To clear past searches, log in to Facebook and go to the upside down triangle in the top-right corner. Click on Activity Log. The Activity Log is where Facebook tracks your actions within the site.
In the left column of the Activity Log, under "Photos, Likes and Comments," click on the "More" link. Then scroll down and click on the "Search" link.
You'll see a list of every Facebook search you've ever made, organized by date. You can clear searches individually by clicking the crossed-circle icon on the right.
Then click "Remove" in the box that appears.