College Algebra: Factoring: Greatest Common Factor

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A major theme in this course is actually how to factor polynomials and just factoring things in general. It's always a good thing to wonder, "Why exactly do we care about factoring? Why is this important?" Well, let me actually show you why. Suppose I asked you to solve this sort of really nasty equation: x^2-4x+4=0. Well, how do you do this? There are x^2's, there's x's, there are a whole bunch of terms in there. How do you do that? Hard problem, so don't do it. Well, what if I asked you this problem to do - what if I said to you that (x-2)^2 = 0, then what does x have to be? Well, then you can look at this and say, "Oh, x has to be two. That's what would make this zero." So this is really easy to solve; this is a lot harder. Well, it turns out this equals that, and so the power of factoring is that within it, it allows us to solve all sorts of complicated equations by breaking things up again - major theme - into little teeny pieces.
Let's actually start to think about factoring and remind ourselves of all the factoring techniques - the plethora of factoring techniques - that are out there.
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Well, let's see. I don't know what the answer is, so let's see if we can figure it out right now live. I see that in terms of the coefficient department I can pull out a three everywhere, but no more. I can pull an a out everywhere, but no more because there's only one a here. And here I can pull out a b^2, but I can't pull out a b^3 because I don't have enough here. So it looks like the 3ab^2 is, in fact, the greatest common factor. And if I factor that out, what am I left with? Well, let's see, what am I left with here? Well, it looks like I'm left with nothing here, so maybe I should write zero. Maybe I should write zero there. Well, that wouldn't work because if I distribute this thing times zero, it would give me zero, and I'd lose that term. So what should I write in here? Well, what I should write in here, of course, is the invisible one that's always everywhere you go. It's always behind you, the invisible one.
So, in fact, there's a one-factor here - maybe I can write it out right in front here - and that one-factor would still remain. Notice now when I take this and multiply it by one, I happily get this. So remember, don't put a zero, always put a one if you take out everything.
Well, here I don't take out everything. I have a negative two, and I need an extra a to top off the a^2, and I need an extra b. So, in fact, you can check this by taking this and multiplying it through by the one, and you get this; multiplying this through by this, and I'd see a 6, -6, a^2, b^3. This is a great technique when every single monomial has a common factor. Always pull out the common factors first. Even if you have to do more factoring later in life, always start by seeing if you can pull out a common factor. It's the easiest thing to do and has the power to actually make the problem easier.
We'll see other techniques, if you want...if you dare.
Prerequisites
Factoring
Factoring Using the Greatest Common Factor Page [1 of 1]