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Resolve-the-system-d-unknow-x-y-z-3-x-y-z-1-x-2-y-2-z-2-1-x-3-y-3-z-3-5-

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Question Number 158855 by LEKOUMA last updated on 09/Nov/21
Resolve the system d′ unknow (x, y,z) ∈ ⊂^3 x+y+z=1 x^2 +y^2 +z^2 =1 x^3 +y^3 +z^3 =−5
Resolvethesystemd′unknow(x,y,z)∈⊂3x+y+z=1x2+y2+z2=1x3+y3+z3=−5
Answered by mr W last updated on 09/Nov/21
x+y+z=1 (x+y+z)^2 =x^2 +y^2 +z^2 +2(xy+yz+zx) 1^2 =1+2(xy+yz+zx) ⇒xy+yz+zx=0 (x+y+z)^3 =x^3 +y^3 +z^3 +3(x+y+z)(xy+yz+zx)−3xyz 1^3 =−5+3×1×0−3xyz ⇒xyz=−2 x,y,z are roots of t^3 −t^2 +2=0 (t+1)(t−2t+2)=0 ⇒(x,y,z)=(−1,1+i,1−i)
x+y+z=1(x+y+z)2=x2+y2+z2+2(xy+yz+zx)12=1+2(xy+yz+zx)⇒xy+yz+zx=0(x+y+z)3=x3+y3+z3+3(x+y+z)(xy+yz+zx)−3xyz13=−5+3×1×0−3xyz⇒xyz=−2x,y,zarerootsoft3−t2+2=0(t+1)(t−2t+2)=0⇒(x,y,z)=(−1,1+i,1−i)
Commented by LEKOUMA last updated on 09/Nov/21
Many thanks
Manythanks

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