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Find the roots α, α2, α3,α4 of the equation x5 - 1 = 0 and show that
(1-α)(1-α2)(1-α3)(1-α4) = 5
1 Answer
written 3.9 years ago by |
We have x5=1=cos0+isin0
∴x5=cos(2kπ)+isin(2kπ)
∴x=[cos(2kπ)+isin(2kπ)]1/5=cos2kπ5+isin2kπ5
Putting k=0,1,2,3,4 we get the five roots,
x0=cos0+isin0=1
x1=cos2π5+isin2π5
x2=cos4π5+isin4π5
x3=cos6π5+isin6π5
x4=cos8π5+isin8π5
Putting x1=a we see that x2=α2, x3=α3, x4=α4
The roots are 1,α,α2,α3,α4 and hence
x5−1=(x−1)(x−α)(x−α2)(x−α3)(x−α4)
(x5−1)/(x−1)=(x−α)(x−α2)(x−α3)(x−α4)
Hence, (x−α)(x−α2)(x−α3)(x−α4)=1+x+x2+x3+x5
Putting, x=1 we get,
(1−α)(1−α2)(1−α3)(1−α4)=5