AIME II #7 2008

[stupidityismygam]

Let [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] be the three roots of the equation

[unparseable or potentially dangerous latex formula]

Find [unparseable or potentially dangerous latex formula]

[stupidityismygam]

Vieta is our friend.

First, by Vieta we see that the sum of the roots is zero, so [unparseable or potentially dangerous latex formula] Similarly [unparseable or potentially dangerous latex formula]

Then, [unparseable or potentially dangerous latex formula]

Now we consider the nifty (yes brad, nifty) factorization [unparseable or potentially dangerous latex formula]

Since [unparseable or potentially dangerous latex formula] we see that [unparseable or potentially dangerous latex formula] By Vieta, [unparseable or potentially dangerous latex formula]

So, [unparseable or potentially dangerous latex formula]

(As an alternative to that factorization you could add the following and note that [unparseable or potentially dangerous latex formula])
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]

[collegebookworm]

[stupidityismygam]

http://mathworld.wolfram.com/VietasFormulas.html

Basically all those formula you use to find sum of roots/product of roots/sum of roots taken 2 at a time/sum of roots taken 3 at a time/etc are Vieta Formula

[collegebookworm]