Tina nailed it. Any
seed that falls can produce more dandelions, any plant can grow back (I think they're the oddball biennials that can live more than 2 years, like dianthus, but not sure.)
I've worked at controlling them for several customers over a period of years (thankfully quit the landscape biz before I was a total cripple) and I found that it's key to dig out the roots, plus do so before they go to
seed, and to get all you can with mowing so you don't have to dig them all and concentrate on digging the ones approaching the flowering or especially seedforming stage, dig a few dozen every chance you get, and each year there are fewer, and less effort required, until they are the exception and not the rule in the lawn.
Oh, ouch, the thought of all that bending reminded me how bad my back aches tonight.
I once read an article in Organic Gardening magazine that claimed dandelions are a sign that your soil is deficient in a specific nutrient, which they access through their deep taproot. It was claimed that dandelions moved the nutrient (I'm thinking magnesium may have been teh mentioned nutrient, but it was years ago that I read this) to the top layers of soil and that they gradually changed the soil content to the extent that they would not grow or grew only sparsely in that soil, so that the "problem" was self correcting over time. In theory, at least, this seems sound reasoning, as it is essentially what one achieves when fields are allowed to lay fallow....plants and their roots secure nutrients from soil and air which have been depleted through heavy cultivation, and nature restores the ideal balance desired for crop growth with little or no effort from us, aside from the agony of watching a plot go sickeningly full of all manner of
weeds for a period of time.