Every little bit of root left in the ground will send up shoots. As you've found out, the roots, though easy to spot, are quite brittle. I've just about got rid of mine after digging out miles of it, but I do have very light free draining soil - heavier soils would make this very difficult. Now I get the odd wispy bit coming up, which I immediately pull out, hoping the chlorophyll deprivation approach will work. Applying glyphosate after luring it up canes is very effective, and helped by bruising the leaves before painting/ spraying. And NEVER put it on your compost heap unless it's rotted down so much it's unrecognisable.