I Have Had A Cough For About 4 Weeks. I

I have had a cough for about 4 weeks. I saw my doctor who prescribed a 5-day Z-pack and 5-day prednisone. Sx persist. Contacted doctor and received Augmenting. No change. Is it possible that this is not a bacterial infection? I do not smoke and I have no other symptoms that seem to be related, but I am noticing that I am shedding more hair than usual.