he polished off an apple pie : EAT, finish, consume, devour, guzzle, wolf down, down, bolt; drink up, drain, quaff, gulp (down); informal binge on, stuff oneself with, get outside of, murder, put away, scoff, shovel down, pig out on, sink, swill, knock back; Brit. informal shift, gollop; N. Amer. informal scarf (down/up), snarf (down/up).
the enemy tried to polish him off : DESTROY, finish off, despatch, do away with, eliminate, kill, liquidate; informal bump off, knock off, do in, take out, dispose of; N. Amer. informal rub out.
I'll polish off the last few pages : COMPLETE, finish, deal with, accomplish, discharge, do; end, conclude, close, finalize, round off, wind up; informal wrap up, sew up.