Factorize-x-3-1-x-3-11-8-

Question Number 191389 by MATHEMATICSAM last updated on 23/Apr/23

Factorize :

x^{3} + \frac{1}{x^{3}} + \frac{11}{8}

Answered by Frix last updated on 23/Apr/23

x^{3} + \frac{1}{x^{3}} + \frac{11}{8} \overset{[t = x + \frac{1}{x}]}{=} t^{3} - 3 t + \frac{11}{8} =

= (t - \frac{1}{2}) (t + \frac{1 - 3 \sqrt{5}}{4}) (t + \frac{1 + 3 \sqrt{5}}{4}) =

= (x + \frac{1}{x} - \frac{1}{2}) (x + \frac{1}{x} + \frac{1 - 3 \sqrt{5}}{4}) (x + \frac{1}{x} + \frac{1 + 3 \sqrt{5}}{4})

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