1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Putnam and beyond prob 121

  1. May 16, 2008 #1
    [SOLVED] putnam and beyond prob 121

    1. The problem statement, all variables and given/known data
    Show that all real roots of the polynomial P(x) = x^5 -10 x +35 are negative.

    2. Relevant 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]

    3. 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.
  2. jcsd
  3. May 16, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper

    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).
  4. May 16, 2008 #3
    OK thanks. Just for the record AM-GM was not my idea but was the title of the section that this problem came from.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook