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Welcome to Algebra I: Factoring Trinomials With a Leading Coefficient of 1

32 minutes

Hi, guys. Welcome to Algebra 1. Today's lesson focuses on factoring trinomials with a leading coefficient of 1. I know it's a mouthful. What you know adding and subtracting and multiplying positive and negative numbers will help you get through this lesson. You ready to get started? Let's go.

(Describer) On a screen, x plus 3 times x plus 4. She uses a stylus.

Okay. To explain to you what I mean by factoring, I first want to jump back to FOILing. If I presented you with this situation and asked you to multiply these binomials together using FOIL, you'd probably write "FOIL" somewhere to keep it straight.

(Describer) She writes F-O-I-L.

Then you'd start multiplying. You'd say, all right, first terms: X times X, that's X squared. I'm going to write that underneath. Outer terms, so X times 4, that's 4X. Inner terms, so 3 times X, that's 3X. Then your last terms, 3 times 4, that's 12. Then you're at the point to combine like terms. Notice they're in the center like they always are. So 4X plus 3X, that's 7X. You'd say, my answer is X squared plus 7X plus 12. We use FOIL to figure out what that product is. Now let's think about how we generated these terms that we got. We know that X squared, we got that when we multiplied the first terms together. Right? That was our first terms. 4X came about when we multiplied the outer terms together. Right? That was our outer. 3X came about by multiplying those inner terms together. And the 12, from multiplying the last terms together. And then when we wrote our final answer, we ended up having to combine the outer and inner term-- or the answer from multiplying the outer and inner terms. The first term was still the first term. The second term was from combining the outer with the inner, and then 12 was our last term. This was our work, and this was our final answer. Now, what about this? What if I gave you this?

(Describer) She erases the work.

Get rid of all that. Even got rid of this.

(Describer) She deletes the original problem.

Get rid of that. There we go. What if I gave you this? What if I gave you this trinomial? This X squared plus 7X plus 12? And I asked you, what were my factors? What did I multiply together to get this trinomial? That is what it means to factor a trinomial, and in this case, the leading coefficient is 1, because the coefficient of this first term is 1. I'll show how we work our way through this problem and factor a trinomial. I'm going to use a different example. I'm going to use X squared plus 7X plus 10 to figure where-- or how we can factor this. Now, to start with, we know that the answer's going to take that format. I'm going to have two sets of parentheses, and I know that there's going to be an X in this one and an X in that one, okay? If you forget, let's flip back. I erased it, but remember this started as this.

(Describer) X plus 3 times x plus 4.

It always takes that type of format, where the X's are first, right? Flip back. What I need to figure out now is what terms will finish out these parentheses. Okay? I need to figure out what two numbers will multiply together to give me 10, but when I combine them I get 7. I'll flip back to show you what I mean by that. Remember working this problem out when it started like this? We said our last term came from multiplying 3 times 4. And this inner term came about when we multiplied-- let's get this work back, that X squared plus 4X plus 3X plus 12. That was where this all began. That last term, like we said, came from multiplying 3 and 4 together. And that 7X came from adding the outer and the inner. So when I'm trying to solve the next problem, I need two numbers that will multiply together to give me the last term, but when I combine them, I get that coefficient of that middle term. Stay with me, I know it's a little funky at first. Let's start by listing the factors of 10. I need to figure out what numbers multiply to give me 10 at first. I'll come over here and factor 10. 1 and 10; 2 and 5. And that's it for 10. I've handled knowing what numbers have a product of 10. Now I need to decide, out of these pairs of factors, how could I combine either pair to get 7? This is what I mean-- it's guess and check. If I combine 1 and 10-- 1 plus 10, that's 11. So that's not going to give me 7. But 2 plus 5, that will give me 7. That means to finish out my factors up here, I would need a positive 2 and a positive 5. I have factored this trinomial completely. If I were to multiply this back out, I would get this as my answer. I'll show you to prove my point.

(Describer) X plus 2 times 2 plus 5.

Those are the factors of that trinomial, so to check your answer, you could multiply that out. So FOIL: X times X... that's X squared. X times 5, that's 5X. 2 times X, that's 2X. 2 times 5, that's 10. Then you're at the point where you combine like terms; you have like terms in the middle there. 5X plus 2X would give you 7X. So X squared plus 7X plus 10. And that is the problem that we started with. That means we chose our factors correctly. Okay? Let's try another one. Factoring is something you have to do a few times to get the hang of it. I might make it look easy, but I've been doing math a long time. When I first learned, it was frustrating. Keep practicing and know your multiplication facts, and that will take you a long way with these. Let's handle this one.

(female narrator) X squared plus 5X plus 6.

(Describer) X-squared plus 5x plus 6.

(instructor) You got to factor it. We know at least it's going to take that format: I'll have two sets of parentheses. The first term is going to be X-- leading coefficient of 1 in this problem, so I'll write X and X. Now I need to figure out what two numbers multiply together to give me 6, but when I combine them, I get 5. Let's start out by getting the factors of 6. So I know 1 times 6 is 6. And I know 2 times 3 is 6. And those are all your factors for 6. With the factors of 6, or the pairs of numbers that multiply to give you 6, you figure out which pair combines to get 5. 1 plus 6, that's 7, so that's not it. Now, 2 plus 3, that is 5. Those are the factors that you need. Back up in my problem, that means I would need X plus 2, X plus 3. And you're all done, you factored it completely. And to check your answer, remember you could FOIL that out and make sure you were right. Let's try some more. All right, I need to factor completely: X squared plus 6X plus 8. So I know I have this to start with; I know my answer's going to take that format. It'll have X and an X out front. Now I need to figure out what numbers would multiply to give me 8, but when I combine them I get 6. Let's start by listing the factors of 8. 1 and 8 will give you 8, as would 2 and 4 if you multiplied them. And that's it for 8. Now, out of these pairs of factors, which pair could you combine and get 6? Well, 1 plus 8, that's 9, so that's not it. But 2 plus 4, that is 6. So that means, to finish out my problem... X plus 2, X plus 4, and you're all done. Let's try another one. So I need to factor completely. X squared plus 7X plus 12. So to start out, I know my problem's going to take that format, with X's out front, okay? I need to figure out what numbers could I multiply together and get 12, but when I combine them, I get 7, okay? Let's start by listing the factors of 12. I know 1 times 12 will give me me 12, and so would 2 times 6, and so would 3 times 4. Those are my factors for 12. Now, which pairs of these factors would combine to give you 7? That's the next thing to figure out. Well, 1 plus 12, that's 13. 2 plus 6, that's 8, and 3 plus 4, that's 7, so that's it. We need X plus 3, X plus 4. Okay? All right. It is your turn. Go ahead and press pause, take a few minutes and work your way through these problems. To compare answers with me, press play.

(Describer) Title: Factor completely. Number One: x-squared plus 9x plus 20. Number Two: x-squared plus 6x plus 9.

(female narrator) Factor completely. Number 1: X squared plus 9X plus 20. Number 2: X squared plus 6X plus 9. Okay, ready to check? Let's check. All right. For the first one, X squared plus 9X plus 20. That factored to X plus 5 times X plus 4. And the second one, X squared plus 6X plus 9, that factored to X plus 3 times X plus 3, okay? To see how I got those, this is what I did. First-- and now I need the pen-- I knew that my answer would take this format. I put the X's in there also. Now I needed to figure out what numbers would multiply together to give me 20, but when I combined them I get 9. Come off to the side. I know that 1 times 20 is 20, so is 2 times 10, and so is 4 times 5. Now I need to see which of these would combine to give me 9. 1 plus 20 is 21. 2 plus 10 is 12. 4 plus 5, that is 9. So X plus 4, X plus 5. And I was all done with that one. And remember, to check-- I will show you--remember, to check, you can multiply your products together to get the trinomial you started with. I'll show you the check to make sure your answers are right before you check your answers against mine, before you turn in your work in class. Always check your work first. If you multiplied that out, that X plus 4 times X plus 5... So FOIL, right? So X times X. That's X squared. Outer: X times 5, that's 5X. Inner: 4 times X, that's 4X. The last: 4 times 5. That's 20. Then you see you've got like terms to combine. The 5X and the 4X will give you 9X. Then you've got your X squared plus 9X plus 20. If you multiplied your factors out, you could check that you got your answer right. Let me show you how I got number two. First, remember, always do this first, write my parentheses in there. And there's an X and an X first in each set. Now I have to figure out what numbers would multiply to give me 9, but combine to give me 6. I know 1 times 9, and I know 3 times 3. Those are factors for 9. Now I have to figure out which pair of factors combines to give me 6. So 1 plus 9, that's 10. And 3 plus 3, that is 6. So that's how I knew I needed X plus 3, and X plus 3, okay?

(Describer) The next problem: x-squared minus 8x plus 15.

(female narrator) The next problem: X squared minus 8X plus 15. All right, so, here... you probably noticed that I have a negative sign here. Your process is the same. I'll set up what my answer will look like, I'll get my factors, etcetera. But the negative puts a spin on things. Now you need to consider you're working with integers, so you could have positive and negative numbers. I wanted to start with positive numbers so you could get the hang of it. Now, as you've got a hang of the process, it's time to step it up a notch. I'm going to start out doing this. I know the answer's going to take that format, with X's first. Now I can figure out my actual numbers. I need to figure out what factors, or what pairs of numbers would multiply together to give me 15, but would combine to get negative 8. For a second just ignore the signs. Let's focus on factoring-- whoo, that was a tongue-twister, "focus on factoring" 15. So factors of 15: 1 and 15, 3 and 5, right? Okay, so here we go. If you multiply a positive number times a positive number, that's positive, right? But if you multiply a negative number times a negative number, that's also positive, right? That's where those integer operations come in place. What I mean is, for example, 1 times 15 is 15. Negative 1 times negative 15 is also 15. So when you multiply numbers together that have the same sign, your answer's always positive. Throwback to pre-algebra on that one. Here, what you have to think about is that your product is a positive 15, but the sum that you're reaching for is negative 8. So the only way you can multiply numbers together and take the same two numbers and get a negative answer is if the numbers themselves were negative. So I have to consider negative 1 and negative 15, and negative 3 and negative 5, because I know there's no way I can combine a 3 and a 5 and get a negative number. I can't combine a positive 1 and positive 15 and get something negative. I have to find the negative cases. For example: negative 1 plus negative 15, that's negative 16-- that's not it. Now, negative 3 plus negative 5, that is negative 8. That's the sum I was looking for. What that means is, over here in my factors, I had X minus 3, and X minus 5. These ones aren't positive; these factors aren't positive. When you see negative signs, you have to start considering negative numbers also when doing your guess and check. Let's try another one with those negatives. So X squared minus 9X plus 20. To start, you'll always get those parentheses up there, have an X in each group. Now I need factors of 20 that would also combine to give me negative 9. We can already start to think about this when we start. If we multiplied two numbers together and our answer was positive, the only way that combining those numbers would give us a negative number was if the numbers were negative. Let's get it on the board so you can see. So I'm factoring 20, so know 1 times 20 is 20. 2 times 10 is 20, 4 times 5 is 20. And that's it for 20. Now looking at these numbers as all positive, there's no way I could combine them and get negative numbers. 1 plus 20 will never equal a negative number. 2 plus 10 will never equal a negative number. 4 plus 5 will never equal a negative number. I'm trying to get negative 9. So the only way that this could work out is if both of my factors were negative. Because multiplying these numbers together would still give me positive 20, but combining them, I would get something that's negative. So let's see. Negative 1 plus negative 20, that's negative 21. That's not it. Negative 2 plus negative 10, that's negative 12. So that's not it. Negative 4 plus negative 5, that's negative 9. So this is the pair that I want. So I need minus 4, minus 5. And that would be my factors. Let's try some more. Remember, I said these, you have to do a few. The more you do, the more comfortable you'll feel, and it'll become second nature. Let's start out with, what? Right, putting those parentheses in there. And putting the X in there. Now I need numbers that will multiply together to give me 4, but when I combine them, I get negative 5. Let's list the factors of 4. I know 1 times 4 is 4, and I know 2 times 2 is 4. Now, combining two positive numbers will never give me a negative number. I know for this situation, I need to consider negative numbers. Multiplying two negative numbers together is still positive, and combining two negative numbers is a negative number. So I need to figure out what's going on here. So negative 1 plus negative 4, that is negative 5; it was the first set this time. I'll show you why it's not the second pair. Negative 2 plus negative 2 is negative 4. We wanted negative 5, so it's not that one. So for my factors, I'd have X minus 1 and X minus 4. Okay? Let's try another one; this one is different. We have two negative signs this time. But same process.

(Describer) X-squared minus 2x minus 8.

Start out with your parentheses. You know there's going to be an X in there. For a minute ignore the sign, and let's just list our factors for 8. So 1 times 8 is 8, and 2 times 4 is 8. But I need a negative 8, so now think back to multiplication of integers. The only way to get a negative number for your answer is if you multiply a negative number times a positive number. This is what I mean. 1 times negative 8 is negative 8. Negative 1 times 8 is negative 8. The only way to get a negative number for your answer is if one of your numbers is negative when you're multiplying. In this situation, because I was trying to get a negative 8, one of these factors is going to be negative. We have to figure out which factors should it be so when you combine those you also get negative 2. You kind of have to do a little guess and check. Use your calculator to avoid mental math. Is there any way you can combine 1 and 8 and get negative 2? Mm-mm, because 1 plus negative 8, that's negative 7. Negative 1 plus 8, that's positive 7. There's no way I can get negative 2 when the numbers are a 1 and an 8; that's not it. Let's consider the 2 and the 4. Is there any way to combine 2 and 4, with one of those being negative, and your answer be negative 2? Yes, there is. If the 4 was negative, because 2 plus negative 4 is negative 2. And that's what we wanted for our sum. That means for my factors, I'd have X plus 2, and X minus 4. Actually, I have a positive and a negative. Let's do another.

(female narrator) X squared plus 2X minus 15.

(Describer) X-squared plus 2x minus 15.

(instructor) I'll start by writing out the format of my answer. I'm going to ignore the sign for a second, and I know I need to factor 15. Let's get the factors of 15. So 1 and 15; and 3 and 5. Now is when we consider the signs. We know we have a negative 15. So when we're multiplying, if you get a negative number for your answer, you multiplied a positive and a negative together. As far as factors go, one of these numbers needs to be negative in the pair. I need to find which number should be negative. Let's think our way through this. I need 2X. Is there any way I could combine a 1 and a 15 and get 2? If either one of them was negative? If the 1 was negative, I'd have negative 1 plus 15, and that's 14. So that wouldn't give it to me. And If the 15 was negative, then I'd have 1 plus negative 15, and that's negative 14. That's not what I need either. It's got to be this pair, but which one of these numbers should be negative? What if the 3 were negative? Then I'd have negative 3 plus 5, which is 2. That's what I need; I need the 3 to be negative. So I need X minus 3, and X plus 5. And those are the factors that I needed. You can always multiply your factors together to make sure what you got for your answer is correct. Okay? It is your turn to try. Take a few minutes and factor these problems, and when you're ready to check your answer, press play. Remember, you can check your answer before that.

(female narrator) Factor completely.

(Describer) Title: Factor completely. Number One: x-squared minus 7x plus 12. Number Two: x-squared plus 3x minus 4. Number Three: x-squared minus x minus 6.

Number 1: X squared minus 7X plus 12. Number 2: X squared plus 3X minus 4. Number 3: X squared minus X minus 6. All right, let's see how you did. I'll factor that first one. My factors were X minus 3, and X minus 4. When I factored the second one, my factors were X plus 4, and X minus 1, and when I factored the third one, my answers were X minus 3, and X plus 2. I'll show you how I got those. First thing I did, I wrote down-- I need my pen-- I wrote down the format of my answers, or the format of my answer. I knew I needed to find numbers that would multiply to give me 12, but combine to give me negative 7. I started out by factoring 12. I know 1 and 12 would give me 12, multiplying. 2 and 6 would, and so would 3 and 4. I know I need a positive 12 for the product. So either both of these numbers are positive, or both are negative, to figure out which pair I need. That's the only way when you're multiplying to get a positive number: multiplying two positive or two negative numbers. Since the sum I need is a negative 7, a negative number, it tells me these were both negative numbers. Because the only way that the product is positive but the sum is negative is if the factors were negative. Let's see which pair it is that we need. If it were this first pair: negative 1 plus negative 12, that's negative 13; that's not it. Negative 2 plus negative 6, that's negative 8. So that's not it. Negative 3 plus negative 4, that is negative 7. That means my factors were X minus 3, and X minus 4. That's how I got the first one, okay? Next, same process. Start out by setting up the format. For this one, I need to factor 4. So I know 1 times 4 gives me 4, and so does 2 times 2. Now I consider the signs. It's a negative 4 that I need. So I know that when I'm multiplying, I get a negative number for my answer if I multiply a positive number times a negative number. In this case, my factors will have two different signs. I need to find which one of these numbers is negative, because when I combine the numbers, I need to get a positive 3. Play around and see what happens. With that first set, if the 1 was negative and the 4 was positive, negative 1 plus 4, that's 3; that was quick. So that's the pair that you need. So X minus 1, and X plus 4. Okay? You're all done with that one. The last one, I started out by setting up the answer. Okay? So I know I need to factor 6. So 1 and 6; 2 and 3. Now, in this case, it's a negative 6. So I know when I'm finding a product, the only way my product is negative is if my factors were a positive number and a negative number. So one of these numbers will be negative in my pair. I have to figure out which number is it. You're trying to combine and get a negative 1-- there's a 1 there you don't see. Play around to see which pair makes negative 1. So with 1 and 6, if the 1 was negative, negative 1 plus 6, that's 5. So that's not it. Now, what if the 6 were negative? 1 plus negative 6, that's negative 5, so that's not it either. For the second pair, if the 2 were negative, negative 2 plus 3, that's a positive 1, but we need a negative 1. So it's not the 2 that should be negative. It's actually the 3. 2 plus negative 3 is negative 1.

That means your factors: (X plus 2), (X minus 3), okay? All right, guys, hope you're feeling confident factoring trinomials where the leading coefficient is 1, and I hope you saw how important multiplication facts and your knowledge of integer operations really help you get through this lesson. Hope to see you back here soon. Bye!

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A trinomial is an algebraic expression consisting of three terms. In this program, students will learn the steps required for factoring a trinomial when the leading coefficient is one. Part of the "Welcome to Algebra I" series.

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Runtime: 32 minutes

Welcome to Algebra I
Episode 1
31 minutes
Grade Level: 7 - 12
Welcome to Algebra I
Episode 2
25 minutes
Grade Level: 7 - 12
Welcome to Algebra I
Episode 3
18 minutes
Grade Level: 7 - 12
Welcome to Algebra I
Episode 4
17 minutes
Grade Level: 7 - 12
Welcome to Algebra I
Episode 5
22 minutes
Grade Level: 7 - 12
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Episode 6
9 minutes
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Welcome to Algebra I
Episode 7
24 minutes
Grade Level: 7 - 12
Welcome to Algebra I
Episode 8
15 minutes
Grade Level: 7 - 12
Welcome to Algebra I
Episode 9
25 minutes
Grade Level: 7 - 12
Welcome to Algebra I
Episode 10
16 minutes
Grade Level: 7 - 12