + i) = z^{2} + 1

= \frac{( z ^{6} + 1 ) ( z + i )}{z ^{2} + 1}

Factor

z^{6} + 1 : (z^{2} + 1) (z^{4} - z^{2} + 1)

= \frac{( z ^{2} + 1 ) ( z ^{4} - z ^{2} + 1 ) ( z + i )}{z ^{2} + 1}

Cancel the common factor:

z^{2} + 1

= (z^{4} - z^{2} + 1) (z + i)

Rewrite

(z^{4} - z^{2} + 1) (z + i)

in standard complex form:

(z^{5} - z^{3} + z) + (z^{4} - z^{2} + 1) i

= (z^{5} - z^{3} + z) + (z^{4} - z^{2} + 1) i

\frac{3}{7.5} \cdot 11.5 - x

\frac{( 50 \cdot e ^{- 1.6 s} )}{s ( 11.4 s + 1 )}

\frac{10 s}{( ( s ^{2} + 4 ) ( s ^{2} + 5 s + 6 ) )}

(- 30 cos (\frac{π x}{6}) + 34) (- 3 sin (\frac{π x}{6}) + 7)

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

What is expand ((z^6+1))/((z-i)) ?
The solution to expand ((z^6+1))/((z-i)) is (z^5-z^3+z)+i(1-z^2+z^4)