toothpaste really is a polisher too! no kidding.
it has small particles that make it work as a "liquid sandpaper".
but to polish a cd / dvd you need to be carefull and ready for a tedious task.
ps: you can polish the bottom of a soda can with chocolate bar and tinfoil, to make a paraboil mirror that can be used to lit fire. (you have to find the spot where the light relfexes, like wheres the reading head in satellite antennas, and put something that ignites easily there.) you need the sun to be up and hot to get this to work. and the more close you are the equator (the line that separates upper and lower hemisphere) the better it works.