Answer: If any roots remain, the ivy will grow back. The best way to get rid of ivy is to dig every last root out. You might want to rototill the area and then rake the roots out. Barring that, you can spray the ivy with an herbicide such as Round Up. The active ingredient (glyphosate) that makes contact with the leaves will be transported down to the roots and will effectively kill most of the roots. They will eventually decompose so no digging is necessary. Glyphosate applications are best made in the fall, but you can try applying now and again in the spring.
Best wishes with your project!
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