Buttercups are perenial weeds, therefore you need to get rid of the whole plant (root and all) to stop it from coming back again and again, bulbous and creeping buttercups are the most common weeds in borders especally in damp soils.
They are not easily pulled up by hand, so its best to loosen the soil first with a fork to make sure you get all the plant.
If you decide to use a weedkiller, a systemic one would do the best job, as it will translocte through the plant systematicly right down to the root. any weedkiller containing 'Glyphosate' should do it at the given rate.
The leaf of the buttercup is not easy to wet and repeat applications may be necessary. Good Luck. T bird+