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Question Number 9648 by Chantria last updated on 22/Dec/16

Answered by sandy_suhendra last updated on 23/Dec/16

x^4 +x^2 −2x+2+(3/x^2 ) > 0 ((x^6 +x^4 −2x^3 +2x^2 +3)/x^2 ) > 0 (((x^6 −2x^3 +1)+(x^4 +2x^2 +1)+1)/x^2 ) > 0 (((x^3 −1)^2 +(x^2 +1)^2 +1)/x^2 ) > 0 the numerator is always positive for the real number x but x^2 ≠ 0 or x≠0

$$\mathrm{x}^{\mathrm{4}} +\mathrm{x}^{\mathrm{2}} −\mathrm{2x}+\mathrm{2}+\frac{\mathrm{3}}{\mathrm{x}^{\mathrm{2}} }\:>\:\mathrm{0} \\ $$$$\frac{\mathrm{x}^{\mathrm{6}} +\mathrm{x}^{\mathrm{4}} −\mathrm{2x}^{\mathrm{3}} +\mathrm{2x}^{\mathrm{2}} +\mathrm{3}}{\mathrm{x}^{\mathrm{2}} }\:>\:\mathrm{0} \\ $$$$\frac{\left(\mathrm{x}^{\mathrm{6}} −\mathrm{2x}^{\mathrm{3}} +\mathrm{1}\right)+\left(\mathrm{x}^{\mathrm{4}} +\mathrm{2x}^{\mathrm{2}} +\mathrm{1}\right)+\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }\:>\:\mathrm{0} \\ $$$$\frac{\left(\mathrm{x}^{\mathrm{3}} −\mathrm{1}\right)^{\mathrm{2}} +\left(\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} +\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }\:>\:\mathrm{0} \\ $$$$\mathrm{the}\:\mathrm{numerator}\:\mathrm{is}\:\mathrm{always}\:\mathrm{positive}\:\mathrm{for}\:\mathrm{the}\:\mathrm{real}\:\mathrm{number}\:\mathrm{x} \\ $$$$\mathrm{but}\:\mathrm{x}^{\mathrm{2}} \neq\:\mathrm{0}\:\mathrm{or}\:\mathrm{x}\neq\mathrm{0} \\ $$

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