We have it in our creek - so much there is no point in tackling it. I haven't tried, but I found a site that talks about it:http://www.hiltonpond.org/ThisWeek070915.html
Specifically, if you scroll down, you'll find:
Common Ragweed has a shallow taproot (above)--easily demonstrated by an upward tug on the main stem that almost always causes the whole plant to pop right out of the earth. This is good news for homeowners who can roam their property plucking ragweed in mid-summer before it has time to make pollen. The other good news is Common Ragweed is an annual, so each plant is destined to die at the end of the current growing season.
If the South Carolina variety is at all like the Texas type, would this work for you, pulling it out before it blooms? I wonder if you're seeing something else if you are finding the roots so tough. Could you post photos of the plants?
The discussion on that page mentions goldenrod and ragweed - we have both plants here. It might be a good idea to be sure which plant it is before tearing it out.
File comment: This photo and information comes from http://www.hiltonpond.org/ThisWeek070915.html .
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Good luck with your research and this project. If it is ragweed, let us know and maybe by then someone will have come by with a more informed answer (as in, they've IDed it and dealt with it!)