Algebra 1 Dividing Polynomials

Welcome back to www.educator.com.0000

In this lesson we are going to take care of dividing polynomials.0003

I like to break this down into two different parts.0009

First, we will look at the division process when you have a polynomial divided by a monomial.0013

We will look at what happens when you divide any two polynomials together.0017

In the very end, I will also show you a very special technique to make the division process nice and clean.0022

To get into the basics of understanding the division of polynomials, I like to take it back to looking at the division process for real-world fractions.0031

Suppose that when you were adding fractions together you know how that process will go.0042

One very important thing that you do when adding fractions is you would find a common denominator.0047

Once you have a common denominator, then you can go ahead and just combine the tops of those fractions together.0053

Think of a quick example like 2/3 and are looking to add 5/3.0060

Since they have exactly the same bottoms then you will only add the 2 and 5 together and get 7/3 as your result.0067

Since this has a giant equal sign in between it, it means you can also follow this process in the other direction.0074

This may look a little unfamiliar, but it does work out if you go the other way.0080

Suppose I had A + B and all of that was dividing by C.0085

The way I could look at this is that both the A and B are being divided by C separately.0090

This is the same equation I had earlier, I just turned it around.0097

The reason why I show this is this will help us understand what happens when we have a binomial divided by a monomial.0102

In fact, that is the example that I have written out.0108

The top is a binomial and the bottom is an example of a monomial.0112

You can see that the way we handle it is we split up that monomial under each of the different parts of the polynomial in the top.0121

Let us see this process with numbers and see what polynomials and you will get an idea of how this works.0134

Working the other direction, if I see (2 + 5) ÷ 3, I want to visualize that as the 2 ÷ 3 and also the 5 ÷ 3.0140

For our polynomials if I have something like (x + 3z) ÷ 2y, then I will put that 2y under both of the parts.0152

In addition, you will notice splitting up over both of the parts in the top, you always want to make sure that you simplify further, if possible.0162

This means if you use your quotient rule for exponents, go ahead and do reduce those powers as much as possible.0172

If you have just any two general polynomials, then you want to think of how the process works with numbers.0183

In fact we are going to go over a long division process so that we can actually keep track of all the parts of what goes into what.0189

To make this process easier, remember to write your polynomial in descending power.0199

Start with the largest power and write it all the way down to the smallest power.0203

An additional thing that will also help in the division process is to make sure you put placeholders for all the missing variables.0214

If I’m looking at a polynomial like 5x² + 1, then I will end up writing it with a placeholder for the missing x.0221

It is not missing but it will help me keep track of where just my x terms go.0233

I'm going to show you this process a little bit later on0239

so you can see how it works with numbers and make some good parallels too doing this with polynomials.0242

Let us look at the division process for numbers.0251

Suppose I gave you 8494 and I told you to divide it by 3 and furthermore I said okay, let us see if you can do this by hand.0254

Some good news that you will probably tell me is that you do not have to take the 3 into that entire number all at once.0263

No, you will just take the 3 into 8494 bit by bit.0269

In fact, the first thing that you will look at is how many times this 3 go into the number 8.0274

That will be the only thing you are worried about.0279

How many times does the 3 go into 8?0282

It goes in there twice and you will write that number on the top.0285

Now after you have that number on the top, you do not leave it up there you go through a multiplication process,0290

2 × 3 and you will write the result right underneath the 8.0296

With the new number on the bottom, you will go ahead and subtract it away, so 8-6 would give you a 2.0305

That would be like one step of the whole division process.0315

You would continue on with the division process by bringing down more terms and doing the process again.0319

At this next stage, we will say okay how many times this 3 go into 24?0329

It goes in there 8 times.0335

Then we could multiply the 8 and 3 together and get a new number for that 3 × 8 = 24.0338

We can go ahead and subtract those away.0352

We will not stop there, keep bringing down our other terms and see how many times 3 goes into 9.0359

Write it onto the top and multiply it by your number out front.0369

Subtract it away and continue the process until you have exhausted the number you are trying to divide.0377

Let us see.0389

3 goes into 4, it looks like it goes in there once.0390

I will get 3, subtract them away and I have remainder of 1.0396

That is a lengthy process but notice the Q components in there.0401

You are only dividing the number bit by bit.0405

You do not have to take care of it all at once.0408

The way you take care of it is you are saying how many times 3 goes into that leading number.0410

You write it on the top, you go through the multiplication process and then you subtract it away from the number.0415

You will see all of those same components when we get into polynomials.0423

We have our answer and I could say it in many different ways but I’m going to write it out.0429

8494 if we divide this by 3 is equal to 2831 and it has a remainder down here of 1.0434

We could say +1 and still being divided by 3.0446

We have many different parts in here that are flying around, and you want to keep track of what these parts are.0451

The part underneath your division bar is the dividend.0458

What you are throwing in there this is your divisor.0466

Your answer would be your quotient.0472

This guy down here is our remainder.0477

Notice how those same parts actually show up in our answer.0483

Dividend, divisor, quotient, and remainder.0488

We put the remainder over the divisor because it is still being divided.0506

Now that we brushed up on the process with numbers, let us take a look at how we do this with polynomials.0510

I want to divide 2x² + 10x + 12 and divide that by x +3.0519

The good news is we do not have to take care of the entire polynomial all at once.0526

We are going to take it in bits and pieces.0530

We will first going to look at x and see how many times it will go into 2x².0532

In fact one thing that I can do to help out the process is think to myself what would I have to multiply x by in order to get a 2x².0537

1 × what would equal to 2? That would have to be 2.0548

What would I have to multiply x by to get an x²?0552

I have to multiply it by x.0555

I will put that on top, just like I did with numbers.0557

After I do that, we will run through a multiplication process.0563

We will take the 2x to multiply it by x, I will multiply it by 3.0568

We will record this new polynomial right underneath the other one.0573

2x × x = 2x².0577

2x × 3 = 6x.0581

Now comes a very important step.0589

Now that we have this new one, we want to subtract it away from the original.0592

Notice how I put those parentheses on there, that will help me remember that I need to subtract away both parts and keep my signs straight.0599

Starting over here, I have 10x – 6x = 4x.0606

Then I have 2x² – 2x², that will cancel out and will give me 0x².0616

If you do this process correctly, these should always cancel out.0622

If they do not cancel out, it means we need to choose a new number up here.0628

That is just one step of the division process, let us bring down our other terms and try this one more time.0635

I want to figure out how many does x goes into 4x?0648

What would I have to multiply x by in order to get 4x?0652

I think I have to multiply it by 4 that is the only way it is going to work out.0658

Now we have the 4, go ahead and multiply it by the numbers out front.0663

4 × x = 4x and 4 × 3 = 12.0669

Once you have them, put on a giant pair of parenthesis and we will go ahead and subtract it away.0678

12 – 12 =0 and 4x – 4x = 0.0687

What that shows is that there is no remainder and that it went evenly.0693

Let us write this out.0699

When I had 2x² + 10x + 12 and I divided it by x + 3, the result was 2x + 4.0699

There was no remainder.0714

We can label these parts as well.0717

We have our dividend, divisor, quotient, and if I did have a remainder I will probably put it out here.0720

Here is our dividend.0739

Here is our divisor and our quotient.0744

Now that we know a lot more about dividing polynomials, let us look at a bunch of examples.0758

Some of them will take a polynomial divided by a monomial.0764

Some of them will take two polynomials and divide them.0767

We will approach both of those cases in two different ways.0770

Example 1, divide a polynomial by a monomial.0774

I will take (50m⁴ -30m³ + 20m) ÷ 10m³.0777

Since we are dividing by a monomial, I will take each of my terms and put them over what I’m dividing them by.0784

50m⁴ ÷ 10m³, 30m³ ÷10m³ and 20m ÷ 10m³.0792

Now that I have done that, I will go through and simplify these one at a time.0811

50 ÷ 10 = 5, m⁴ ÷ m³ = I can use my quotient rule and simply subtract the exponents and get m¹.0816

Continuing on 30 ÷ 10 = 3, if I subtract my exponents for m³ and m³, I have m⁰.0830

Onto the last one, 20m ÷ 10m³, there is 2.0844

Let us see, if I subtract the exponents I will m^-2.0852

Then I can go through and just clean this up a little bit.0857

5m – anything to the 0 power is 1, 1 × 3 + and I will write this using positive exponents, 2/m².0860

This will be the final result of dividing my polynomial by a monomial.0872

Let us try this scheme with something a little bit more complicated.0881

This one is (45x⁴ y³ + 30x³ y² - 60x² y) ÷ 50x² y.0884

We have to put our thinking caps for this one.0894

We will start off by taking all of our terms in the top polynomial and putting them over our monomial, only one term.0897

Once we have this all written out we simply have to simplify them one at a time.0917

Let us start at the very beginning.0925

15 goes into 45 3 times, now I will simplify each of my variables using the quotient rule.0929

4 – 2 =2 and 3 – 1= y².0939

There is my first term, moving on.0946

15 goes into 30 twice.0950

Using my quotient rule on the x, 3 – 2 = 1 and y² – y¹ = y¹.0955

Both of these have an exponent of 1 and I do not need to write it.0964

One more, 15 goes into 60 four times.0969

I have x²/x² which will be x⁰ and y/y will be y⁰.0975

Anything to the 0 power is 1.0983

I can end up rewriting that term 3x2 y² + 2xy – 4.0987

There is my resulting polynomial.1000

In this next example, we are going to take one polynomial divided by another polynomial.1007

I would not be able to split it up quite like I did before, now we will have to go through that long division process.1012

Be careful you want to make sure that you line up your terms in descending order.1018

If you look at the powers of the top polynomial you would not mix up.1024

I want to start with that 3rd power then go to the 2nd power, then go to the 1st power, just to make sure I have it all lined up.1027

Let us write it out.1034

2x³ + x² + 5x + 13, now it is in a much better order to take care of.1035

We will take all of that and we will divide it by 2x + 3.1048

That looks good.1055

It is time to get into the division process.1057

Our first terms there, and what would I need to multiply 2x by in order to get 2x³?1060

The only thing that will work would be an x².1068

Once we find our numbers up top, we will multiply them by the polynomial out front.1072

2x × x² = 2x³, that is a good sign, it is the same as the number above it.1079

X² × 3 =3x², now comes the part that is tricky to remember.1088

Always subtract this away and do not be afraid to use this parenthesis to help you remember that.1097

x² – 3x², 1 – 3 = -2x².1106

Then I have 2x³ – 2x³ and those will be gone, cancel out like they should.1116

The first iteration of this thing looks pretty good.1122

Let us try another one.1126

I want to figure out how may times does 2x go into -2x²?1129

I have to multiply it by –x.1137

Let us bring down some more terms.1146

I will get onto our multiplication process.1149

-x × 2x =-2x².1153

-x × 3 =-3x.1158

We have our terms in there, it is time to subtract it away and be careful with all of our signs.1164

5x - -3x when we subtract a negative that is the same as addition.1171

I’m looking at 5x + 3x = 8x.1179

-2x² - -2x² = that is a lot of minus signs.1188

That would be the same as -2x + 2x² and they do cancel out like they should.1193

That was pretty tricky keeping track of all those signs but we did just fine.1201

We have 8x + 13 and I’m trying to figure out what would I have to multiply 2x by in order to get that 8x.1207

There is only one thing I can do I need to multiply by 4.1218

We will multiply everything through and see what we get.1224

4 × 2x = 8x, 4 × 3 =12.1228

I can subtract this away and let us see what we get.1238

13 – 12=1, 8x – 8x = 0.1245

Here I have a remainder of 1.1250

I could end up writing my quotient and I can put my remainder over the divisor.1257

There is our answer.1269

In some polynomials you want to make sure you put in those placeholders.1275

That way everything lines up and works out good.1280

It is especially what we will have to do for some of those missing powers in this one.1283

I’m missing an x² and a single x.1287

Let us write this side and put those in.1291

x³, I have no x², no x – 8.1294

All of that is being divided by x – 2.1305

I’m going to go through and let us see what I need to multiply x by in order to get x³.1310

I think I’m going to need x².1318

Now that I found it, I will go ahead and multiply it through.1323

x² × x =x³ and x² × -2 = -2x².1327

We have our terms, let us go ahead and subtract it away.1336

0x² – 2x², this is one of those situations where if we subtract a negative is the same as adding.1345

0x² + 2x² =2x².1354

x³ – x³, that is completely gone.1360

Bring down our other term here and we will keep going.1366

What would I have to multiply x by in order to get 2x²?1374

I’m going to need 2x.1381

Let us multiply that through, 2x × x =2x².1386

2x × -2 =-4x.1392

Now we found that, let us subtract that away.1399

Starting on the end, 0 – -4, that is the same as 0 + 4x = 4x.1405

Then 2x² – 2x², they are completely gone, you do not have to worry about it.1415

We will bring down our -8 and continue.1422

What would I have to multiply x by in order to get 4x?1428

That will have to be 4.1433

4 × x = 4x and 4 × -2 = -8.1437

It looks like it is exactly the same as the polynomial above it.1447

I know when I subtract, I would get 0.1452

There is no remainder for this one.1454

I will take (x³ – 8) ÷ x -2 and the result is x² + 2x + 4.1457

Division process can take a bit but as long as you do the steps very carefully, you should turn out okay.1467

Let us try this giant one.1476

This is (2m⁵ + m⁴ + 6m³ – 3m² – 18) ÷ m² + 3.1478

There are a lot of things to consider in here.1489

One thing that I will be careful of is putting those placeholders for this guy down here.1493

Notice that it is missing an m, let us give it a try.1498

I will have m² + 0m + 3 and all of that is going into our other polynomial 2m⁵ + m⁴ + 6m³ – 3m² – 18.1501

Lots of things to keep track of but I think we will be okay.1523

What would I have to multiply the m² by in order to get a 2m⁵?1527

That would be 2m³.1535

I can run through the multiplication and write it here.1539

2m³ × m² = 2m⁵.1543

Now we can multiply it by our 0 placeholder but anything times 0 will give us 0 so 0m⁴.1549

By my last one, let us see 2m³× 3 + 6m³.1558

Let us subtract that away.1568

6m³ – 6m³ those are gone.1572

I have m⁴ – 0m⁴ I still have m⁴.1575

2m⁵ – 2m⁵ those are gone.1580

I dropped away quite a bit of terms.1584

Let me go ahead and write in my 0m³ as one of those placeholders so I can keep track of it.1587

Let us try this again.1598

m² goes into m⁴, if I multiply it by another m².1600

Multiplying through I have m² × m² = m⁴.1609

m² × 0m = 0m³ and m² × 3 = 3m².1614

We will take that and subtract it away.1626

I need to go ahead and subtract these.1636

Be very careful on the signs of this one.1638

I have –m² and I’m subtracting -3m², the result here will be -6m².1641

The reason why it is happening is because of that negative sign out there.1652

Now I have 0m³ – 0m³, 0 – 0 =0, m⁴ – m⁴ = 0.1659

It looks like I forgot an extra placeholder.1669

I need 0m and then I need my 18.1676

Let us bring down both of these.1684

I need to figure out what I have to what would I have to multiply m² by in order to get -6m².1691

-6 will do it, -6m² 6 × 0 = 0m and -6 × 3 =-18.1699

It is exactly the same as the polynomial above it.1718

Since they are exactly the same and I’m subtracting one from the other one, 0 is the answer.1723

There is no remainder, it went evenly.1729

The quotient for this one would be 2m³ + m² – 6.1732

In this polynomial, I have (3x³ + 7x² + 7x + 11) ÷ 3x + 6.1744

The reason why I put this one is because it can get a little bit difficult figuring out what you need to multiply to get into that second polynomial.1753

I’m going to warn you, this involves a few fractions.1762

Let us give it a shot.1767

(3x³ + 7x² + 7x + 11) ÷ 3x + 6.1770

Let us start off at the very beginning.1788

What do I need to multiply my 3x by in order to get 3x³?1790

The only thing that will work will be an x².1797

I will go through and I will multiply and get the result.1804

3x³ + 6x² and let us subtract that away.1809

7x² – 6x² = 1x².1821

Now comes the tricky part, I need to figure out what I need to multiply 3x by in order to get x².1837

If I’m looking at just the variable part of this, I have to multiply and x by another x in order to get an x².1845

We will go ahead and put that as part of our quotient.1851

What do I have to multiply 3 by in order to get 1 out front?1854

That is a little bit trickier.1859

3 × what = 1.1861

That is almost like an equation onto itself.1865

What we see is that x would have 1/3, a fraction.1869

It is okay, we can use fractions and end up multiplying by those.1874

3x × 1/3x = 1x².1881

Let us multiply that through.1887

1/3x × 3x = 1x² I will write it down and 1/3x × 6 = 2x.1888

Now we can take that and subtract it away.1900

7x – 2x = 5x.1904

Bringing down our extra terms and I think this one is almost done.1913

What would I have to multiply 3x to get 5x?1917

Let us see.1922

he only way I can get an x into another x is to multiply by 1, but I’m going to think of how do I get 3 and turn it into 5?1924

Let us do a little bit of scratch work on this one.1933

3 × what = 5?1939

If we divide both sides by 3 I think we can figure out it is 5/3.1945

That I can write on top 5/3.1951

We can go through multiplying.1956

5/3 × 3 = 5x.1958

5/3 × 6 =10.1966

We will go ahead and subtract this away.1977

11 – 10 = 1 and 5x – 5x = 0.1980

We have a remainder of 1.1985

Now that we have all of the quotient and the remainder, let us go ahead and write it out.1993

We have (x² + 1/3x + 5/3 + 1) ÷ 3x + 6.1998

Definitely do not be afraid some of those fractions to make sure it goes into that second polynomial.2010

This process can get a little messy as you can definitely see from those examples.2018

I have a nice clean way that you can go through the division process known as synthetic division.2022

This is a much cleaner way for the division process so that you can keep track of all the variables.2027

It is much clean but be very careful in how you approach this.2033

It works good when dividing by polynomials of the form x + or – number.2037

It will work especially with my little example right here (5x³ - 6x² +8) ÷ x -4.2043

First watch how I will set this up.2051

I'm going to create like a little upside down division bar and that is where I will end up putting the polynomial that I'm dividing.2054

But I will not put the entire thing I’m only going to put the coefficients of all of the terms.2062

The coefficient of the x³ is a 5.2069

The coefficient of my x² is -6.2073

I will put a 0 placeholder in for my missing x and then my last coefficient will be 8.2077

Once I have all of those I will put another little line.2086

I want to put in the value of x that would make this entire polynomial 0, if x was 4 that would be 0.2090

I’m going to write 4 out here.2105

That turns to be a tricky issue and many students remember what to put out over here because it will be the opposite of this one.2109

If you see x – 4 put in a 4.2118

If you see something like x + 7 then put in -7.2121

We got that all set let us go through this synthetic division process.2127

It tends to be quick watch very carefully how this works.2131

The very first thing that you do in the synthetic division process is you take the first number here and you simply copy it down below.2137

This will be a 5.2146

Once you get that new number on the bottom, go ahead and multiply it by your number out front, 4×5 = 20.2149

That is one step of the synthetic division process.2161

To continue from there simply add the column -6 + 20 and get the result.2165

This would be a 14.2173

Once you have that feel free to multiply it out front again.2177

14 × 4 = 56.2181

There are 2 steps now we will take the 56 and we will add 0, 56.2191

When we get our new number on the bottom, go ahead and multiply it right out front.2200

4 × 50 = 200.2205

4 × 6 = 24, 224.2212

One last part to this we got to do some addition.2222

8 + 224 = 232.2225

It does not look like we did much of any type of division.2234

We did a lot of adding and we did a lot of multiplying but do you know what these new numbers stand for on the bottom.2236

That is the neat part, these new numbers I have here in green stand for the coefficients of our result.2243

You know what happens after the division.2249

The way you interpret these is the last number in this list will always be your remainder.2252

I know that my remainder is 232.2260

As for the rest of the values, the 5, 14, and 56, those are the coefficients on our variables.2264

What should they be? Let me show you how you can figure that out.2270

Originally we had x³ as the polynomial that we are dividing and these new ones will be exactly one less in power.2274

That 5 goes with 5x² and the 14 goes with the 14x and 56 has no x on it.2282

The result for this one is 5x² + 14x + 56 with a remainder of 232 which you can write over x – 4.2293

Since it is still being divided2312

It is a much cleaner and faster method for division.2315

Let us go ahead and practice it a few times just to make sure got it down.2318

We will go ahead and do (10x⁴ - 50x³ – 800) ÷ x – 6.2325

It is quite a large problem.2332

We will definitely remember to put in some of those placeholders to keep track of everything.2334

First I will write in all of the coefficients of my original polynomial.2339

I have a 10x⁴ - 50x³ I need to put in a placeholder for my x² and another placeholder for my x.2344

We will go ahead and put in that -800.2356

What shall we put on the other side?2364

Since I'm dividing by x – 6, the value of 6 will be divided that would make that 0.2367

I will use 6 and now onto the synthetic division process.2376

The first part we will drop down to 10, just as it is.2381

Then we will multiply by the 6 out front 60.2387

Now that we have that, let us add the -50 and the 60 together, 10 again.2396

We will multiply this out front.2405

That result will be 60.2409

We will go ahead and add 0 + 60 = 60 and multiply 60 × 6= 360.2417

Now we will add 0 + 360 = 360.2430

I have to take 360 × 6.2435

think I have to do a little bit of scratch work for that one.2442

6 × 0 = 6 × 6 = 36 and then 3 × 6 = 18 + 3 = 21.2443

I have 2160, let us put it in.2455

Only one last thing to do is we need to add -800 to the 2160 and then let us do a little bit of scratch work to take care of that.2465

0 – 0= 0, 6 – 0= 6, 20 and we will breakdown and will say 11 – 8 =3.2478

I have 1, I have 1360.2489

Now comes the fun part, we have to interpret exactly what this means.2493

Keep in mind that this last one out here that is our remainder.2498

Originally our polynomial was x⁴ so we will start with x³ in our result.2505

10 x³ + 10x² + 60x + 360 and then we have our remainder 1360.2513

It is all still being divided by an x – 6.2532

That was quite a bit of work, but it was a lot clean than going through the long division process.2536

Let us see this one more time.2540

In this last one we will use (5 - 3x + 2x² – x³) ÷ x + 1.2545

We will first go ahead and put the coefficients of our top polynomial in descending order.2555

Be very careful as you set this one up.2560

On that side you can write out the polynomial first in descending order and then go ahead and grab its coefficients.2564

-x³ would be the largest, then I have 2x² and then I have 3x, and 5.2570

I need -1, 2, -3 and 5.2579

I’m dividing by x + 1 so the number I will use off to the left will be -1.2589

I think we have it all set up now let us run through that process.2597

The first I’m going to bring down is -1 then we will go ahead and multiply.2600

Negative × negative is positive.2608

2 + 1 =3, 3 × -1= -3.2615

-3 + -3 = - 6, -6 × -1 = 6 and 5 + 6 =11.2630

Now we have our remainder, we can finally write down the resulting polynomial.2647

We started with x³ so I know this would be x².2657

-x² + 3x – 6 and I still have my 11 being divided by x + 1.2662

Now you know pretty much everything that there is to know about dividing polynomials.2676

If you divide by a monomial, make sure you split it out among all the terms.2680

If you divide a monomial by a polynomial, you can go through the long division process2685

or use this synthetic division process to make it nice and clean.2690

Thank you for watching www.educator.com.2694