Chemoprevention therapy is the use of certain drugs to try to prevent the development of cancer. The use of celecoxib has been approved for use in reducing the number of adenomatous colorectal polyps in familial adenomatous polyposis (FAP). It is not known whether there is a clinical benefit from a reduction in the number of colorectal polyps in FAP patients. The use of celecoxib may be an effective way to prevent the development of sporadic adenomatous polyps, precursors of colorectal cancer. This randomized phase III trial is studying celecoxib to see how well it works compared to a placebo in preventing the development of adenomatous colorectal polyps in patients who have had at least one polyp removed.