Definitely file it. What will happen is that you will be monetarily determined...as swalsh says, you must have sufficient covered wages in the base period, the last eighteen months roughly, to set up a claim.
The week's worth of work you did with the last place will be considered, but it will pretty much be "no fault" as far as if you say you showed up and did your best, and the employer determined you were "not a good fit" that wasn't a disqualifying reason to be let go.
But then they'll drop back onto the issue of why you were terminated from the next to last place. How long you worked there is definitely an issue, as your wages to set up the claim will be from this place, probably, and if they can show that they had a valid misconduct reason to terminate you (such as pointing out on their absence or tardiness policy, or violation of a company policy with progressive discipline) some kind of genuine misconduct, then you would probably be denied benefits based on this termination.
If they can't show this, then you probably will be approved for benefits. It's a toss up, and we really don't have enough information to make even a guess.
If you are denied, of course you will have the option to appeal the decision.
But you have absolutely no reason not to file for unemployment and see what happens.