League of Legends (LoL) Question: Why do you go triforce on Jax?

First let's look at what Trinity Force gives:

25 Attack Damage
40% Attack Speed
20% Cooldown Reduction
250 Health
250 Mana
5% Movement Speed

UNIQUE Passive: Rage: Basic attacks grant 20 Movement Speed for 2 seconds on hit. Minion, monster, and champion kills grant 60 (half for ranged champions) Movement Speed for 2 seconds.

UNIQUE Passive: Spellblade: After using an ability, your next basic attack deals bonus physical damage equal to 200% of your base Attack Damage. 1.5 second cooldown. Does not stack with Sheen or Lich Bane.

Now why does this matter? Well let's take a look at Jax's kit and base stats.

Passive: Relentless Assault: This gives Jax increased attack speed after every auto attack, stacking up to 8 times.

Q: Leap Strike: This allows Jax to leap to an enemy or allied unit or ward. If it's an enemy unit, Jax attacks and the damage scales with 100% of Jax's bonus attack damage.

W: Empower: This ability gives Jax extra attack range and increases his damage on his next auto attack by a flat amount plus a percentage of Ability Power.

E: Counter Strike: This allows Jax to avoid auto attacks for a short time while also applying a stun at the end and dealing damage scaling with 50% of Jax's bonus attack damage.

R: Grandmaster-At-Arms: This is Jax's ultimate ability and it has 2 parts. Passively it grants bonus damage on every 3rd auto attack he makes. On activation, Jax gains armor and magic resist based on his bonus AD and AP, respectively.

Jax also has the 3rd highest Base Attack Damage of any champion at level 1, and has the 21st highest Base Attack Damage of any champion at level 18.

Why does all of that matter? Let me explain.

Jax has 3 abilities that directly benefit from the bonus AD given by Trinity Force, seeing as his Q, E, and Ult all gain a direct boost.

Trinity Force also gives attack speed, which gives further benefit to Relentless Assault and the passive portion of Grandmaster-At-Arms directly, as with the passive ability, the attack speed gained is percentile, which means the more attack speed you build, the more attack speed you get from the passive. And with the passive portion of the ultimate, you get to hit the 3rd attack bonus faster and more often. Attack speed is also helpful for Counter Strike because while you are unaffected by auto attacks, you can still auto attack during that time. The attack speed allows you to get more auto attacks in during the first cast of the ability.

Jax also gets a major benefit from the 20% cooldown reduction, as all of his basic abilities can be reduced to between 1.8 and 4.8 seconds, with his ultimate at a 48 second cooldown. Obviously this all means that his abilities will be up more often.

The flat health and mana is also beneficial, but that's true for any champion who uses mana.

And lastly let's look at the most important passive of Trinity Force, the Spellblade passive. This means that after using an ability, Jax's next attack will deal bonus damage equal to 200% of his Base Attack Damage. I mentioned earlier that Jax has one of the highest Base Attack Damage stats in the game, so this translates into quite a bit of damage. That damage can be added onto by the damage from Empower, further increasing the amount of burst damage possible.

All-in-all, Jax, has some of the best synergy with Trinity Force of any champion in the game.

DMG

Its an all round (multiple stats) item, and the passive damage goes well with buying Guinsoo's Rageblade. The 40% attack speed is great earlier on with the cooldown reduction. With Iceborn Gauntlet as the alternative if you are behind while you still get armor instead of health you still get spellblade and the CDR.

I am not a Jax player, but each of your use of abilities give you a proc, Tri-force procs works with base AD, and Jax have the 3rd most AD early and one of the highest late game in addition Tri-force is a big item that give a lot of stats and take only one item slot, and Sheen give a good spick early game. So over all its a very good item.