Video Transcript: Factoring Quadratics - Part 2

Intro to grouping 

In this video, I want to focus on a few more techniques for factoring polynomials. And in  particular, I want to focus on quadratics that don't have a 1 as the leading coefficient. For  example, if I wanted to factor 4x squared plus 25x minus 21. Everything we've factored so far, or all of the quadratics we've factored so far, had either a 1 or negative 1 where this 4 is sitting. All  of a sudden now, we have this 4 here. So what I'm going to teach you is a technique called,  factoring by grouping. And it's a little bit more involved than what we've learned before, but it's a neat trick. To some degree, it'll become obsolete once you learn the quadratic formula,  because, frankly, the quadratic formula is a lot easier. But this is how it goes. I'll show you the  technique. And then at the end of this video, I'll actually show you why it works. So what we  need to do here, is we need to think of two numbers, a and b, where a times b is equal 4 times  negative 21. So a times b is going to be equal to 4 times negative 21, which is equal to negative  84. And those same two numbers, a plus b, need to be equal to 25. Let me be very clear. This is  the 25, so they need to be equal to 25. This is where the 4 is. So we go, 4 times negative 21.  That's a negative 21. So what two numbers are there that would do this? Well, we have to look  at the factors of negative 84. And once again, one of these are going to have to be positive. The other ones are going to have to be negative, because their product is negative. So let's think  about the different factors that might work. 4 and negative 21 look tantalizing, but when you  add them, you get negative 17. Or, if you had negative 4 and 21, you'd get positive 17. Doesn't  work. Let's try some other combinations. 1 and 84, too far apart when you take their  difference. Because that's essentially what you're going to do, if one is negative and one is  positive. Too far apart. Let's see you could do 3-- I'm jumping the gun. 2 and 42. Once again, too far apart. Negative 2 plus 42 is 40. 2 plus negative 42 is negative 40-- too far apart. 3 and-- Let's  see, 3 goes into 84-- 3 goes into 8 2 times. 2 times 3 is 6. 8 minus 6 is 2. Bring down the 4. Goes  exactly 8 times. So 3 and 28. This seems interesting. And remember, one of these has to be  negative. So if we have negative 3 plus 28, that is equal to 25. Now, we've found our two  numbers. But it's not going to be quite as simple of an operation as what we did when this was  a 1 or negative 1. What we're going to do now is split up this term right here. We're going to  split it up into positive 28x minus 3x. We're just going to split that term. That term is that term  right there. And of course, you have your minus 21 there, and you have your 4x squared over  here. Now, you might say, how did you pick the 28 to go here, and the negative 3 to go there?  And it actually does matter. The way I thought about it is 3 or negative 3, and 21 or negative 21 , they have some common factors. In particular, they have the factor 3 in common. And 28 and 4  have some common factors. So I grouped the 28 on the side of the 4. And you're going to see  what I mean in a second. If we, literally, group these so that term becomes 4x squared plus 28x.  And then, this side, over here in pink, it's plus negative 3x minus 21. Once again, I picked these.  I grouped the negative 3 with the 21, or the negative 21, because they're both divisible by 3.  And I grouped the 28 with the 4, because they're both divisible by 4. And now, in each of these  groups, we factor as much out as we can. So both of these terms are divisible by 4x. So this  orange term is equal to 4x times x-- 4x squared divided by 4x is just x-- plus 28x divided by 4x is  just 7. Now, this second term. Remember, you factor out everything that you can factor out.  Well, both of these terms are divisible by 3 or negative 3. So let's factor out a negative 3. And  this becomes x plus 7. And now, something might pop out at you. We have x plus 7 times 4x 

plus, x plus 7 times negative 3. So we can factor out an x plus 7. This might not be completely  obvious. You're probably not used to factoring out an entire binomial. But you could view this  could be like a. Or if you have 4xa minus 3a, you would be able to factor out an a. And I can just  leave this as a minus sign. Let me delete this plus right here. Because it's just minus 3, right?  Plus negative 3, same thing as minus 3. So what can we do here? We have an x plus 7, times 4x.  We have an x plus 7, times negative 3. Let's factor out the x plus 7. We get x plus 7, times 4x  minus 3. Minus that 3 right there. And we've factored our binomial. Sorry, we've factored our  quadratic by grouping. And we factored it into two binomials. Let's do another example of that, because it's a little bit involved. But once you get the hang of it's kind of fun. So let's say we  want to factor 6x squared plus 7x plus 1. Same drill. We want to find a times b that is equal to 1  times 6, which is equal to 6. And we want to find an a plus b needs to be equal to 7. This is a  little bit more straightforward. What are the-- well, the obvious one is 1 and 6, right? 1 times 6  is 6. 1 plus 6 is 7. So we have a is equal to 1. Or let me not even assign them. The numbers here  are 1 and 6. Now, we want to split this into a 1x and a 6x. But we want to group it so it's on the  side of something that it shares a factor with. So we're going to have a 6x squar ed here, plus--  and so I'm going to put the 6x first because 6 and 6 share a factor. And then, we're going to  have plus 1x, right? 6x plus 1x equals 7x . That was the whole point. They had to add up to 7 .  And then we have the final plus 1 there. Now, in each of these groups, we can factor out as  much as we like. So in this first group, let's factor out a 6x. So this first group becomes 6x  times-- 6x squar ed divided by 6x is just an x. 6x divided by 6x is just a 1. And then, the second  group-- we're going to have a plus here. But this second group, we just literally have a x plus 1.  Or we could even write a 1 times an x plus 1. You could imagine I just factored out of 1 so to  speak. Now, I have 6x times x plus 1, plus 1 times x plus 1. Well, I can factor out the x plus 1. If I  factor out an x plus 1, that's equal to x plus 1 times 6x plus that 1. I'm just doing the distributive  property in reverse. So hopefully you didn't find that too bad. And now, I'm going to actually  explain why this little magical system actually works. Let me take an example. I'll do it in very  general terms. Let's say I had ax plus b, times cx-- actually, I'm afraid to use the a's and b's. I  think that'll confuse you, because I use a's and b's here. They won't be the same thing. So let  me use completely different letters. Let's say I have fx plus g, times hx plus, I'll use j instead of i.  You'll learn in the future why don't like using i as a variable. So what is this going to be equal to? Well, it's going to be fx times hx which is fhx. And then, fx times j. So plus fjx. And then, we're  going to have g times hx. So plus ghx. And then g times j. Plus gj. Or, if we add these two  middle terms, you have fh times x, plus-- add these two terms-- fj plus gh x. Plus gj. Now, what  did I do here? Well, remember, in all of these problems where you have a non-1 or non-negative 1 coefficient here, we look for two numbers that add up to this, whose product is equal to the  product of that times that. Well, here we have two numbers that add up-- let's say that a is  equal to fj. That is a. And b is equal to gh. So a plus b is going to be equal to that middle  coefficient. And then what is a times b? a times b is going to be equal to fj times gh. We could  just reorder these terms. We're just multiplying a bunch of terms. So that could be rewritten as  f times h times g times j. These are all the same things. Well, what is fh times gj? This is equal to  fh times gj. Well, this is equal to the first coefficient times the constant term. So a plus b will be  equal to the middle coefficient. And a times b will equal the first coefficient times the constant  term. So that's why this whole factoring by grouping even works, or how we're able to figure 

out what a and b even are. Now, I'm going to close up with something slightly different, but just to make sure that you have a well-rounded education in factoring things. What I want to do is  to teach you to factor things a little bit more completely. And this is a little bit of a add-on. I was going to make a whole video on this. But I think, on some level, it might be a little obvious for  you. So let's say we had-- let me get a good one here. Let's say we had negative x to the third,  plus 17x squared, minus 70x. Immediately, you say, gee, this isn't even a quadratic. I don't know  how to solve something like this. It has an x to third power. And the first thing you should  realize is that every term here is divisible by x. So let's factor out an x. Or even better, let's  factor out a negative x. So if you factor out a negative x, this is equal to negative x times--  negative x to the third divided by negative x is x squared. 17x squared divided by negative x is  negative 17x. Negative 70x divided by negative x is positive 70. The x's cancel out. And now, you have something that might look a little bit familiar. We have just a standard quadratic where  the leading coefficient is a 1. We just have to find two numbers whose product is 70, and that  add up to negative 17. And the numbers that immediately jumped into my head are negative 10 and negative 7. You take their product, you get 70. You add them up, you get negative 17. So  this part right here is going to be x minus 10, times x minus 7. And of course, you have that  leading negative x. The general idea here is just see if there's anything you can factor out. And  that'll get it into a form that you might recognize. Hopefully, you found this helpful. I want to  reiterate what I showed you at the beginning of this video. I think it's a really cool trick, so to  speak, to be able to factor things that have a non-1 or non-negative 1 leading coefficient. But to some degree, you're going to find out easier ways to do this, especially with the quadratic  formula, in not too long. 

Factoring quadratics by grouping 

We're asked to factor 4y squared plus 4y, minus 15. And whenever you have an expression like  this, where you have a non-one coefficient on the y squared, or on the second degree term-- it  could have been an x squared-- the best way to do this is by grouping. And to factor by  grouping we need to look for two numbers whose product is equal to 4 times negative 15. So  we're looking for two numbers whose product-- let's call those a and b-- is going to be equal to  4 times negative 15, or negative 60. And the sum of those two numbers, a plus b, needs to be  equal to this 4 right there. So let's think about all the factors of negative 60, or 60. And we're  looking for ones that are essentially 4 apart, because the numbers are going to be of different  signs, because their product is negative, so when you take two numbers of different signs and  you sum them, you kind of view it as the difference of their absolute values. If that confuses  you, don't worry about it. But this tells you that the numbers, since they're going to be of  different size, their absolute values are going to be roughly 4 apart. So we could try out things  like 5 and 12, 5 and negative 12, because one has to be negative. If you add these two you get  negative 7, if you did negative 5 and 12 you'd get positive 7. They're just still too far apart. What  if we tried 6 and negative 10? Then you get a negative 4, if you added these two. But we want a  positive 4, so let's do negative 6 and 10. Negative 6 plus 10 is positive 4. So those will be our  two numbers, negative 6 and positive 10. Now, what we want to do is we want to break up this  middle term here. The whole point of figuring out the negative 6 and the 10 is to break up the  4y into a negative 6y and a 10y. So let's do that. So this 4y can be rewritten as negative 6y plus 

10y, right? Because if you add those you get 4y. And then the other sides of it, you have your 4y squared, your 4y squared and then you have your minus 15. All I did is expand this into these  two numbers as being the coefficients on the y. If you add these, you get the 4y again. Now,  this is where the grouping comes in. You group the term. Let me do it in a different color. So if I  take these two guys, what can I factor out of those two guys? Well, there's a common factor, it  looks like there's a common factor of 2y. So if we factor out 2y, we get 2y times 4y squared,  divided by 2y is 2y. And then negative 6y divided by 2y is negative 3. So this group gets factored into 2y times 2y, minus 3. Now, let's look at this other group right here. This was the whole  point about breaking it up like this. And in other videos I've explained why this works. Now  here, the greatest common factor is a 5. So we can factor out a 5, so this is equal to plus 5 times  10y, divided by 5 is 2y. Negative 15 divided by five is 3. And so we have 2y times 2y minus 3, plus 5 times 2y minus 3. So now you have two terms, and 2y minus 3 is a common factor to both. So  let's factor out a 2y minus 3, so this is equal to 2y minus 3, times 2y, times that 2y, plus that 5.  There's no magic happening here, all I did is undistribute the 2y minus 3. I factored it out of  both of these guys. I took it out of the parentheses. If I distribute it in, you'd get back to this  expression. But we're done, we factored it. We factored it into two binomial expressions. 4y  squared plus 4y, minus 15 is 2y minus 3, times 2y plus 5. 

Factoring quadratics: common factor + grouping 

We're asked to factor 35k squared plus 100k, minus 15. And because we have a non-1  coefficient out here, the best thing to do is probably to factor this by grouping. But before we  even do that, let's see if there's a common factor across all of these terms, and maybe we can  get a 1 coefficient, out there. If we can't get a 1 coefficient, we'll at least have a lower  coefficient here. And if we look at all of these numbers, they all look divisible by 5. In fact their  greatest common factor is 5. So let's at least factor out a 5. So this is equal to 5 times-- 35k  squared divided by 5 is 7k squared. 100k divided by 5 is 20k. And then negative 15 divided by 5 is negative 3. So we were able to factor out a 5, but we still don't have a 1 coefficient here, so  we're still going to have to factor by grouping. But at least the numbers here are smaller so it'll  be easier to think about it in terms of finding numbers whose product is equal to 7 times  negative 3, and whose sum is equal to 20. So let's think about that. Let's figure out two  numbers that if I were to add them, or even better if I were to take their product, I get 7 times  negative 3, which is equal to negative 21. And if I were to take their sum, if I add those two  numbers, it needs to be equal to 20. Now, once again, because their product is a negative  number, that means they have to be of different signs, so when you add numbers of different  signs, you could view it as you're taking the difference of the positive versions. So the  difference between the positive versions of the number has to be 20. So the number that  immediately jumps out is we're probably going to be dealing with 20 and 21, and 1 will be the  negative, because we want to get to a positive 20. So let's think about it. So if we think of 20  and negative 1, their product is negative 21. Sorry. If we take 21 and negative 1, their product is  negative 21. 21 times negative 1 is negative 21. and if you take their sum, 21 plus negative 1,  that is equal to 20. So these two numbers right there fit the bill. Now, let's break up this 20k  right here into a 21k and a negative 1k. So let's do that. So let's rewrite the whole thing. We  have 5 times 7k squared, and I'm going to break this 20k into a-- let me do it in this color right 

here-- I'm going to break that 20k into a plus 21k, minus k. Or you could say minus 1k if you  want. I'm using those two factors to break it up. And then we finally have the minus 3 right  there. Now, the whole point of doing that is so that we can now factor each of the two groups.  This could be our first group right here. And so what can we factor out of that group right  there? Well, both of these are divisible by 7k, so we can write this as 7k times-- 7k squared  divided by 7k, you're just going to have a k left over. And then plus 21k divided by 7k is just  going to be a 3. So that factors into that. And then we can look at this group right here. They  have a common factor. Well, we can factor out a negative 1 if we like, so this is equal to  negative 1 times-- k divided by negative 1 is k. Negative 3 divided by negative 1 is positive 3.  And, of course, we have this 5 sitting out there. Now, ignoring that 5 for a second, you see that  both of these inside terms have k plus 3 as a factor. So we can factor that out. So let's ignore  this 5 for a second. This inside part right here, the stuff that's inside the parentheses, we can  factor k plus 3 out, and it becomes k plus 3, times k plus 3, times 7k minus 1. And if this seems a  little bizarre to you, distribute the k plus 3 on to this. K plus 3 times 7k is that term, k plus 3  times negative 1 is that term. And, of course, the whole time you have that 5 sitting outside.  You have that 5. We don't even have to put parentheses there. 5 times k plus 3, times 7k minus  1. And we factored it, we're done. 

Factoring quadratics: negative common factor + grouping 

We need to factor negative 12f squared minus 38f, plus 22. So a good place to start is just to see if, is there any common factor for all three of these terms? When we look at them, they're all  even. And we don't like a negative number out here. So let's divide everything, or let's factor  out a negative 2. So this expression right here is the same thing as negative 2 times-- what's  negative 12f squared divided by negative 2? It's positive 6f squared. Negative 38 divided by  negative 2 is positive 19, so it'll be positive 19f. And then 22 divided by negative 22-- oh, sorry,  22 divided by negative 2 is negative 11. So we've simplified it a bit. We have the 6f squared plus  19f, minus 11. We'll just focus on that part right now. And the best way to factor this thing, since we don't have a 1 here as the coefficient on the f squared, is to factor it by grouping. So we  need to look for two numbers whose product is 6 times negative 11. So two numbers, so a  times b, needs to be equal to 6 times negative 11, or negative 66. And a plus b needs to be  equal to 19. So let's try a few numbers here. So let's see, 22, I'm just thinking of numbers that  are roughly 19 apart, because they're going to be of different signs. So 22 and 3, I think will  work. Right. If we take 22 times negative 3, that is negative 66, and 22 plus negative 3 is equal  to 19. And the way I kind of got pretty close to this number is, well, you know, they're going to  be of different signs, so the positive versions of them have to be about 19 apart, and that  worked out. 22 and negative 3. So now we can rewrite this 19f right here as the sum of negative 3f and 22f. That's the same thing as 19f. I just kind of broke it apart. And, of course, we have the  6f squared and we have the minus 11 here. Now, you're probably saying, hey Sal, why did you  put the 22 here and the negative 3 there? Why didn't you do it the other way around? Why  didn't you put the 22 and then the negative 3 there? And my main motivation for doing it, I like  to put the negative 3 on the same side with the 6 because they have the common factor of the  3. I like to put the 22 with the negative 11, they have the same common factor of 11. So that's  why I decided to do it that way. So now let's do the grouping. And, of course, you can't forget 

this negative 2 that we have sitting out here the whole time. So let me put that negative 2 out  there, but that'll just kind of hang out for awhile. But let's do some grouping. So let's group  these first two. And then we're going to group this-- let me get a nice color here-- and then  we're going to group this second two. Well, that's almost an identical color. Let me do it in this  purple color. And then we can group that second two right there. So these first two, we could  factor out a negative 3f, so it's negative 3f times-- 6f squared divided by negative 3f is negative  2f. And then negative 3f divided by negative 3f is just positive f. Actually, a better way to start,  instead of factoring out a negative 3f, let's just factor out 3f, so we don't have a negative out  here. We could do it either way. But if we just factor out a 3f, 6f squared divided by 3f is 2f. And  then negative 3f divided by 3f is negative 1. So that's what that factors into. And then that  second part, in that dark purple color, can factor out an 11. And if we factor that out, 22f divided by 11 is 2f, and negative 11 divided by 11 is negative 1. And, of course, once again, you have that negative 2 hanging out there. Now, inside the parentheses, we have two terms, both of which  have 2f minus 1 as a factor. So we can factor that out. This whole thing is just an exercise in  doing the reverse distributive property, if you will. So let's factor that out, so you have 2f minus  1, times this 3f, and then times that plus 11. Let me do that in the same shade of purple right  over there. And you know, you can distribute it if you like. 2f minus 1 times 3f will give you this  term, 2f f minus 1 times 11 will give you that term. And we can't forget that we still have that  negative 2 hanging out outside. I want to change the colors on it. And we're done factoring it.  Negative 12f squared minus 38f, plus 22 is negative 2 times 2f minus 1, times 3f plus 11.