I'm not sure how to be more specific. It's a very old Italian plum tree that apparently spread its roots quite far, and it now has suckers popping far from where its trunk stood. 

I want to know how to kill the suckers/roots.  

Thank you

Hi, thanks for reaching out. I don't know of a product that can be injected but there are several that can be painted on. This may be the best solution for this tricky situation.

Here is an example of one product that contains the herbicide triclopyr. (Glyphosate is another common ingredient which can be used as described below.) This is a common ingredient in brush clearing products. It is non-selective so you wouldn't want to spray because it will kill all the plants it comes in contact with. But you can clip the plum suckers and then paint this herbicide on the cut end. The herbicide will be absorbed into the plum and should provide control. Many of these products come with a built-in applicator brush or you can use a small disposable foam brush. As with any pesticide wear gloves, long sleeves, etc. to keep off your skin. Make sure to carefully follow the label instructions of the specific product that you purchase for safety and application instructions. 

Please let me know if you have additional questions and best of luck!