Hi jax main here i started to feel jax items i usually build are bad so i thought about making guide about how to utilize runes/items that are meant for other roles.
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If it comes for ap jax u can play him 2 ways maybe more. Burst is best vs squishe enemy. For runes you have some options like
Electrute for faster dmg
Dark Harvest for better scaling
Predator for chase
First strike for faster items
Aery for teamfights
Every ap item excluding riftmaker are up to your choice.

You can also play more bruiser style ap jax
For runes in my opinion you have 2 options
Conqueror for bigger dmg and more omnivamp
Fleet footwork for sustain and mobility
For items rush Riftmaker/Nashor Tooth then get the other one.Then priorotize items for survivality or tankiness. Consider demonic embrace.

I was very unlucky bcs i couldnt try shoijn on Jax so i tried to recreate it going for crit build. Problem is its expensive and take a lot of your space. But i still think it can work.
For runes you can go:
Conqueror for more ad and omnivamp
Lethal Tempo for more as
First Strike for faster items
For items rush reaver or shieldbow then take the other one. 3rd item u have 4 options
Navori Quickblades for spam abilities
Infinity edge for more dmg
Bloodthrister for better sustain
Phantom dancer for mobility/attack speed
Then build your defensive items like wits end or frozen heart.

My farvorite style of playing jax.
The tank jax its always good when you need to play more for teamfight style.
They are 2 ways of playing tank Jax

DMG Tank tanking your enemy while dealing them big dmg for this style you always go lethal tempo
For items u rush sunfire cape HP and Resistances follow up by ramping dmg. Then you go Titanic Hydra. These 2 items buffs your dmg real big.
Then go tank and normal defensive jax items.

If you dont feel as DMG tank you can go for more utility Tank Jax. You can take any runes byt u find it good with aftershock. Your items are either Chemtank for chase or icefist for slow.
Then you can go any tank or support items.

One of the weirdest build for jax.
I belive your runes should be
Electrute for faster dmg
Conqueror for longer fights
Lethal tempo for more attack speed
Hail of Blades for 3 fast auto (using W doafter first auto makes it 4 autos instead of 3)
First Strike for faster items
Your Items Can Be
Prowels Crawl for some unsuspected mobility.
Eclipse for better duels.
Darksword for teamfight.
Next you can go BoRK or reaver
Other items you can go for more dmg or for normal defensive items.

Hybrid Jax has been forgoten sadly bcs in my opinion its one of fun style that i like to sometimes play. Hybrid Jax for these who dont know is an AD and AP build. So for my build you can go either
Conqueror for more dmg and omnivamp
Lethal Tempo for more attack speed

Your build is BoRK for the duel power.
Riftmaker for better longer fight/omnivamp and then reaver for more ad burst and mana regen. Then go for any defensive items.

I also enjoy playing full ah jax with coqueror build you can get in items.