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Question Number 30592 by abdo imad last updated on 23/Feb/18

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

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

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

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

Answered by sma3l2996 last updated on 23/Feb/18

1) l e t s a y p (x) = 0

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

= {(x^{n} - c o s α)}^{2} + {s i n}^{2} (α) = 0

x^{n} - c o s α = i s i n α o r x^{n} - c o s α = - i s i n α

x_{1}^{n} = s i n α + i c o s α = e^{i α}; x_{2}^{n} = c o s α - i s i n α = e^{- i α}

s o r o o t s o f p (x) a r e :

x_{1, k} = e^{i (α + 2 k π) / n}; x_{2, k} = e^{- i (α + 2 k π) / n} ∖ k = (0, 1, 2, \dots, n - 1)

2)

p (x) = \prod_{k = 0}^{n - 1} (x - e^{i (α + 2 k π) / n}) \times \prod_{k = 0}^{n - 1} (x - e^{- i (α + 2 k π) / n})

p (x) = \prod_{k = 0}^{n - 1} (x - e^{i (α + 2 k π) / n}) (x - e^{- i (α + 2 k π) / n})

3)

p (x) = \prod_{k = 0}^{n - 1} (x^{2} - (e^{i (α + 2 k π) / n} + e^{- i (α + 2 k π) / n}) x + e^{i (α + 2 k π - α - 2 k π) / n})

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