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Bournemouth gifted Arsenal the lead just 12 minutes in when a poor pass from Steve Cook rolled to Alexis Sanchez, setting him up for an easy goal.
Theo Walcott had the Gunners 2-1 up just eight minutes into the second half when he nodded in a Nacho Monreal cross at the back post.
And the icing on the cake was added late on as Sanchez got his second of the game.