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### Adding & Subtracting Rational Expressions

- To add or subtract rational expressions, they must have a common denominator. If they do have a common denominator, then we simply add or subtract the numerators of each.
- To find a common denominator for the rational expression begin by factoring each denominator. Then list all the factors that are different, followed by the ones that are the same.
- When writing a rational expression so that it has the common denominator, multiply its missing factor on the top and bottom.
- When subtracting rational expressions, remember to subtract the entire second expression. Often times using parenthesis and distributing the sign will help in this process
- Once you are done adding or subtracting, check to see if the new polynomial in the numerator can be factored. If it can, you may be able to simply a bit further.

### Adding & Subtracting Rational Expressions

- [(13 + 8)/3c]
- [21/3c]

- [(22 + 10)/6t]
- [32/6t]

- [(15 âˆ’ 9)/2q]
- [6/2q]

- [(4k + 3 + 6k âˆ’ 11)/(2k âˆ’ 1)]

- [(8s + 9 âˆ’ ( 5s âˆ’ 2 ))/(3s + 2)]

- [(7d + 5 + 11d âˆ’ 5)/3d]
- [18d/3d]

- [(6y + 12)/(y + 2)]
- [(6( y + 2 ))/(y + 2)]

- [(6x + 4)/(3x + 2)]
- [(2( 3x + 2 ))/(3x + 2)]

- [(21z âˆ’ 18)/(7z âˆ’ 6)]
- [(3( 7z âˆ’ 6 ))/(7z âˆ’ 6)]

^{2}+ 11)/(5w âˆ’ 4)] âˆ’ [(w

^{2}âˆ’ 12)/(4 âˆ’ 5w)]

- [(w
^{2}+ 11)/(5w âˆ’ 4)] âˆ’ [(w^{2}âˆ’ 12( âˆ’ 1 ))/(4 âˆ’ 5w( âˆ’ 1 ))] - [(w
^{2}+ 11)/(5w âˆ’ 4)] âˆ’ [(( âˆ’ w^{2}+ 12 ))/(5w âˆ’ 4)] - [(w
^{2}+ 11w + w^{2}âˆ’ 12)/(5w âˆ’ 4)] - [(5w
^{2}+ 11w âˆ’ 12)/(5w âˆ’ 4)] - [(( 5w âˆ’ 4 )( w + 3 ))/(5w âˆ’ 4)]

^{2}âˆ’ 11x + 30 and x

^{2}âˆ’ 25

- Factor: x
^{2}âˆ’ 11x + 30 = (x âˆ’ 5)(x âˆ’ 6) - Factor: x
^{2}âˆ’ 25 = (x + 5)(x âˆ’ 5)

^{2}+ 6x âˆ’ 16 and x

^{2}âˆ’ 4

- Factor: x
^{2}+ 6x âˆ’ 16 = (x âˆ’ 2)(x + 8) - Factor: x
^{2}âˆ’ 4 = (x + 2)(x âˆ’ 2)

^{2}âˆ’ 3x âˆ’ 70 and x

^{2}âˆ’ 49

- Factor: x
^{2}âˆ’ 3x âˆ’ 70 = (x âˆ’ 10)(x + 7) - Factor: x
^{2}âˆ’ 49 = (x + 7)(x âˆ’ 7)

[2x/(4x âˆ’ 4)] + [6/(5x âˆ’ 5)]

- [2x/(4(x âˆ’ 1))] + [6/(5(x âˆ’ 1))]
- ( [5/5] )[2x/(4(x âˆ’ 1))] + ( [4/4] )[6/(5(x âˆ’ 1))]
- [10x/(20(x âˆ’ 1))] + [24/(20(x âˆ’ 1))]
- [(10x + 24)/(20(x âˆ’ 1))]
- [(2(5x + 12))/(2(10)(x âˆ’ 1))]

- [3x/(5(x + 3))] + [18/(7(x + 3))]
- [21x/(35(x + 3))] + [90/(35(x + 3))]

- [4n/(2(n âˆ’ 7))] âˆ’ [10/(9(n âˆ’ 7))]
- [36n/(18(n âˆ’ 7))] âˆ’ [20/(18(n âˆ’ 7))]
- [(36n âˆ’ 20)/(18(n âˆ’ 7))]
- [(4(9n âˆ’ 5))/(18(n âˆ’ 7))]

- [4c/(6(c âˆ’ 6))] âˆ’ [( âˆ’ 8)/(4(c âˆ’ 6))]
- [8c/(12(c âˆ’ 6))] âˆ’ [( âˆ’ 24)/(12(c âˆ’ 6))]
- [(8c + 24)/(12( c âˆ’ 6 ))]
- [(4( 2c + 6 ))/(4g3( c âˆ’ 6 ))]

^{2}âˆ’ 8x âˆ’ 10)]

- [(x + 1)/(2x + 2)] âˆ’ [(x + 3)/(( 2x + 2 )( x âˆ’ 5 ))]
- [(( x âˆ’ 5 )g( x + 1 ))/(( x âˆ’ 5 )g( 2x + 2 ))] âˆ’ [(x âˆ’ 3)/(( 2x + 2 )( x âˆ’ 5 ))]
- [(x
^{2}âˆ’ 4x âˆ’ 5)/(( 2x + 2 )( x âˆ’ 5 ))] âˆ’ [(x + 3)/(( 2x + 2 )( x âˆ’ 5 ))] - [(x
^{2}âˆ’ 4x âˆ’ 5 âˆ’ x âˆ’ 3)/(( 2x + 2 )( x âˆ’ 5 ))]

^{2}âˆ’ 5x âˆ’ 8)/(( 2x + 2 )( x âˆ’ 5 ))]

^{2}âˆ’ 11x âˆ’ 4)]

- [(x + 1)/(3x âˆ’ 1)] âˆ’ [(x + 12)/(( 3x âˆ’ 1 )( x + 4 ))]
- [(( x + 4 )g( x + 1 ))/(( x + 4 )g( 3x âˆ’ 1 ))] âˆ’ [(3x + 12)/(( 3x âˆ’ 1 )( x + 4 ))]
- [(x
^{2}+ 5x + 4)/(( 3x âˆ’ 1 )( x + 4 ))] âˆ’ [(3x + 12)/(( 3x âˆ’ 1 )( x + 4 ))] - [(x
^{2}+ 5x + 4 âˆ’ 3x âˆ’ 12)/((3x âˆ’ 1)( x + 4 ))] - [(x
^{2}+ 2x âˆ’ 8)/(( 3x âˆ’ 1 )( x + 4 ))]

^{2}âˆ’ 29y âˆ’ 42)] + [(y âˆ’ 8)/(( 5y + 6 ))]

- [(4y + 2)/(( 5y + 6 )(y âˆ’ 7)] + [(y âˆ’ 8)/(5y + 6)]
- [(4y + 2)/(( 5y + 6 )(y âˆ’ 7))] + [((y âˆ’ 8)g( y âˆ’ 7 ))/((5y + 6)( y âˆ’ 7 ))]
- [(4y + 2)/(( 5y + 6 )(y âˆ’ 7))] + [(y
^{2}âˆ’ 15y + 56)/((5y + 6)( y âˆ’ 7 ))] - [(4y + 2 + y
^{2}âˆ’ 15y + 56)/((5y + 6)( y âˆ’ 7 ))]

^{2}âˆ’ 11y + 58)/((5y + 6)( y âˆ’ 7 ))]

*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 Rational Expressions

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
- Objectives
- Adding and Subtracting Rational Expressions
- Example 1
- Example 2
- Adding and Subtracting Rational Expressions Cont.
- Least Common Denominators
- Transitioning from Fractions to Rational Expressions
- Identifying Least Common Denominators for Rational Expressions
- Subtracting vs. Adding
- Example 3
- Example 4
- Example 5
- Example 6

- Intro 0:00
- Objectives 0:07
- Adding and Subtracting Rational Expressions 0:41
- Common Denominators
- Common Denominator Examples
- Steps to Adding and Subtracting Rational Expressions
- Example 1 3:34
- Example 2 5:27
- Adding and Subtracting Rational Expressions Cont. 6:57
- Least Common Denominators
- Transitioning from Fractions to Rational Expressions
- Identifying Least Common Denominators for Rational Expressions
- Subtracting vs. Adding
- Example 3 11:19
- Example 4 12:36
- Example 5 15:08
- Example 6 16:46

### Algebra 1 Online Course

### Transcription: Adding & Subtracting Rational Expressions

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

*In this lesson we are going to take a look at adding and subtracting rational expressions.*0003

*In order for this process to workout correctly I have to spend a little bit of time working on finding a least common denominator.*0009

*This will help us so we can rewrite our rational expressions and actually get them together.*0017

*Then we will get into the addition process.*0022

*We will look at the some examples that have exactly the same denominator.*0025

*We will look at others which have different denominators.*0029

*Once we know more about the addition process, the subtraction process will not be too bad.*0032

*You will see that it will involve many of the similar things.*0036

*So far we have covered a lot about multiplying and dividing rational expressions,*0044

*but we also need to pick up how we can add and subtract them.*0048

*One key that will help us with this is you want to think of how you add and subtract simple rational number.*0053

*Think of those fractions, how do you add and subtract fractions?*0059

*One key component is that we need a common denominator before we can ever put those fractions together.*0063

*Think of how that will play a part with your rational expressions.*0070

*Let us first look at some numbers okay.*0076

*Let us suppose I had 2/1173 and 5/782 and I was trying to add these or subtract them.*0078

*No matter what I'm trying to do, I have to get a common denominator.*0088

*One thing that will make this process very difficult is that I chose some very large numbers in order to get that common denominator.*0093

*One thing that could help out when searching for this common denominator is not to take the numbers directly since they are so large.*0101

*You have to break them down into their individual factors.*0108

*Down here I have the same exact numbers, but I have broken them down into 3 Ã— 17 Ã— 23.*0112

*I have broken down the 782 into 2 Ã— 17 Ã— 23.*0119

*What this highlights is that the numbers may not be that different after all.*0123

*After all, they both have 17 and 23 in common and the only thing that is different is this one has a 3 and this one has a 2.*0128

*When trying to find that common denominator, I want to make sure it has all the pieces necessary*0139

*or I could have built either one of these denominators.*0145

*It must have the 2, 3 and also those common pieces of the 17 and the 23.*0147

*This is the number that you would have been interested in for getting these guys together.*0153

*When working with the rational expressions, we will be doing the same process.*0161

*We want them to have exactly the same denominator, but I would not be entirely obvious to look at the denominator and know what that is.*0166

*We will have to first factor those denominators, we can see that the pieces present in which ones are common and which ones are not.*0176

*We will go ahead and list out the different factors in the denominator first.*0184

*We will also list out the variables, when it appears the greatest number of times.*0192

*We will list the factors multiplied together to form what is known as the least common denominator.*0200

*Think of the least common denominator as having all the factors we need and we could have built either one of those original denominators.*0206

*Let us take a quick look at how this works with some actual rational expressions.*0216

*I want to first look at factoring the bottoms of each of these.*0221

*8 is the same as 2 Ã— 2 Ã— 2 and y ^{4} we will have bunch of y all multiplied by each other.*0225

*For 12 that would be 3 Ã— 2 Ã— 2 and then a bunch of y, 6 of them.*0236

*When building the least common denominator, I will first gather up the pieces that are not the same.*0245

*Here I have the 3 and this one has an extra 2.*0254

*I need both of these in my least common denominator 2 and 3.*0259

*This one has a couple of extra y.*0267

*Let us put those in there as well.*0270

*Once we have spotted all the differences between the two then we can go ahead and highlight everything that is the same.*0274

*We have a couple more twos and 1, 2, 3, 4 y.*0282

*Let us package this altogether.*0290

*2 Ã— 2 Ã— 2 = 8 Ã— 3 = 24 and then I have 6y, y ^{6}.*0292

*Let us do some shortcuts here.*0303

*One, you could have just figured out the least common denominator of the 8 and the 12 that will help you get the 24.*0305

*You can take the greatest value of y ^{6} and gotten the y^{6}.*0312

*We use techniques like that to help us out when looking for that least common denominator.*0321

*Some of our expressions may get a little bit more complicated than single monomials on the bottom.*0330

*Let us see how this one would work.*0335

*This is 6 /x ^{2} - 4x and 3x â€“ 1/ x^{2} â€“ 6.*0336

* In order to figure out what our common denominator needs to be, we are going to have to factor first.*0343

*Let us start with that one on the left and see if we can factor the bottom.*0350

*It looks like it has a common x in there.*0353

*We will take out and x from both of the parts.*0356

*For the other rational expression that looks like the difference of squares.*0363

*x + 4 and x â€“ 4*0370

*We just have to look with these individual factors.*0376

*I can see that what is different is this x + 4 piece and the x piece.*0381

*Let us put both of those into our common denominator first.*0388

*I have x and x + 4 then we can go ahead and include the pieces that are common.*0392

*Any common pieces will only include once.*0402

*This down here represents what our least common denominator would be.*0406

*We need an x, x + 4, and x -4.*0411

*Finding the least common denominator is only half the battle.*0419

*Once you find the least common denominator, you have to change both of your rational expressions*0422

*so that they contain this least common denominator.*0428

*Once you have identified it go one step farther and rewrite the expressions so that they have this least common denominator.*0433

*Let us watch how this works with our numbers.*0441

*That way we could get a better understanding before we get into the rational stuff.*0443

*Here are these fractions that I had earlier and you will notice that the bottom is already factored.*0447

*Our LCD in this case was 2 Ã— 3 Ã— 17 Ã— 23.*0454

*Now suppose I want them to both have this as their new denominator.*0463

*2 Ã— 3 Ã— 17 Ã— 23, 2 Ã— 3 Ã— 17 Ã— 23.*0472

*When looking at the fraction on the left here the only difference between this and the new LCD that I wanted to have is it is missing a 2.*0483

*I could give it a 2 on the bottom but just to balance things out I will also have to give it a 2 on the top.*0495

*On the top of this one will be 2 Ã— 2.*0504

*I better highlight that this 2 was the one we put in there.*0510

*For the other one, it needs to have that 3, I will give it a 3 on the top and there is where the 3 came from in the bottom.*0515

*You can see that we give the missing pieces to each of the other fractions.*0526

*If I was looking to add or subtract these I will be in pretty good shape since they have exactly the same denominator.*0532

*We want to do the same process with our rational expressions, give to the other rational expressions its missing pieces*0538

*so it can have that least common denominator.*0545

*We can find a least common denominator now, which means we can get to the process of adding and subtracting our rational expressions.*0551

*Think of how this works with our fractions.*0560

*If I have two fractions and I have exactly the same denominator then I will leave that denominator exactly the same*0562

*and I will only add the tops together.*0570

*This works as long as my bottom is not 0.*0574

*This will be the exact same thing that we will do for our rational expressions, the only difference is that this P, Q, and R*0578

*that you see is my nice little example, all of those represent polynomials instead of individual numbers.*0585

*As soon as we get our common denominator we will just add the tops together.*0591

*If they do not already have a common denominator, we have to do a little bit of work.*0599

*It means we want to find a common denominator and often times we will have to factor first before we can identify what that is.*0603

*Then we will have to rewrite the expression so that both of them have this least common denominator.*0613

*Once we have that then we will go ahead and add the numerators together and leave that common denominator in the bottom.*0619

*Even after that we are not necessarily done.*0627

*Always factor at the very end to make sure that you are in the lowest terms.*0629

*Sometimes when we put these together we can do some additional canceling and make it even simpler.*0633

*The subtraction process is similar to the addition process.*0644

*You will go through the process of finding the common denominator.*0650

*Make sure they both have it, and then you will end up just subtracting the tops.*0652

*Remember though you want to subtract away the entire top of the second fraction.*0660

*Often to do this, it is a good idea to use parentheses and distribute through by your negative sign on the top part.*0665

*That way we would not forget any of your signs.*0671

*It is usually a very common mistake when subtracting these rational expressions.*0674

*Let us get to business.*0681

*Let us look at this example and add the rational expressions.*0682

*I have (3x / x ^{2} â€“ 1) + (3 / x^{2} â€“ 1).*0686

*The good news is our denominators are already exactly the same.*0691

*I will simply keep that as my common denominator in the bottom and we will just add the tops.*0697

*Even though we have added this and put into a single rational expression, we are not done.*0707

*We want to make sure that it is in lowest terms.*0711

*Let us go ahead and factor the top and bottom, see if there are any extra factors hiding in there.*0716

*As I factor the top, this will factor into 3 Ã— x +1 and then we can factor the bottom, this will be x + 1 and x â€“ 1.*0722

*I can definitely say yes there is a common piece in there, it is an x +1 and we can go ahead and cancel that out.*0735

*I'm left with a 3 / x - 1 and now I have not only added the rational expressions, this is definitely in lowest terms.*0743

*Let us try this on another one.*0758

*We want to add together the two rational expressions I have (-2/w + 1) + (4w/w ^{2} -1).*0760

*This one is a little bit different.*0769

*The denominators are not the same.*0771

*Let us see if we can figure out what the denominator should have in the bottom by factoring them out and seeing what pieces they have.*0775

*w ^{2} - 1 is the difference of squares which would break down into w + 1 and w -1.*0785

*It looks like that first fraction is missing a w -1.*0796

*We have to give it that missing piece.*0802

*We will write it in blue.*0808

*I will give it an extra w -1 on the bottom and on the top just to make sure it stays the same.*0809

*Our second fraction already has our least common denominator, so no need to change that one.*0819

*Now that it has a common denominator we will keep it on the bottom and we will simply add the numerators together.*0826

*I got -2w -1 + 4w.*0835

*We cannot necessarily leave it like that, Iâ€™m going to go ahead and continue combining the top,*0841

*maybe factor and see if there is anything else I can get rid of.*0846

*Let us distribute through by this -2.*0851

*-2w + 2 + 4w / (w +1) (w -1)*0853

*(-2w + 4w) (2w + 2)*0868

*I think I already see something that will be able to cancel out.*0876

*Let us factor out a 2 in the top.*0880

*Sure enough, we have a w + 1 in the top and bottom that we can get rid of.*0887

*That is gone.*0895

*Our final expression here is 2 / w -1 and now we have added the two together and brought it down to lowest terms.*0898

*Let us do a little bit of subtraction.*0910

*This one involves (5u /u â€“ 1 â€“ 5) + (u /u -1).*0912

*This is one of our nice examples and that we are starting off and has exactly the same denominator.*0919

*That is good so we can go ahead and just subtract the tops.*0925

*I will have 5u - the other top 5 + u.*0931

*Note what I did there, I still have the entire second top and I put it inside parentheses and I'm subtracting right here.*0940

*One common mistake is not to put those parentheses in there and you will only end up subtracting the 5.*0950

*You do not want to do that.*0956

*You want to subtract away the entire second top.*0956

*To continue on, I want to see if there is anything that might cancel.*0961

*Iâ€™m going to try and crunch together the top a little bit and see if I can factor.*0964

*Let us distribute through by that negative sign.*0969

*5u - 5 - u / u -1*0974

*That will give me 5u â€“ 1 = 4u â€“ 5/ u â€“ 1.*0981

*It looks like the top does not factor anymore.*0995

*This guy is in lowest terms.*0997

*Let us tackle one more last one and these ones involve denominators that are not the same.*1008

*We are going to have to do a lot of work on factoring and seeing what pieces they have before we even get into the subtraction process.*1019

*Let us go ahead and factor the rational expression on the left.*1026

*On the bottom I can see that there is an (a) in common.*1032

*Over on the other side I can factor that into a - 5 and another a â€“ 5.*1042

*They almost have the same denominator.*1056

*They both have that common a - 5 piece and the one on the left has an extra a.*1059

*And one on the right has an additional a â€“ 5.*1063

*Let us give to the other one the missing pieces.*1066

*Here is our rational expression on the left, we will give it an additional a â€“ 5.*1078

*With this one it already has a - 5 twice, we will give it an additional a on the bottom and on the top.*1091

*Now that they have exactly the same denominator we can focus on the tops 3a aâ€“ 5 - 4a /(a â€“ 5) (a- 5).*1104

*It looks like we can do just a little bit of combining on the top.*1123

*I have (3a ^{2} - 15a â€“ 4a) / (a)(a â€“ 5)(a-5)*1127

*When combined together the -15a and the -4a, 3a ^{2} â€“ 19a.*1142

*We have completely subtracted these we just need to worry about factoring and canceling out any extra terms.*1156

*On the top I can see that they both have an extra a in common, 3a-19.*1168

*Let us take that out of the top.*1174

*We will cancel out that a and now we brought it down into lowest terms.*1183

*I have 3a - 19 / (a â€“ 5) (a -5)*1189

*Whether you are adding or subtracting rational expressions make sure that you have your common denominator first.*1202

*Once you do, you just have to focus on the tops of those rational expressions by putting them together.*1209

*Once you do get them together remember you are not done yet, feel free to factor one more time and reduce it to lowest terms.*1215

*Thank you for watching www.educator.com.*1222

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