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Factorised-the-polynom-z-4-1-be-polynom-with-lower-degree-but-have-real-coefficient-




Question Number 55070 by gunawan last updated on 17/Feb/19
Factorised the polynom z^4 +1   be polynom with lower degree,  but have real coefficient
$$\mathrm{Factorised}\:\mathrm{the}\:\mathrm{polynom}\:{z}^{\mathrm{4}} +\mathrm{1}\: \\ $$$$\mathrm{be}\:\mathrm{polynom}\:\mathrm{with}\:\mathrm{lower}\:\mathrm{degree}, \\ $$$$\mathrm{but}\:\mathrm{have}\:\mathrm{real}\:\mathrm{coefficient} \\ $$
Commented by mr W last updated on 17/Feb/19
z^4 +1=(z^2 +1)^2 −((√2)z)^2   =(z^2 +(√2)z+1)(z^2 −(√2)z+1)
$${z}^{\mathrm{4}} +\mathrm{1}=\left({z}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} −\left(\sqrt{\mathrm{2}}{z}\right)^{\mathrm{2}} \\ $$$$=\left({z}^{\mathrm{2}} +\sqrt{\mathrm{2}}{z}+\mathrm{1}\right)\left({z}^{\mathrm{2}} −\sqrt{\mathrm{2}}{z}+\mathrm{1}\right) \\ $$

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