That wheel is almost certainly undertensioned, probably not stress-relieved, both typical of a low-cost machine-built wheel.

With three spokes broken in a short time, I suspect many of the remaining spokes have been stress-cycled to the point they're ready to fail. You could try finding a good, local wheel builder -- one who has a spoke tensiometer and isn't afraid to use it! -- and have him or her tension and stress-relieve the wheel, replacing any more spokes that fail during the process. Or you could get a quote to do that, send it off with a warrantee claim, get a new wheel, then have that done with your new wheel. The second option should cost around $25, since you'd be starting with a "known good" collection of parts.

Could you do that yourself? In theory, yes. But since you've been checking tension and re-tensioning, apparently you haven't been getting the wheel tight enough.