# Putnam and beyond prob 121

[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.