Resiliency and an extensive root system make an unwanted poplar tree a source of serious frustration when it comes to proper termination. The sucker roots of many varieties of poplar trees have been known to survive having asphalt and concrete laid on top of them, eventually sprouting and destroying those hardy materials overhead, so it isn't hard to image what one of them might do to the foundation of a house or other structure if not fully removed. Thankfully, there are a few steps that can be taken to assure that the poplar tree stays gone for good.
Using a chainsaw, fell the poplar tree, cutting it down as close to the ground as possible. Be sure to wear protective gear (gloves and goggles) at all times. Only attempt take down a tree if you are experienced and properly trained.
Remove the newly felled tree and dispose of it in a manner approved by the guidelines set forth by your local municipality. Make sure to police the area for any seedlings or shoots that may have broken loose as the tree came down; these can take root and leave you with yet another unwanted poplar tree down the road.
Paint the stump of the poplar tree with a nonselective herbicide -- glyphosate, triclopyr or ammonium sulphamate are recommended. It's better to paint the stump rather than spraying it, to protect nearby growth; the herbicide will kill any other greenery it lands on. This step should be repeated every other day for a minimum of two weeks.
Dig up the area around the base of the poplar stump. Digging down to the top of the root system in a five-foot radius around the now-dead tree stump will make removal much easier and help ensure that the root system is thoroughly eradicated.
Pull out the stump and remaining root system with the stump puller. Dispose of the remnants of the tree. Again, be sure to wear proper protective gear when using this heavy machinery.