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Question

If x2x+1=0 then the value of (x+1x)2+(x2+1x2)2+(x3+1x3)2+(x5+1x5)2 is

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Solution

Given : x2x+1=0
x+1x=1x2+1x2+2=1x2+1x2=1
Cubing x+1x=1, we get
x3+1x3+3(x+1x)=1x3+1x3=2

Also,
(x2+1x2)(x3+1x3)=x5+1x5+x+1x2=x5+1x5+1x5+1x5=1

Therefore,
(x+1x)2+(x2+1x2)2+(x3+1x3)2+(x5+1x5)2=12+(1)2+(2)2+12=7

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