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### Multiplying Polynomials

- When multiplying polynomials together we want to make sure that every term of one polynomial, gets multiplied by every term in the second polynomial.
- If we have a monomial (one term) multiplied by a polynomial, the multiplication process is just the distributive property.
- If we have two binomials multiplied together we can use FOIL to ensure that we multiply the First, Outside, Inside, and Last terms together.
- If we have larger polynomials being multiplied together it is useful to organize into a table, or stack them on top of one another.

### Multiplying Polynomials

2m

^{2}( 4m

^{3}− 5m

^{4}+ 6m

^{2})

- 2m
^{2}( 4m^{3}) + 2m^{2}( − 5m^{4}) + 2m^{2}( 6m^{2})

^{5}− 10m

^{6}+ 12m

^{4}

5h

^{3}( 3h

^{6}+ 7h

^{3}− 12h

^{4})

- 5h
^{3}( 3h^{6}) + 5h^{3}( 7h^{3}) + 5h^{3}( − 12h^{4})

^{9}+ 35h

^{6}− 60h

^{7}

6y

^{7}( 11y

^{2}− 14y

^{7}− 9y

^{10})

- 6y
^{7}( 11y^{2}) + 6y^{7}( − 14y^{7}) + 6y^{7}( − 9y^{10})

^{9}− 84y

^{14}− 54y

^{17}

4k

^{4}( 9k

^{3}+ 11k

^{6}− 7k

^{5})

- 4k
^{4}( 9k^{3}) + 4k^{4}( 11k^{6}) + 4k^{4}( − 7k^{5})

^{7}+ 44k

^{10}− 28k

^{9}

2m( 5m

^{4}− 9m + 7m

^{5}) − 4m

^{3}( 6m

^{3}+ 4m

^{2})

- ( 2m )( 5m
^{4}) + ( 2m )( − 9m ) + ( 2m )( 7m^{5}) + ( − 4m^{3})( 6m^{3}) + ( − 4m^{3})( 4m^{2}) - 10m
^{5}− 18m^{2}+ 14m^{6}− 24m^{6}− 16m^{5} - ( 10m
^{5}− 16m^{5}) − 18m^{2}+ ( 14m^{6}− 24m^{6}) - − 6m
^{5}− 18m^{2}− 10m^{6}

^{6}− 6m

^{5}− 18m

^{2}

5j

^{2}( 7j − 3j

^{2}) + 6j( 4j

^{4}− 8j

^{2}+ 2j )

- ( 5j
^{2})( 7j ) + ( 5j^{2})( − 3j^{2}) + ( 6j )( 4j^{4}) + ( 6j )( − 8j^{2}) + ( 6j )( 2j ) - 35j
^{3}− 15j^{4}+ 24j^{5}− 48j^{3}+ 12j^{2} - ( 35j
^{3}− 48j^{3}) − 15j^{4}+ 24j^{5}+ 12j^{2} - − 13j
^{3}− 15j^{4}+ 24j^{5}+ 12j^{2}

^{5}− 15j

^{4}− 13j

^{3}+ 12j

^{2}

6x(3x

^{2}+ 4x − 10x

^{5}) + 4x(7x

^{2}− 8x + 2x) − (3x

^{3}− 2x

^{5}+ 5x)

- ( 6x )( 3x
^{2}) + ( 6x )( 4x ) + ( 6x )( − 10x^{5}) + ( 4x )( 7x^{2}) + ( 4x )( 2x ) + ( − 3x )( x^{3}) + ( − 3x )( − 2x^{5}) + ( − 3x )( 5x ) - 18x
^{3}+ 24x^{2}− 60x^{6}+ 28x^{3}− 32x^{2}+ 8x^{2}− 3x^{4}+ 6x^{6}− 15x^{2} - ( 18x
^{3}+ 28x^{3}) + ( 24x^{2}− 32x^{2}+ 8x^{2}− 15x^{2}) + ( − 60x^{6}+ 6x^{6}) − 3x^{4} - 46x
^{3}− 15x^{2}− 54x^{6}− 3x^{4}

^{6}− 3x

^{4}+ 46x

^{3}− 15x

^{2}

3y

^{2}( y

^{2}+ 4y − 9y

^{3}) − 7y( − 10y

^{2}+ 3y

^{3}) + 4y

^{3}( 4y

^{11})

- ( 3y
^{2})( y^{2}) + ( 3y^{2})( 4y ) + ( 3y^{2})( − 9y^{3}) + ( − 7y )( − 10y^{2}) + ( − 7y )( − 3y^{3}) + ( 4y^{3})( 4y^{11}) - 3y
^{4}+ 12y^{3}− 27y^{5}+ 70y^{3}− 21y^{4}+ 16y^{14}

^{14}− 27y

^{5}− 18y

^{4}+ 82y

^{3}

4k( 2k + 3 ) − 5( k ) = − 2k( 3k − 6 ) − 10

- 8k
^{2}+ 12k − 5k^{2}= − 6k^{2}+ 12k − 10 - 3k
^{2}+ 12k = − 6k^{2}+ 12k − 10 - 9k
^{2}= − 10 - [(9k
^{2})/9] = [10/9] - k
^{2}= [10/9]

5m( 2m − 3 ) + 8 = 3m( 2m + 8 ) − m( − 4m + 1 )

- 10m
^{2}− 15m + 8 = 6m^{2}+ 24m + 4m^{2}− m - 10m
^{2}− 15m + 8 = 10m^{2}− 23m - − 15m + 8 = − 23m
- 8 = − 8m

( 2x + 4y )( 6x

^{2}− 3xy + 5y

^{2})

- ( 2x )( 6x
^{2}) + ( 2x )( − 3xy ) + ( 2x )( 5y^{2}) + ( 4y )( 6x^{2}) + ( 4y )( − 3xy ) + ( 4y )( 5y^{2}) - 12x
^{3}− 6x^{2}y + 10xy^{2}+ 24x^{2}y − 12xy^{2}+ 20y^{3}

^{3}+ 18x

^{2}y − 2xy

^{2}+ 20y

^{3}

( 3j − 7k )( 5j

^{2}+ 2jk − k

^{2})

- ( 3j )( 5j
^{2}) + ( 3j )( 2jk ) + ( 3j )( − k^{2}) + ( − 7k )( 5j^{2}) + ( − 7k )( 2jk ) + ( − 7k )( − k^{2}) - 15j
^{3}+ 6j^{2}k − 3jk^{2}− 35j^{2}k − 14jk^{2}+ 7k^{3}

^{3}− 29j

^{2}k − 17k

^{2}+ 7k

^{3}

( 6x

^{2}− 4x + 10 )( 4x

^{2}+ 3x + 5 )

- ( 6x
^{2})( 4x^{2}) + ( 6x^{2})( 3x ) + ( 6x )( 5 ) + ( − 4x )( 4x^{2}) + ( − 4x )( 3x ) + ( − 4x )( 5 ) + ( 10 )( 4x^{2}) + ( 10 )( 3x ) + ( 10 )( 5 ) - 24x
^{4}+ 18x^{3}+ 30x − 16x^{3}− 12x^{2}− 20x + 40x^{2}+ 30x + 50

^{4}+ 2x

^{3}+ 28x

^{2}+ 40x + 50

( 8r

^{2}+ 10r − 4 )( 3r

^{2}− 2r − 1 )

- ( 8r
^{2})( 3r^{2}) + ( 8r^{2})( − 2r ) + ( 8r^{2})( − 1 ) + ( 10r )( 3r^{2}) + ( 10r )( − 2r ) + ( 10r )( − 1 ) + ( − 4 )( 3r^{2}) + ( − 4 )( − 2r ) + ( − 4 )( − 1 ) - 24r
^{4}− 16r^{3}− 8r^{2}+ 30r^{3}− 20r^{2}− 10r − 12r^{2}+ 8r + 4

^{4}+ 14r

^{3}− 40r

^{2}− 2r + 4

( 4x − 5 )( 8x + 7 )

- Foil:( 4x )( 8x ) + ( 4x )( 7 ) + ( − 5 )( 8 ) + ( − 5 )( 7 )
- 32x
^{2}+ 28x − 40 − 35

^{2}+ 28x − 75

( 6p + 12 )( 10p − 8 )

- ( 6p )( 10p ) + ( 6p )( − 8 ) + ( 12 )( 10p ) + ( 12 )( − 8 )
- 60p
^{2}− 48p + 120p − 96

^{2}+ 120p − 144

( c − 12 )( 3c + 2 )

- ( c )( 3c ) + ( c )( 2 ) + ( − 12 )( 3c ) + ( − 12 )( 2 )
- 3c
^{2}+ 2c − 36c − 24

^{2}− 34c − 24

( 5a + 6b )( 7a − 9b )

- ( 5a )( 7a ) + ( 5a )( − 9b ) + ( 6b )( 7a ) + ( 6b )( − 9b )
- 35a
^{2}− 45ab + 42ab − 54b^{2}

^{2}− 3ab − 54b

^{2}

( 12x − 8y )( 9x − 11y )

- ( 12x )( 9x ) + ( 12x )( − 11y ) + ( − 8y )( 9x ) + ( − 8y )( − 11y )
- 108x
^{2}− 132xy − 72xy + 88y^{2}

^{2}− 204xy + 88y

^{2}

( 4m − 9n )( 7m + 6n )

- ( 4m )( 7m ) + ( 4m )( 6n ) + ( − 9n )( 7m ) + ( − 9n )( 6n )
- 28m
^{2}+ 24mn − 63mn − 54n^{2}

^{2}− 39mn − 54n

^{2}

*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

### Multiplying Polynomials

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
- Multiplying Polynomials 0:41
- Distributive Property
- Example 1 2:49
- Multiplying Polynomials Cont. 8:22
- Organize Terms with a Table
- Example 2 13:40
- Multiplying Polynomials Cont. 16:33
- Multiplying Binomials with FOIL
- Example 3 18:49
- Example 4 20:04
- Example 5 21:42

### Algebra 1 Online Course

### Transcription: 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 polynomial*0015

*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 multiplied*0044

*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 2x ^{4} that monomial one term and multiplied by 3x^{2} + 2x – 5.*0075

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

*We will have 2x ^{4} × 3x^{2}, then we will have 2x^{4} × 2x, and 2x^{4} × -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 x ^{4} and x^{2} by adding their exponents together x^{6}.*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 x ^{4} and x^{1} power = x^{5}.*0137

*At the very end, 2 × -5 = -10 and we will just keep the x ^{4}.*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 m ^{3} 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 m ^{3} gets multiplied by 2m^{2}.*0198

*Then I will have m ^{3} × 4m and m^{3} × 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 × 2m ^{2}, 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 × 2m ^{3}, 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 m ^{3} × 2m^{2}, adding exponents that would be 2m^{5}.*0302

*Now I have m ^{3} × 4m so add those exponents, and you will get 4m^{4}.*0313

*Onto m ^{3} × 3 = 3m^{3} and now we continue down the list over here.*0324

*-2m × 2m ^{2}, well 2 × -2 = -4 then add the exponents and I will get m^{3}, that will take care of that.*0333

*We will go to this guy -2m × 4m = -8m ^{2}.*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 2m ^{2}, 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 2m ^{5} and it looks like that is the only m^{5}, it has no other like terms to combine.*0387

*We will go on to m ^{4}, let us see what do we got for that.*0397

*I think that is the only one, so m ^{4}.*0403

*3m ^{3} looks like I have a couple of m^{3}, 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

*2m ^{5}, since it is the only one.*0436

*4m ^{4}, since this the only one, let us check this after them.*0440

*Now I have 2m ^{3}, 3 – 4=-1m^{3} and that will take care of those ones.*0446

*-8m ^{2} + 2m = -6m^{2}.*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 together*0503

*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 x ^{2} + 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 × x ^{2} = -4x^{2}.*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 = 3x ^{2}.*0629

*And one more x × x ^{2} = x^{3}.*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 - 1x ^{2} and at the very beginning x^{3}.*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 x ^{2}, 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 × x ^{2}.*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 × x ^{2} = x^{3}.*0722

*x × 3x = 3x ^{2} and x × 5 = 5x.*0728

*It looks pretty good.*0738

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

*-4 × x ^{2} = -4x^{2}, - 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 x ^{3} that is my only x^{3} so I will just write it all by itself.*0768

*But I have a couple of x ^{2}'s so I will write both of those and combine them together, -4x^{2} + 3x =-x^{2}.*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 m ^{3} – 2m and 1, 2m^{2}, 4m and 3.*0848

*Let us fill in the boxes.*0864

*2m ^{2} + m^{3} = 2m^{5}, 2 × -2 =-4m^{3}, 1 × 2m^{2} = 2m^{2}.*0865

*On to the next row, 4m ^{4}, 4 × -2 = -8m^{2}, 4m × 1 = 4m.*0880

*Last row, 3 × m ^{3} = 3m^{3}, 3 × -2 =-6m and 3 × 1 =3.*0896

*Let us go through and start combining everything.*0909

*I have a 2m ^{5} I will write that as our first term, 2m^{5}.*0911

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

*We can call that one done.*0930

*3m ^{3} – 4m^{3}, two of those I need to combine, that will be -1m^{3}.*0934

*I’m onto my squares, -8m ^{2} + 2m^{2} = -6m^{2}, -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 - m ^{3} 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 x ^{2}.*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 terms*1101

*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 x ^{2} - 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 = y ^{2}.*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, -3x ^{2}.*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, 8y ^{2} + 14xy - 3x^{2}.*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 x ^{2} – 2xy and 3y^{2}.*1340

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

*x ^{2} × x =x^{3}, x^{2} × -5y = -5x^{2}y.*1364

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

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

*One more row, 3y ^{2} × x =3xy^{2}.*1401

*I have 3y ^{2} × -5y -15y^{3}.*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 x ^{3} that we can put with this one.*1429

*It look like it is all by its lonesome, we will just say x ^{3}.*1435

*We are looking for x ^{2}y, they must have x^{2} 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 x ^{2}y terms.*1453

*Continuing on, I have an xy ^{2}.*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= 13xy ^{2}.*1468

*That takes care of those terms.*1482

*One more -15y ^{3}, I will put it in -15y^{3}.*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

1 answer

Last reply by: Professor Eric Smith

Tue Dec 30, 2014 3:45 PM

Post by Mohammed Jaweed on December 30, 2014

how do you combine the terms correctly

0 answers

Post by patrick guerin on July 11, 2014

Thank you for the lecture.