Factoring by Grouping - Negative Sign
In order to factor by grouping with a negative sign in the middle, meaning this symbol right here, we still have to group the first two terms and the last two terms together. The most important thing is (remember from the last video "factoring by grouping"), we have to have a plus sign there. That's what we want because otherwise, it is going to end up looking like this: some people are going to put parentheses around the first two terms and the last two terms. Remember what we started off with. We started off with this negative being out here and this positive 5 without the parentheses, but when we put parentheses around it, it means that (let's just write this down real quick) 2x cubed minus x squared minus 10x minus 5. Of course, that's not what we started with. So, this would be an improper case of factoring. Let's go ahead and clear this and people are like, "Well, I guess I can go ahead and include the negative sign. So, I'll do that", but from this point, now it looks like a multiplication problem. If you are on a test or a quiz, what's going to happen is that you are going to FOIL this out and it will look horrible. It's definitely not going to look like what you started off with. So, let's talk about the proper way of doing this to maintain everything in terms of quantity and what we started off with. So, let's clear this and let's rewrite it with the plus sign in there. We have the 2x cubed minus x squared, plus, and then we want to write negative 10x plus 5. Now, notice. There is a plus there which makes this a lot easier to handle. So, now, we can go ahead and group the first two together and then the last two together. Notice, we didn't change any numbers around and it definitely doesn't look like a multiplication problem. Now, what we're going to do is go ahead and factor each grouping. With this first grouping here, it looks like we have an x squared in common. We've got 2x minus 1. Then bring down our plus sign. And in this term, we actually have a negative 5 in common. So what we want to have a negative 5 out here. The reason why we pull out a negative 5, I will get to in a minute, but if we have negative 5 then that means we've got a 2x left over and a minus 1. So, of course, if you want to check yourself about this, you can always multiply that back out, and you should get this answer. So, with this whole idea of plus minus what we're going to do is just change that to a negative. We have x squared times 2x minus 1 minus 5 times 2x minus 1. Now, what we need to do here is we have these terms. All we have to do is notice that 2x minus 1 is actually the greatest common factor. So, let's take out the 2x minus 1, and what we have left over is the x squared and this negative 5. So this is fully factored. Now let's go back to this idea of "Why did we factor out this negative 5 here and not a positive 5?" What would happen if we'd factor that out as a positive 5, this is what it would have looked like: 2x squared times 2x minus 1 plus 5 times negative 2x plus 1. In order to factor the rest of this, these two guys have to be the same. Although, they don't look the same. But, they would look the same if we went ahead and did the extra step of factoring out that negative as well. Then, we would have the negative 5 times 2x minus 1. So, remember. You always want to get these two guys to be exactly the same so that you can factor it out.