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If 1, ω 1 , ω 2 , ......, ω 99 be the 100th roots of unity, then (2 – ω) (2 – ω 2 ) (2 – ω 3 ) .... (2 – ω 99 ) is

If 1, ω1 , ω2 , ......, ω99 be the 100th roots of unity, then (2 – ω) (2 – ω2) (2 – ω3) .... (2 – ω99) is

Grade:12

1 Answers

Ravi
askIITians Faculty 69 Points
6 years ago
Consider 2ndroot of unity
x2-1=0
x=+1,-1

Now, forming an equation using these roots
(x-1)(x+1)
=x2-1

Now, Consider 4throot of unity
x4-1=0
(x2+1)(x2-1)=0
x= +i,-i,+1,-1

Now, if an equation is formed using these roots altogether,

(x+1)(x-1)(x+i)(x-i)=(x2+1)(x2-1)= x4-1

which is the root equation from which these roots were derived.
Hence,

for the 100throot, conversely
(x–ω)(x–ω2)(x–ω3)....(x–ω99)= x100-1 ...(asω1,2,3,........99are again roots of unity)
Substituting x=2, we get,
2100-1

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