# 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.

(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.

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

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

(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.

(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