What we've had to do in situations like this is scale the entire application down if the available browser width or height is below a certain threshold. The app doesn't always look as nice scaled down, but if it is a requirement to avoid scroll bars you don't have much choice. Here is the code we use currently.
You can listen to JavaScript resize event, and then call your flex resize function using ExternalInterface. If you want to find out what is your monitor size you can simply find it in the Capabilities class. Also consider reading this article about flash attributes at spatially optional scale attribute.
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