Do Digital Antennas Work?
Wow - I am jealous - a 40 foot LCD TV - that is what you call an mancave or NBA Arena (No Boys Allowed).
Seriously, the main board for a 40" Polaroid runs about $25-75 depending on model - bear in mind these are not typically slip-in socket boards - you are likely to have to disconnect and reconnect a number of wire connectors, and may have to do some soldering of connections. Google this search term - there are a number of companies selling them on the web - you will have to expand the search phrase with your exact model number to narrow the search - 40" Polaroid LCD TV main board
One thing to do first is call Polaroid - there are a number of comments on the web about a pending class action suit about Polaroid LCD TV main boards catching fire, and several commenters say that Polaroid sent a repairman to their home and replaced the main board for free when it failed, so you might try that first.