# Welcome to Algebra I: Factoring Trinomials With a Leading Coefficient Not Equal to 1

43 minutes

(Describer) Titles: Welcome to Algebra 1: Factoring Trinomials with a Leading Coefficient Not Equal to 1.

Hi, guys. Welcome to Algebra 1. Today's lesson will focus on factoring trinomials when your leading coefficient is not 1. What you know about factoring trinomials with a leading coefficient of 1 will help you get through these problems. You ready? Let's go.

(Describer) On screen, x-squared plus 5x plus 6.

Okay, so to jog your memory, let's first think back about how you would factor this trinomial, for example. You'd probably start out by setting up what the answer is going to look like: with your sets of parentheses. You know you'll have some X's out front, like that.

(Describer) She writes on the screen.

Then you need the factors of 6. I know 1 times 6 will give me 6, and so will 2 times 3. And I need the pair of factors that, in this case, will combine to give me 5. So when I combine 1 and 6, I get 7. That's not the pair. 2 and 3, when I combine those, I get 5. That is the pair that I want. I'd say my factors are X plus 2, and X plus 3. Here, we factored a trinomial that had a leading coefficient of 1. There's an invisible 1 there, for that first term. What happens when you have something like this?

(Describer) Three x-squared plus 10x plus 8.

(Describer) ...in the first parentheses.

Now I need to divide 4X by X. So 4X divided by X, or like a 1X. 4 divided by 1, that's just 4. X to the 1st divided by X to the 1st, subtracting those exponents, that's X to the 0, and anything to the 0 power is just 1. This is really just like 4 times 1, which is just 4. That means the second group-- or the second spot-- would have a 4.

(Describer) 3x plus 4.

(Describer) She writes X.

I'm going to open up the parentheses. And now it's time to divide each of these terms by X. We'll come off over here. So 3X squared divided by X, which is like a 1X, really, to the 1st because we don't see that exponent. So 3 divided by 1, that's 3. X squared divided by X to the 1st, that's just X because we subtract our exponents here. For my first term in here, I know I have a 3X; it'll be plus, so get that in there. For the second term, I need to divide X by X. Well, X divided by X, that's just 1. That means I have a 1 in here, for that second group of parentheses, or that second term in the parentheses. It's time to do the same thing with 6X plus 2. Let me get this work out of our way.

(Describer) So far, the answer is x times 3x plus 1.

All righty. I need the greatest common factor between 6X and 2. I see that that's just a 2; it's a constant term. They won't have any X's in common. I need to focus on these numbers here. As far as 6 goes and 2 goes-- 2 is prime, so the only factors for 2 are 1 and 2. And 2 is also a factor of 6, so between these two terms, my greatest common factor is going to be 2. So I'll write plus 2. Go ahead and open my parentheses. I know I'm going to have a plus sign in between. Now it's time to divide each of these terms by our greatest common factor, over here. So 6X divided by 2. So 6 divided by 2, that's 3. This X is just hanging along for the ride, so 3X. Then I have 2 divided by 2... which is 1. So then you look. Here you notice you have the same group in those parentheses, so we're at the point of rewriting this. Here we have our X plus 2, which is out front. We bring that together, so X plus 2. And then that factor that was repeated: 3X plus 1. And you are all done. You factored that trinomial. There are a few steps to it, but do it a few times and you get the hang of it. Let's try some more. Factor completely: 2X squared plus 5X plus 3. Again, we need to start out by multiplying 2 times 3. 6 just keeps popping up here. So 2 times 3 is 6. So factors of 6: 1 and 6... 2 and 3. Now we need the factors of 6 that are going to combine to give us 5. Let's see what we got here. 1 and 6, that's 7. 2 and 3, that is 5. So I'll split this middle term and rewrite this as 2X squared plus 2X plus 3X plus 3. Okay? Do you remember what we do next? We're at the point where we take this first group, we take the second group, and factor each group. We pull out the greatest common factor in each group. Let's get this work out of the way. So 2X squared and 2X. They both have that 2 in common. The first term has X squared, so there's a product of two X's with that one. The second term is 2X-- one X for that one. For the numbers, 2 was the greatest common factor. For the variables, X was the greatest common factor. So my greatest common factor: 2X. Open up my parentheses. I know I have a plus in between. Now I'm going to divide each of these terms by that greatest common factor. So 2X squared divided by 2X. So 2 divided by 2, that's 1. X squared divided by X to the 1st, that's X, so basically it's X. When I divide over there, I get X. Then I have 2X divided by 2X. Well, anything divided by itself is 1, so we know that 2X divided by 2X is 1. We've handled that first set. Let's go on to the next group, this 3X plus 3. Let me get this work out of the way. All righty. So 3X and a 3. They both have 3-- there's a 3 involved in each term. For my number, the greatest common factor is 3. The first term has an X, the second term doesn't, so they don't have an X in common. They only have a 3 in common. I'll write my greatest common factor, 3, for this group, open up my parentheses, go ahead and put the plus inside, and start dividing. So 3X divided by 3. Well, 3 divided by 3, I know that's 1. X on top and no X on the bottom, so that carries along for the ride. I have, basically, 1X, or just X. Now, 3 divided by 3... that's 1, so you got a 1.

(Describer) 2x times, open parentheses, x plus 1, close parentheses, plus 3 times, open parentheses, x plus 1, close parentheses.

(female narrator) 2X times open parentheses, X plus 1, close parentheses, plus 3 times, open parentheses, X plus 1, close parentheses. All right, remember what we do next? We're at the point now to clean it up and write our final answer. We've got the 2X out here, the 3 out front here. In one set of parentheses you'd have 2X plus 3.

(Describer) The first factor in the answer.

Then you'd have that factor that was in both of those-- the repeated one, that X plus 1.

(Describer) The other factor.

And you are all done, you've factored completely. Okay? Let's try another.

(female narrator) 4X squared minus X minus 3.

(Describer) 4x-squared minus x minus 3.

Okay, so this one's involving some negatives. That'll be a little different. Other than that, the process is exactly the same. Start out by multiplying A and C. So 4 times 3, that's 12. And I know I need factors of 12 that will combine to give me negative 1. Once I list these factors, they'll have different signs: One number must be positive and the other negative, if when I combine them, I get negative 1. Let's list the factors: 1 and 12; 2 and 6; 3 and 4, right? Now we have to find which one of these pairs of numbers will work and which should be negative. We know we're trying to get negative 1, so throw the first pair out. There's no way to combine a 1 and a 12 and get negative 1, no matter which were negative. Negative 1 plus 12 would be 11. 1 plus negative 12 would be negative 11. So there's no way it's the first pair. 2 and 6? If the 2 were negative, it'd be negative 2 plus 6, and that's 4. If the 6 were negative, then it'd be a negative 4. So that's not it; it has to be this last pair. I need to find which number needs the negative. Negative 3 plus 4 would be positive 1. And I needed a negative 1, so that means it's the 4 that's negative. 3 plus negative 4 will give me negative 1. Now I'm ready to go ahead and split my middle term. I'm going to rewrite it as-- I've got my 4X squared-- I'm going to write plus 3X minus 4X minus 3, okay? Do you remember what comes next? Go ahead, you take your groups... and you start factoring the pairs, okay? Let's get some more room to work up here. If I look at these first terms, the 4X squared and the 3X. 4 and 3 have nothing in common but 1, but I do see I have an X squared and an X. These terms each have an X as a factor. So I'm going to write X as the greatest common factor, okay? Let's get the parentheses up, and get the plus sign in there. So 4X squared divided by X-- I'm ready to start dividing these terms by my greatest common factor. That's essentially a 1X to the 1st over there. So 4 divided by 1. That's 4X squared divided by X, that's just X. So now I have 4X over here. And then 3X divided by X. So 3X to the 1st divided by 1X to the 1st. So 3 divided by 1, that's just 3. X to the 1st divided by X to the 1st is X to the 0 power, which is just 1. Basically, 3 times 1, that's not changing anything, so I'll just leave it as 3. Bring it over here. Now I'm at the point to go ahead and look at these factors. I have a negative 4X and a negative 3. The only thing these two terms have in common is a negative 1, that's it. Both of these terms are negative. We knew 3 and 4 had no factors in common. And this is the only term that has the X. The only thing they have in common is a negative 1. Now it's time to go through. Just go ahead and get that in there.

(Describer) The second set of parentheses.

And let's get this work out of the way. Here we go. Now, I'm not going to write the sign in here just yet because I want to show you something over here. Now, I have negative 4X divided by negative 1. So negative 4X divided by negative 1. Negative 4 divided by negative 1 is just 4. X on top, but no X on the bottom, so it carries along for the ride. So I get 4X over here. Then I have a negative 3 divided by negative 1. So negative 3 divided by negative 1, that's positive 3. So plus 3, okay?

(Describer) It's another 4x plus 3.

Remember what comes next? You're at the finish line, just about. It's time to just write that final answer. So I see I have the X out front here and the negative 1 out front here. So X minus 1.

(Describer) The first factor.

And both of these have the 4X plus 3. So 4X plus 3, and you're all done.

(Describer) The second factor.

Exactly like we saw when we were factoring trinomials where the leading coefficient was 1, check your answer by multiplying your factors out. At the last step, you write your factors. To check your answer, multiply your factors together. Use FOIL and multiply them together to get the starting problem. You can still check your answers that way.

(Describer) Next, 2x-squared plus x minus 15.

(Describer) X times 2x minus 5.

(Describer) Title: Factor completely. Number One: 5x-squared plus 11x plus 2. Number Two: 3x-squared minus 13x plus 4.

(female narrator) Factor completely. Number 1: 5X squared plus 11X plus 2. Number 2: 3X squared minus 13X plus 4. You ready to check? Let's see. When I factored that first trinomial, I got 5X plus 1, and X plus 2. And when I factored the second one, I got 3X minus 1, and X minus 4. This is how I did it.

(Describer) She goes to the first problem by itself.

First thing, remember, we're following our steps. We multiply the A and the C. So 5 times 2, that's 10. Now I need factors of 10. I need to find the pair of factors that, when I combine them, I get 11. As far as 10 goes, I know 1 times 10 will give me 10. So will 2 times 5. I need the pair that gives me 11. You probably spotted that pretty quickly--the 1 and 10. 1 plus 10 is 11. You know that's the pair of factors you want. Now jump back to your trinomial, and you can split that middle term. So 5X squared plus X plus 10X plus 2. Now you're ready to take it in groups, and find the greatest common factor in each group. As far as 5X squared and 5 goes, the only factor they have in common is X.

(Describer) She writes x and the first parenthesis.

(Describer) ...parentheses.

I need to divide each of those terms by that negative 4. Let's get this work out of the way. So I have negative 12X divided by negative 4... which--ooh, 12X. So negative 12 divided by negative 4, that's 3. X up top but nothing on the bottom, so it just carries along. You see you've got your 3X. Then I have positive 4 divided by negative 4. Positive 4 divided by negative 4 is negative 1. You see we have our minus 1. Now we're at the last step, to go ahead and write the final answer. So I'll bring these together in a group of parentheses. I'd have X minus 4. Then for my second set, 3X minus 1, that factor that was in both of those. Remember, you can always multiply these factors together to check your answer and get that trinomial that we started with. I hope you're feeling confident about factoring trinomials where you leading coefficient is not 1, and that knowing how to factor trinomials where your leading coefficient is 1 helped you through this lesson. Hope to see you back here soon. Bye!

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In this program, students will learn the steps for factoring a trinomial when the leading coefficient is not one. Part of the "Welcome Algebra I" series.

## Media Details

Runtime: 43 minutes