Question

Daca $x + \frac{1}{x}=5$   si x≠0, sa se afle valoarea expresiei : 
$E= x+ x^{2} + x^{3} + x^{4} + \frac{1}{x^{4}} + \frac{1}{x^{3}} +\frac{1}{x^{2}} + \frac{1}{x}$

Answer 1

$(x+ \frac{1}{x} ) ^{2} = x^{2} + 2+\frac{1}{ x^{2} } \\ x^{2} +\frac{1}{ x^{2} } = 25-2=23 \\ \\ (x+ \frac{1}{x} ) ^{3}= x^{3} + \frac{1}{ x^{3} } + 3(x+ \frac{1}{x} ) \\ x^{3} + \frac{1}{ x^{3} }=125-15=110 \\ \\ ( x^{2} +\frac{1}{ x^{2} } )^{2}=x^{4} + 2+\frac{1}{ x^{4} } \\ x^{4} +\frac{1}{ x^{4} }=529-2=527 \\ \\ E=x +\frac{1}{ x }+x^{2} +\frac{1}{ x^{2} }+x^{3} +\frac{1}{ x^{3} }+x^{4} +\frac{1}{ x^{4} } \\ E=5+23+110+527=665$