I just did a quick googling of this and found that what you have is Lucky Bamboo, Not true bamboo.. There is a mass of info on the web about it. It is derived from some sort of lilly plant.. google "lucky bamboo"
Yea i was reading online about how hardy this plant is. One site even said that if it wasnt so slow growing it would be an invasive species! anyways, I read that you can chop it just about anywhere and the smaller piece will grow just fine. Its all chopped up, in the tank, with a little lower than usual water level. Lets hope my fish lasts longer than the plant.