We discuss step 0 of factoring, extracting any common factors to simplify the process going forward.
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Factoring out a common factor is the process of finding the greatest common divisor and then pulling it out of each term. Unfortunately, when this is typically taught, it is taught by an expert that has done this thousand upon thousands of times. This seems like a good thing (and it certainly is!) but it comes with one notable downside - with that much practice, the instructor typically sees exactly what to factor out quickly and completely. Again, this may seem like a good thing (and it’s certainly good for the instructor) but it can lead to a misconception that the student should always look for the greatest common divisor and factor it out all at once - and this is not only untrue, it is actually more likely to cause algebraic errors.
Consider the following function: . On the one hand, we could try to figure out a greatest common divisor, but instead lets try a different approach. Instead of looking for the best common factor, try looking for any common factor, and start there. For instance, at a glance we can see each of the constants is even, which means that divides each of them, so is a common factor. Let’s factor that out!
To factor out a common factor we start by writing the thing we’re factoring out, out front, then divide that value out of each term that we are pulling it out of. So we would have something like the following: Notice here that, despite the fact we included the middle calculation where we had fractions, the fact that it is a common factor means that everything divides out evenly - leaving us with a nice expression at the end with the common factor of factored out.
But, you might say, that isn’t our greatest common factor, so we didn’t do it right! Indeed, we haven’t factored out the greatest common factor, but that doesn’t mean we screwed up, it means we didn’t finish yet. The key here is to iterate the process. In other words, now that we’ve factored out , we look at the result and see if there is another common factor that we can factor out of what remains. In this case, maybe we see that is in each term, so we can factor that out.
We keep iterating this, taking out whatever we can find as a common factor until we can’t find any more common factors. Note that, although I have been pulling out simple values so far, there is no reason you can’t do multiple terms at once if you happen to see them. The key idea here is that, you should pull out whatever you see that is a common factor, and don’t worry about if it is the greatest common factor - just check your result after you factor out your common factor to see if there is more to do. Eventually you’ll get to the greatest common factor.
Returning to our example, you might see now that you can pull out a , which gives:
At this point we kind of luck out, because one of our terms has been reduced to a - which has no common factor other than . In other words, as soon as we get a we have to be done. This doesn’t always happen, but if it does, then we immediately know we are done. In order to know that we are done however, all that is really required, is that there is no remaining common factor between the inner terms.
Now, had I happened to see it, I could have done all this in one step by pulling out the greatest common factor of , but that is likely to be difficult to see, so instead we pulled out chunks of that common factor one after another until we got everything.