how to kill Bermuda grass

how to kill Bermuda grass in my garden my vegetable garden it has hard red clay soil I tiller it with a tiller still comes back it have even tried weedkiller it is very hard to dig up and always comes back fast

Bermuda can be extremely difficult to erradicate, especially if it's growing where you have desireable plants.
It is extremely deep rooted and will keep coming back unless you are majorly diligent about pulling it up when you see it.

IF you can bear the thought of removing all the plants in your vegetable garden, you could douse it with Round Up, then hand remove the runners and allow the soil to "rest" for a week or two., and cover the soil with black plastic or wet newspapers to keep the sun out. In the meantime, don't water the soil...the heat of the sun will bake the soil and hopefully kill any remaining runners while they die of thirst.

If you have a lot of bermuda near the vegetable garden, (as in a lawn full of bermuda) dig out a path about two feet wide between it and the garden, and dig a trench a few inches deep between the two. Keep this space clear and don't allow the bermuda to get near the garden.

Bermuda will also start from seed that forms on any that is growing nearby, so keep it mowed and don;t allow it to go to seed, otherwise, you're back to the same old problem again.

You HAVE to be diligent at pulling it up the minute you see it.
It CAN be done, but you have to stick with it, and eventually, it will give up.

this type of grass can grow up and through concrete etc, when you till the soil, you are cutting the roots up, this is like propagating new plants, each bit that the tiller breaks off the plant, the tiny bits regrow roots of their own, it is a hard job to eradicate it but as
Jasperdale has said, it can be done, just not in one season as far as I am concerned,
your right, it has really deep roots and they are so fleshy, they snap easy when you are digging them out, for now, I would let the bits grow to about 2/3 inches, get your strongest weed killer you can and a paint brush (old small ladies make up brush will do nicely) rubber gloves, you then need to bruise the grass leaves, dont break them off, but after this, paint the weedkiller onto the grass and let this be taken down to the roots, this takes a couple of weeks before you see the grass discolour as it takes that long to reach the roots that feed the grass, using the weedkiller this way is safer than spraying as it wont come into contact with your veg or other plants, once the weedkiller dries on the grass, it cant contaminate anything else either, it is a much safer way to do it this way as your only applying the killer where it's needed.
I am sorry to say, but you may have to do this a few times before you notice less grass growing back, even next year, there could be new bits show up that were hidden under ground this season, but if you have the patience to do this, you really will get rid faster than tilling. I dont know how it works, but my old Dad had this trouble and he swore that heavy planting of turnip seeds late winter, got rid of this problem, he scattered the seeds thickly, then as they grew, the grass grew threw the seeds and the roots of the turnips (swedes) and when he lifted the turnips using his garden fork, most of the grass roots came out with the turnips, I was really young and have a feeling the grass actually grew into the turnip ball shaped veg and the grass came away also. Hope this all helps you out as you must be round the bend with hard work up till now, good luck. WeeNel.