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Welcome to Algebra I: Factoring Binomials

31 minutes

Hi guys, welcome to Algebra 1. Today's lesson focuses on factoring binomials. Now you're going to have to think back and use your knowledge of greatest common factor to get through this lesson. You ready? Let's go. Let me take you back a bit so you can remember what it meant to be the greatest common factor, how to find the greatest common factor. It's all about knowing which numbers-- in this case 15, 18. I need to find out what are the factors of 18? What numbers can I multiply together to get 18-- I'm sorry, to get 15--and then do the same thing for 18. What numbers can I multiply together to get 18? And then I just look for the largest number that they have in common. Let me show you what I mean. So 15, I'm gonna come... right under here. The factors of 15-- What can I multiply together to get 15? You have to think back to your multiplication facts. I know I can use 1 and 15. 1 times 15 will give me 15. 15 is odd, so I can't use 2 to multiply by anything to get anything 15, but I can use 3. 3 times 5 is 15. As far as 15 goes, that's it for its factors. Now let's look at 18; we need to factor 18. What pairs of numbers could you multiply together to get 18? Well, 1 times 18, definitely-- that's always a pair of factors. One times anything always gives whatever number you start it with. It's even, so I know 2 would work. 2 times 9, that gives me 18. Will 3 times anything give you 18? It will. 3 times 6 will give you 18. As far as 18 goes, that's it. If you don't feel confident about your multiplication facts and you think you're leaving one out, just use your calculator and start trying things out. Or you can go back and get a multiplication table; you may have used that back in elementary school. Let that help you get your facts stronger. Eventually you won't need them at all, and you'll just remember what they are. Okay, so look back here. Since I've listed the factors of 15 and the factors of 18, I look for the largest number they have in common-- their greatest common factor. If you scan both sets, you'll notice it's 3. The greatest common factor-- often abbreviated as GCF-- for these two terms, it's 3. That's the biggest factor they have in common. Okay? Let's try another one. 30x and 10. You saw the x-- maybe it threw you, but just ignore the x for a second. Let's focus on the number part of this for a minute. Factors of 30, because I'm ignoring the x for a minute. I know that 1 times 30 will give me 30. It's even, so you start thinking about 2. 2 times 15, that'll give you 30. Okay? 3 times 10. There's actually one more pair-- 5 and 6. 5 times 6 will also give you 30. Then you come over here. Actually, let's bring that x back now. There's an x here; I'll teach you a little trick. I'm just going to write 1x is involved over here so that I know there's a factor of x that's included in that 30x term. That'll just help me keep that in my head. For the 10, I know 1 times 10, that'll give me 10. 2 times 5, that'll give me 10. That's about it as far as getting 10. Now we're at the point that we can scan our list and see what's the greatest common factor here. I see they both have a 1 in common, and they both have a 5 in common, and they both have a 10 in common. 10 as far as my number is concerned-- that's the greatest common factor so far. Remember I said ignore the x for a second and we'd come back to it? The 30x term, there's an x involved there. X is a factor of that term, but this is just a plain old 10; it's just a constant term. There's no variable involved. As far as this problem is concerned, the greatest common factor is just 10. I can't write 10x because that 10 didn't have an x attached to it. All right, let's keep going. I've got 22x squared and 11x, need the greatest common factor. I'm going to ignore the x squared for a second and just focus on that 22 to get started. I know 1 times 22 would give me 22. It's even, so 2 times something; in this case, 11 will give me 22. As far the 22 part is concerned, those are the only ways to get 22. Now let's look at the variable part of this--that x squared. I know that x squared means that I multiply x times x together to get x squared, so I can say this also has an x times x as a pair of factors. That 22x squared includes a pair of "x"s as its factors, that x and that x. So keep that in mind, move over here to the 11x. 11 is a prime number, so its only factors are 1 and itself. So 1 times 11, that's the only way you can get 11 when multiplying whole numbers. I see that I have an x here, and there's just one x; I don't have an x squared term. Like we did in that problem a second ago, I'll write there's a factor of x involved in this. I'm going scan my list and see what they have in common. They do have a 1 in common, but that's not a big number here. The biggest number they have in common, 11. As far as the number part is concerned, 11 is the greatest common factor as far as the coefficients go. I have two "x"s over here, x times x, and there's just one x for this term. The greatest number of "x"s that these terms have in common is just one, because both of these terms have at least one x. For this problem, I'd say the greatest common factor is 11x. 11 was the largest coefficient they had in common, the largest regular number, and they had at least one x-- both of them had at least one-- so I can write 11x, greatest common factor. Let's keep moving. Now, greatest common factor is going to help you know how you factor binomials-- when you're adding together two monomials. The first step is to figure out the greatest common factor. I wanted make sure that you remembered how to do that. After you've got it, divide each term by your greatest common factor. Okay? Let me show you what I mean.

(female narrator) Example 1: Factor completely. 15x squared minus 50x. Okay, here we go. "Factor completely." When you see directions like that, all that's telling you to do is factor this problem in the simplest way, so that when you've written your answer, there's nothing else you could do to simplify it. You want it in its simplest form. What I want to do here is first identify the greatest common factor. I'm going to split these terms apart and think about them separately for a second to help me figure out the greatest common factor. I'm going to do a little scratch work over here and this is to get the GCF. Right now, I'm focusing on 15x squared. I know there's a minus here to keep in mind, but to find the greatest common factor, I'll ignore it for a minute and just say, "I've got 50x as the term I want to factor." We start breaking this down. For 15x squared think about how you can get 15, factors of 15. You have 1 and 15, 3 and 5-- we handled 15 before so you probably remember those terms, those factors. You have x squared part of this, so that means you have a product of two "x"s, so x times x-- that's also going to be a pair factors for you. Done with that first term, so now let's handle the second one. Let's first handle the 50 and ignore the x for a second. You know 1 times 50, 2 times 25, you think about quarters. 3 times anything, 4 times anything, but 5 times 10. That pretty much sums up your factors for 50. Move that out of the way. There's one x here, so I'll write down that I have one factor of x. Now that I've got my list, I'll pull out my greatest common factor. I scan the numbers first and I see 5 is the greatest common factor as far as the numbers go. The 15x squared has two "x"s, the 50x just has one. So they have at least one x, both of them. So that means my greatest common factor between these two terms is 5x. All right? We've completed step one. We know what the greatest common factor is for these terms. Step 2 is to divide each of these terms by your greatest common factor to get the rest of your answer. I'll set up what format your answer is going to look like. You're going to have your 5x and there'll be a set of parentheses right here. We're trying to factor this, or figure out what we'll multiply together to get this binomial. We multiply 5x times something, and we need to figure out that something. That minus sign, so I don't forget about it, I'm going to bring it in my parentheses, so I remember that is separated by subtraction. Now that I've got that, get us some more room to work. Let's get this out of the way. We found the greatest common factor. Let's go ahead and handle step two, which is to divide each term in your binomial by your greatest common factor. That's how you figure out what goes in your parentheses. Okay? Let's start. You've got that 15x squared. We're going to divide it by our greatest common factor, that 5x. Now you got to think back to those rules of exponents, remember how you handle situation like this. You divide the coefficients, so 15 divided by 5, that's 3. Remember when you don't see an exponent, there's a 1-- invisible, okay? Because to divide that x squared by x, remember you need to subtract those exponents, that rule of exponents. 2 minus 1, that's just 1. So this is 3x to the first, or I could just write 3x, which is what I'm going to do, write 3x. I know that this first part of what goes in my parentheses is 3x, which is what I get when I divide that first term by the greatest common factor. Now I'm going to move on and divide 50x by the greatest common factor, and that's how I figure out what goes in this empty space, what term goes here. So now 50x divided by 5x-- still remembering our laws of exponents, divide the coefficients. So 50 divided by 5, that's 10. Remember, there's some invisible ones that I'm not seeing right there. So x to the first divided by x to the first-- remember, when you're dividing you subtract your exponents; so 1 minus 1, that's zero. I am going to write that down so you know what I mean. It's x to the zero power. Remember, anything we raise to the zero power is just 1. This is essentially saying that my answer is 10 times 1 because that's what I get when I handle the "x"s, but just write that as 10-- 10 times 1 is just 10. And 10 is the number that's going to go right here. When I factor completely, 15x squared minus 50x, you get 5x times 3x minus 10. What that means is if you multiply this out, using the distributive property, this would be your answer. Factoring reverses multiplication. It's like you're answering the question, "What did I multiply together to get that first problem?" That's basically what we're doing every time, trying to figure out what do we multiply together to get whatever they're asking us about. Let's try another one to make sure you got it. "Factor completely. 3x plus 15." It looks a little different but the steps are the same. It's just a binomial; you're just going to factor it. First you know you need to find your greatest common factor. I'm going to write that over here, that I'm finding the greatest common factor between 3x and 15. As far as the 3x is concerned, the coefficient-- 3 is prime. So we know the only factors are 1 and 3. There is that one little x hanging out with that 3, so I know I have a factor of x, and I'm done as far as 3x concerned. Now the 15, we've handled 15 a few times. We're trying to give you numbers that are small and familiar so that you start to feel confident about your factors. So 15: 1 and 15, 3 and 5. And it's just a constant term; there's no x, no variable attached. I don't have any "x"s to write as factors of 15. I'm at the point now where I can find the greatest common factor. You look at your list. All they have in common is 3. The greatest common factor between 3x and 15 is 3. Step one, all done. Remember, this is when, on the last problem, I set up the format of my answer? I knew I'd have 3 and then something in parentheses. This one's a plus, 3x plus 15, so I'm going to put the plus sign in between just to keep my signs straight. Now we're ready for step two. Go ahead and divide our terms in the binomial by the greatest common factor. Okay. We need to divide 3x by 3. 3x divided by 3. Remember, when you divide the same number on the top and bottom, it just cancels out because it just equals 1. All you're left with that first division problem is x. The first thing that goes into my parentheses, the first term is x, and now I need to divide 15 by the greatest common factor, by 3. So 15 divided by 3, that's 5. And so 5 is the second term that goes in your parentheses. You're all done. When you factor 3x plus 15 completely, your answer is 3 times x plus 5. Remember, you can check that by multiplying that out and ensuring you get what you started with.

(narrator) Example 3: Factor completely. 10x squared plus 4xy. All right, here we go. This one's involving more than one variable because I know you saw the y, but the process is exactly the same. We're going to find the greatest common factor, and then we're going to divide each of these terms by it. Let's get the greatest common factor first. I've got 10x squared, and my other term is 4xy. You think to yourself, "Okay, she said ignore the variable, look at the coefficient," so ignore that x squared and focus on the number first. What are the factors of 10? So 1 and 10, 2 and 5-- that's it for 10. Here you have x squared, so you remember how you represented that? That is the product of x and x. Okay? All done for 10x squared. Let's factor 4xy. Ignore the variables; just focus on the 4 to start with. You got 1 and 4, that'll give you 4, and 2 times 2 will give you 4. Then you have an x and a y, so you have a product of x and y. That means you have two separate variables, that's all. Now we look for the greatest common factor. You scan the numbers and the largest one they have in common is 2. They both have at least an x, but they don't each have a y, so that's all I can deal with, really. My greatest common factor is 2x for that problem. Okay? Step one is all done. Now you're ready for step two. You're going to take your greatest common factor. Let's set up the answer over here. I know I'll have 2x and I'll have a set of parentheses, and the sign in this one is going to be a plus. Now I'm ready to divide each of these terms by my greatest common factor; let's see what I get. Let's get this out of our way and let's start dividing. I need to divide 10x squared by 2x. So 10x squared divided by 2x. So, 10 divided by 2, that's 5. This is just 2x. Remember, there's an invisible 1 there. So x squared divided by x, you're really just subtracting these exponents. So 2 minus 1, that's 1. So 5x to the first-- remember we don't generally write when it's a 1 as an exponent-- so 5x. That's what will go up there. Okay? Now same thing. Divide the next term by your greatest common factor. So 4xy divided by 2x. Let's handle the numbers first, the coefficients. So 4 divided by 2 is 2. There's the invisible 1 here, an invisible 1 here. Let's go ahead and put the invisible 1 with the y too. If I handle my "x"s, x to the first divided by x to the first, you subtract your exponents. So 1 minus 1, that's zero. Really this is x to the zero power. Remember the 1 when we first started, anything to the zero power is 1. Basically when you run into this situation and you're finding your greatest common factor, you can consider the "x"s just wiped out, because if the answer ends up being 1, you don't really need to write "times 1." It's not going to change anything. That 2 times 1 is just 2. When I see that's 1, all right, I'm done with the "x"s. Let's look at this y. I've got y to the first and I don't have any y's on the bottom, so that means that that y just keeps on hanging out. It's 2y. When I handle this division problem, the answer is 2y. That's what's going in my parentheses. And you are all done. When you factor that binomial completely, you get 2x times 5x plus 2y. Do you remember all along that I've been saying that if you wanted to check it, you can multiply what you got for your answer, actually distribute this throughout, and make sure you get what you started with? I'm going to show you what I mean now. This is a way that you can check your answers and know if you're right before you press play again to see what I got. You can do this to check and see if what you got is the right answer. I'm going to rewrite this. I need the pen back first. So, 2x and then 5x plus 2y. This is what I mean by, "You can just multiply it out to check your answers." Just use the distributive property. 2x times 5x, that's 10x squared, and then 2x times 2y-- well, 2 times 2 is 4, and x times y is just xy. The answer we got was the problem that we started with. You can always do that to check your answer. Like I said, factoring just reverses multiplication. If you multiply your factors, you'll know if what you did was right. Okay? It's your turn to try. Go ahead and factor these completely, and if you want, I encourage you to check your answers and see if what you got is correct before I reveal what these answers are. Press play when you're ready to see what I did.

(narrator) Factor completely: Number 1: 3x squared minus 21x. Number 2: 8y squared plus 6xy. Number 3: 15x plus 9. All right, here's what I did. For the first one, my factors were 3x and x minus 7. 3x times x minus 7. For number two, I got 2y times 4y plus 3x. For number three, I got 3 times 5x plus 3. Let me show you how I did it. So, I followed our process. First thing I did was find our greatest common factor. So, greatest common factor-- need the pen back. Here we go. I wrote down that I had a 3x squared, and I wrote down I had a 21x, and I just started factoring. I ignored the x squared for a second and just focused on the 3. So 3 is prime, so that's an easy one to factor. It's just 1 and 3, and then you had x squared, so you know that's the product of an x and an x. Okay? That's it for 3x squared. For 21x--21-- 1 times 21 will give you 21 and so will 3 times 7. You only have one x over here. So, one x as a factor. Then I looked for my greatest common factor. They had a 3 in common-- that's the largest number. And they each had at least an x. That meant my greatest common factor was 3x. Now that I knew that, I set up my answer, so I knew I'd have a 3x, I knew I had a parentheses, and I see there should be subtraction in between. Let's get some more room to work up here. Now divide each of these terms by your greatest common factor, exactly what I did. 3x squared divided by 3x-- so, 3 divided by 3, that's just 1, so basically, these are cancelling out. Then I have x squared times x to the first, so subtract these exponents-- 2 minus 1, that's just 1. So, it was just x to the first, or just x. Okay? Then divide your 21x by 3x. 21x divided by 3x. Handle the coefficients first. 21 divided by 3, that's 7. X to the first divided by x to the first, so I know I need to subtract those exponents-- 1 minus 1 is zero. Remember anything to the zero power is 1. Multiplying 7 by 1 doesn't change anything so I know in my mind, "Okay, I've handled this one." So it's 7. That's how I got the first one, 3x times x minus 7. In the second one, did the same exact process. My greatest common factor... I factored 8y squared and I factored 6xy. My factors of 8-- 1 and 8, 2 and 4, and that's it for 8. For the y squared part, you know that's the product of 2 "y"s. You have a y and a y as factors. For the 6, you think of factors of 6. 1 and 6, 2 and 3, and here you have an x and a y. So x times y-- that's going to be a factor of that term. You scan: What's your greatest common factor here? The largest number they have in common is 2 and they each have at least one y. Your greatest common factor here, 2y. Now I'm going to go over here and set up my answer. I know I'll have a 2y, I know I'll have parentheses, and I know there's going to be a plus sign in between. Let's get some more room to work up here. Eraser, here we go. All righty. And I need to divide each of these terms by the greatest common factor. So 8y squared divided by 2y. So, 8 divided by 2, that's 4. Y squared divided by y to the first, subtracting my exponents-- so that's just y. I have 4y right here. And then now 6xy divided by 2y-- coefficients first, handle those-- so 6 divided by 2, that's 3. I have an x up top, but none on the bottom, so x just carries along for the ride. Then I have y to the first divided by y to the first. 1 minus 1, that's zero. Anything to the zero power is 1, so basically it never happened, because multiplying 3x times 1 isn't going to change anything. Over here, I have 3x. That's how I made my way to the answer on that one. On the last one, same process. Greatest common factor. I'm factoring 15x, and I'm factoring 9. You're probably a pro at factoring 15 at this point. So 1 and 15, 3 and 5. And you know you have one factor of x. You're done with 15x. Nine-- 1 times 9 will give you 9, and so will 3 times 3. And it's a constant term so I don't have any variable factors. I look and I find my greatest common factor. They each have a 3 in common, and that's it. So 3 is the greatest common factor, so now I'm done with that scratch work. Come over here and set up my answer. I know I have a 3, I know I have a set of parentheses, and I know I will have addition in between, right? Let's get some work space here and get that out of the way. All right. Now I need to divide each of these terms by that 3, by my greatest common factor. So 15x divided by 3-- so 15 divided by 3, that's 5. X--and I don't have any "x"s on the bottom so that's just carrying along, so 5x. And the 9 divided by 3 is 3. That second term is 3. All right. I hope you're feeling confident about factoring binomials and I hope you saw how important knowing the greatest common factor was in these problems. Hope to see you soon. Bye, guys.

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In this program, students learn how to factor a binomial. An expression is completely factored when no further factoring is possible. The possibility of factoring by grouping exists when an expression contains four or more terms. Part of the "Welcome to Algebra" series.

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Runtime: 31 minutes

Welcome to Algebra I
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31 minutes
Grade Level: 7 - 12
Welcome to Algebra I
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Grade Level: 7 - 12