let-p-x-cos-2n-arccos-x-with-x-1-1-find-the-roots-of-p-x-and-factorize-p-x-

Question Number 34020 by prof Abdo imad last updated on 29/Apr/18

l e t p (x) = c o s (2 n a r c c o s (x)) w i t h x \in [- 1, 1]

f i n d t h e r o o t s o f p (x) a n d f a c t o r i z e p (x)

Commented by math khazana by abdo last updated on 02/May/18

l e t p u t a r c c o s x = t \Leftrightarrow x = c o s t

p (x) = 0 \Leftrightarrow c o s (2 n t) = 0 \Leftrightarrow 2 n t = \frac{π}{2} + k π k \in Z

\Leftrightarrow t_{k} = \frac{π}{4 n} + \frac{k π}{2 n} k \in Z b u t c a r d {t_{k}} < + \infty w e c a n

p r o v e t h a t k \in [0, 2 n - 1]] s o t h e r o o t s o f p (x) a r e

x_{k} = c o s (t_{k}) = c o s (\frac{π}{4 n} + \frac{k π}{2 n}) a n d

p (x) = λ \prod_{k = 0}^{2 n - 1} (x - c o s (\frac{π}{4 n} + \frac{k π}{2 n})) .