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### Multiply & Divide Rational Expressions

- A rational expression involves one polynomial being divided by another polynomial
- To simplify a rational expression we want to cancel out common factors in the numerator and denominator. This is similar to simplifying fractions.
- To multiply rational expressions, we multiply across the top and bottom of the fractions.
- To divide rational expressions, flip the second rational expression and multiply.
- It is often best to factor the polynomials first before combining using multiplication. This makes the simplification process a bit easier.

### Multiply & Divide Rational Expressions

[(40a

^{2}b

^{3}c

^{5})/(20a

^{3}b

^{5}c

^{4})] Ã—[(15a

^{4}b

^{3}c

^{2})/(24a

^{2}b

^{5}c

^{3})]

- [(1c
^{2})/(4b^{2}c^{2})] Ã—[3a/(6b^{2})] - [1/(4b
^{2})] Ã—[a/(2b^{2})] - [a/(4b
^{2}Ã—2b^{2})]

^{4})]

[(18u

^{4}v

^{3}w

^{5}x

^{8})/(26v

^{4}w

^{6}x

^{2})] Ã—[(14v

^{5}w

^{2}x

^{4})/(54u

^{3}v

^{6}w

^{7})]

- [(1ux
^{8})/(13w^{4})] Ã—[(7vx^{2})/(3v^{3}w^{2})] - [(ux
^{8})/(13w^{4})] Ã—[(7x^{2})/(3v^{2}w^{5})]

^{10})/(39v

^{2}w

^{9})]

[(42c

^{3}d

^{2}e

^{4})/(60c

^{5}d

^{6}e

^{2})] Ã—[(45c

^{6}e

^{2}f

^{3})/(35d

^{4}ef

^{2})]

- [(6c
^{3})/(4d^{6})] Ã—[(3cf^{3})/(5d^{2}ef^{2})] - [(3c
^{3})/(2d^{6})] Ã—[3cf/(5d^{2}e)]

^{4}f)/(10d

^{8}e)]

[(6y + 36)/(4y âˆ’ 14)] Ã—[(16y + 30)/(2y + 12)]

- [(6( y + 6 ))/(2( 2y âˆ’ 7 ))] Ã—[(2( 8y + 15 ))/(2( y + 6 ))]
- [3/(2y âˆ’ 7)] Ã—[(8y + 15)/1]

[(24n âˆ’ 36)/(10n âˆ’ 35)] Ã—[(16n + 48)/(18n âˆ’ 27)]

- [(12( 2n âˆ’ 3 ))/(5( 2n âˆ’ 7 ))] Ã—[(16( n + 3 ))/(9( 2n âˆ’ 3 ))]
- [12/(5( 2n âˆ’ 7 ))] Ã—[(16( n + 3 ))/9]

[(16b + 44)/(27b âˆ’ 45)] Ã—[(15b âˆ’ 25)/(16b âˆ’ 72)]

- [(4( 4b + 11 ))/(9( 3b âˆ’ 5 ))] Ã—[(5( 3b âˆ’ 5 ))/(8( 2b âˆ’ 9 ))]

[(9x + 12)/(4x + 20)] Ã—[(x

^{2}+ 3x âˆ’ 10)/(3x

^{2}+ x âˆ’ 4)]

- [(3( 3x + 4 ))/(4( x + 5 ))] Ã—[(( x + 5 )( x âˆ’ 2 ))/(( 3x + 4 )( x âˆ’ 1 ))]
- [3/4] Ã—[(x âˆ’ 2)/(x âˆ’ 1)]

[(6x âˆ’ 21)/(6x âˆ’ 24)] Ã—[(5x

^{2}+ 25x + 20)/(2x

^{2}+ x âˆ’ 28)]

- [(3( 2x âˆ’ 7 ))/(6( x + 4 ))] Ã—[(5( x
^{2}+ 5x + 4 ))/(2x^{2}+ x âˆ’ 28)] - [(3( 2x âˆ’ 7 ))/(6( x + 4 ))] Ã—[(5( x + 4 )( x + 1 ))/(( 2x âˆ’ 7 )( x + 4 ))]
- [3/6] Ã—[(5( x + 1 ))/(( x + 4 ))]
- [1/2] Ã—[(5( x + 1 ))/(( x + 4 ))]

[(x

^{2}âˆ’ 12x + 32)/(4x

^{2}âˆ’ 16x âˆ’ 20)] Ã—[(2x

^{2}âˆ’ 50)/(2x

^{2}âˆ’ 15x âˆ’ 8)]

- [(( x âˆ’ 4 )( x âˆ’ 8 ))/(4( x
^{2}âˆ’ 4x âˆ’ 5 ))] Ã—[(2( x^{2}âˆ’ 25 ))/(2x^{2}âˆ’ 15x âˆ’ 8)] - [(( x âˆ’ 4 )( x âˆ’ 8 ))/(4( x + 1 )( x âˆ’ 5 ))] Ã—[(2( x + 5 )( x âˆ’ 5 ))/(( 2x + 1 )( x âˆ’ 8 ))]
- [(x âˆ’ 4)/(4( x + 1 ))] Ã—[(2( x + 5 ))/(2x + 1)]
- [(x âˆ’ 4)/(2( x + 1 ))] Ã—[(x + 5)/(2x + 1)]

[(2x

^{2}âˆ’ 10x âˆ’ 28)/(x

^{2}âˆ’ 2x âˆ’ 80)] Ã—[(5x

^{2}âˆ’ 36x âˆ’ 32)/(6x

^{2}âˆ’ 12)]

- [(2( x
^{2}âˆ’ 5x âˆ’ 14 ))/(x^{2}âˆ’ 2x âˆ’ 80)] Ã—[(5x^{2}âˆ’ 36x âˆ’ 32)/(6( x^{2}âˆ’ 2 ))]

[(12x

^{2}y

^{3}z

^{4})/(18xy

^{4}z)] Ã·[(14x

^{4}yz

^{2})/(81x

^{3}y

^{3}z

^{3})]

- [(12x
^{2}y^{3}z^{4})/(18xy^{4}z)] Ã—[(81x^{3}y^{3}z^{3})/(14x^{4}yz^{2})] - [(6y
^{2}z^{2})/2y] Ã—[(9x^{2}z^{2})/(7x^{2})] - [(3yz
^{2})/1] Ã—[(9z^{2})/7]

^{4})/7]

[(36a

^{2}b

^{4}c)/(27a

^{4}b

^{3}c

^{2})] Ã·[(48a

^{4}b

^{2}c

^{3})/(54a

^{3}b

^{5}c

^{2})]

- [(36a
^{2}b^{4}c)/(27a^{4}b^{3}c^{2})] Ã—[(54a^{3}b^{5}c^{2})/(48a^{4}b^{2}c^{3})] - [(3b
^{2})/1a] Ã—[(2b^{2})/(4a^{2}c^{2})]

^{4})/(4a

^{3}c

^{2})]

[(24j

^{3}k

^{5}i

^{6})/(8j

^{5}k

^{6}i

^{4})] Ã·[(60j

^{5}k

^{2}i

^{3})/(40j

^{5}k

^{2}i)]

- [(24j
^{3}k^{5}i^{6})/(8j^{5}k^{6}i^{4})] Ã·[(40j^{5}k^{2}i)/(60j^{5}k^{2}i^{3})] - [(2i
^{3})/(1k^{4}i^{3})] Ã—[1/jk]

^{5})]

[(4r âˆ’ 28)/(10r âˆ’ 40)] Ã·[(7r âˆ’ 49)/(3r + 15)]

- [(4r âˆ’ 28)/(10r âˆ’ 40)] Ã—[(3r + 15)/(7r âˆ’ 49)]
- [(4( r âˆ’ 7 ))/(10( r âˆ’ 4 ))] Ã—[(3( r + 5 ))/(7( r âˆ’ 7 ))]
- [4/(10( r âˆ’ 4 ))] Ã—[(3( r + 5 ))/7]

[(16t + 40)/(21t âˆ’ 35)] Ã·[(10t âˆ’ 8)/(6t âˆ’ 10)]

- [(16t + 40)/(21t âˆ’ 35)] Ã—[(6t âˆ’ 10)/(10t âˆ’ 8)]
- [(8( 2t + 5 ))/(7( 3t âˆ’ 5 ))] Ã—[(2( 3t âˆ’ 5 ))/(2( 5t âˆ’ 4 ))]
- [(8( 2t + 5 ))/7] Ã—[1/(5t âˆ’ 4)]

[(26x âˆ’ 39)/(14x + 30)] Ã·[(24x âˆ’ 36)/(5x âˆ’ 90)]

- [(26x âˆ’ 39)/(14x + 30)] Ã—[(5x âˆ’ 90)/(24x âˆ’ 36)]
- [(13( 2x âˆ’ 3 ))/(2( 7x + 15 ))] Ã—[(5( x âˆ’ 18 ))/(12( 2x âˆ’ 3 ))]
- [13/(2( 7x + 15 ))] Ã—[(5( x âˆ’ 18 ))/12]

[(6y âˆ’ 20)/(16y + 18)] Ã·[(36y âˆ’ 120)/(8y âˆ’ 36)]

- [(6y âˆ’ 20)/(16y + 18)] Ã—[(8y âˆ’ 36)/(36y âˆ’ 120)]
- [(2( 3y âˆ’ 10 ))/(2( 8y + 9 ))] Ã—[(4( 2y âˆ’ 9 ))/(12( 3y âˆ’ 10 ))]
- [1/(( 8y + 9 ))] Ã—[(2( 2y âˆ’ 9 ))/6]
- [(2( 2y âˆ’ 9 ))/(6( 8y + 9 ))]

[(2h âˆ’ 10)/(4h + 8)] Ã·[(2h

^{2}âˆ’ 8h âˆ’ 10)/(2h

^{2}+ 7h + 6)]

- [(2( h âˆ’ 5 ))/(4( h + 2 ))] Ã—[(2h
^{2}+ 7h + 6)/(2( h^{2}âˆ’ 4h âˆ’ 5 ))] - [(2( h âˆ’ 5 ))/(4( h + 2 ))] Ã—[(( 2h + 3 )( h + 2 ))/(2( h âˆ’ 5 )( h + 1 ))]
- [1/4] Ã—[(2h + 3)/(h + 1)]

[(2x âˆ’ x âˆ’ 15)/(3x + 21)] Ã·[(5x âˆ’ 15)/(2x

^{2}+ 8x âˆ’ 42)]

- [(2x âˆ’ x âˆ’ 15)/(3x + 21)] Ã—[(2x
^{2}+ 8x âˆ’ 42)/(5x âˆ’ 15)] - [(( 2x + 5 )( x âˆ’ 3 ))/(3( x + 7 ))] Ã—[(2( x
^{2}+ 4x âˆ’ 21 ))/(5( x âˆ’ 3 ))] - [(( 2x + 5 )( x âˆ’ 3 ))/(3( x + 7 ))] Ã—[(2( x + 7 )( x âˆ’ 3 ))/(5( x âˆ’ 3 ))]
- [(2x + 5)/3] Ã—[2/5]

[(14y + 42)/(14y + 35)] Ã·[(2y

^{2}+ 4y âˆ’ 6)/(2y

^{2}+ 13y + 20)]

- [(14y + 42)/(14y + 35)] Ã·[(2y
^{2}+ 13y + 20)/(2y^{2}+ 4y âˆ’ 6)] - [(14( y + 3 ))/(7( 2y + 5 ))] Ã—[(( 2y + 5 )( y + 4 ))/(2( y
^{2}+ 2y âˆ’ 3 ))] - [(14( y + 3 ))/(7( 2y + 5 ))] Ã—[(( 2y + 5 )( y + 4 ))/(2( y + 3 )( y âˆ’ 1 ))]
- [7/7] Ã—[(y + 4)/(y âˆ’ 1)]

*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

### Multiply & Divide 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 0:00
- Objectives 0:09
- Multiply and Divide Rational Expressions 0:44
- Rational Numbers
- Dividing by Zero
- Canceling Extra Factors
- Negative Signs in Fractions
- Multiplying Fractions
- Dividing Fractions
- Example 1 8:04
- Example 2 14:01
- Example 3 16:23
- Example 4 18:56
- Example 5 22:43

### Algebra 1 Online Course

### Transcription: Multiply & Divide Rational Expressions

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

*In this lesson we are going to take a look at how you can multiply and divide rational expressions.*0003

*There is a lot to know when it comes to rational expressions.*0011

*We will first start off by looking at what values would be undefined or restricted in some of these rational expressions.*0013

*We will also take quite a bit of time to figure out to how simplify them and put them into lowest terms.*0021

*There will be a few situations where you have to recognize that there are equivalent forms of these different rational expressions.*0028

*They may look different but they are the same.*0035

*Finally we will get into that multiplication and division process so you can see it all put together.*0038

*A lot of things that we will do these rational expressions ties into rational numbers in general.*0046

*In terms of the techniques and how we simplify them.*0052

*Remember that a rational number is any number that can be written as a fraction.*0056

*The number 2/3 is a good example of one of these rational numbers.*0061

*We are working with rational expressions you can consider those of the form p/q when p and q are both going to be polynomials.*0066

*Let me give you some examples of what these rational expressions look like.*0076

*2x + 3 Ã· 5x ^{2} - 7.*0080

*Since both of those are polynomials and they are simply being divided that will be a good rational expression.*0086

*How about x ^{3} - 27 Ã· 4x + 3? That will also be a good example of another rational expression.*0093

*What this rational expression since we have some variables in them, we are always concerned about dividing by 0.*0107

*We do not want use any values for x that would make the bottom 0.*0113

*Think of this expression up here x + 5 Ã· 2x - 4.*0117

*I can plug in many, many different values for my x as long as I do not get a 0 on the bottom.*0123

*In fact if I try and plug in a 2 into this that would give me 0 on the bottom.*0130

*2 Ã— 2 - 4 and that is bit of a problem.*0136

*Since it gives me that 0 the bottom this expression is undefined at 2 or I might simply make a note somewhere and say x can not equal 2.*0141

*For many of the expressions that you will see on the future slides here,*0151

*we will assume that it can take on many different values, but not those restricted values.*0156

*When we are into the multiplying and dividing we will always be concerned with taking our answer down to its simplest form.*0165

*How is it that we go ahead and reduce a fraction to it simplest terms?*0173

*Remember how you do this process with some similar type of rational expressions.*0178

*We will try and cancel out a lot of the extra pâ€™s that are present.*0183

*If Iâ€™m looking at 6x ^{2} Ã· 2x^{2} I could look of this as 2 Ã— 3 Ã— x Ã— x Ã— x and in the bottom would be 2 Ã— x Ã— x.*0188

*As long as I have multiplication there I can simplify it by canceling out a lot of these extra factors and get something like 3x.*0204

*The good news is that type of simplifying is the same process we want to do for rational expressions.*0215

*We want to look for their common factors.*0222

*These are polynomials we will probably have to factor first, so that we have the actual factors in their pâ€™s that are multiplied.*0226

*Watch how that works for this guy 4y + 2 Ã· 6y + 3.*0233

*Let us factor the top and bottom.*0238

*On the top there is a 2 in common and I can take it out from both parts.*0241

*That would be 2y + 1 and on the bottom it looks like there is a 3 in common, we will take that out, 2y + 1.*0246

*Notice we have this common piece 2y + 1 that is multiplied on the top and bottom.*0259

*We can cancel that out and we would just be left with 2/3.*0266

*Remember that we can only cancel out factors, pieces that are multiplied.*0272

*If you attempted to do some canceling at the very beginning, you can not do that, not yet because you are dealing with addition.*0277

*Be very careful as we try and simplify your expressions.*0287

*Some answers may look a little different from other answers, but actually they may just be equivalent.*0294

*One thing that can make equivalent expression is where you put the negative sign.*0301

*You could put it in the top, the bottom or out front of one of these rational expressions.*0306

*In all cases, they represent the same exact quantity.*0313

*You will say that they are all equivalent.*0317

*This also applies to your rational expressions, but sometimes it is not quite as obvious.*0320

*For example, maybe I'm looking at x - 7 Ã· 2.*0328

*Let us go ahead and put a negative sign on the top.*0334

*If I do that, that would be -x and may be distribute it through with that negative + 7 Ã· 2.*0339

*I could have also given that negative sign to the bottom x â€“ 7 Ã· 2 and give that to the bottom and that would give me x - 7 Ã· -2.*0350

*These two expressions are equivalent.*0367

*They do equal the same thing, even though they look a little different.*0369

*Be careful if you are working on your homework, working with other people and you do not get exactly the same answer,*0375

*you might actually have equivalent expressions just inside a different form.*0380

*Now that we know about simplifying and some things to watch out for how do you get into the multiplication and division process.*0389

*I want to think back on how you do this with fractions.*0396

*Before multiplying fractions together and it is a nice process of multiplying across the top and multiplying across the bottom.*0400

*The same thing applies to your rational expressions.*0409

*We will be dealing with polynomials for sure, but will just multiply across the top and across the bottom.*0412

*In order for this to work, we must know how to factor our polynomials.*0418

*That way we can end up just multiplying their factors together.*0423

*When we are all done, we want to make sure that we have written it in lowest terms.*0427

*Try and cancel out any extra factors after you are done multiplying.*0431

*If you know how the multiplication process works, then you will also know a lot about the division process.*0438

*Think about how this works when you divide fractions.*0445

*From looking at something like 2/3 Ã· 5/7 and we have been taught to flip the second fraction and then multiply.*0448

*Which is of course, multiply across the top and multiply across the bottom.*0456

*There is some good news and this also applies to our rational expressions.*0460

*If you want to multiply them together, flip your second rational expression and then multiply the two.*0465

*Factoring will definitely help in this process that way we have to keep track of the individual factors, and where they go.*0472

*Always write your answer in lowest terms when you are all done.*0479

*That is quite a bit of information just on simplifying and multiplying and dividing.*0485

*We will look at some quick examples and see how this works out.*0491

*We want to take all of these rational expressions and put them into lowest terms.*0497

*We will be going through a simplification process.*0501

*Notice how in a lot of these we are dealing with addition and subtraction, do not cancel out yet until you get it completely factored.*0505

*Let us start with the first one.*0512

*I have (x ^{2} - y^{2}) Ã· (x^{2} + 2xy + y^{2}).*0514

*These look like some very special formulas that we had earlier.*0520

*I have the difference of squares on top.*0523

*I have a perfect square trinomial on the bottom so I can definitely factor these.*0525

*I have x + y x â€“ y on the top and on the bottom x + y, x + y.*0533

*Notice how we have a common factor the x + y.*0549

*We can go ahead and cancel that out.*0554

*This will leave us with an x - y Ã· x + y.*0558

*I can be assured that this is in the lowest terms because there are no other common factors to get rid of.*0565

*The next ones are very tricky.*0571

*Notice how the top and bottom almost look like the same thing.*0574

*It is tempting to try and cancel out.*0578

*Be careful, we cannot cancel them out unless they are exactly the same thing.*0581

*One thing to notice here is a -5 and here is 5.*0586

*Those are not the same thing, they are different in sign.*0590

*And same thing over here, this is w ^{2} and this one is â€“w^{2}.*0593

*Those are not the same in sign.*0597

*If you end up with a situation like this where they are almost the same, you are dealing with subtraction and the order is just reversed.*0600

*You can factor out a -1 from either the top or bottom.*0607

*If I factor out a -1 from the top then what is left over?*0614

*-1 Ã— what will give me a w ^{2}, is a -w^{2} and let us see if I take out 5, that should do it.*0620

*-1 Ã— -w = w, -1 Ã— 5 = -5.*0633

*All of that is on top and I still have my 5 - w ^{2} on the bottom.*0641

*We are getting a little bit closer and things are starting a matchup in sign a little bit better.*0646

*Iâ€™m just simply going to reverse the order of these and you will see that they are common factors.*0652

*-1 is still out front, (5 - w ^{2}) (5 + w^{2}) and now I can go ahead and cancel these out.*0657

*The only thing left here is a -1.*0677

*That looks much nicer than what we started with.*0680

* In the next one I have 25q ^{2} - 16/12 â€“ 15q.*0684

*This one is going to take a little bit more work but I see I have one of those special cases on the top.*0691

*That is another difference of squares.*0697

*(5q + 4) (5q â€“ 4)*0700

*Let us see if we can do anything with the bottom.*0712

*Does anything go into 12 and 15?*0714

*These both have a 3 in common, let us take that out.*0718

*We are looking pretty good and we can see that this is getting pretty somewhere to the previous example.*0725

*These look almost the same we are dealing with subtraction, but the order is just reversed.*0732

*We are going to take out a negative from the bottom so that they will be exactly the same.*0736

*(5q + 4) (5q â€“ 4)*0741

*There you will take out the 3, let us take out -1 as well.*0750

*That will give us -4 + 5q.*0753

*It is better starting to match the top we just have to reverse the order.*0758

*Reading on the bottom 5q - 4 and now we can see we have a common factor to go ahead and get rid of.*0773

*The answer to this one would be 5q + 4 Ã· -3..*0782

*The last one involves 9 â€“ t Ã· 9 + t.*0791

*In this one, another one that looks very close.*0796

*Unfortunately there is not a whole lot we can do to simplify it.*0800

*It is already simplified.*0802

*You might be wondering why cannot we just cancel out some 9 and call it good from there.*0805

*We can only cancel out common factors, things that are multiplied.*0810

*We cannot cancel out the 9 nor we can cancel out the tâ€™s.*0814

*Unfortunately there is nothing to factor from the top or factor from the bottom.*0818

*This one is simplified just as it is.*0822

*Be on the watch out for cases like this and know when you can cancel out those extra terms.*0834

*In this next few we are going to go through the multiplication process and then try and bring it down to lowest terms.*0844

*We just have to multiply across the top and then multiply across the bottom.*0850

*That will make our lives a little bit easier.*0854

*Let us start off with this first one.*0856

*Multiplying across the top I will have 8x ^{2} Ã— 9 / 3xy^{2}.*0859

*From here I can cancel a lot of my extra stuff.*0873

*I will cancel out extra 3 that it is the 9.*0876

*I can cancel out one of these xâ€™s here and I can cancel out one of these yâ€™s.*0880

*Let us see what is left over.*0886

*I still have 8x, 3, a single y on the bottom.*0888

*This is 24x Ã· y and that only multiply the two together, but I brought it down to lowest terms.*0895

*Onto the next one, multiplying the top together will give me 3t â€“ (u Ã— u) / (t Ã— 2) Ã— (t â€“ u).*0905

*One obvious common term is that t â€“ u, let us go ahead and get rid of that.*0926

*We would have left over 3u t Ã— 2 or just make my brain feel better, 2 Ã— t.*0933

*We have multiplied those together and reduce it to its lowest terms.*0944

*I want to point out something, back here it is tempting to go through the distribution process and put the 3 and t and the 3 and u,*0949

*but actually you do not want to do that just yet.*0958

*Go ahead and leave them into your factors because it will make it much easier to cancel them out.*0961

*If you do end up distributing them, you have to pull them back a part into their factors later on.*0966

*You are not saving yourself any work.*0971

*Leave the factors in there or if is not factored already go ahead and factor it so you can easier multiply and reduce.*0974

*Let us get into a much bigger one.*0984

*In this one we want to multiply together and then put into its lowest terms.*0987

*(x ^{2} + 7x + 10 / 3x + 6) Ã— (6x â€“ 6 / x^{2} + 2x â€“ 15)*0992

*This is a rather large one, but I'm not going to multiply together the tops first or the bottoms just yet.*1005

*Iâ€™m going to work on factoring just for little bit.*1011

*How would my first polynomial here factor?*1017

*My first terms would need to be an x that will give me x ^{2} and my other terms would have to be 2 and 5.*1023

*That will be the only way that I can get that 10 and they would add to be the 7 in the middle.*1031

*That would factor that and we will also factor the bottom.*1037

*It looks like there is a common 3, let us pull that out.*1041

*We have just factored that first rational expression.*1047

*Onto the next one I see it has a common 6, x â€“ 1 and on the bottom I see a trinomial.*1051

*Let us go ahead and break that down into two binomials.*1061

*X ^{2} + 2x â€“ 15.*1065

*What would that factor into?*1068

*How about 5 and -3?*1069

*Now that I have all the individual factors, I can multiply them together much easier.*1075

*I will simply write all of the factors on the top and multiply that out.*1079

*All the factors in the bottom, 3x + 2 x + 5 and x â€“ 3.*1092

*In that step they are all multiplied together.*1102

*Let us cancel out a lot of those extra terms.*1105

*x + 5 were gone, x + 2 those were gone.*1109

*Let us cancel out one of the 3 and 6.*1116

*The only thing we have left over is 2 and x â€“ 1 / x -3.*1120

*Since there are no more common factors I know that this one is finally in lowest terms.*1129

*This takes care of a lot of multiplication let us get into dividing these and also putting them into lowest terms.*1138

*With this one needs one need one extra step, we have to flip the second rational expression and then multiply and reduce.*1144

*Let us start off with this first one.*1154

*I have 9x ^{2} Ã· 3x + 4.*1156

*I will flip the second one, 3x + 4 Ã· 6x ^{3}.*1162

*If Iâ€™m going to multiply the two together, I need to write all of their factors, the top ones together.*1173

*We will write all the factors of the bottom one together.*1184

*Now they are multiplied.*1189

*We can go ahead and cancel a lot of those extra terms.*1192

*We have a common 3x + 4 in the top and bottom, let us get rid of that, we do not need that.*1195

*We cancel out an extra 3 in the top and bottom.*1201

*On top we have x ^{2} and in the bottom we have x^{3} so an x^{2} will cancel out leaving me x in the bottom.*1207

*That is quite a bit of canceling.*1216

*Let us see if we can figure out what is left over.*1217

*I still have a 3 on top, we still have a 2 and x on the bottom.*1219

*3 / 2x is the only thing left when we reduce it to lowest terms.*1225

*The next one looks like a do- see, let us go ahead and take this one nice and easy.*1231

*We will first write the first rational expression, just as it is.*1237

*We will go ahead and flip the second rational expression.*1247

*4r -1 ^{2} and -r^{2} r + 3*1250

*If I'm going to multiply this together we will go ahead and write all of the factors on the top.*1263

*4r ^{2} â€“ 1 on the bottom 2r + 1 -r^{2} and r + 3*1272

*If we multiply things together let us see what we can cancel out.*1290

*One initial thing, I see an r + 3, let us get rid of that.*1294

*I got an r on top and r ^{2} so let us get rid of some r.*1299

*I think there is even a little bit more we can cancel out as long as we recognize that we have a very special polynomial on top.*1305

*Notice how we have the difference of squares on top the 4r ^{2} -1.*1318

*We can write that as 2r + 1 and 2r - 1.*1325

*We can see that we do have an extra factor hiding in there that we can get rid of.*1333

*We can get rid of that 2r + 1.*1338

*What is left over, I have 4 Ã— 2r -1 on the top / -r in the bottom.*1346

*This one is completely factored.*1354

*I got one more example and this is another large one.*1365

*We will have to just walk through it very carefully.*1369

*I have xw - x ^{2} / x^{2} â€“ 1 then we are dividing that by x - w / x^{2} + 2x +1.*1371

*Let us see what we can do.*1381

*Let us go ahead and rewrite it and have the second one flipped over.*1383

*Multiply it by x ^{2} + 2x + 1x â€“ w.*1395

*We will go ahead and start putting things together.*1407

*Let us multiply (xw - x ^{2}) (x^{2} + 2x + 1) Ã· (x^{2} â€“ 1) (x â€“ w)*1410

*This one looks like it is loaded with lots of things that we can go ahead and factor.*1428

*Let us go ahead and do that and we are going to do that bit by bit.*1432

*Iâ€™m looking at this very first factor here and notice how they both have terms in there that have x.*1436

*We can factor out an x from each of those.*1443

*That will leave us with w â€“ x.*1449

*In this one over here we can factor it into an x + 1 and x + 1.*1454

*It is one of our perfect squares.*1460

*(x + 1)(x +1)*1465

*On to the bottom, this one right here is the difference of squares x + 1 x -1.*1473

*Unfortunately this guy we do not have anything special about it, I will just write it.*1486

*That will allow me to at least cancel out a few things.*1493

*I have an extra x + 1 that I will go ahead and get rid of.*1496

*It looks like I can almost get rid of something else.*1500

*Notice how we have this w â€“ x and x â€“ w.*1503

*It is almost the same thing, and it is involving subtraction.*1507

*I need to factor out a -1 from one of these.*1511

*Let us do it to the top -w + x.*1515

*I still have an x + 1 on top / x â€“ 1 and x â€“ w.*1522

*-x will rearrange the order of these guys and switch them around.*1537

*That will give us (x â€“ w) (x + 1) / (x â€“ 1) (x â€“ w).*1544

*We can see sure enough, there is another piece that we can go ahead and get rid of.*1556

*We only have one thing left, -x Ã— x +1 / x â€“ 1.*1563

*Since we have no other common factors, this is finally in its lowest terms.*1575

*When it comes to working with these rational expressions, remember how all of these processes work, which is normal fractions.*1580

*Whether that means multiplying fractions by going across the top and bottom*1587

*or dividing fractions by flipping the second one.*1591

*Then cancel out your extra factors so you can be assured that is brought down to lowest terms.*1593

*Thank you for watching www.educator.com.*1599

0 answers

Post by GERARDO MORALES on October 3, 2013

If theres an option to fast forward your lecture?

1 answer

Last reply by: Professor Eric Smith

Thu Oct 3, 2013 5:27 PM

Post by GERARDO MORALES on October 3, 2013

If theres an option to fast forward your lecture?