How do you politely ask for a raise at work?

The first thing you should do is write a list of accomplishments/successes you've had in the last 3-6 months. If your company has an employee manual, check to see what the policy is for employee evaluations. There might be a chance that you're actually due an evaluation (it seems like HR and bosses always drag their heels on this).

If your company doesn't have annual/regular employee reviews and you've been there more than 6 months, ask for a meeting with your boss. You can use that opportunity to go over the accomplishments you've had and ask if he'd consider giving you a salary bump.

"Hello sir, may I have a raise?"