Clean the polished tile in your normal fashion. Use the method recommended by the manufacturer.
Rub a melamine foam eraser back and forth across the scratch. This helps to remove lingering dirt and debris that settle in the crack. It also buffs the area, allowing the damaged area to regain the shine. Wipe away any foam debris with a soft cleaning cloth.
Squeeze a small amount of white toothpaste -- about the size of a pea -- on a cleaning cloth. Rub the toothpaste on the scratch in the polished tile. Use a circular rubbing motion. This is helpful in buffing out the scratch. Wipe off any toothpaste residue with a clean section of the cloth.
Hold a soft cleaning cloth over the mouth of a bottle of white distilled vinegar. Invert the bottle to dampen the cloth. Rub the cloth over the scratch on the polished tile until the scratch is gone. The vinegar is mildly acidic and therefore is effective in removing the scratch.