Algebra 1 Multiplying Polynomials

Welcome back to www.educator.com.0000

In this lesson we are going to take care of multiplying polynomials.0002

Specifically we will look at the multiplication process in general, so you can apply that to many different situations.0009

We will look at some more specific things like how you multiply a monomial by any type of polynomial0015

and how can you multiply two binomials together.0021

I will also show you some special techniques like how you can organize all of this information into a nice handy table.0026

I will show you the nice way that you can multiply two binomials using the method of foil.0033

Watch for all of these things to play a part.0038

Now in order to multiply two polynomials together, what you are trying to make sure is that every term in the first polynomial gets multiplied0044

by every single term in the second polynomial.0052

That way you will know every single term gets multiplied by every other term.0056

If you have one of your polynomials being a monomial, it only has one term and this looks just like the distributive property.0061

Let us do one real quick so you can see that it is just the distributive property.0071

We are going to take to 2x⁴ that monomial one term and multiplied by 3x² + 2x – 5.0075

We will take it and multiply it by all of these terms right here.0083

We will have 2x⁴ × 3x², then we will have 2x⁴ × 2x, and 2x⁴ × -5.0091

Each of these needs to be simplified, but it is not so bad.0115

You take the 2 and 3, multiply them together and get 6.0119

And then we will use our product rule to take care of the x⁴ and x² by adding their exponents together x⁶.0124

We will simply run down to all of the terms doing this one by one.0134

2 × 2 =4, then we add the exponents on x⁴ and x¹ power = x⁵.0137

At the very end, 2 × -5 = -10 and we will just keep the x⁴.0147

This represents our final polynomial after the two of them multiplied together.0156

Remember that we are looking so that every term in one is multiplied by every term in the other one.0161

What makes this a little bit more difficult is, of course, when you have more terms in your polynomials.0171

As long as you make sure that every term in one gets multiplied by every term in the other one should work out just fine.0177

Be very careful with this one and see how that turns out.0183

First, I’m going to take this first m³ term and make sure it gets multiplied by all three of my other terms.0187

Let us put that out here.0194

I need to make sure m³ gets multiplied by 2m².0198

Then I will have m³ × 4m and m³ × 3.0206

Now if I stop to there I would not quite have the entire multiplication process down.0221

We also want to take the -2m and multiply that by all 3.0226

Let us go ahead and put that in there as well.0232

We will take -2m × 2m², then we will take another -2m × 4m and then -2m × 3.0234

That is quite a bit but almost there.0264

Now you have to take the 1 and multiplied by all 3 as well.0266

1 × 2m³, 1 × 4m,1 × 3.0272

That is a lot of work and we still have lots of simplifying to do,0292

but we made sure that everything got multiplied so now it is just matter of simplifying.0295

Let us take it bit by bit.0301

Starting way up here at the beginning I have m³ × 2m², adding exponents that would be 2m⁵.0302

Now I have m³ × 4m so add those exponents, and you will get 4m⁴.0313

Onto m³ × 3 = 3m³ and now we continue down the list over here.0324

-2m × 2m², well 2 × -2 = -4 then add the exponents and I will get m³, that will take care of that.0333

We will go to this guy -2m × 4m = -8m².0345

Now this -2m × 3 = -6m and that takes care of those.0354

Onto the last where we multiplied one by everything.,0363

Unfortunately, 1 × anything as itself we will have 2m², 4m and 3.0366

I have all of my terms and the resulting polynomial, but it still not done yet.0376

Now we have to combine our like terms.0380

Let us go through and see if we can highlight all the terms that are like.0383

I will start over here with 2m⁵ and it looks like that is the only m⁵, it has no other like terms to combine.0387

We will go on to m⁴, let us see what do we got for that.0397

I think that is the only one, so m⁴.0403

3m³ looks like I have a couple of m³, I’m going to highlight those.0408

I have some squares, I will highlight those.0415

Let us see what else do we have in here, it looks like we have single m’s.0420

There is that one and there is that one and there is a single 3 in the m.0428

We can combine all these bit by bit.0433

2m⁵, since it is the only one.0436

4m⁴, since this the only one, let us check this after them.0440

Now I have 2m³, 3 – 4=-1m³ and that will take care of those ones.0446

-8m² + 2m = -6m².0459

That one is done and that one is done.0467

-6m + 4m = -2m done and done.0471

And then we will just put our 3 in the end.0478

You can see it is quite a process when your polynomials get much bigger,0483

but it is possible to take every term and multiply it by every other term.0487

Now watch for later on how I will show you some special techniques to keep track of all of these terms that show up.0492

They will actually not be quite as bad as this one.0498

One way we can deal with much larger polynomials and keep track of all of those terms that multiplied together0503

is try and organize all of those terms in a useful way.0508

I’m going to show you two techniques that you can actually organize all that information.0512

One of them we will be using a table and another one we will look like more standard multiplication where you stack one on top of the other.0517

What I'm trying to with each of these methods is ensure that every term in one polynomial gets multiplied by every term in the other polynomials.0524

I’m not are changing the rule while we are doing a shortcut.0531

We are just organizing information in a better way.0534

No matter which method you use, make sure you do not forget to combine your like terms at the end so you can see the resulting polynomial.0536

Let us give it a try.0543

I want to multiply x² + 3x + 5 × x -4.0545

The way I’m going to do this is first I’m going to write the first polynomial right on top of the second polynomial.0550

From there I’m going to start multiplying them term by term and I'm starting with that -4 in the bottom,0566

now multiply it by all the terms in that top polynomial.0573

Let us give it a try.0580

First I will do -4 × 5 = -20 then I will take a -4 × 3x = -12x and I have -4 × x² = -4x².0582

That takes care of that -4 and make sure that it gets multiplied by all of the other terms.0603

We will do the same process with the x.0609

We will take it and we will multiply it by everything in that top polynomial.0612

x × 5 = 5x and I’m going to write that one right underneath the other x terms.0618

This will help me combine my like terms later.0626

x × 3x = 3x².0629

And one more x × x² = x³.0635

I have all of my terms it is a matter of adding them up and I will do it column by column.0644

This will ensure I get all of my like terms -20 - 7x - 1x² and at the very beginning x³.0650

That is my resulting polynomial.0665

Now another favorite way that I like to combine the terms of my polynomial is to use a table structure.0669

Watch how I set this one up.0676

First, along the top part of my table I'm going to write the terms of the first polynomial.0679

My terms are x², 3x and 5.0687

Along the side of it I will write the terms of the other polynomials, so x, -4.0697

Now comes the fun part, we are going to fill in the boxes of this table by multiplying a row by a column.0706

In this first one we will take an x × x².0713

It feels like you are a completing some sort of word puzzle or something, only guesses would be a math puzzle.0717

Also x × x² = x³.0722

x × 3x = 3x² and x × 5 = 5x.0728

It looks pretty good.0738

I will take the next row and do the same thing.0739

-4 × x² = -4x², - 4 × 3x = -12x and -4 × 5 = -20.0743

You will get exactly the same terms that you do know using the other method in a different way of looking at them.0757

We need to go through and start combining our like terms.0764

Looking at my x³ that is my only x³ so I will just write it all by itself.0768

But I have a couple of x²'s so I will write both of those and combine them together, -4x² + 3x =-x².0776

Here I have -12x + 5x =- 7x and of course the last one -20.0789

Oftentimes you will find your like terms are diagonals from each other, but it is not always the case that seems to be very common.0800

A good important thing to recognize in the very end is that you get the same answer either way.0807

Use whichever method works the best for you, and that you are more comfortable with.0813

Now that we have some good methods and above, let us try multiplying these polynomials again and see how it is a little bit easier.0821

I will use my table method and we will take the terms of one polynomial write along the top.0832

I will take the terms of the second polynomial and write them alongside.0843

You will see this will go much quicker m³ – 2m and 1, 2m², 4m and 3.0848

Let us fill in the boxes.0864

2m² + m³ = 2m⁵, 2 × -2 =-4m³, 1 × 2m² = 2m².0865

On to the next row, 4m⁴, 4 × -2 = -8m², 4m × 1 = 4m.0880

Last row, 3 × m³ = 3m³, 3 × -2 =-6m and 3 × 1 =3.0896

Let us go through and start combining everything.0909

I have a 2m⁵ I will write that as our first term, 2m⁵.0911

I’m onto my 4m⁴ and I think that is the only one I have floating around in there, 4m⁴.0918

We can call that one done.0930

3m³ – 4m³, two of those I need to combine, that will be -1m³.0934

I’m onto my squares, -8m² + 2m² = -6m², -6m + 4m =-2m and the last number, 3.0949

The great part is that it goes through and combines all of your like terms and I know I got them off because they are all circle.0977

I’m going to fix this -1, so it is just a - m³ but other than that I will say that this is a good result right here.0984

Some other nice techniques you can use to multiply polynomials together is if both of those polynomials happen to be binomials.0995

Remember that they have exactly two terms, this method is known as the method of foil.1003

That stands for a nice little saying it tells you to multiply the first terms together, the outside terms, the inside terms and the last terms.1010

It is a great way of helping you memorize and get all of those terms combined like they should.1020

It also saves you from creating a large structure like a table when you do not have to.1027

Let us see how it works with this one.1032

I have x -2 × x - 6 I’m going to take this bit by bit.1035

The first terms in each of these binomials would be the x and the other x.1042

Let us multiply those together and that would give us an x².1047

Then we will move on to the outside terms.1056

By outside that would be the x and -6 we will multiply those together, - 6x.1060

Continuing on, we are on inside terms, -2 and the x, they need to multiply together -2x.1072

And then our last terms -2 × -6 = 12.1084

We do get all of our terms by remembering first outside and inside last.1096

With this method, oftentimes your outside and inside terms will be like terms1101

and you will be able to combine them, and this one is no different.1106

They combined to be 8x.1108

Once you have all of your terms feel free to write them out.1112

This is x² - 8x – 4 + 12 and the more you use this method, it will come in handy for a factoring a little bit later on.1115

Let us try out our foil method as we go through some of these examples.1129

Here I want to multiply the following binomials, 5x - 6 × 2y + 3.1134

First, I'm going to take the first terms together that will be the 5x and 2y, 5 × 2 = 10x × y.1142

That is as far as I can put those together since they are not like terms.1154

Outside terms that would be 5x and the 3 = 15x.1159

Onto inside terms, -6 × 2y = 12y and the last terms -6 × 3 = -18.1168

We got our first outside, inside, last and it looks like none of these are like terms.1186

I will just write them as they are 10xy + 15x -12y – 18 and we will call this one done.1190

Let us try another one, and in this one you will see it has few more things that we can combine.1206

We are going to multiply - 4y + x and all of that will be multiple by 2y -3x.1211

Starting off with our first terms let me highlight them.1218

- 4y × 2y = -8 and y × y = y².1223

That is the case here of our first terms.1232

Now we will do our outside terms, -4 × -3 = 12 and x × y.1235

Onto the inside terms, x × 2y = 2xy.1250

Of course our last terms, -3x².1261

Now that we have all of our terms notice how our outside and inside terms, they happen to be like terms so we will put them together.1273

That will give us our final polynomial, 8y² + 14xy - 3x².1280

We can say that this one is done.1294

One more example and this one is a little bit larger one.1303

In fact, the second polynomial in here is a trinomial so we will not be able to use the method of foil.1307

That is okay, we will still be able to multiply it together,1315

but I will definitely use something like a table to help me organize my information a little bit better.1318

Okay, along the top of this table, let us go ahead and write our first polynomial, x – 5y.1330

Then along the rows we will put our second polynomial, I have an x² – 2xy and 3y².1340

Here comes the fun part, just fill in all of those blanks by multiplying a row and a column.1358

x² × x =x³, x² × -5y = -5x²y.1364

Onto the next row, -2xy × x, the x’s we can put those together as an x² and the a y.1378

The last part here -2xy × -5y, let us put the y’s together, -2 × -5 =10xy².1389

One more row, 3y² × x =3xy².1401

I have 3y² × -5y -15y³.1412

We have all of our terms in there, now we need to combine the like terms.1422

Let us start here on the upper corner.1427

If we have any single x³ that we can put with this one.1429

It look like it is all by its lonesome, we will just say x³.1435

We are looking for x²y, they must have x² and they must have y, I think I see two of them, here is one and here is that other one.1441

Let us put these together, -2 + -5 = -7 and they are x²y terms.1453

Continuing on, I have an xy².1464

I have two of those so let us put them together, we will take this one and we will take that one, 10 + 3= 13xy².1468

That takes care of those terms.1482

One more -15y³, I will put it in -15y³.1483

Now I have the entire polynomial.1491

Remember, at its core when you multiply polynomials you just have to make sure that every term gets multiplied by every other term.1493

Use these techniques such as foil or a table to help you organize all of those terms.1500

Thank you for watching www.educator.com.1506