Growing French Tarragon

I have bought French Tarragon plants from you last year and again this year. I live in the Chicago area (zone 5). Can French Tarragon live through our winters if I leave the plants in my garden? Last year I dug up the tarragon and transplanted them in pots and brought indoors. Once indoors I started to get small plants coming up from the dirt...as if the plant was sprouting new plants. I tried to separate and transplant these new shoots but they failed to take. How do I start new tarragon plants from an existing plant?

Tarragon will over winter in zone 5. Make sure that you get the French Tarragon that grows only from cuttings, since it does not set seeds. Tarragon, like other temperate climate herbs goes dormant during the winter months and. the old stems die down when you take the plant indoors. It sends out new shoots as you have noticed yourself. When you purchase new plants it would be better to leave the plants in ground over the winter months. Harvest the tender tips through the summer and fall and keep the tips (3 inches) in zip lock bags for winter use. Tarragon freezes very well and retains its flavor for months.