I have a broken tooth, but the root is still there. Will I need a root canal? (photo)

From the photo you provided it looks like there is minimal tooth remaining above the gum tissue. It may be a better treatment option to have the root that remains extracted and then an implant placed. There are options to restore the root that is left but often times they are costly and involve multiple appointments and the tooth will still have a poor long term prognosis.

When teeth are broken off at the gum line the treatment, maybe surprisingly, is not root canal and build ups any more.  There is not enough tooth structure to hold on to when they are broken off at the gum.  The best option is to remove the root and place an implant.  An implant, or a man made tooth root, is a titanium screw that is twisted into the space where your tooth was.  Once the bone has healed around it, the implant is restored with a connector called an abutment (like a post and core inside a cap on a natural tooth) followed by a crown (or cap).  This is the now traditional way to handle a tooth broken off at the gum line.  The other options for replacing a missing tooth are: removable partial denture or a bridge. These have there limitations and from the picture you submitted not what I would recommend. Best of luck to you!