we-are-in-C-3-S-x-y-z-2i-1-and-xyz-2-xy-yz-xz-2-1-i-P-z-z-3-1-2i-z-2-2-1-i-2-show-that-a-b-c-is-solution-of-S-if-and-only-a-b-c-are-roots-of-P-

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Question Number 122658 by mathocean1 last updated on 18/Nov/20

we are in C^3 . (S): { ((x+y+z=2i−1 and xyz=2)),((xy+yz+xz=−2(1+i) )) :} P(z)=z^3 +(1−2i)z^2 −2(1+i)−2. show that (a,b,c) is solution of (S) if and only a;b;c are roots of P.

$${we}\:{are}\:{in}\:\mathbb{C}^{\mathrm{3}} . \\ $$$$\left({S}\right):\:\:\begin{cases}{{x}+{y}+{z}=\mathrm{2}{i}−\mathrm{1}\:{and}\:{xyz}=\mathrm{2}}\\{{xy}+{yz}+{xz}=−\mathrm{2}\left(\mathrm{1}+{i}\right)\:}\end{cases} \\ $$$${P}\left({z}\right)={z}^{\mathrm{3}} +\left(\mathrm{1}−\mathrm{2}{i}\right){z}^{\mathrm{2}} −\mathrm{2}\left(\mathrm{1}+{i}\right)−\mathrm{2}. \\ $$$${show}\:{that}\:\left({a},{b},{c}\right)\:{is}\:{solution} \\ $$$${of}\:\left({S}\right)\:{if}\:{and}\:{only}\:{a};{b};{c}\:{are}\: \\ $$$${roots}\:{of}\:{P}. \\ $$

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