Possible solutions for Facebook authentication failed
1. Check connection - It may sound little silly but whenever you are facing Facebook Authentication Failed error, first you should check the internet connection. If you are using mobile data, switch to Wifi and see what happens.
2. Clear Cache and Data - If you see that internet connection is not a reason for Facebook Authentication Failed error, then next step is to clear both Cache and Data of the Facebook app. To do that, go to Settings > Application Manager > All tab > Facebook > Clear Cache and then Clear Data.
3. Reinstall the Facebook app - If you are still stuck with the error then uninstall the Facebook app. Then re-install the current version of Facebook going to Play Store if you are an android user. After the installation is done, reboot your phone and try to login. It should fix the issue. In most of the cases, Facebook Authentication Failed problem occurs because of obsolete version of the Facebook app.
Hope the above readines resolve your problem.
Else feel free to call me to help you with your query.