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Question Number 7511 by Obenfo last updated on 01/Sep/16

Factorise the expression 9x^4 + (1/x^4 ) +2.

$$\mathrm{Factorise}\:\mathrm{the}\:\mathrm{expression}\:\mathrm{9}{x}^{\mathrm{4}} +\:\frac{\mathrm{1}}{{x}^{\mathrm{4}} }\:+\mathrm{2}. \\ $$

Answered by sandy_suhendra last updated on 01/Sep/16

because (3x^2 +(1/x^2 ))^2 = 9x^4 +6+(1/x^4 )  so   9x^4 +2+(1/x^4 ) = (3x^2 +(1/x^2 ))^2 − 4 = (3x^2 +(1/x^2 )+2)(3x^2 +(1/x^2 )−2)

$${because}\:\left(\mathrm{3}{x}^{\mathrm{2}} +\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right)^{\mathrm{2}} =\:\mathrm{9}{x}^{\mathrm{4}} +\mathrm{6}+\frac{\mathrm{1}}{{x}^{\mathrm{4}} } \\ $$$${so}\:\:\:\mathrm{9}{x}^{\mathrm{4}} +\mathrm{2}+\frac{\mathrm{1}}{{x}^{\mathrm{4}} }\:=\:\left(\mathrm{3}{x}^{\mathrm{2}} +\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right)^{\mathrm{2}} −\:\mathrm{4}\:=\:\left(\mathrm{3}{x}^{\mathrm{2}} +\frac{\mathrm{1}}{{x}^{\mathrm{2}} }+\mathrm{2}\right)\left(\mathrm{3}{x}^{\mathrm{2}} +\frac{\mathrm{1}}{{x}^{\mathrm{2}} }−\mathrm{2}\right) \\ $$

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