simplify (iz+1)^3

Solution

simplify

(i z + 1)^{3}

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

Solution steps

(i z + 1)^{3}

Apply Perfect Cube Formula:

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

(i z + 1)^{3} = (i z)^{3} + 3 (i z)^{2} \cdot 1 + 3 i z \cdot 1^{2} + 1^{3}

= (i z)^{3} + 3 (i z)^{2} \cdot 1 + 3 i z \cdot 1^{2} + 1^{3}

(i z)^{3} = - z^{3} i

3 (i z)^{2} \cdot 1 = - 3 z^{2}

3 i z \cdot 1^{2} = 3 z i

1^{3} = 1

= - z^{3} i - 3 z^{2} + 3 z i + 1

Rewrite in standard complex form:

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

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

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