factorize-inside-C-x-the-polynom-x-n-y-n-

Question Number 46881 by maxmathsup by imad last updated on 02/Nov/18

f a c t o r i z e i n s i d e C [x] t h e p o l y n o m x^{n} + y^{n}

Commented by maxmathsup by imad last updated on 02/Nov/18

l e t c n s i d e r t h e p o l y n o m e p (x) = x^{n} + y^{n} i f y = 0 p (x) = x^{n}

i f y \neq 0 l e t f i n d r o o t s o f p (x)

p (x) = 0 \Leftrightarrow x^{n} = - y^{n} \Leftrightarrow {(\frac{x}{y})}^{n} = - 1 = e^{i (2 k + 1 π} \Rightarrow \frac{x}{y} = e^{i \frac{(2 k + 1) π}{n}} \Rightarrow

t h e r o o t s o f p (x) a r e x_{k} = y e^{i \frac{(2 k + 1) π}{n}} w i t h k \in [[0, n - 1]] \Rightarrow

p (x) = λ \prod_{k = 0}^{n - 1} (x - x_{k}) b u t λ = 1 \Rightarrow p (x) = \prod_{k = 0}^{n - 1} (x - y e^{i \frac{(2 k + 1) π}{n}})

= (x - {y e}^{i \frac{π}{n}}) (x - y e^{i \frac{3 π}{n}}) \dots . . (x - y e^{i \frac{(2 n - 1) π}{n}}) .