My parents own a flower shop, but I kill green things.
Anyway, as far as I know, a cut sunflower will not sprout roots. You could, however, germinate some seeds and use those, but really, your best bet would just be to either buy a sunflower plant, or but some already-germinated seeds.
If you are interested just in growing something for fun, do a spider-plant or avacado seed. It's about the easiest thing in the world to do.
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