When applying to grad school, how can I address having failed a core course before retaking it?
You can address this in your statement of purpose.

If you failed an intro-level course during your freshman year (e.g., intro to physics), no one is likely to care; straight As in all the core classes prove that you know your stuff. You can mention this in your statement of purpose, but I would do so only as an aside -- maybe even humorously. 

If this course was a junior-level course (e.g., quantum), then this is a bit more serious. You should definitely devote ~3 sentences in your statement of purpose explaining what happened. It is a bit strange that you would get straight As in all the other courses and an F in a separate course, so they'll be glad to see a reasonable explanation. 

My concern is that they may assume the anomalous F was due to cheating -- so I would definitely give an explanation that they can buy, rather than letting them make their own conclusions..