The militant islamist have had the unfortunate experience of being exposed to the warrior nation of the USA. That's not to say that the US is a warrior nation per se, but that they have only been exposed to that side of our culture. Afghanistan, Iraq, Iran, Palestine, etc. really only have a sense for how badly we can eff them up. If only they knew how good we have it here they might try a little liberty and relaxation with the whole jihad thing.
I mean I grew up with ice cream and air conditioning. They're growing up with midnight raids and mine fields. They are not like us in very many ways.
I think that eradication is impossible, irresponsible, elitist, and wrong. But we can help these places assimilate to a more western culture.
It doesn't seem likely that a shooting/bombing/droning strategy is working. I think we need to start employing drastically different tactics in this "war on terror."
We ought to introduce them to some comfort and try to ween them off the whole death to America (and it's allies) trip that they seem to be on. We should drop playboy magazines on their villages.