factorize-inside-C-x-x-2-y-2-z-2-

Question Number 46882 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] x^{2} + y^{2} + z^{2}

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

w e h a v e {(x + y + z)}^{2} = x^{2} + y^{2} + z^{2} + 2 (x y + y z + z x) \Rightarrow

x^{2} + y^{2} + z^{2} = {(x + y + z)}^{2} - {(\sqrt{2 (x y + y z + z x})}^{2} l e t 2 (x y + y z + z x) = r e^{i θ} \Rightarrow

x^{2} + y^{2} + z^{2} = {(x + y + z)}^{2} - {(\sqrt{r} e^{i \frac{θ}{2}})}^{2}

= (x + y + z - \sqrt{r} e^{i \frac{θ}{2}}) (x + y + z + r e^{i \frac{θ}{2}}) .

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

a n o t h e r m e t h o d w e h a v e

x^{2} + y^{2} + z^{2} = x^{2} - {(i \sqrt{y^{2} + z^{2}})}^{2} = (x - i \sqrt{y^{2} + z^{2}}) (x + i \sqrt{y^{2} + z^{2}}) o r

x^{2} + y^{2} + z^{2} = x^{2} + y^{2} - {(i z)}^{2} = (\sqrt{x^{2} + y^{2}} - i z) (\sqrt{x^{2} + y^{2}} + i z)