havent tried it yet was just messing around on leaguecraft's item builder and while it looks like it could work im not sure. heres the item build.

Lichbane
Rylai's Crystal Scepter
Rabadon's Deathcap
Zephyr
Nashor's Tooth
Sorcerer's Shoes

It gives 30% cd reduction, can spam q and w to hell and with his innate passive his attack speed should be about 2.0 so his util passive can proc like crazy. this is without runes or masteries havent done them yet just looking for some input tho. was thinking about defensive runes and masteries to make his kinda tanky.

AP Jax has generally been too expensive to work. AD is just too good on him to ignore, too cost effective.
A hybrid tank is a good build nowadays.

Merc Treads
Iceborn Gauntlet
Black Cleaver
Trinity Force
Guinsoo's Rageblade
Wit's End

Iceborn Gauntlets and Trinity Force are redundant. I'd replace IG with Frozen Mallet.

They actually are NOT redundant. The TF deals 150% to the target and the IG deals 125% to those caught in the AOE. Jax uses every bit of the stats on both of them.

Just uses every bit of stats on most all items; but spending that much gold to gain +25% of your -base- AD without items/itemisation on one target makes no sense. Your overall gain per gold is demonstrably lower and 3500g is a lot to place on IG when you already have trinity at 4200g. You would be better off simply taking Iceborn with a focused item; Iceborn, Phage and any Zeal upgrade (or Zephyr) will give you the same bonuses as Trinity force, because trinity does not improve the Phage proc

Frozen Mallet + Iceborn Gauntlets will yield far better results than Trinity + Iceborn Gauntlets

Vera: Having Trinity + Frozen Mallet doesn't make much sense tbh, you're making the Phage component of Trinity Force redundant by having Frozen Mallet. You'd be better off with two Trinity Forces, given the chance to proc stacks ;D

If that was all that you gained then I would agree with you. You are omitting the long list of stats that you get from IG + TF that are outside the Spellsword passive.
I disagree. You are tougher by 450 health but you don't need the FM slow. You only gain another 5% on the slow but it lasts half the duration. With a 2s cooldown on IG, the slow is constant as long as you use Jax's W every time it is off cooldown. You lose out on TF's AP, AS, MS and mana. If you don't like maneuverability for better initiates and escapes but you want to be more of a tank (and less of a bruiser) then go with the FM.

Running any two of IG, TF, or FM together is a waste of gold. Pick one and build something else worthwhile so you're not doubling down on procs when you only need one to be effective (IG or FM's proc are more reliable than TF anyway).

Quite frankly, AP/AS Jax should run:

Sorceror's Shoes/Mercury's Treads - 1100-1200 gold
Iceborn Gauntlet (15% CDR) 40 AP - 3400 gold
Nashor's Tooth (20% CDR) 65 AP - 2500 gold
Guinsoo's Rageblade 40 AP (+32 with maxed item stacks) - 2600 gold

The above four are core items on AP/AS Jax and easily max out CDR (@ 39%, 1% shy of 40%) with 4% in masteries, providing a potential 177 AP from items for 9600/9700 gold. The final two items are situational.

One of Hextech Gunblade, Will of the Ancients, Zephyr, or Manamune into Muramana
One of Wit's End, Abyssal Scepter, or Twin Shadows for magic resist purposes if necessary, or alternately a Zhonya's Hourglass if that AD Carry is causing lots of grief.

How about Malady? Anyone?

Malady is good for AP on hit champs. It rules pretty hard on Diana.

more redundant. I would replace trinity force with a frozen mallet. Now your slow stacks, and sometimes slows other people.