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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
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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

x51=(x1)(xα)(xα2)(xα3)(xα4)

(x51)/(x1)=(xα)(xα2)(xα3)(xα4)

Hence, (xα)(xα2)(xα3)(xα4)=1+x+x2+x3+x5

Putting, x=1 we get,

(1α)(1α2)(1α3)(1α4)=5

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