Polish plastic with a fine abrasive, either by using one found in a plastic polishing kit or by using several ordinary household materials. Toothpaste, baking soda and silicon-carbide sandpaper can all yield excellent results.
Prepare the plastic before polishing it. If it has been in a harsh environment, it may have a greasy coating that can interfere in the polishing process. This coating can usually be removed by wiping the plastic with a mild detergent solution and a microfiber or cotton cloth. Vinegar also works well to smooth the surface of plastic to ready it for polishing.
Both baking soda and toothpaste are mild abrasives that can be used to polish plastic. If the plastic is discolored or deeply etched, squeeze non-gel toothpaste onto the surface, and rub it in a circular motion with the microfiber or cotton cloth. Alternatively, make a paste of baking soda and water and apply it in the same way.
Another good way to polish plastic is using progressively finer sheets of wet/dry or silicone-carbide sandpaper. Immerse each sheet in water for 10 minutes prior to use. Sand the plastic lightly in a circular motion, and wipe off any reside with the microfiber or cotton cloth.