decompose-sur-R-x-x-2n-1-1-

Question Number 30590 by abdo imad last updated on 23/Feb/18

d e c o m p o s e s u r R [x] x^{2 n + 1} - 1 .

Answered by sma3l2996 last updated on 23/Feb/18

l e t P (x) = x^{2 n + 1} - 1

α_{k} = e^{\frac{2 i k π}{2 n + 1}} a r e r o o t s o f P (x)

P (x) = \prod_{k = 0}^{2 n} (x - e^{\frac{2 i k π}{2 n + 1}})

α_{1} = e^{\frac{2 i π}{2 n + 1}}; α_{2 n} = e^{\frac{2 i (2 n) π}{2 n + 1}} = e^{\frac{2 i (2 n + 1 - 1) π}{2 n + 1}} = e^{\frac{- 2 i π}{2 n + 1}} \times e^{2 i π} = {\overset{__}{α}}_{1}

(x - α_{1}) (x - {\overset{__}{α}}_{1}) = x^{2} - 2 c o s (\frac{2 π}{2 n + 1}) + 1

α_{2} = {\overset{__}{α}}_{2 n - 1}

α_{n - 1} = {\overset{__}{α}}_{n + 2}

I n g e n e r a l α_{j} = {\overset{__}{α}}_{2 n - (j - 1)}

a n d (x - α_{j}) (x - α_{2 n - (j - 1)}) = x^{2} + 2 c o s (\frac{2 j π}{2 n + 1}) + 1

s o P (x) = (x - 1) \prod_{k = 1}^{n} (x^{2} + 2 c o s (\frac{2 k π}{2 n + 1}) + 1)