Hello, my name is Kritya and I have played Jax since early season 1, seeing the ups and downs he has been through. With the recent changes in the 3.10a patch, Jax has seen a substantial increase in power, even if the number changes were relatively small.
What made Jax so strong?
Well the changes on phage are the first thing to spring to mind. The 60 movement speed on killing a creep suits Jax's trading style in lane perfectly, along with the 20 movement speed on attack. Early game you won't be killing many champions all too easily, even with a slow, which is why I feel the change from Icy to Rage benefits Jax, I will be going over this in more detail in a later chapter.
Then there is the upgrade to Trinity Force, upon using a skill, your next auto attack does bonus damage equal to 200% (up from 150%) of your base attack damage. Seeing as Jax procs triforce so much, this change massively benefits his mid/late game. there is also the fact that it is 215 gold cheaper than before.
But BotRK was nerfed?
Despite BotRK being Jax's first item in more or less every scenario, I have found that by using lifesteal quints, it is possible to go triforce into BotRK and still have decent sustain early on. You'll end up buying a few more potions, but with all the improvements on phage and triforce Jax becomes a large threat once completing it. Also, the BotRK nerf only increased cooldown, and decreased range. The decreased range doesn't affect Jax too greatly as most of the time you would want to use it you will be in melee range. As for the cooldown increase, it went from 60 seconds to 90 seconds, honestly it won't affect you that much, late game team fights rarely occur twice within 90 seconds and by the time you buy it with this build you won't be fighting every 60 seconds in lane, your opponent's turret will probably be gone way before then.