Why was general john burgoyne's defeat at the battle of saratoga a decisive moment in the revolutionary war?

The defeat of General John Burgoyne at the Battle of Saratoga was a critical moment in the Revolutionary War because it brought France into the war on the side of the patriots

France realized that the United States could be victorious after Burgoyne’s army surrendered to American forces after Saratoga. Vergennes was directed by the king to negotiate an alliance with the Americans.

answer; ///i believe that the correct answer is (b); ///the candidate must be a native born american citzen;