### Adding & Subtracting Polynomials

- A polynomial is a term or a finite sum of terms in which all variables have whole number exponents and no variables appear in the denominator.
- Polynomials are classified by how many terms they have, and the largest power present.
- To evaluate a polynomial we substitute a value into all variables, and then simplify.
- To add or subtract polynomials together, we add or subtract the like terms of each of them.
- When adding or subtracting it is often helpful to line up the like terms.

### Adding & Subtracting Polynomials

( 3x

^{2}âˆ’ 5x âˆ’ 8 ) + ( 5x

^{4}âˆ’ 9x

^{2}+ x )

- ( 3x
^{2}âˆ’ 9x^{2}) + ( âˆ’ 5x + x ) + 5x^{4}âˆ’ 8 - âˆ’ 6x
^{2}âˆ’ 4x + 5x^{4}âˆ’ 8

^{4}âˆ’ 6x

^{2}âˆ’ 4x âˆ’ 8

( 6y

^{3}+ 3y

^{2}âˆ’ 7y + 2 ) + ( 4y

^{2}+ 10y )

- ( 3y
^{2}+ 4y^{2}) + ( âˆ’ 7y + 10y ) + 6y^{3}+ 2 - 7y
^{2}+ 3y + 6y^{3}+ 2

^{3}+ 7y

^{2}+ 3y + 2

( x

^{4}+ x

^{3}âˆ’ x

^{2}+ 2x ) + ( 12x

^{4}âˆ’ 3x

^{3}+ 6x + 14 )

- ( x
^{4}+ 12x^{4}) + ( x^{3}âˆ’ 3x^{3}) + ( 2x + 6x ) âˆ’ x^{2}+ 14 - 13x
^{4}âˆ’ 2x^{3}+ 8x âˆ’ x^{2}+ 14

^{4}âˆ’ 2x

^{3}âˆ’ x

^{2}+ 8x + 14

( 10x

^{2}âˆ’ 4x

^{4}+ 5x

^{3}âˆ’ 18 ) âˆ’ ( 5x

^{3}+ 9x âˆ’ x

^{2})

- ( 10x
^{2}âˆ’ 4x^{4}+ 5x^{3}âˆ’ 18 ) + ( âˆ’ 5x^{3}âˆ’ 9x + x^{2}) - ( 10x
^{2}+ x^{2}) + ( 5x^{3}âˆ’ 5x^{3}) + ( âˆ’ 4x^{4}) + ( âˆ’ 9x ) âˆ’ 18 - ( 11x
^{2}) + 0 âˆ’ 4x^{4}âˆ’ 9x âˆ’ 18

^{4}+ 11x

^{2}âˆ’ 9x âˆ’ 18

( 3d

^{2}âˆ’ 4d

^{3}+ 6d ) âˆ’ ( 7d + 5d

^{3}âˆ’ 11d

^{2})

- ( 3d
^{2}âˆ’ 4d^{3}+ 6d ) + ( âˆ’ 7d âˆ’ 5d^{3}+ 11d^{2}) - ( 3d
^{2}+ 11d^{2}) + ( âˆ’ 4d^{3}âˆ’ 5d^{3}) + ( 6d âˆ’ 7d ) - 13d
^{2}âˆ’ 9d^{3}âˆ’ d

^{3}+ 13d

^{2}âˆ’ d

( 4g

^{3}+ 2g

^{5}âˆ’ 8g

^{2}) âˆ’ ( 6g

^{4}+ 14g

^{2}âˆ’ 18g

^{3})

- ( 4g
^{3}+ 2g^{5}âˆ’ 8g^{2}) + ( âˆ’ 6g^{4}âˆ’ 14g^{2}+ 18g^{3}) - ( 4g
^{3}+ 18g^{3}) + ( âˆ’ 8g^{2}âˆ’ 14g^{2}) + 2g^{5}âˆ’ 6g^{4} - 22g
^{3}âˆ’ 22g^{3}+ 2g^{5}âˆ’ 6g^{4}

^{5}âˆ’ 6g

^{4}+ 22g

^{3}âˆ’ 22g

^{3}

( 3m

^{2}âˆ’ m + 5 ) âˆ’ ( 6m âˆ’ 12m

^{2}) + ( 20m

^{2}âˆ’ 10m âˆ’ 1 )

- ( 3m
^{2}âˆ’ m + 5 ) + ( âˆ’ 6m + 12m^{2}) + ( 20m^{2}âˆ’ 10m âˆ’ 1 ) - ( 3m
^{2}+ 12m^{2}+ 20m^{2}) + ( âˆ’ m âˆ’ 6m âˆ’ 10m ) + ( 5 âˆ’ 1 )

^{2}âˆ’ 17m + 4

( 4s

^{3}âˆ’ 16s

^{4}+ 3s ) + ( 7s

^{2}âˆ’ 9s

^{3}+ s ) âˆ’ ( 11s

^{4}âˆ’ 2s

^{2}+ 8s

^{3}+ 15 )

- ( 4s
^{3}âˆ’ 16s^{4}+ 3s ) + ( 7s^{2}âˆ’ 9s^{3}+ s ) + ( âˆ’ 11s^{4}+ 2s^{2}âˆ’ 8s^{3}âˆ’ 15 ) - ( 4s
^{3}âˆ’ 9s^{3}âˆ’ 8s^{3}) + ( âˆ’ 16s^{4}âˆ’ 11s^{4}) + ( 3s + s ) + ( 7s^{2}+ 2s^{2}) âˆ’ 15 - âˆ’ 13s
^{3}âˆ’ 27s^{4}+ 4s + 9s^{2}âˆ’ 15

^{4}âˆ’ 13s

^{3}+ 9s

^{2}+ 4s âˆ’ 15

( 3c

^{2}âˆ’ 18c

^{3}+ 14c ) âˆ’ ( 12c + 15c

^{2}âˆ’ 16c ) + ( 7c

^{3}âˆ’ 10c

^{2})

- ( 3c
^{2}âˆ’ 18c^{3}+ 14c ) + ( âˆ’ 12c âˆ’ 15c^{2}+ 16c ) + ( 7c^{3}âˆ’ 10c^{2}) - ( 3c
^{2}âˆ’ 15c^{2}âˆ’ 10c^{2}) + ( âˆ’ 18c^{3}+ 7c^{3}) + ( 14c âˆ’ 12c + 16c ) - âˆ’ 22c
^{2}âˆ’ 9c^{3}+ 18c

^{3}âˆ’ 22c

^{2}+ 18c

( 5t

^{5}âˆ’ 8t

^{3}+ 2t âˆ’ 9 ) âˆ’ ( 4t

^{2}âˆ’ 7t + 13 + 21t âˆ’ t

^{2}) + ( 9t

^{3}+ t

^{5}âˆ’ 17 + 16t )

- ( 5t
^{5}âˆ’ 8t^{3}+ 2t âˆ’ 9 ) + ( âˆ’ 4t^{2}+ 7t âˆ’ 13 âˆ’ 21t + t^{2}) + ( 9t^{3}+ t^{5}âˆ’ 17 + 16t ) - ( 5t
^{5}+ t^{5}) + ( âˆ’ 8t^{3}+ 9t^{3}) + ( 2t + 7t âˆ’ 21t + 16t ) + ( âˆ’ 4t^{2}+ t^{2}) + ( âˆ’ 9 âˆ’ 13 âˆ’ 17 ) - 6t
^{5}+ t^{3}+ 4t âˆ’ 3t^{2}âˆ’ 39

^{5}+ t

^{3}âˆ’ 3t

^{2}+ 4t âˆ’ 39

*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

### Adding & Subtracting 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 0:07
- Adding and Subtracting Polynomials 0:25
- Terms
- Coefficients
- Leading Coefficients
- Like Terms
- Polynomials
- Monomials, Binomials, Trinomials, and Polynomials
- Degrees
- Evaluating Polynomials
- Adding and Subtracting Polynomials Cont. 9:25
- Example 1 11:48
- Example 2 13:00
- Example 3 14:41
- Example 4 16:15

### Algebra 1 Online Course

### Transcription: Adding & Subtracting Polynomials

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

*In this lesson we are going to take a look at how you can add and subtract polynomials.*0002

*Before we get too far though or have to say what polynomials are and how we can classify them.*0009

*I think we will finally get into adding and subtracting them.*0015

*To watch for along the way, I will cover how you can evaluate polynomials for several different values.*0018

*Recall some vocabulary that we had earlier.*0029

*A term is a piece that is either connected using addition or subtraction.*0033

*In this little example down below, this expression I have four terms.*0040

*A coefficient is the number in front of the variable.*0051

*The coefficient of this first term would be a 4 and I have a coefficient of 6, - 5 and 8 would be a coefficient.*0055

*We often like to organize things from the highest power to the smallest power.*0068

*The highest power here, it is a special name we will call this the leading coefficient.*0073

*This will be important terms you will often here me use later on when talking about polynomials.*0083

*Recall that things are like terms if they have the exact combination of variables with the same exponents.*0092

*In this giant list here, I have lots and lots of examples of like terms.*0100

*This first one is an example of like terms because they both have an m ^{3} and notice it has no difference what the coefficient out front is.*0106

*I have 19, 14 but it does not matter that part.*0117

*What does matter is that I they both have an m ^{3}.*0121

*The next two have both y ^{9} and if you have more than one variable in there then both of those variables better matchup.*0127

*Both of them have a single x and they both have a y ^{2}.*0136

*If we understand polynomials and understand terms then you can start understanding polynomials.*0144

*A polynomial is a term or a finite sum of terms in which all the variables have whole number exponents and no variables up here in the denominator.*0150

*To make this a little bit more clear, I have many different examples of polynomials and many different examples of things that are not polynomials.*0160

*I will pick these over and make sure that they fit the definition.*0168

*This first one here, I know that it is a polynomial as I can see that has a finite number of terms that means that stops eventually.*0174

*If I look at all the exponents present like the two, then all of those are nice whole numbers and I do not see anything in the denominator.*0182

*In fact, there are no fractions there and we do not have any variables in the denominator.*0191

*That is why that one is a polynomial.*0196

*I will put in this next example to highlight that you could have coefficient that are fractions*0201

*but the important part is that we do not have variables in the denominator, that would make it not a polynomial.*0206

*It is okay if we have more than one variable like start mixing around m and p, just as long as the exponents on those state nice whole numbers.*0216

*Even the next one is a good example of this.*0227

*I have y and x, but I have a finite number and eventually stops and it looks like I do not have any variables in the bottom.*0229

*Now a lot of things get to be a polynomial, even very small things.*0239

*For example, something like 4x is a type of polynomial, it is not a very big polynomial and has exactly one term, but is finite.*0245

*All of the exponents are nice whole powers and there are no variables in the denominator.*0253

*You can have even very, very short polynomials.*0259

*This one is just the number 5 has no variables, or even consider it x ^{0}.*0261

*Compare these ones to things that are not polynomials and watch how they break the definition in some sort of way.*0267

*In this first one is not very big, it is 1 Ã· x and it is not polynomial because we are dividing by x.*0276

*We do not want that x in the bottom.*0282

*This next one is not a polynomial because of its exponents.*0286

*It has an exponent of Â½ and another exponent of 1/3 and because of those exponents, it is not a polynomial.*0290

*The next one is a little tricky.*0299

*It looks like it should be a polynomial, I mean I have 1, 2x to the first and 3x ^{2}.*0301

*The reason why this is not a polynomial has to do with this littleâ€¦*0307

*That indicates that this keeps going on and on forever.*0311

*In order to be a polynomial, it should stop.*0315

*It should be finite somewhere.*0318

*That one is not a polynomial.*0319

*Other things that we want to watch out for is we do not have any of our variables and roots and none of those exponents should be negative.*0323

*That should give you a better idea of when something is a polynomial or not a polynomial.*0334

*We can start to classify the types of polynomials we have by looking at two aspects.*0342

*One is how many terms they have.*0348

*If it only has one term we would call that a monomial.*0351

*In example 3x, it only has a single term, it is a monomial.*0357

*If it has two terms then we can call that one a binomial, think of like a bicycle or something like that, two terms.*0363

*If I have 3 terms we will go ahead and call it a trinomial.*0374

*Usually if it has four or more terms you will get a little bit lazy we just usually call those as polynomials.*0384

*Technically, all of these are examples of polynomials but they are just a very specific type of polynomial.*0391

*Let me write that one with a bunch of different terms, 5x ^{4} - 3x^{3} + x^{2} - x +7 that would be a good example of a polynomial.*0398

*In addition to talking about the monomial and binomial, that fun stuff, you can also talk about the degree of a polynomial.*0413

*What the degree is, it is the highest power of any nonzero terms.*0421

*You are looking for that biggest exponent.*0426

*In this first one here, you can see that the largest exponent is 2, we would say that this is a 2nd degree polynomial.*0429

*In the next one, just off to the right, the largest power in there is a 3, so this would be a 3rd degree polynomial.*0440

*Now you can combine these two schemes together and get specific on the types of polynomials you are talking about.*0452

*Not only for that first one can I say it is a 2nd degree because it has the largest power of 2 in there,*0459

*but I can say it is a 2nd degree binomial because it has two terms in it.*0465

*With the other one in addition to saying it is a 3rd degree polynomial, I can take a little further and say it is a 3rd degree trinomial.*0473

*That is because we have 1, 2, 3 terms present in a polynomial.*0484

*To evaluate a polynomial it is a lot like evaluating functions.*0494

*We simply take the value that we are given and we substitute it for all copies of the variables.*0498

*Let us give that a try with 2y ^{3} + 8y - 6 and we want to evaluate it for y = -1.*0504

*I will go through and everywhere I see a copy of y, we will put in that -1.*0513

*Let us go ahead and work on simplifying this.*0527

*-1 ^{3} would be -1 Ã— -1Ã— -1,that would simply be -1.*0532

*8 Ã— -1 would be -8 and then we have -6.*0539

*Continuing on, I have -2 â€“ 8 â€“ 6 = -16.*0546

*If you are evaluating it, just take that value and substitute it in for all copies of that variable.*0557

*Onto what we wanted to, adding and subtracting polynomials.*0568

*When we get into addition and subtraction, what we are looking to do is add or subtract the like terms present in the polynomial.*0573

*There are two ways you should go about this and both of them are perfectly valid so you will use whichever method you are more comfortable with.*0581

*Here I want to add the following two polynomials.*0590

*The way Iâ€™m going to do this is I'm going to simply highlight which terms are like terms.*0593

*Here is 4x ^{3} and 6x^{3} those are like terms - 3x^{2} 2x^{2}, those are like.*0599

*2x and - 3x and so one by one we will take these like terms and simply put them together.*0610

*4x ^{3} and 6x^{3} = 10x^{3}.*0619

*-3x ^{2} and 2x^{2} would be -1x^{2}.*0626

*2x - 3x =- 1x.*0633

*You can see I have all of the parts there and it looks like I am left with a 3rd degree trinomial.*0640

*The other way that you can do this, if you are a little bit more comfortable with it, is you can take one polynomial over the other polynomial.*0645

*If you do it this way, you want to make sure you line up what are your like terms.*0660

*Put your x ^{3} on your x^{3}, your x^{2} on your x^{2} and you x on your x.*0665

*It is essentially the same idea and that you go through adding all of the terms as you go along.*0672

*I will start over on the right side of this one, 2x and I'm adding - 3x, there is - x for that one.*0678

*Onto the next set of terms, - 3x ^{2} + 2x^{2} = - x^{2} and 4x^{3} and 6x^{3} = 10x^{3}.*0688

*You can see you get exactly the same answer but just use whatever method you are more familiar with.*0701

*Let us try some examples.*0711

*In this one we want to determine if these are polynomials or not.*0712

*Let us try that first one.*0718

*I see I have 2x + 3x ^{2} â€“ 8x^{3}.*0719

*The things we are watching out for is, one does it stop.*0723

*I do not see anyâ€¦ out here, I know this is finite, that looks pretty good.*0727

*I do not see any variables in the denominator, so that is good.*0732

*There are no fractions with the x's in the bottom.*0736

*All of the exponents here, the 1, 2, and the 3 all of those are nice whole numbers.*0740

*This one is looking good.*0747

*I will say that this one is a polynomial.*0749

*2/x + 5/4x ^{2} â€“ x^{3}/6.*0757

*In this one I can think I can see a problem right away.*0762

*Notice how we have variables in the bottom and because of that I will say that this one is not a polynomial.*0765

*Be careful and watch out for that criteria.*0776

*This next example, we want to just go through and classify what types of polynomials these are.*0782

*We use two criteria for this, we look at its degree and we will see how many terms it has.*0788

*The first one, the largest power I can see in here is this 3.*0794

*I will say this is a 3rd degree.*0798

*It is a polynomial but let us be a little more specific.*0811

*It has 1, 2, 3 terms, so I will say this is a 3rd degree trinomial.*0814

*Let us try another one, the largest power here is 4, 4th degree.*0826

*It only has 1, 2 terms, so it will be our binomial.*0838

*One more to classify, this one has a bunch of different exponents, but of course the largest one is the only one we are interested in.*0847

*This is a 3rd degree and now we count up all of its terms, 1, 2, 3, 4.*0856

*Since it has four terms, I will just keep calling this one a polynomial.*0870

*Let us get into adding the following polynomials together.*0883

*The way Iâ€™m going to do this is Iâ€™m going to line them up, one on top of the other.*0886

*Starting with the first one, I have 3x ^{3}, I do not have any x^{2}, I have 4x and I have 1.*0892

*All of that would represent my first polynomial there.*0905

*Below that I want to write the second polynomial but I want to lineup the terms, -3x ^{2} I will put it under the other x^{2}.*0910

*I have a 6x I will put that under the other x and the 6 I will go ahead and put that with the 1.*0922

*You can now have things all nice and lined up.*0931

*Let us go ahead and add them one at a time.*0934

*1 + 6 = 7, -4 + 6x =2x, 0x ^{2} + -3x^{2} = -3x^{2}.*0938

*One more, this has nothing to add to it so just 3x ^{2}.*0961

*This would be our completed polynomial after adding the two together.*0967

*One last example is we are going to work on subtracting polynomials.*0977

*With these ones, what I suggest is being very careful with your signs.*0981

*You will see that Iâ€™m going to start with lining one on top of the other one.*0985

*But Iâ€™m going to end up distributing my negative signs, I will turn this into an addition problem.*0989

*Let us give it a try.*0995

*The polynomial on the left is 7y ^{2} -11y + 8 and right below that is -3y^{2} + 4y + 6.*0996

*Now comes the important part, we are subtracting these polynomials so I will put a giant minus sign up front.*1015

*Before getting too far, I could go through and try and subtract this term by term,*1023

*but it is much easier to distribute my negative sign and just look at this like an addition problem.*1028

*My top polynomial will stay unchanged and we will leave that as it is.*1036

*After distributing the bottom, here is what we get negative Ã— - 3y = 3y ^{2}, negative Ã— 4y = -4y and for the very last one -6.*1043

*We can take care of this as an addition problem and we know that the subtraction is taken care of because we put it into all of our terms.*1060

*8 + -6 = 2, -11y + -4y = -15y and then I have a 7y ^{2} + 3y^{2} = 10y^{2}.*1071

*This final polynomial represents the two being subtracted.*1098

*Now you know a little bit more about polynomials and have put them together.*1105

*Remember that you are just combining your like terms.*1109

*Thank you for watching www.educator.com.*1112

1 answer

Last reply by: Professor Eric Smith

Fri Aug 26, 2016 7:12 PM

Post by Delores Sapp on January 2, 2016

I am having difficulty understanding how the answer to the following was derived:

The instructions say to state the degree Â for these polynomials:

4r^4 - 2r^5+1

Â

6x^2 +2x -4

The answer in the math book has the answer to the first example as degree 4 and the answer to the second example as degree 3. Please explain the process and rationale for determining the answers.

I thought that the largest exponent determined the degree except in monomials where the exponents are added.. Clarify my confusion. Thanks.

2 answers

Last reply by: patrick guerin

Sun Jul 13, 2014 7:08 PM

Post by patrick guerin on July 6, 2014

Why is it that variables can only be in the numerator? Is there a reason for that or is that just the definition for the a polynomial. Also thanks for the lecture.

4 answers

Last reply by: Khanh Nguyen

Fri Oct 9, 2015 5:19 PM

Post by Khanh Nguyen on June 11, 2014

You do teach very well but in "Evaluating Polynomials" you forgot to move the exponent down.

Could you fix that?

Thx

1 answer

Last reply by: Professor Eric Smith

Mon Aug 19, 2013 1:30 PM

Post by Jeremy Canaday on August 16, 2013

I must say that you teach this area of algebra extremely well. I bought the book Practical Algebra a few weeks ago and it ruined and confused my confidence in Math. You have restored my faith in this subject.