Stand Alone Router

I know this may sound like a stupid question for this in the know but I need to ask it anyway as it may be my only alternative. Can a cable modem and router work without being connected to a computer? The reason I ask is because my cable company suggested that I would get much better perfromance out of my wireless connections if I connected to a cleaner cable jack. Now I know this is true but I cannot put the computer by the jack so my question is can I just put a splitter right out of the jack, connect it to the modem, them conect the modem to the router and have my desktop which used to be connected directly to the router work if I install a pci wireless card that would connect to the router? Sounds simple but before I do it I was hoping for some input.

Yes.. in easy terms, when you hook that router up, the modem sees that as your computer. Your connection is establish. At that point, you just need to establish a connection from your PC to the Router.. the router will hold the connection to the modem.
So setup your Computer to connect wirelessly to the router and you should be set once you connect the modem to the router.