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if w is a complex cube root of unity show that (2-w) (2-w^2)=7

if w is a complex cube root of unity show that (2-w) (2-w^2)=7

Grade:11

1 Answers

Arun
25750 Points
6 years ago
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

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