Understanding Factor by Grouping Techniques

In this video, we're going to work on a factoring technique called factor by grouping. This isn't a super common technique, but it can come in handy. Depending on how your teacher teaches you how to factor trinomials, you may end up using factor by grouping to do that. So let me start with a pretty easy example here so we can just go through the steps. And factor by grouping usually comes in most handy when you have four turns. Although it can be used for other scenarios such as six terms. Yeah, six terms is another, or five. Anyway, most of the time it's four terms. I could maybe show you one that has more than four terms. But let's start with this and see how far we get. Okay, so we have these four terms. And if you've done anything with factoring, you know the first thing you always want to do is look for the greatest common factor. And if I look at these four terms, I see some twos, but it's not in all the terms. I see a bunch of Ms, but it's not in the last term. So none of these terms have a common factor, or there is no factor that is common to all of these terms. So I can't take out a greatest common factor. Alright, this technique of factor by grouping, the first thing that you're going to do is actually group some of the terms together. And usually if there's four terms, you're going to group the first two and the last two terms together and leave the plus sign in the middle there. Now if this is a minus sign right here, we're going to have a little bump in the road we're going to have to talk about. We'll do that in a minute. Okay, so the first step is to group the first two and the last two terms together. And then we'll just look at the first two terms and say, do they have a common factor? Just the first two terms. I got an m squared and a 2m. Well, they have a common factor of m, so I can take that m out just of these first two terms here. And what am I left with? I'm left with m plus 2. And remember, you can check that by distributing. m times m is m squared, and m times 2 is 2m, so I'm good. I'm going to bring my plus down. Now I'm going to look at this second set of parentheses here. Is there a common factor in this second set of parentheses? M times n and 2 times n. Well, they both have a factor of n, so I can pull that out. And what am I left with? I'm left with m plus 2. Now, here's how you know factor by grouping is going to work. Factor by grouping is going to work if, after you take the greatest common factor out of your groupings, what's left in the parentheses is exactly the same. If that's exactly the same, I know this factor by grouping technique is going to work. Now if I just show you the next step, it's probably going to be kind of confusing, so I'm going to build up to it a little bit, okay? So bear with me here for a minute. Let's say we have m times x plus n times x, okay? And I wanted to take out a common factor. What would the common factor be there? Well, hopefully you see it would be x, right? I could take that x out, and I'd be left with m plus n. Right? Everybody comfortable with that? Great. Okay, let's take it a step further. Let's say this isn't an X. Let's say this is a baseball. That sounds like fun and something I can draw. Let's say this is M times a baseball. A smiley face might have been more fun. And N times a baseball. Does that look like a baseball? Alright, what do they have in common? They have a baseball in common, so we can factor that out. I know, you're starting to think I'm a little crazy right now. Okay, let's get rid of this baseball. Let's say instead of a baseball, we have, oh, I don't know, how about m times, oh, some m plus 2 thing. And then I have n times this m plus 2 thing. They have a common factor of m plus 2. So I could take that m plus 2 out. And that's what's happening up here. That's how you know that you can use factor by grouping. You need to look at this as like two things multiplied together plus two things multiplied together and say, well, this m plus 2 factor is the same. So I can literally factor out this m plus 2 and I'm going to underline it twice in orange just to highlight that. And what am I left with? I'm left with m plus 2. N. Okay? Hopefully you can see that, that we're taking out this greatest common factor here, and we're moving it in front, and we're left with, oh, let's use green, we're left with M. Plus n, and that becomes this factor right here, what you're left with. Okay, so the keys, group the first two, group the last two. Take out the greatest common factor in each of your groupings. What you have left in the parentheses should be exactly the same. If it's not exactly the same, factor by grouping will not work for you. You'll have to try something else. Okay, whatever that, if that's... thing is in the parentheses exactly the same, whatever that thing is, you can pull that out in front, take it, factor it out, and then write what you have left as your second factor. So you can actually check this. This is kind of cool. Let's take our answer, which was m plus 2 times m plus n, and we could actually check this by multiplying it out. Let's do that. If we multiply it out, we should end up with what we started with. Let's see. m times m is m squared, m times n is mn, 2 times m, and 2 times n, and that's basically what we started with right here, just rearrange the terms to look like that. That's how you can check it and see if it's right. Alright, let's have you try one, and then we really need to talk about what to do if there's a minus in there, but let's try one more of these that you can try before we go to a tougher one. Oh, come on. Let's try y squared minus 6y plus yw minus 6w. So maybe pause it and try that one real quick and then start the video again. Alright, first step, factor by grouping. Group the first two. Group the last two. Now we're going to take out the greatest common factor of each set of parentheses. So here is a y, and I'm left with y minus 6. And in the second parentheses, the common factor is a w, so I'm left with a y minus 6. And I'm very happy because this y minus 6 factor is exactly the same. So I'm going to pull that y minus 6 factor out front. Underline it in orange to highlight that fact. And what am I left with when I pull that out? If I take out that y minus 6, I'm going to be left with this y plus w. And that's my answer. y minus 6 times y plus w. If I want to check that, I can multiply it out. y times y is y squared. y times w is wy. I just put them in alphabetical order. 6 times y, negative 6 times y, and negative 6 times w. And that is what we started with. If you rearrange the terms, you get the same thing. And I have yw instead of wy, but that's the same thing because of the commutative property. All right, let's try one where you have the... minus between the two terms and let me show you what you have to watch out for. This is the big, this is the kind of pothole in these problems. Okay, let's say we have 7y minus 9x minus 3xy plus 21. Alright, so if I just follow the guidelines that I gave at the beginning and group the first two and group the last two, can you see the problem there? Can you see what we did by putting in those parentheses? We actually changed the problem because if I put these parentheses in with a minus in front, that minus sign is now going to apply to each term in the parentheses and it changed my positive 21 to a negative 21, okay? I don't want to do that because that is not what... the problem says. The problem is positive 20. So how do I deal with this? You have to have a plus between your parentheses. So what you do to deal with it, we know that subtraction is the same as adding a negative. So I'm just going to change this to plus a negative 3 and then put my parentheses in. You know what? I'm looking at this problem here and This is an interesting problem. I'm glad I picked this problem. This is a good one. So let's see what happens here. What do I have in this first set of parentheses as a common factor? I don't have one. Alright, so I'll just write 1, and that leaves me 7y minus 9x. Alright, how about this second set of parentheses? Well, I got a 3 and a 21, so 3 goes into both of those. And you could either take out a 3 or a negative 3. Let's just practice taking out a negative 3. So if I took out a negative 3, I'd be left with x, y, and then positive 21 divided by negative 3 is negative 7. Oh, no. Look at this. These are not the same. That's no good. That means my factor by grouping didn't work. Alright, back to the drawing board. Let's go back to the original problem and see what could we do. This was a minus 3xy, right? What could we do to maybe make this work? Well, we grouped them the first two and the last two, because that's how I gave it to you. You can rearrange these terms if you want, right? You can rearrange them in any order you want and then try grouping them again. So let's see. Maybe I like the 3 and the 9 together and the 7 and the 21 together. Now, if I wasn't wanting to work you a little bit harder, I would put the negative 9x and the negative 3y first, so I wouldn't have to worry about this minus in the middle. But let's just practice that minus in the middle. Just in case. Alright, so I'm going to rearrange these, and I'm going to put the 21 up here with the 7y, like this. Is that going to make a difference? Is that going to make a difference? Let's try it. So now, what if I group the first two together? Alright, remember I can't have a minus here when I group, so I'm going to change that to plus a negative and group. Now let's see what happens. We have a common factor here of 7. What are we left with? y plus 3. Okay, let's see what happens here. What is our common factor in the second parentheses? Well, we have a negative 9 and a negative 3. Obviously, 3 goes into both of those. Remember, we're trying to get this second set of parentheses to look like this first set of parentheses. So, I'd really like everything to be positive. Let's take out a negative 3. We can divide negative 9 by negative 3 and negative 3 by negative 3. And what else do they have in common? They both have an x term, right? So we can take that x out. Okay, what are we left with? What's negative 9 divided by negative 3? That's positive 3. And then the x came out. Remember, you can distribute here to see that negative 3x times 3 is negative 9x. So we're good. Negative 3 divided by negative 3 is positive 1. You could write the 1 if you want. And the x came out and so we're just left with a y. Everybody see that? So you could take negative 3x times y and end up with negative 3xy. Well look at that! That's so much better. Now we have the y plus 3 and 3 plus y. Those are the same, right? Let me get rid of this part up here just so we have a little bit of room. And we'll rewrite this up here and we'll finish. So now I've got 7 times y plus 3 minus 3x, I'm just copying this, times, and instead of writing 3 plus y, I'm going to write y plus 3, just to get those looking the same. Those are the same, so I can take that out, y plus 3, and I'm going to be left with a 7 minus 3x. So in the second parentheses, I'm going to have 7 minus 3x. And that is my answer. And if you foiled that out, you would end up with what we started with. So that problem was kind of good. It showed us a couple things that can happen when you're doing factor by grouping, which is, number one, you could have a negative or a minus sign. and you in between your two parentheses and you have to fix that by changing it to plus and negative making sure that you've got this plus sign between your two parentheses and then right here and then the second thing that can happen is sometimes the order that the terms are given in don't produce factor by grouping right they don't produce the scenario that works in factor by grouping and you can fix that by rearranging the terms. Alright so that's a little introduction to factor by grouping. 15 minutes is long enough so if I do different kinds of factor by grouping, I'll save that for a different video. I hope that helped.

So let me start with a pretty easy example here so we can just go through the steps. And factor by grouping usually comes in most handy when you have four turns. Although it can be used for other scenarios such as six terms.

Yeah, six terms is another, or five. Anyway, most of the time it's four terms. I could maybe show you one that has more than four terms.

But let's start with this and see how far we get. Okay, so we have these four terms. And if you've done anything with factoring, you know the first thing you always want to do is look for the greatest common factor.

And if I look at these four terms, I see some twos, but it's not in all the terms. I see a bunch of Ms, but it's not in the last term. So none of these terms have a common factor, or there is no factor that is common to all of these terms. So I can't take out a greatest common factor. Alright, this technique of factor by grouping, the first thing that you're going to do is actually group some of the terms together.

And usually if there's four terms, you're going to group the first two and the last two terms together and leave the plus sign in the middle there. Now if this is a minus sign right here, we're going to have a little bump in the road we're going to have to talk about. We'll do that in a minute.

Okay, so the first step is to group the first two and the last two terms together. And then we'll just look at the first two terms and say, do they have a common factor? Just the first two terms. I got an m squared and a 2m. Well, they have a common factor of m, so I can take that m out just of these first two terms here.

And what am I left with? I'm left with m plus 2. And remember, you can check that by distributing. m times m is m squared, and m times 2 is 2m, so I'm good.

I'm going to bring my plus down. Now I'm going to look at this second set of parentheses here. Is there a common factor in this second set of parentheses? M times n and 2 times n. Well, they both have a factor of n, so I can pull that out.

And what am I left with? I'm left with m plus 2. Now, here's how you know factor by grouping is going to work. Factor by grouping is going to work if, after you take the greatest common factor out of your groupings, what's left in the parentheses is exactly the same. If that's exactly the same, I know this factor by grouping technique is going to work.

Now if I just show you the next step, it's probably going to be kind of confusing, so I'm going to build up to it a little bit, okay? So bear with me here for a minute. Let's say we have m times x plus n times x, okay? And I wanted to take out a common factor.

What would the common factor be there? Well, hopefully you see it would be x, right? I could take that x out, and I'd be left with m plus n. Right?

Everybody comfortable with that? Great. Okay, let's take it a step further. Let's say this isn't an X.

Let's say this is a baseball. That sounds like fun and something I can draw. Let's say this is M times a baseball. A smiley face might have been more fun. And N times a baseball.

Does that look like a baseball? Alright, what do they have in common? They have a baseball in common, so we can factor that out. I know, you're starting to think I'm a little crazy right now. Okay, let's get rid of this baseball.

Let's say instead of a baseball, we have, oh, I don't know, how about m times, oh, some m plus 2 thing. And then I have n times this m plus 2 thing. They have a common factor of m plus 2. So I could take that m plus 2 out.

And that's what's happening up here. That's how you know that you can use factor by grouping. You need to look at this as like two things multiplied together plus two things multiplied together and say, well, this m plus 2 factor is the same. So I can literally factor out this m plus 2 and I'm going to underline it twice in orange just to highlight that. And what am I left with?

I'm left with m plus 2. N. Okay? Hopefully you can see that, that we're taking out this greatest common factor here, and we're moving it in front, and we're left with, oh, let's use green, we're left with M. Plus n, and that becomes this factor right here, what you're left with. Okay, so the keys, group the first two, group the last two.

Take out the greatest common factor in each of your groupings. What you have left in the parentheses should be exactly the same. If it's not exactly the same, factor by grouping will not work for you.

You'll have to try something else. Okay, whatever that, if that's... thing is in the parentheses exactly the same, whatever that thing is, you can pull that out in front, take it, factor it out, and then write what you have left as your second factor.

So you can actually check this. This is kind of cool. Let's take our answer, which was m plus 2 times m plus n, and we could actually check this by multiplying it out.

Let's do that. If we multiply it out, we should end up with what we started with. Let's see. m times m is m squared, m times n is mn, 2 times m, and 2 times n, and that's basically what we started with right here, just rearrange the terms to look like that. That's how you can check it and see if it's right.

Alright, let's have you try one, and then we really need to talk about what to do if there's a minus in there, but let's try one more of these that you can try before we go to a tougher one. Oh, come on. Let's try y squared minus 6y plus yw minus 6w.

So maybe pause it and try that one real quick and then start the video again. Alright, first step, factor by grouping. Group the first two.

Group the last two. Now we're going to take out the greatest common factor of each set of parentheses. So here is a y, and I'm left with y minus 6. And in the second parentheses, the common factor is a w, so I'm left with a y minus 6. And I'm very happy because this y minus 6 factor is exactly the same. So I'm going to pull that y minus 6 factor out front. Underline it in orange to highlight that fact.

And what am I left with when I pull that out? If I take out that y minus 6, I'm going to be left with this y plus w. And that's my answer. y minus 6 times y plus w.

If I want to check that, I can multiply it out. y times y is y squared. y times w is wy.

I just put them in alphabetical order. 6 times y, negative 6 times y, and negative 6 times w. And that is what we started with. If you rearrange the terms, you get the same thing.

And I have yw instead of wy, but that's the same thing because of the commutative property. All right, let's try one where you have the... minus between the two terms and let me show you what you have to watch out for.

This is the big, this is the kind of pothole in these problems. Okay, let's say we have 7y minus 9x minus 3xy plus 21. Alright, so if I just follow the guidelines that I gave at the beginning and group the first two and group the last two, can you see the problem there? Can you see what we did by putting in those parentheses? We actually changed the problem because if I put these parentheses in with a minus in front, that minus sign is now going to apply to each term in the parentheses and it changed my positive 21 to a negative 21, okay?

I don't want to do that because that is not what... the problem says. The problem is positive 20. So how do I deal with this?

You have to have a plus between your parentheses. So what you do to deal with it, we know that subtraction is the same as adding a negative. So I'm just going to change this to plus a negative 3 and then put my parentheses in. You know what? I'm looking at this problem here and This is an interesting problem.

I'm glad I picked this problem. This is a good one. So let's see what happens here.

What do I have in this first set of parentheses as a common factor? I don't have one. Alright, so I'll just write 1, and that leaves me 7y minus 9x. Alright, how about this second set of parentheses?

Well, I got a 3 and a 21, so 3 goes into both of those. And you could either take out a 3 or a negative 3. Let's just practice taking out a negative 3. So if I took out a negative 3, I'd be left with x, y, and then positive 21 divided by negative 3 is negative 7. Oh, no. Look at this. These are not the same.

That's no good. That means my factor by grouping didn't work. Alright, back to the drawing board.

Let's go back to the original problem and see what could we do. This was a minus 3xy, right? What could we do to maybe make this work?

Well, we grouped them the first two and the last two, because that's how I gave it to you. You can rearrange these terms if you want, right? You can rearrange them in any order you want and then try grouping them again. So let's see. Maybe I like the 3 and the 9 together and the 7 and the 21 together.

Now, if I wasn't wanting to work you a little bit harder, I would put the negative 9x and the negative 3y first, so I wouldn't have to worry about this minus in the middle. But let's just practice that minus in the middle. Just in case.

Alright, so I'm going to rearrange these, and I'm going to put the 21 up here with the 7y, like this. Is that going to make a difference? Is that going to make a difference? Let's try it.

So now, what if I group the first two together? Alright, remember I can't have a minus here when I group, so I'm going to change that to plus a negative and group. Now let's see what happens.

We have a common factor here of 7. What are we left with? y plus 3. Okay, let's see what happens here. What is our common factor in the second parentheses?

Well, we have a negative 9 and a negative 3. Obviously, 3 goes into both of those. Remember, we're trying to get this second set of parentheses to look like this first set of parentheses. So, I'd really like everything to be positive. Let's take out a negative 3. We can divide negative 9 by negative 3 and negative 3 by negative 3. And what else do they have in common? They both have an x term, right?

So we can take that x out. Okay, what are we left with? What's negative 9 divided by negative 3?

That's positive 3. And then the x came out. Remember, you can distribute here to see that negative 3x times 3 is negative 9x. So we're good. Negative 3 divided by negative 3 is positive 1. You could write the 1 if you want.

And the x came out and so we're just left with a y. Everybody see that? So you could take negative 3x times y and end up with negative 3xy. Well look at that!

That's so much better. Now we have the y plus 3 and 3 plus y. Those are the same, right?

Let me get rid of this part up here just so we have a little bit of room. And we'll rewrite this up here and we'll finish. So now I've got 7 times y plus 3 minus 3x, I'm just copying this, times, and instead of writing 3 plus y, I'm going to write y plus 3, just to get those looking the same. Those are the same, so I can take that out, y plus 3, and I'm going to be left with a 7 minus 3x.

So in the second parentheses, I'm going to have 7 minus 3x. And that is my answer. And if you foiled that out, you would end up with what we started with. So that problem was kind of good.

It showed us a couple things that can happen when you're doing factor by grouping, which is, number one, you could have a negative or a minus sign. and you in between your two parentheses and you have to fix that by changing it to plus and negative making sure that you've got this plus sign between your two parentheses and then right here and then the second thing that can happen is sometimes the order that the terms are given in don't produce factor by grouping right they don't produce the scenario that works in factor by grouping and you can fix that by rearranging the terms. Alright so that's a little introduction to factor by grouping. 15 minutes is long enough so if I do different kinds of factor by grouping, I'll save that for a different video. I hope that helped.

Transcript for:Understanding Factor by Grouping Techniques

Transcript for:
Understanding Factor by Grouping Techniques