Grade 11Algebra
if w is a complex cube root of unity show that (2-w) (2-w^2)=7
1 Answer
Arun
Since w is a non-real complex root of unity, then w ≠ 1, so
w² + w + 1 = 0
So when w is a non-real complex root of unity, we have:
w³ = 1
w² + w + 1 = 0
(2−w)(2−w²)
= w³ − 2w² − 2w + 4
= w³ − 2w² − 2w − 2 + 6
= w³ − 2(w² + w + 1) + 6
= 1 − 2(0) + 6
= 7
w² + w + 1 = 0
So when w is a non-real complex root of unity, we have:
w³ = 1
w² + w + 1 = 0
(2−w)(2−w²)
= w³ − 2w² − 2w + 4
= w³ − 2w² − 2w − 2 + 6
= w³ − 2(w² + w + 1) + 6
= 1 − 2(0) + 6
= 7
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