Possible keylogger installed, restore factory settings?

Hi there. So So i have Sony Xperia and i am afraid a friend (no more friend acctualy) of mine may have installed keylogger, other friends told me he used to to this to his girlfriends and then mess up with their lives. Anyways i've noticed some strange things on my facebook account, and since i only use facebook through my phone i decided to reset to factory settings. Now i am pretty sure its just paranoia, but a friend told me to check if my phone was rooted beacuse it could be a problem if it is, so i;ve red in the web that i should use Root Checker Basic to see if Root if there is root access, but the app says root access in not propery installed. My question is, can keylogger survive factory reset, and is it possible the phone to be rooted and then unroted again, will the program show that toot access is not installed then or once the phone is rooted it will always show up. Is there other ways to completely wipe clean my phone to be sure? Thanks!

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