1-decompose-inside-C-x-p-x-x-2n-2-cos-x-n-1-2-decopose-p-x-inside-R-x-

Question Number 36904 by prof Abdo imad last updated on 07/Jun/18

1) d e c o m p o s e i n s i d e C [x]

p (x) = x^{2 n} - 2 (c o s α) x^{n} + 1

2) d e c o p o s e p (x) i n s i d e R [x]

Commented by math khazana by abdo last updated on 11/Jun/18

1) l e t p u t x^{n} = t \Rightarrow p (x) = t^{2} - 2 c o s (α) t + 1

Δ^{^{'}} = {c o s}^{2} - 1 = {(i s i n α)}^{2} \Rightarrow t_{1} = e^{i α} a n d t_{2} = e^{- i α}

p (x) = (t - e^{i α}) (t - e^{- i α}) = (x^{n} - e^{i α}) (x^{n} - e^{- i α}) r o o t s

o f z^{n} - e^{i α} = 0 \Rightarrow z^{n} = e^{i α} \Rightarrow r = 1 a n d

n θ = α + 2 k π \Rightarrow θ = \frac{α + 2 k π}{n} k \in [[0, n - 1]] s o

t h e r o o t s a r e z_{k} = e^{i \frac{α + 2 k π}{n}}

r o o t s o f x^{n} - e^{- i α} = 0 \Rightarrow x^{n} = e^{- i α} \Rightarrow r = 1

a n d n θ = - α + 2 k π \Rightarrow θ_{k} = \frac{- α + 2 k π}{n} \Rightarrow

t h e r o o t s a r e λ_{k} = e^{i \frac{- α + 2 k π}{n}} a n d k \in [[0, n - 1]]

\Rightarrow p (x) = \prod_{k = 0}^{n - 1} (x - z_{k}) \prod_{k = 0}^{n - 1} (x - λ_{k})

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

k \in [[0, n - 1]] .