So one end of the cable sits near the splitter, but isn't connected to the splitter, and the other is connected to an outlet in a bedroom? There's no reason to terminate it, it isn't generating any noise. If you're having problems with noise, make sure you cap all your coaxial outlets in the home and remove any unnecessary splitters/replace splitters with unused ports with splitters that have less ports. (If you need 2 splits, replace a 3 branch with a 2 branch).
The one time I had trouble with noise, the cable guy used a 4 port splitter when I had only two devices connected to it (TV & modem).