Wash the plastic in warm water with mild dish detergent to remove any surface dirt or oils. Dry thoroughly.
Buff the plastic with a mixture of water and either toothpaste or baking soda on a soft rag. Rub the scratched area very gently -- moving the cloth in small circles -- to polish the scratch without harming the undamaged areas. Rinse and wipe dry. Repeat if necessary.
Treat the plastic with furniture polish, automobile wax or brass polish. Apply the polish with a clean, dry cloth. Buff the polish gently into the plastic until the scratches are no longer visible. Wipe away any excess polish.
If you prefer commercial products over household cleaners, purchase a glass etching compound, which is available at many hardware and discount stores. This is a good option for treating plastic eyeglass lenses. Apply a thick coating of the etching compound to the plastic, let it sit for five minutes then wipe it dry. This method works well for coated plastic eyeglass lenses, but may not work for uncoated plastic components. If you proceed with this step, do not allow the etching compound to touch the frames of the glasses, and do not rub it into the plastic lenses.
Consider purchasing a scratch removal kit, if at-home remedies do not work. Follow the kit's instructions carefully.