jlg@lanl-a.UUCP (10/18/83)
A problem was recently submitted that was supposed to be so simple that a
high school student could solve it in 10-15 minutes. The problem was to
find the sum of the 17th powers of the roots of:
17 2
x + 3 x + 2 x - 1
Well I can see that the sum of the roots is zero. I can also see that the
product of the roots is one. But as for the sum of the seventeenth powers!
I'd like to see the high schooler that can solve this problem in 10-15 minutes!kenner@cmcl2.UUCP (10/23/83)
#R:lanl-a:-299100:cmcl2:27800002:000:849 cmcl2!kenner Oct 22 17:32:00 1983 Maybe a present-day high-schooler can't do it in 10-15 minutes but a real old-timer should be able to do it in far less time. See sections 562 and 563 (pages 468-9) in Hall and Knight for the method. Put simply, the sum of the nth powers of the roots of f(x)=0 is the coefficient of x**(-n) in f'(x)/f(x). If I haven't made a mistake, the sum of the 17th powers is 17 (the sum of the 15th powers is -45, the 16th is -32, the 30th is -135 (I haven't computed the ones higher, the lower ones that I haven't given are all zero)). Hall and Knight is a very interesting book in which to find things of this sort. For those not familiar with it, it was first published in 1887. The edition I have was from 1957. It's title is "Higher Algebra: A sequel to Elementary Algebra for Schools" and it is published by Macmillan & Co. and St. Martins Press.