Algebra 1 Adding & Subtracting Rational Expressions

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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 × 17 × 23.0112

I have broken down the 782 into 2 × 17 × 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 necessary0139

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 × 2 × 2 and y⁴ we will have bunch of y all multiplied by each other.0225

For 12 that would be 3 × 2 × 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 × 2 × 2 = 8 × 3 = 24 and then I have 6y, y⁶.0292

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 y⁶ and gotten the y⁶.0312

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 /x² - 4x and 3x – 1/ x² – 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 – 40370

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 expressions0422

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 × 3 × 17 × 23.0454

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

2 × 3 × 17 × 23, 2 × 3 × 17 × 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 × 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 pieces0538

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 same0562

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 R0578

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 / x² – 1) + (3 / x² – 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 × x +1 and then we can factor the bottom, this will be x + 1 and x – 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/w² -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

w² - 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, I’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 – 1 – 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

I’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 -10974

That will give me 5u – 1 = 4u – 5/ u – 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 – 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 – 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 – 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 a– 5 - 4a /(a – 5) (a- 5).1104

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

I have (3a² - 15a – 4a) / (a)(a – 5)(a-5)1127

When combined together the -15a and the -4a, 3a² – 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 – 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

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