1. To remove Zynga Slingo application requests from appearing on your Facebook page, type "Zynga Slingo" into the search field at the top of your page. Select the resulting Zynga Slingo App Page and NOT their Game page.
2. Go to the right side of the "Play Game" drop down menu and click on the down arrow. Select Block App.
3. Click Okay to block the app. You will no longer see your friends' progress in the game nor will you be pestered by their Zynga Slingo app requests.
4. This technique can apply to any Facebook application. Some software developers name the app the same as the page so when you try to directly block the app by clicking on the X next to the request, it goes to the App page which assumes you want to install the program instead of removing the request.
