If-a-b-c-d-e-are-thr-roots-of-2x-5-3x-3-2x-7-0-find-the-value-of-cyc-a-3-1-

Question Number 201857 by cortano12 last updated on 14/Dec/23

If a, b, c, d, e are thr roots of

{2 x}^{5} - {3 x}^{3} + 2 x - 7 = 0, find the

value of \prod_{cyc} (a^{3} - 1)

Commented by Frix last updated on 14/Dec/23

I found 108

Answered by Frix last updated on 14/Dec/23

f (x) = x^{5} + {b x}^{3} + d x + e = 0

\prod_{cyc} (x^{3} - 1) = - f (1) f (- \frac{1}{2} + \frac{\sqrt{3}}{2} i) f (- \frac{1}{2} - \frac{\sqrt{3}}{2} i) =

= {(b + e)}^{3} + d^{3} - 3 d (b + e) + 1

[Also works for f (x) = x^{5} + {a x}^{4} + {b x}^{3} + {c x}^{2} + d x + e]

Commented by Frix last updated on 14/Dec/23

Works for any polynome

\prod_{cyc} (x^{m} - 1) = - \underset{n = 0}{\prod^{m - 1}} f (e^{i \frac{2 π n}{m}})

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