Proving All Real Roots of a Polynomial are Negative Using the AM-GM Inequality

[ehrenfest]

[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

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

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.

 

[morphism]

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).

 

[ehrenfest]

OK thanks. Just for the record AM-GM was not my idea but was the title of the section that this problem came from.