So I have xfinity wifi and internet, it works great and all but the signal doesn't reach everywhere in my house. Range extenders don't really cut it since the speed it halved, will a tri band router increase the range of the internet?
Tri-bandedness itself does not increase range, but currently (up to mid-2015, the time of this writing), tri-band 802.11ac routers are top-of-the-line, state-of-the-art, flagship routers from the major players in the consumer wireless router industry, and as such, besting each other at rate-at-range is important for this subcategory of wireless routers, so they usually have great PAs (Power Amplifiers) on their transmit paths, great LNAs (Low-noise amplifiers) on their receive paths, great high-gain omnidirectional antennas, and they make use of beamforming when they can. It's really those things, not the fact that they're tri-band, that makes them have great range.
Whenever you decide to buy a new wireless router, always buy a top-of-the-line model, because those always have the best no-compromise designs. On everything else they cut corners in ways that harm the speed, range, and robustness of the network. The farther you move down the price curve, the worse the corner-cutting becomes.
It depends on the type of router. If the router supports 802.11ac it should extend the range as it can use more simultaneous bandwidth (in Hz), avoid interference and more effective MIMO reception. Tri band 802.11ac devices should provide the longest possible range without extenders, but your computer/phone/device also needs to support it.