let-p-x-x-10-1-1-find-roots-of-p-x-2-factorize-i-nside-C-x-p-x-3-factorize-inside-R-x-p-x-

Question Number 42195 by math khazana by abdo last updated on 20/Aug/18

l e t p (x) = x^{10} - 1

1) f i n d r o o t s o f p (x)

2) f a c t o r i z e i n s i d e C [x] p (x {

3) f a c t o r i z e i n s i d e R [x] p (x) .

Answered by maxmathsup by imad last updated on 21/Aug/18

1) p (z) = 0 \Leftrightarrow z^{10} - 1 = 0 \Leftrightarrow z^{10} = 1 i f z = r e^{i θ} w e g e t r^{10} e^{i 10 θ} = e^{i 2 k π} \Rightarrow

r = 1 a n d θ = \frac{2 k π}{10} = \frac{k π}{5} s o t h e r o o t s o f p (x) a r e z_{k} = e^{i \frac{k π}{5}} w i t h k \in [[0, 9]]

2) p (x) = \prod_{k = 0}^{9} (x - z_{k}) = \prod_{k = 0}^{9} (x - e^{\frac{i k π}{5}})

3) g e n e r a l l y l e t d e c m p o s e q (x) = x^{2 n} - 1 t h e r o o t s a r e z_{k} = e^{\frac{i k n}{n}} w i t h

k \in [[0, 2 n - 1]] \Rightarrow q (x) = \prod_{k = 0}^{2 n - 1} (x - e^{\frac{i k π}{n}}) b u t z_{0} = 1 z_{1} = e^{\frac{i π}{n}}

z_{2} = e^{\frac{i 2 π}{n}} z_{n - 1} = e^{\frac{i (n - 1) π}{n}} z_{n} = - 1 z_{n + 1} = e^{\frac{i (n + 1) π}{n}} \dots z_{2 n - 1} = e^{\frac{i (2 n - 1) π}{n}}

\Rightarrow {\bar{z}}_{1} = e^{- \frac{i π}{n}} = e^{2 i π - \frac{i π}{n}} = e^{i \frac{(2 n - 1) π}{n}} = {\bar{z}}_{2 n - 1} \dots . . \Rightarrow z_{2 n - p} = {\bar{z}}_{p} \Rightarrow

q (x) = (x^{2} - 1) \prod_{k = 1}^{n - 1} (x - z_{k}) (x - {\bar{z}}_{k}) = (x^{2} - 1) \prod_{k = 1}^{n - 1} (x^{2} - 2 c o s (\frac{k π}{n}) x + 1)

f o r n = 5 w e g e t p (x) = (x^{2} - 1) \prod_{k = 1}^{4} (x^{2} - 2 c o s (\frac{k π}{5}) x + 1)

= (x^{2} - 1) (x^{2} - 2 c o s (\frac{π}{5}) x + 1) (x^{2} - 2 c o s (\frac{2 π}{5}) x + 1) (x^{2} - 2 c o s (\frac{3 π}{5}) x + 1) (x^{2} - 2 c o s (\frac{4 π}{5}) x + 1) .

= (x^{2} - 2 c o s (\frac{π}{5}) x + 1) (x^{2} + 2 c o s (\frac{π}{5}) x + 1) (x^{2} - 2 c o s (\frac{2 π}{5}) x + 1) (x^{2} + 2 c o s (\frac{2 π}{5}) x + 1) .

Commented by maxmathsup by imad last updated on 21/Aug/18

p (x) = (x^{2} - 1) (x^{2} - 2 c o s (\frac{π}{5}) x + 1) (x^{2} + 2 c o s (\frac{π}{5}) x + 1) (x^{2} - 2 c o s (\frac{2 π}{5}) x + 1) (x^{2} + 2 c o s (\frac{2 π}{5}) x + 1) .