let p(x)=(1+x^2 )(1+x^4 )....(1+x^2^n ) with n integr 1) find the roots of p(x) 2) factorize p(x) inside C[x]


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Question Number 33743 by prof Abdo imad last updated on 23/Apr/18

$l e t p (x) = (1 + x^{2}) (1 + x^{4}) . . . . (1 + x^{2^{n}}) w i t h n i n t e g r$ $1) f i n d t h e r o o t s o f p (x)$ $2) f a c t o r i z e p (x) i n s i d e C [x]$
Commented by caravan msup abdo. last updated on 24/Apr/18

$w e h a v e p r o v e d t h a t p (x) = \frac{1 - x^{2^{n + 1}}}{1 - x^{2}}$ $i f x^{2} \neq 1 s o p (x) = 0 \Leftrightarrow x^{2^{n + 1}} = e^{i 2 k π}$ $i f x = r e^{i θ} w e g e t r = 1 a n d 2^{n + 1} θ = 2 k π$ $\Rightarrow θ_{k} = \frac{2 k π}{2^{n + 1}} = \frac{k π}{2^{n}} a n d k \in [[1, 2^{n + 1} - 1]]$ $t h e r o o t s o f p (x) a r e t h e c o m p l e x$ $z_{k} = e^{i \frac{k π}{2^{n}}} w i t h 1 ⩽ k ⩽ 2^{n + 1} - 1 a n d k \neq 2^{n}$ $2) p (x) = \prod_{k = 1 a n d k \neq 2^{n}}^{2^{n + 1} - 1} (z - e^{i \frac{k π}{2^{n}}})$
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