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

factorize inside C[x] the polynom x^n +y^n

factorizeinsideC[x]thepolynomxn+yn

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

let cnsider the polynome p(x)=x^n +y^n if y=0 p(x)=x^n if y≠0 let find roots of p(x) p(x)=0 ⇔x^n =−y^n ⇔((x/y))^n =−1 =e^(i(2k+1π) ⇒(x/(y )) = e^(i(((2k+1)π)/n)) ⇒ the roots of p(x) are x_k =y e^(i(((2k+1)π)/n)) with k ∈[[0,n−1]] ⇒ p(x) =λ Π_(k=0) ^(n−1) (x−x_k ) but λ =1 ⇒p(x) =Π_(k=0) ^(n−1) (x−y e^(i(((2k+1)π)/n)) ) =(x−ye^(i(π/n)) )(x−y e^(i((3π)/n)) ).....(x−y e^(i(((2n−1)π)/n)) ) .

letcnsiderthepolynomep(x)=xn+ynify=0p(x)=xnify≠0letfindrootsofp(x)p(x)=0⇔xn=−yn⇔(xy)n=−1=ei(2k+1π⇒xy=ei(2k+1)πn⇒therootsofp(x)arexk=yei(2k+1)πnwithk∈[[0,n−1]]⇒p(x)=λ∏k=0n−1(x−xk)butλ=1⇒p(x)=∏k=0n−1(x−yei(2k+1)πn)=(x−yeiπn)(x−yei3πn).....(x−yei(2n−1)πn).

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