If weeds return they are coming from either seed or roots left in the ground after pulling. Knowing what weeds you are fighting would be helpful. Preen biodegrades into some questionable materials and carbon dioxide. It doesn't take care of all weeds. It doesn't work well in wet soils, so if you are irrigating frequently it doesn't do it's job. The first thing I would do is quit the Preen and try another horticultural tact. The first thing to do is ID the weed(s) and let's go from there.
hortster

Preen should keep plants from germinating and growing for several weeks and if it is not doing so it is not being applied properly. Without using that environmentally unsound material my planting beds usually need attention for unwanted plant growth 3 or 4 times a year, if I don't mulch well. Rather then spending money on a questionable environmentally unsound material to control "weed" growth apply a good mulch to your garden.

How can I "ID the weed(s)"? Can I submit an image to you? How do I do that?

Preen is mostly for preventing germination of seeds. It won't kill the root on existing plants. Roundup (glyphosate) works better on these.

For most weed prevention, heavy mulching is indeed your best bet. And yes, you can post images. On this, I am no pro.... Hopefully, you will get a good answer on this.

You could post a picture of your plant for identification. You could post a picture on one of the picture sites and post a link to that here. Sometimes people post pictures here that are just too small to see, so be sure your picture is one that will allow us to see what you see.

Hope this makes sense. An easy way is to open a free account at Photobucket.com. Upload your pictures there. Click on the HTML line (third line beneath your picture in Photobucket) and it should say "copied", then paste here [this gardenweb forum] in the message area - not in a link [You can paste multiple pictures regardless of size and you won't see the picture, but only computer lingo until you click on "Preview Message" below]. You will be able to post pictures on any other forum, too. After pasting your photos here [this gardenweb forum], you should see them when you hit "Preview Message" before you actually post the message.

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I see the same plant growing in my planting beds and since it is a plant I do not want growing where I do not want it to grow it is a "weed" and I pull them. Since many plants spread seeds on the wind or via birds or other animals (think burs) there is no way to permantently stop weeds form growing. Mulches can help however. Ma Nature does not like soil left bare and exposed to the elements so she plants lots of plants and we consider many of them to be "weeds".

What are the names of the two weeds pictured?

There is a "name that plant" forum where you might get a quicker answer since your pictures are "buried" in the middle of this discussion.

The second one is likely hawkweed, Hieracium species (H. vulgatum?). I'm not sure on the first one but it has characteristics of some of the Euphorbias.
hortster

The first is a Euphorbia. To verify, when the stem is broken, it exudes milky sap.

Both weeds are annuals. (grow for just one season) so remove prior to setting seed. Easy to pull out, roots and all. Caution with the first one as some people are sensitive to the sap -- for your protection, wear gloves.

OP said "But, within a couple of weeks, my garden is again full of weeds. (I go through this exercise three or four times a year)."

Yes, because many, MANY seeds are in the soil -- it's called the Soil Seed Bank -- and will sprout when conditions are suitable.

As for "(I go through this exercise three or four times a year)." -- hey! consider yourself among the majority. We all do. And we all will, forever and ever. As long as weeds exist.

Weeding is a never ending job because of the seed bank in the soil. Your best control is to keep the soil well mulched. Otherwise, you have to hoe or otherwise get rid of the weeds every couple of weeks. If you use surface irrigation and can't heavily mulch the soil or lay down black plastic, then repeated mechanical or chemical control are your main options.

Using a product like Preen that contains trifluralin will work very well and I use trifluralin on the suitable parts of my garden. It must be applied properly or it won't work, however. First, it is only effective against germination of susceptible seeds. Anything already growing when you apply it won't be controlled. Second, it is sensitive to sunlight. Just applying it to the surface will allow the sun to degrade it and within a couple days it will no longer be effective. Third, to be effective you must get it into the germination zone. Shallow incorporation, such as by hoeing, into the top 1-2 inches of soil will get it out of the sunlight and into the germination zone. You can also use a sprinkler to apply about 1/2" of water after the herbicide is applied to carry it into the soil. Finally, not all weeds are susceptible to trifluralin, and they will grow just fine even if it is applied, just like your garden beans, peas, onions, and others will grow.

...garden beans, peas, onions, and others will grow. No way I would put that stuff anywhere near my food.

Thanks, Beeone, for the info on using Preen. I've casually looked at it once or twice for using on mulched paths in various ornamental gardens, but much of what I deal with there is tree seeds (buckthorn, wild cherry, buckthorn, redbud, buckthorn....), and I always figured that they might be pretty resistant.

The additional work to get it into the upper mulch levels, and into the dark, has settled me. I'll take the money I've thought of using for Preen, and do something else with it.

Any "weed" seed would germinte really well where the soil is left bare and exposed to the elements. Putting a mulch down can help with "weed" control because mulches 1) aide in keeping "weed" seeds from germinating, 2) aide in retaining soil moisture, 3) aide in soil temperature control and, 4) if appropriate materials are used can aide in adding needed organic matter to soils.

How do I pull out the weeds "ROOTS and ALL"?
How can I be sure that I've gotten EVERY root?

How do I pull out the weeds "ROOTS and ALL"?
When you pull a weed, if you don't have a root ball attached, a shovel, hand trowel, or dandelion fork can help you get the roots.

How can I be sure that I've gotten EVERY root?
Usually, you can't be sure so you just keep an eye on things. If your weed starts to regrow, dig deeper or wider and keep trying.

linda, I know what you mean. Most tree seedlings hard to pull and go very deep very quickly. These and grass consume most my weeding time.

Mine's another vote in the "use more mulch" contingent.

Reference the two posted pictures, the second one is Groundsel, the first might be black medic.