What is (a-ib)(a+ib)^2 ?

The solution to (a-ib)(a+ib)^2 is (a^3+ab^2)+(a^2b+b^3)i

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simplify (a-ib)(a+ib)^2

simplify

(a - ib) (a + ib)^{2}

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

Solution steps

(a - ib) (a + ib)^{2}

= (a + ib)^{2} (a - ib)

Expand

(a + ib)^{2} : a^{2} + 2 abi - b^{2}

= (a^{2} + 2 abi - b^{2}) (a - ib)

Distribute parentheses

= a^{2} a + a^{2} (- ib) + 2 abia + 2 abi (- ib) - b^{2} a - b^{2} (- ib)

Simplify

a^{2} a + a^{2} (- ib) + 2 abia + 2 abi (- ib) - b^{2} a - b^{2} (- ib) : a^{3} + a^{2} bi + a b^{2} + i b^{3}

= a^{3} + a^{2} bi + a b^{2} + i b^{3}

Rewrite in standard complex form:

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

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

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What is (a-ib)(a+ib)^2 ?
The solution to (a-ib)(a+ib)^2 is (a^3+ab^2)+(a^2b+b^3)i