Putnam and beyond prob 121

  • Thread starter ehrenfest
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  • #1
ehrenfest
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[SOLVED] putnam and beyond prob 121

Homework Statement


Show that all real roots of the polynomial P(x) = x^5 -10 x +35 are negative.


Homework Equations


the AM-GM inequality:

If x_1,...,x_n are nonnegative real numbers, then

[tex]\frac{\sum x_i}{n} \leq \left( \Pi x_i\right)^{1/n}[/tex]


The Attempt at a Solution


I know this should be really easy. But I can't figure out what to do. Its not hard to show that all of the real roots are less than 2. I am guessing that if y is nonnegative real root, then I should apply AM-GM to c_1 y, c_2 y, c_3 y, c_4 y, c_5 y where the c_i are nonnegative but I cannot figure out what the c_i are.
 

Answers and Replies

  • #2
morphism
Science Advisor
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AM-GM is a good idea. Notice that we have x^5, 35=2^5+3, and 10x=(2^5x^5)^(1/5) * 5. So if x>0, then P(x)>0 (details left to you).
 
  • #3
ehrenfest
2,020
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OK thanks. Just for the record AM-GM was not my idea but was the title of the section that this problem came from.
 

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