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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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written 3.0 years ago by
teamques10
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We have $x^5=1=\cos 0+i \sin 0$
$\therefore x^5 = \cos (2k\pi) + i\sin (2k\pi)$
$\therefore x = [\cos (2k\pi) + i\sin (2k\pi)]^{1/5} = \cos \dfrac {2k\pi}{5}+i\sin \dfrac {2k\pi}{5}$
Putting k=0,1,2,3,4 we get the five roots,
$x_0 = \cos 0 + i\sin 0=1$
$x_1 = \cos \dfrac {2\pi}{5}+ i\sin \dfrac {2\pi}{5}$ …
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