Hey, everyone. So in math and algebra, so far, we've seen how to multiply numbers and polynomials. I could take, like, 2 times 3, and I know that multiplies to 6. And I can take this expression and do use the distributive property and know that it multiplies to x2 + 3x. Well, in some problems now, they're actually going to give you the right side of the equation, and they're going to ask you for the left side. They're asking you for what things you have to multiply to get here. And that's a process called factoring, and we're going to take a look at how to do that for polynomials. And what I'm going to show you here is that, basically, whereas multiplication was taking simpler terms and multiplying into more complicated expressions, now we're going to do the opposite. Factoring is the opposite of multiplication. We're going to take a complicated expression and break it down into its simpler factors. So I'm going to show you how to basically factor something like 6 into just 2 times 3 or this expression into x×x+3. Alright?

So it turns out there are actually 4 ways to factor polynomials, and we'll be discussing each one of them in great detail in the next couple of videos. We're going to take a look at the first ones. I'm going to show you how this works. The first thing you should do is you should always look for greatest common factors inside your expressions. This has an acronym, the greatest common factor. It's called the GCF, and so I'm going to show you how to factor out this GCF using this example over here. So we're going to take a look at this example of 2x2+6 and, basically, what the greatest common factor is, is it's the largest expression that evenly divides out of each of the terms in the polynomial. I have to figure out the largest thing that divides out of everything in that polynomial. Alright?

And here's how to do this. Here's a step by step process. The first thing I like to do is write what's called a factor tree for each one of the terms. In factor trees, you may have seen them before. It's basically just a way to break down larger things into things that multiply. So, for example, 12, there's two numbers that multiply to 12. I have 2 times 6. Right? I could also have used 4×3 and would have been perfectly fine. So if I do 2 times 6, I can't break down 2 anymore, but I can break down 6, which actually just breaks down into 2 times 3. So in other words, the 2 and the 2 and the 3, these are all the factors of 12. I can also do the same thing for things with variables. In other words, the 6x2 breaks down to 6 times x times x, but then I can also just keep breaking down the 6 into 2 times 3, and I've already seen that.

So in other words, the 2, the 3, and the x and the x, those are all just the factors of 6x2. We're going to be doing this exact same thing, but now for each one of these terms in this expression. I'm going to do some color coding over here. I'm going to do the 2x2, and I'm going to do the 6 over here. So what does the 2x2 break down to? It just breaks down into 2 times x times x. Can I break down anything else? No. Because 2 can't be broken down any further. What about the 6? I've seen that the 6 can break down into 2 times 3.

So once I've done all the factor training for each one of the terms, I'm going to put a big parenthesis around them and include the sign that it was in the original expression. And now what I'm going to do is I want to figure out the largest thing that is evenly divisible out of each one of the terms or the largest thing that pops up in between both of the terms. So if I take a look at this expression, what are the common items that appear in these terms? Well, I see a 2 that pops up in the left term, and it also pops up in the right term.

Are the x's common between both the terms? No. Because they only appear on the left side. What about the 3? Is that common? Well, no. Because that only appears on the right term. The one thing that pops up in both is the 2, so that is the greatest common factor. That leads us to now the second step. What do you do with this 2? Well, basically, what we're going to do is we're basically just going to move it and extract it outside of the parentheses and kind of just remove it from this whole entire expression, and then we're going to leave everything inside that we had from step 1. So here's how this works. I take the 2 and I pull it to the outside of parentheses. And then what was left over?

What was left over were the 2 factors of x and then one factor of 3. So, in other words, what I've done here is I've done 2, and then I have x2+3. That was what was left over on the inside, and this was my greatest common factor. One easy way to check this, just in case you ever worry that you didn't do it correctly, is if you do the distributive property, you should basically just get back to your original expression. That's basically what the GCF is. It's like the opposite of the distributive property. Alright? That's all there is to it. Let's take a look at a couple more examples here.

So I have 7x2 and 5x. Here what happens is I'm going to write down the factor tree again. So I have 7x2 and 5x. This just becomes 7 times x times x, and this just becomes 5 times x. Now I keep a minus sign over here, and then I just put a big parenthesis.

Can I break down the 7 or 5 any further? Well, actually, I can't because 7, the only two things that multiply 7 are 7 and 1. Same thing with 5. Only things that multiply are 5 and 1. So these are all prime factors. What are the common items between the two? Well, it's not 7 and 5, but I do see one power of x that's between both of them. Why is it x? Well, it's basically just because there's only x in the left term, but there's only one power of x in the right term. So the thing that's the largest thing that's common between both of them is just one power of x.

So now what do I do with this x? I pick it up, and I basically just move it to the outside of the expression over here. So I write an x, and then I have a parenthesis. And then what was left over in my expression? I have a 7x and then a 5, and then you always have to include the sign that was in here. So this just becomes 7x-5. If you distribute this, you should get back to your original expression.

Now let's take a look at the last one over here. So this, 8x this negative 8x3 plus 16x. Alright. So let's do the factorization of this, and let's do the factorization of this. Now one of the things you actually can notice here is that 8 and 16 are multiples of each other. We didn't have to worry about those two in the first two examples, but 8 is just a multiple of 16. So one of the shortcuts that you can use is instead of having to factor completely,