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Help for II-A Algebra 9.18

 
 
Picture of Fiona Lu
Help for II-A Algebra 9.18
by Fiona Lu - Tuesday, August 6, 2019, 10:29 PM
 

Questions 9.18 is Find the sum of the 55th powers of the 55 roots of x5+2x−1=0x5+2x−1=0.

Currently, I'm having trouble figuring out how to use different groupings of the roots (taken by 2, taken by 3, etc.) to isolate the fifth powers of the roots. 

Does anyone have any good hints or help for this problem?

 
Picture of David Reynoso
Re: Help for II-A Algebra 9.18
by David Reynoso - Monday, August 12, 2019, 1:17 PM
 

Since all five roots of $x^5 + 2x -1 = 0$ satisfy the equation, for each of them we have that $x_{k}^5 = -2x_{k} + 1$, ($k=1,2,3,4,5$). Using this, what is $x_{1}^5 + x_{2}^5 + x_{3}^5 + x_{4}^5 + x_{5}^5$ equal to?

 
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