Question

If w is a complex cube root of unity, show that (a+b)^2+(aw+bw^2)^2+(aw^2+bw)^2=6ab​

Answer 1

A knowledge of enough algebra, at least to know why these three things are true.a+b+c=0⟹a3+b3+c3=3abc“Any equation of nth degree has at most n roots, therefore unity or for that matter any number has 3 cube roots”And thata3±b3=(a±b)(a2∓ab+b2)The cube roots of unity are solutions to the equationx3=1⟹x3−1=0⟹(x−1)(x2+x+1)=0which means a cube root of unity either followsx−1=0or x2+x+1=0HINT: This second property shall help you solve this; do not look at what follows before trying it yourself.

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