Question

Daca x este un numar real nenul astfel incat $x+ \frac{1}{x}=5$, aflati $x^{2} + \frac{1}{ x^{2} }$ si $x^{3}+ \frac{1}{ x^{3} }$.
Si rezolvareea!!!!!

Answer 1

Ridicam prima relatie la patrat:
$(x+ \frac{1}{x}) ^{2} = x^{2} +2+ \frac{1}{ x^{2} } =25 => x^{2} + \frac{1}{ x^{2} }=23$
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$(x+ \frac{1}{x})( x^{2} + \frac{1}{ x^{2} })= x^{3}+ (\frac{1}{x}+x)+ \frac{1}{ x^{3} } = x^{3}+ \frac{1}{ x^{3} }+5$
$(x+ \frac{1}{x})( x^{2} + \frac{1}{ x^{2} } )=5*23=115$
$( x^{3}+ \frac{1}{ x^{3} }+5=115 => x^{3}+ \frac{1}{ x^{3} }=110$