Algebra 1 Factoring Trinomials

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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 x².0042

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 terms0129

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 x² + 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 x².0169

We are looking to multiply two things together and get x².0177

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 y² + 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 y²?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 y², outside terms 10y, inside terms 2y and last term is 20.0524

These middle ones would combined giving us y² + 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 x² - 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 x².0566

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 x², outside terms -11x, inside terms 2x, and last terms -22.0643

Since I have a positive × negative, these middle terms combined I will get x² - 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 r² + 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 r².0691

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 =r², the outside terms should be 2r, inside terms 1r, and the last term is 2.0735

Combining the two middle terms here, I get r² + 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 t² – 6tu + 8u².0827

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 t².0841

That must be t and another t.0848

I need two things to multiply and give us 8u².0853

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 u².0867

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 t², outside terms -4tu, inside terms -2tu and my last terms – 2 × -4 = 8u².0912

We can combine our middle terms giving us t² – 6tu + 8u².0931

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 3x⁴ -15x³ + 18x².0978

Look with this one, everything here is divisible by 3 and I can pull out an x² from all of the variables.0985

Let us take it out of the very beginning, I have 3x² and let us write down what is left.0996

3x⁴ ÷ 3x² = x², =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 x²?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 x² - 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 3x² 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 2x³ - 18x² - 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

2x³ ÷ 2x = x², -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 x².1180

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

x² - 11x + 2x – 22.1229

Combining these middle guys x² - 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

2x³ - 18x² - 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 terms1292

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

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