simplify (z-2i)^3

Solution

simplify

(z - 2 i)^{3}

(z^{3} - 12 z) + (- 6 z^{2} + 8) i

Solution steps

(z - 2 i)^{3}

Apply Perfect Cube Formula:

(a - b)^{3} = a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}

(z - 2 i)^{3} = z^{3} - 3 z^{2} \cdot 2 i + 3 z (2 i)^{2} - (2 i)^{3}

= z^{3} - 3 z^{2} \cdot 2 i + 3 z (2 i)^{2} - (2 i)^{3}

Simplify

z^{3} - 3 z^{2} \cdot 2 i + 3 z (2 i)^{2} - (2 i)^{3} : z^{3} - 6 z^{2} i - 12 z + 8 i

= z^{3} - 6 z^{2} i - 12 z + 8 i

Rewrite in standard complex form:

(z^{3} - 12 z) + (- 6 z^{2} + 8) i

= (z^{3} - 12 z) + (- 6 z^{2} + 8) i

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s (t - π)

\frac{a ^{2} - b}{2 ( a - b )}

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