I personally don't like laying down synthetic ground covers like that but, it is commonly practised in organic circles. Permaculturists tend to like to use black and white newspaper. I always caution people against using any sort of plastics because polyvinyl chloride (PVC) plastics contain plasticizers that leak into the environment. Anyway, just ask what the plastic is made of.
I have found that what works best with invasive weeds is to:
1) pull all weeds out before they have gone to seed and place in hot compost
2) dig up any roots and hot compost or burn these as well
3) cover area with mulch or (newspaper) and then with mulch (Leaves, manure, leaves, coffee grinds and so on).
4) Finally plant as you will
Soem good ground covers include:
Clover ( a legume so, N fixing bacteria associate with the roots)
Vetch (another legume)
These three have fairly shallow root systems and will grow above a plastic but, it is best for your plants to work with what you have in the soil already, including all the wonderful worms, insects, fungi and bacteria.