Welcome
@Jenster100 
You have your work cut out for you there! Those roots will continue to send up new growth, and they will be really hard to remove. Even if you think you can dig every bit out - it will probably pop up again somewhere. Your best bet is to drill into the roots as much as possible and fill all the holes with brush killer. It will need to be a strong herbicide to deal with the problem.
The other method that might succeed after time is the use of epsom salts in large quantity. Although the salts are often used to feed plants, it is possible to kill awkward stumps with an overdose of the stuff. This method will just feed any surrounding plants in the area if it should seep out. Again, you would need to drill nice big holes into the roots, fill them up with salts and cover with black plastic or similar to keep the rain off - tie in well and leave for at least a year or two, cutting off any bits that try to grow back.
Good luck