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### Factoring Trinomials

- When factoring a trinomial, always begin by looking for a greatest common factor first.
- Some trinomials can be factored by using the FOIL process in reverse.
- Determine what your first terms should be
- Determine a good choice for your last terms
- Check to see that the outside and inside terms combine to give the original middle term
- Repeat the process if when multiplying the two binomials together, you do not get the original
- If the middle and last term in the trinomial is positive, then your two last terms of the binomials must also be positive
- If the middle term is negative and last term positive in the trinomial, then your two last terms of the binomials must be negative.
- If the last term of the trinomial is negative, then the two last terms of the binomial must be different in sign.

### Factoring Trinomials

x

^{2}+ 16x + 60

- ( x + p )( x + q )
- p + q = b = 16
- pq = c = 60
- p = 6,q = 10
- ( x + 6 )( x + 10 )
- Foil to check your work.
- ( x + 6 )( x + 10 )
- x
^{2}+ 10x + 6x + 60

^{2}+ 16x + 60

w

^{2}+ 11w + 24

- p + q = b = 11
- pq = c = 24
- p = 3,q = 8

k

^{2}+ 8k + 16

- p + q = b = 8
- pq = c = 16
- p = 4,q = 4

y

^{2}− 7y + 12

- ( x − p )( x − q )
- p + q = b = − 7
- pq = c = 12
- p = − 3,q = − 4

g

^{2}− 6g + 8

b

^{2}− 11b + 18

8n − 20 + n

^{2}

- n
^{2}+ 8n − 20 - c〈0 .
- ( x + p )( x − q )

r

^{2}+ 17r − 18

s

^{2}− 36 = − 9s

- s
^{2}+ 9s − 36 = 0

2

^{2}+ 40 = 132

- 2
^{2}− 132 + 40 = 0

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Factoring Trinomials

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Objectives 0:06
- Factoring Trinomials 0:25
- Recall FOIL
- Factor a Trinomial by Reversing FOIL
- Tips when Using Reverse FOIL
- Example 1 7:04
- Example 2 9:09
- Example 3 11:15
- Example 4 13:41
- Factoring Trinomials Cont. 15:50
- Example 5 18:42

### Algebra 1 Online Course

### Transcription: Factoring Trinomials

*Welcome back to www.educator.com.*0000

*In this lesson we are going to work on factoring trinomials.*0003

*We are not going to tackle up all types of trinomials just yet.*0006

*For the first part we are going to focus on ones where the squared term has a coefficient of 1.*0010

*We will also look at polynomials where we can factor out a greatest common factor.*0015

*In future lessons we will look at the more complicated trinomials.*0020

*One of our first techniques we have to dig back in our brains and recall how we used foil in order to multiply two binomials together.*0028

*For example, what did we do when we are looking at x -3 × x +1.*0037

*Using the method of foil we would multiply our first terms together and get something like x ^{2}.*0042

*We would multiply our outside terms together 1x then we would multiply our inside terms.*0049

*And finally we would multiply our last terms together.*0060

*Sometimes we had to do a little bit of work to clean this up.*0067

*As long as we made sure that everything got multiplied by everything else.*0070

*We where assured that we could multiply these two binomials out.*0075

*Since we are working with factoring and breaking things down into a product you want to think of this process, but do it in reverse.*0079

*If you had a trinomial to begin with, how could you then break this down into two binomials?*0087

*Since this is the exact same one as before I will simply write down the two binomials that it will break up into.*0096

*This is the process that we are after of taking a trinomial and breaking it down into two binomials.*0105

*In doing so it is not quite a straightforward process.*0114

*The way we are going to attack this is to think of that foil process in our minds.*0119

*This will help us determine our first and last terms in those binomials.*0124

*Now, after we have chosen something for those first and last terms*0129

*we will have to check to make sure that the outside and inside terms combined to be our middle term.*0133

*Sometimes we will have to do some double checking just to make sure that it does combine and give us that middle term.*0140

*Sometimes there are lots of different options.*0146

*We may have to do this more than once until we find just the right values that make it work.*0149

*Watch how that works with this trinomial.*0154

*I have x ^{2} + 2x -8 and what we are looking to do is break this down into two binomials.*0157

*I’m going to go ahead and write down the parentheses just to get it started.*0165

*I first want to determine what should my first terms be in order to get that x ^{2}.*0169

*We are looking to multiply two things together and get x ^{2}.*0177

*The only two things that will work is x and x.*0180

*We have a good chance that those are our first terms.*0184

*Rather than worrying about the outside and inside just yet, we jump all the way to the last terms.*0191

*That will be here and here and I'm looking to multiply them and get -8.*0197

*Here is the thing there are lots of different options that we could have, it could be 1 and 8, 2 and 4.*0203

*We could also look at the other order of these maybe 4 and 2, 8 and 1.*0211

*We want to choose the proper pair that when combined together will actually give us that 2x in the middle.*0217

*Let us go ahead and try something.*0224

*Watch how this process works.*0226

*Suppose I just tried the first thing on the list, this 1 and 8.*0228

*1, 8, with this combination I can be sure that my first terms work out and that my last terms work out.*0233

*I'm not completely confident until I check those outside and inside terms to make sure that they work out.*0242

*I will do some quick calculations.*0250

*Let us check our outside terms, 8 × x = 8x and inside terms 1 and x, and these would combine to give us a 9x.*0252

*If you compare that to the original it is not the same.*0265

*What that is indicating is that pair of 1 and 8, those are not the ones we want to use.*0270

*We can backup a little bit and try another pair of numbers.*0277

*Let me try something different.*0287

*I'm going to try 4 and -2.*0290

*Now when I do my outside and inside terms I will get -2x on the outside,*0298

*4x on the inside and those combine to give me 2x, which is the same as my middle term.*0305

*I know that this is how it should be factored.*0313

*If you want you can go through the entire foil process, just to double check that all the rest of terms work out.*0319

*In fact it is not a bad idea when you are done with the factoring process, just to make sure it is okay.*0324

*Now, there are a few tips to help you along the way when doing this reverse foil method.*0333

*It works good, as long as you are leading coefficient is 1 and you can take a look at the signs of the other two coefficients.*0339

*In general, here is what you are trying to do.*0351

*You are looking for two integers whose product will give you c.*0354

*They multiply and give you c but whose sum is b.*0358

*It is what we did in that last example.*0363

*Now you can get a little bit more information if you look at the signs of b and c.*0367

*If b and c are both positive then those two integers you are looking for must also be positive.*0371

*One situation that might happen is you know both integers that you are looking for will be negative if c is positive,*0383

*that is this one in the end and d is negative.*0391

*Watch for that to happen.*0396

*Of course one last thing, you will know that the integers you are looking for are different in sign, one positive and one negative if c is negative.*0400

*That is the only way they could multiply together and give you a negative number here on the end.*0410

*Watch for me to use these shortcuts here in just a little bit.*0416

*We want to use this reverse foil method in order to factor the following polynomial y ^{2} + 12y + 20.*0426

*I’m going to start off by writing set of parentheses this will break down into some binomials.*0435

*Let us start off for those first terms.*0444

*What times what would give us a y ^{2}?*0447

*One thing that will do it is just y and y.*0451

*Let us look for two values that would multiply and give us a 20.*0457

*It is okay for me to write down some different possibilities like 1 and 20, 2 and 10, 4 and 5.*0462

*You can imagine the same values just flipped around.*0470

*Let us see if we can use any information to help us out.*0474

*Notice how this last term out here is positive and so is my middle term, both of them are positive.*0479

*Now what is that telling me about my signs, I know that the two numbers I'm looking for will both be positive.*0488

*That actually is quite a bit of information because now when we look at our list I can pick two things that will add to be the middle term 12.*0496

*I know they multiply between.*0504

*Let us drop those in there, 2 and 10.*0507

*I have factored the trinomial.*0511

*Let us quickly go through the foil process just to make sure that this is the one we are looking for.*0514

*We are looking to make sure that this matches up with the original.*0520

*First terms would be a y ^{2}, outside terms 10y, inside terms 2y and last term is 20.*0524

*These middle ones would combined giving us y ^{2} + 12y + 20 and that shows that our factorization checks out.*0534

*We know that this is the proper way to factor it.*0544

*Let us try another one, this one is x ^{2} - 9x – 22.*0551

*Let us start off in much the same way.*0557

* Let us write down a set of parentheses and see if we can fill in the blanks.*0559

*We need two numbers that when multiplied together will give us x ^{2}.*0566

*That must be an x and another x.*0571

*Now we need a pair of numbers that will multiply and give us a -22.*0577

*Let us write down some possibilities like 1 and 22, 2 and 11, I think that is it.*0581

*Let us get some information about the signs of these numbers.*0589

*I’m looking at this last number here and notice how it is negative,*0594

*I know that these two numbers I’m looking for on my last terms, one of them must be positive and one of them must be negative.*0598

*The question is which one is positive and which one is negative?*0607

*We will look to our middle term to help out.*0613

*I need the larger term to be negative, so that I will get a negative in the middle, that -9.*0617

*We have plenty of information it should be pretty clear that it is actually the 2 and 11 off my list that will work.*0624

*2 and 11.*0632

*Let us just check it real quick by foiling things out to make sure that this is how it should factor.*0637

*First terms x ^{2}, outside terms -11x, inside terms 2x, and last terms -22.*0643

*Since I have a positive × negative, these middle terms combined I will get x ^{2} - 9x -22.*0656

*That is the same as my original, so I know that I have factored it correctly.*0666

* Let us use the reverse foil method for this polynomial.*0679

*It is not very big, it is r ^{2} + r + 2.*0682

*We are going to start off by setting down those two parentheses.*0686

*We are hunting for two first terms that will multiply to give us r ^{2}.*0691

*There is only one choice that will do that, just r and r.*0697

*We turn our attention to the last terms and we need them to multiply to be 2.*0704

*Unfortunately, there is only one possibility for that, 1 and 2.*0711

*Let us hunt down our signs.*0717

*The last term is positive, the middle term is positive that says both of the terms that we are looking for must both be positive.*0720

*Let us put them in and now we can go and check this using the foil process.*0728

*r × r =r ^{2}, the outside terms should be 2r, inside terms 1r, and the last term is 2.*0735

*Combining the two middle terms here, I get r ^{2} + 3r + 2.*0746

*Something very interesting is happening with this one, let us take a closer look.*0755

*If we look at the resulting polynomial that we got after foiling out and we compare that with the original, they are not the same.*0759

*That tells us something. It tells us that this is not the correct factorization.*0769

*Now if this is not how it should be factored then what other possibilities do we have?*0783

*If we look at all of our possibilities for those last terms, it must contain 1 and 2 if it is going to multiply and give us 2.*0789

*Since that did not work and I have no other possibilities, it tells us that this does not factor into two binomials using 1 and 2.*0798

*This is an indication that our original is actually prime.*0807

*Watch out for ones like this where when you try and factor it, it simply does not factor into those two binomials.*0813

*Let us try something with a few more variables in it.*0824

*This one is t ^{2} – 6tu + 8u^{2}.*0827

*Even though we have a few more variables in here, you will see that this process works out the same as before.*0833

*Starting off with those first terms, something × something will give us t ^{2}.*0841

*That must be t and another t.*0848

*I need two things to multiply and give us 8u ^{2}.*0853

*Well I'm not quite sure about that 8 yet because it could be 1 and 8, 2 and 4, or could be those reversed.*0860

*I did know about the u, you better have a u and another u in order to get that u ^{2}.*0867

*Let us just focus on numbers for bit and see what information we can get from there.*0874

*Looking at the sign of my last term it is positive, but my middle term is negative.*0881

*The information I’m getting from there is that both of my numbers I’m looking for must both be negative.*0887

*I think we can pick it out from our list now and it looks like we must use the 2 and 4.*0896

*Finally let us check that to make sure this one works.*0906

*Let us be careful since we have both t’s and u’s.*0909

*First terms t ^{2}, outside terms -4tu, inside terms -2tu and my last terms – 2 × -4 = 8u^{2}.*0912

*We can combine our middle terms giving us t ^{2} – 6tu + 8u^{2}.*0931

*And now we can see that yes, this has been factored correctly since the resulting polynomial is the same as my original.*0940

*This one is good.*0947

*One thing to watch out for is to make sure you pull out any common factors at the very beginning.*0953

*That is what we will definitely need to do with this example before we even start the forming process.*0959

*Always check out for a good common factor.*0966

*This is also good idea because it can potentially make your number smaller, so that you do not have to think of as many possibilities.*0970

*We are going to quickly factor 3x ^{4} -15x^{3} + 18x^{2}.*0978

*Look with this one, everything here is divisible by 3 and I can pull out an x ^{2} from all of the variables.*0985

*Let us take it out of the very beginning, I have 3x ^{2} and let us write down what is left.*0996

*3x ^{4} ÷ 3x^{2} = x^{2}, =15 ÷ 3 = -5x, 18 ÷ 3 + 6 and I think that is all my leftover parts.*1004

*With this one now, I will go and do the reverse foil process on that.*1024

*I need two binomials, let us break it down.*1030

*What should my first terms be in order to get my x ^{2}?*1035

*That must be an x and an x.*1042

*I have to look at my last terms and then multiply together to give us a 6.*1047

*1 and 6, possibly 2 and 3, but you can use the signs to help you out.*1052

*The last term is positive, the middle term is negative, so both of these will be negative.*1058

*It looks like I need to use that 2 and 3.*1066

*Some quick checking to make sure this is the correct factorization.*1077

*I have x ^{2} - 3x - 2x + 6 and looks like my outside and inside terms do combine and give us that - 5x.*1080

*This is the correct factorization and let us go head and write out that very first 3x ^{2} that we took out at the very beginning.*1093

*Now that we have all the pieces, we can say that this is the correct factorization.*1103

* Always look for a common factor that you could pull out from the very beginning before starting the foil method.*1109

*Sometimes you can pull something out, sometimes you can not but it will make your life easier if you can find something.*1116

*Keep that in mind for this next example.*1125

*This one is 2x ^{3} - 18x^{2} - 44x.*1128

*Let us come at this over.*1134

*It looks like everything is divisible by 2 and they all have an x in common.*1136

*Let us take out a 2x at the very beginning.*1142

*What do we have left?*1148

*2x ^{3} ÷ 2x = x^{2}, -18 ÷ 2 = -9x and -44x ÷ 2x = -22.*1150

*Now we want to factor that into some binomials.*1168

*Let us go ahead and copy over this 2x just we can keep track of it.*1174

*I need my first terms to multiply together and get an x ^{2}.*1180

*That will be an x and another x.*1186

*I need to look at my last terms so that they multiply together to give me a -22.*1190

*Some of our possibilities are 1 and 22, 2 and 11.*1197

*Since the last one is negative and my middle term is negative, I know that these will be different in sign.*1204

*Let us take the 2 and 11 off of our list, those are the ones we need, -11 and 2.*1214

*Let us quickly combine things together and make sure that it is the correct factorization.*1224

* x ^{2} - 11x + 2x – 22.*1229

*Combining these middle guys x ^{2} - 9x -22, so that definitely checks with this polynomial right here.*1236

*Now if you want to go ahead and put in the 2x as well, this will take you back all the way to the original one.*1249

*Remember to use your distribution property so you can see how that will work out.*1258

*2x ^{3} - 18x^{2} - 44x and sure enough that is the same as the original.*1264

*Everything checks out. I know that this is the correct factorization for our polynomial.*1275

*Just a few things, make sure that when you are using this method, always check for common factor to pull it out.*1286

*Make sure you set down your first terms and then your last terms*1292

*and definitely check those signs to help you eliminate some of your possibilities.*1297

*Thank you for watching www.educator.com.*1303

1 answer

Last reply by: Professor Eric Smith

Mon Jul 10, 2017 4:51 PM

Post by Jiaming Xu on June 7, 2017

What would be the easiest method? Is it grouping, because that's what I usually use.