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Lecture Comments (5)

2 answers

Last reply by: Peter Ke
Tue Nov 10, 2015 8:11 PM

Post by Peter Ke on November 8, 2015

At 11:40 "Quick Example", I understand at the left side of the equation, how you got rid of the denominator by multiplying but then how you got the right side though?

The right side was A(x^2 - x + 2) + (Bx+C)(x+3). How did you get this!?
I really don't get it...

1 answer

Last reply by: Professor Selhorst-Jones
Tue Sep 9, 2014 9:52 AM

Post by David Llewellyn on September 8, 2014

Your explanation that an expression with all exponents of x <=0 does not qualify as a polynomial (see my question in Polynomials lecture) explains why you stop the polynomial division with a remainder once you hit a constant in this lecture.
BTW, presumably, you can also have irreducible factors of any even power (x^2n) n>= 1  with 2n terms in the denominator. I can see that all polynomials will have, at least, one real factor as this polynomial will generate an odd function.

Partial Fractions

    Note:Before starting, it should be noted that this is a rather difficult concept to explain just with writing. It helps a lot to see it in action, so it's strongly recommended that you watch the video if you find any of this confusing.
  • Long ago, you learned how to combine fractions: you put them over a common denominator, then combine them. But what if we wanted to do the reverse? What if we had a fraction that we wanted to break up into multiple, smaller fractions? We call these smaller fractions partial fractions and the process partial fraction decomposition.
  • To understand this lesson, you'll need some familiarity with solving systems of linear equations. Previous experience from past Algebra classes will probably be enough, but if you want a refresher, check out the lesson Systems of Linear Equations. Knowing how to factor polynomials and understanding them in general will also be necessary, along with the ability to do polynomial division.
  • If we have a polynomial fraction of the from [N(x)/D(x)], there are two possibilities in regards to the degrees of the top and bottom polynomials:
    • Proper: degree of N(x) < degree of D(x);
    • Improper: degree of N(x) ≥ degree of D(x).
    To decompose a polynomial fraction, it must first be proper. If the fraction is improper and we want to decompose it, it must first be made proper through polynomial division. [Remember that the remainder from the division goes back on the original denominator, which we will then decompose.]
  • Once the polynomial fraction is proper, the next step is to factor the denominator. After it's broken down into its smallest factors, we're ready to decompose. Notice that there are two types that these smallest possible factors can come in:
    • Linear factors raised to a power: (ax+b)m;
    • Irreducible quadratic factors raised to a power: (ax2+bx+c)m. [Remember, `irreducible' means it can't be broken up further. That means quadratics like x2+1 or 5x2−3x+20.]
  • The partial fraction decomposition must include the following for each linear factor (ax+b)m:




    + …+Am


    where A1, …, Am are all constant real numbers.
  • The partial fraction decomposition must include the following for each irreducible quadratic (ax2+bx+c)m:




    + …+Amx+Bm


    where A1, …, Am and B1, …, Bm are all constants.
  • If we have multiple of a given type or mixed types, we just decompose each of them based on their rules and put them all together.
  • Finally, we need to figure out what goes on the numerators. To do this, have the original fraction on one side of an equation and the partial fraction decomposition on the other. Then multiply both sides of the equation by the denominator of the original fraction. This gives an equation that we can transform into a system of linear equations and solve to find the values for each of the constants.

Partial Fractions

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
  • Introduction: Idea 0:04
  • Introduction: Prerequisites and Uses 1:57
  • Proper vs. Improper Polynomial Fractions 3:11
  • Possible Things in the Denominator 4:38
  • Linear Factors 6:16
    • Example of Linear Factors
    • Multiple Linear Factors
  • Irreducible Quadratic Factors 8:25
    • Example of Quadratic Factors
    • Multiple Quadratic Factors
  • Mixing Factor Types 10:28
  • Figuring Out the Numerator 11:10
    • How to Solve for the Constants
    • Quick Example
  • Example 1 14:29
  • Example 2 18:35
  • Example 3 20:33
  • Example 4 28:51

Transcription: Partial Fractions

Hi--welcome back to

Today, we are going to talk about partial fractions.0002

Consider if we had two quotients of polynomials that we wanted to add together, like, say, 7/(x - 5) and 3/(x + 2).0005

We could put them over a common denominator, then combine them; we have x - 5 here and x + 2 here,0014

so we can multiply the one on the left by x + 2, so we get (x - 5)(x + 2) on the bottom.0019

Its top will get also multiplied by that (x + 2).0026

For the other one, we will have (x - 5) multiplied on it, so it has (x - 5) multiplied on the top,0030

so we have (x - 5)(x + 2)--a common denominator--on the bottom now.0035

We multiply this out; we get 7x + 14 and 3x - 15; we combine those, and we get 10x - 1 on top.0039

divided by (x - 5)(x + 2), expanded to x2 - 3x - 10.0049

So, it is not a bad idea; we can get from here to here by putting it over a common denominator, and then just adding things out and simplifying.0054

But if we wanted to do the reverse process--what if we started with a fraction involving large polynomials,0064

and we wanted to break it into smaller fractions made of the polynomials' factors?0070

If we wanted to do this process in reverse, if we started at (10x - 1)/(x2 - 3x - 10),0074

and we somehow wanted to be able to get that into 7/(x - 5) and 3/(x + 2), what process could we go through?0080

We call the smaller fractions on the right partial fractions, because they are parts of that larger fraction.0088

So, each of these here (7/(x - 5) and 3/(x + 2))--they are each called a partial fraction.0095

And the process to break it up is called partial fraction decomposition.0101

So, whatever this method is that we haven't explored yet, to get from that larger polynomial quotient0105

into these smaller partial fractions, is called partial fraction decomposition.0111

So, how do we do it? To understand this lesson--to understand how we do it--you will need some familiarity with solving systems of linear equations.0116

Previous experience from past algebra classes will probably be enough to get through this.0124

But if you want a refresher, if you are a little confused by how this stuff is working later on,0128

check out the lesson Systems of Linear Equations--that will help explain this stuff if it confuses you right now.0132

You also need to know how to factor polynomials, and you will need to understand them in general,0138

along with the ability to do polynomial division; you will need all of that for some of the stuff in here, as well.0141

Now, sadly, we won't be able to see the application of partial fraction decomposition in this course.0147

However, it is quite useful in calculus, where it will allow us to solve otherwise impossible problems.0152

Partial fraction decomposition is this really useful thing that lets us break up these complicated things0157

that we couldn't solve, and turn them into a form that is actually pretty easy to solve.0162

It is really handy in calculus.0166

We won't be able to see its use in this course; we won't see it any time soon.0167

But it is helpful to practice it now, just like we practiced stuff about factoring complicated polynomials in algebra,0171

before we really had a great understanding of what it was all about.0176

We are practicing something so that we can use it later on.0179

And also, it just is a great chance to flex our brain and get some mental "muscles" that are kind of difficult ideas, but really require some analytic thought.0182

All right, let's get to actually figuring out how to do this.0191

If we have a polynomial fraction in the form numerator polynomial divided by denominator polynomial,0194

a normal rational function format, there are two possibilities in regards to the degrees of the top and bottom polynomials.0199

We call them proper (proper is when the degree of the numerator polynomial is less than the degree of the denominator polynomial--0206

the degree of n is less than the degree of the denominator) and improper (when the degree of our numerator0214

is greater than or equal to the degree of the denominator).0222

To decompose the fraction, it must be proper; we have to be in this proper format.0226

We have to make sure that the degree of our numerator is less than the degree of our denominator, if we want to do partial fraction decomposition.0231

So, what do we do if the fraction is improper?0239

We have to turn it into a proper thing; so if the fraction is improper, and we want to decompose it,0241

we have to make it proper through polynomial division.0245

We can make that numerator smaller by being able to break it off and divide out the part that isn't smaller and the part that is smaller.0249

Remember, when you divide your polynomial division, the remainder from the division0256

goes back onto our original denominator, which we will then decompose.0261

We will be able to decompose the part that is the remainder.0265

The other part that comes out cleanly--well, that is just going to be there.0267

That is just going to be a polynomial that is then going to be added to whatever ends up coming out of our decomposition.0271

We will see an example of this in Example 3.0276

Once we have a proper polynomial fraction to decompose into partial fractions0280

(remember, proper: our numerator has to be smaller than our denominator before we can enact this process),0284

once we have done that, we can then factor the denominator.0289

Our next step is to factor the denominator.0293

After the denominator is broken into its smallest possible factors, we are ready to decompose.0295

Now, there are two types that the smallest possible factors can come in.0300

They can come in linear factors, which is forms ax + b, and they will be raised to some power, because we might have a multiplicity of these factors.0302

There might be multiple of a given factor, so if we have multiple ax + b's, we will have m of them, (ax + b)m.0309

So, we will have linear factors in this; or we could have irreducible quadratic factors,0316

that is, things ax2 + bx + c that can't be broken up further into linear factors.0320

There is no way to break them up further in the reals.0325

So, ax2 + bx + c...and once again, it will be raised to some power;0328

there will be m of them, so it is raised to the m, because there are m of them multiplying together.0331

Remember: "irreducible" means it can't be broken up further in the reals.0336

That means quadratics like x2 + 1 or 5x2 - 3x + 20, because they have no roots.0339

x2 + 1 has no roots; x2 + 1 = 0...there are no solutions, in the reals, at least.0346

If we allow for the complex, there are; but we are not working with the complex.0355

There are no solutions in the reals; so, since there are no solutions in the reals, it can't be factored any further.0358

x2 + 1 is irreducible, therefore.0364

The next step of decomposition will behave differently, depending on which flavor of factor we have--0366

whether we are at a linear factor or we are at an irreducible factor; and we will look at the two, one after another.0371

Linear factors: the partial fraction decomposition must include the following for each linear factor that is in this form (ax + b)m.0377

So, it is going to have this in its decomposition: A1/(ax + b) + A2/(ax + b)2 +...0385

notice that we are doing this where it is going to keep stepping up with this exponent on the bottom,0394

as we keep putting these things in, until eventually we get up to our mth step,0398

because we had m of them to begin with, so we have to have m of them in the end.0402

And each one of these A's on top (A1 up until Am) are all just constant real numbers.0406

We are using A with a subscript, this A1, A2, A3 business,0411

because we just need a way of being able to say m of them, and we are not quite sure that we are going to go only out to m.0415

Anyway, the point is that they are all going to be constant real numbers.0420

For example, if we had (x + 7)3, then we have some stuff up top.0423

We don't really care what the stuff is right now, because we just want to see how it will break out.0428

We will deal with the stuff later.0431

If we have stuff over (x + 7)3, then it is going to break into 3 of these, because of this "cubed."0433

We will have A/(x + 7) + B (we can switch into just using letters in general, because we know0439

that each one of these capital letters just means some constant real number; we will figure them out later--0444

that will come up later--don't worry)...A/(x + 7) + B/(x + 7)2 + C/(x + 7)3.0449

Notice: we started cubed, and so we have three of these; we step up each time with the exponent,0457

until we have gotten to the number of our multiplicity that we originally started with--0462

whatever the exponent was originally on the factor.0465

If we have multiple linear factors, we do it for each one of these; so if we have (2x + 3)2, x10469

(since there is nothing there), (x - 5)1 (since there is nothing there)...we have A/(2x + 3) + B/(2x + 3)20475

(because remember: it was squared to begin with, so it has to have 2 of itself) + C/x.0483

And there is only to the 1, so there is only going to be one of it; plus D/(x - 5).0490

And it is only 1, because there is only one of them to begin with.0495

And each one of these A, B, C, D...they are each just constant numbers.0498

We are going to figure them out later on; don't worry.0503

Irreducible quadratic factors: the method is very similar for irreducible quadratic factors.0507

The decomposition must include the following for each irreducible quadratic: ax2 + bx + c to the m.0511

Once again, this is the same thing of stepping out m times.0516

A1x + B1 over ax2 + bx + c: notice how, previously, it was A over some x plus a constant,0520

because we had that the degree of the top was one below the degree on the bottom.0528

So, the degree on the top here is a linear factor on top, because we have a quadratic factor on the bottom.0532

Previously, we had a constant factor on top, because we had a linear factor on the bottom.0537

So, A1x + B1 + A2x + B2, over...that should be a capital...0541

ax2 + bx + c, now squared, because we are doing our second step...0548

we do this out m steps, until we are at our m constants, and we are at the mth exponent on the bottom.0552

So, A1...all of these capital letters...B1 to Bm...they are all just constants.0558

And we will figure them out later.0564

For example, if we had (x2 - 4x + 5)2 on our bottom, then we have some stuff--0565

it doesn't really matter right now--x2 - 4x + 5, squared; so we are going to have this happen twice.0571

The first one will be just to the one; the second time, it will be squared.0576

And on the top, it is going to be Ax + B and Cx + D.0580

So, A, B, C, D...these are all just constants; we will deal with them later.0585

If we have multiple irreducible quadratics, we just do each of these.0590

For example, if we have x2 + 2x + 3, that is an irreducible; and x2 + 4, squared...0593

then we have Ax + B, x2 + 2x + 3...there is only one of them, so it only shows up once,0599

+ Cx + D over x2 + is going to show up twice because of that 2, so it shows up a second time here.0605

And we have Ex + F; so we just keep doing this process with capital letter, x, + capital letter, plus capital letter, x, plus capital letter,0612

until we have worked out all of the times that we had things showing up--a total of 3, because we had one there and two there.0620

So, we have a total of 3.0627

If we have both types, a linear and an irreducible quadratic, mixed together in the denominator, that is OK--perfectly fine.0629

We just decompose based on both of the rules.0634

So, for example, if we had (x + 9)2, then the x + 9's are going to follow this A, this single-constant format,0637

because linears just had one constant on the top; x + 9, and then (x + 9)2.0645

And then, we switch to the other rule, x2 + 1; now we are dealing with an irreducible quadratic.0650

So, x2 + 1...and we have Cx + D on top, because we have to be able to have...0656

since we have a quadratic on the bottom, we now have linear factors on the top.0661

When we are dealing with linear factors on the bottom, even if it is multiple linear factors, it is just constants on the top.0665

All right, now, finally: how do we figure out what those numerators are?0671

We have mentioned that the numerators are built out of constants.0674

Remember: A for linear factors (constants for linear factors); Bx + C...linear factors for irreducible quadratic factors.0677

But we haven't talked about how we actually find out what these capital letters' values are.0685

We solve for the constants by doing partial fraction decomposition.0690

We set them up in this format, and then we multiply each side of the equation by the original denominator.0693

It will make more sense as we work through examples; let's look at this example.0698

If we have (3x2 + 3x - 4)/(x + 3), one factor, times (x2 - x + 2), an irreducible quadratic factor,0701

then we would be able to break it into A/(x + 3) (we get this right here)...and x2 - x + 2 would be Bx + C/(x2 - x + 2).0709

So, we know, from what we just talked about (this process that we just went through) that that is how it breaks up.0721

But how do we figure out what A and Bx + C are?0726

Well, notice: we can just work things out the way we would a normal algebra thing.0728

We multiply both sides, because we want to get rid of this denominator.0734

We will multiply by x + 3 on the left side, and x2 - x + 2 on both sides.0737

So, we do the same thing on this side, as well: (x + 3)...we will color-code this...and (x2 - x + 2).0746

So, the (x + 3) will distribute to the one on the left and the one on the right.0759

It has to hit both of them, because it is distributed, because there is this plus sign right here.0764

So, (x + 3) distributes to both of them; similarly, (x2 - x + 2) will distribute to both of these, as well.0768

Now, notice: on the left side, we have (x + 3) on the bottom, and we are multiplying by (x + 3); so the (x + 3)'s cancel out.0776

(x2 - x + 2) is right here and here; they cancel out.0783

What about on the left side? Well, we have (x + 3) and (x + 3); so here, it is going to cancel this and this.0788

But it will still have the (x2 - x + 2) coming through.0796

What about the Bx + C over (x2 - x + 2)?0799

Well, the (x + 3) is still going to come through--it doesn't cancel out any factors here.0801

But (x2 - x + 2) is the same thing, so it will cancel out that part of the factor coming in.0805

So, we will see that we have A times (x2 - x + 2) and Bx + C times (x + 3) coming through.0809

This part right here manages to come through on the Bx + C, and this part right here manages to come through on the A.0819

And on the left, since we canceled out everything on the bottom, we just have what we had on our numerator originally.0828

That just makes its way down; so we have 3x2 + 3x - 4 = A times x2 - x + 2, plus Bx + C times (x + 3).0834

Which...we can then expand everything on the right, and we can work things out by creating a system of linear equations.0845

The examples are going to help greatly in clarifying how this works, so check them out,0851

because we will actually see how this process of setting up our system of linear equations works,0855

and solving for what these values are, in each of these examples.0859

Not Example 2, but we will see it in a lot of the examples.0863

And it will make a whole lot more sense as we see it in practice.0866

So, let's go check it out: the first example is a nice, simple example to get things started with.0868

Find the partial fraction decomposition for (4x + 3)/x(x + 3).0873

So, we have 4x + 3, over x(x + 3); that is going to be equal to...we have two linear factors on the bottom, so A/x, plus our other linear factor, B/(x + 3).0878

Now, at this point, we can multiply everything by x and (x + 3).0900

Great; so, on the left side, that is just going to cancel this stuff out; and on the right side, it will end up distributing.0912

So, on the left side, we have 4x + 3 =, notice: we have the x multiplying here; that is going to cancel out.0917

So, the x's parts get canceled out; but we are left with x + 3 multiplying on it.0926

Plus B times (x + 3) the x + 3 part cancels out; I will switch colors here.0931

x gets through; but the (x + 3) cancels out the denominator; great--we have managed to get rid of all of our denominators.0939

That is great; it will make things easier to see what is going on.0946

And we have some stuff multiplying through; so at this point, 4x + 3 on the left equals Ax + 3A + Bx.0949

Now, notice: we see here that we have a 3 constant.0965

Now, what constants do we have on the right side?0970

We have to have all of our x's match up and all of our constants match up.0972

Well, the only thing that is actually a constant on the right side is this 3A, because we have Ax and Bx.0976

But we don't have any B just as B; we don't have any constants of B.0982

So, it must be that 3 and 3A are equal, because they are all of the constants that show up on either side of our equation.0986

The same thing has to be on the left and the right; otherwise, it is not an equation.0992

That tells us that 3 = 3A; so what is our A?0996

We divide both sides by 3; we get 1 = A; so now we have figured out what A is.1001

What about the other part? Well, we have that 4x; 4x is equal to...let's give it a special thing; we will give it a curly around...1007

4x is equal to all of our x's on the right side, put together.1019

So, that is our Ax, combined with our Bx; so it must be the case that 4x is equal to Ax + Bx,1023

because the same number of x's must be on the right side as on the left side.1034

So, 4x must equal Ax + Bx, because these are our sources of x on the right side.1039

4x = Ax + Bx; we just figured out what A is, so we can plug that in over here; so we have 4x = Ax + Bx...1045

Now, notice...actually, before we do that, even, we have 4x = Ax + Bx; so if that is the case, let's just get rid of these x's.1055

All right, we know that it must be the case that, since 4x = Ax + Bx...well, that has to be true for any x that we plug in, at all.1063

So, it must be the case that 4 = A + B; we can divide the x out on both sides, and we cancel that to 4 = A + B.1072

Now, we can substitute in that 1; that will make it even clearer what is going on: 4 = 1 + B.1082

Subtract the 1; we get 3 = B; so at this point, we have figured out what A is; we have figured out what B is.1088

So, we see that we can now go back to our partial fraction decomposition.1094

We can plug in actual numbers, and we can have what that is equivalent to; and we have 1 over (x + 3) over (x + 3).1098

And there is our answer; great.1111

All right, the next one: Write out the form of the partial fraction decomposition, but do not solve for the constant.1114

That is good, because this would be really hard, if we actually had to solve it.1120

So, first we have x, and then (x - 7)2, and then 2x2 + 1, an irreducible quadratic, cubed.1123

All right, we have A/x, plus...the next one is a linear, so it is also just one constant up top, (x - 7);1131

but it is squared in our original thing, so it gets squared; so another constant...C/(x - 7)...this time,1142

it is two of them; it is squared; plus Dx + E, because now we are dealing with an irreducible quadratic.1149

The first time it showed up, so 2x2 + 1, plus Fx + G...just keep going with letters;1158

I am just counting off the alphabet at this point, Fx + G...A, B, C, D, E, F, G...I don't want to make a mistake with my alphabet!...1166

2x2 + we are at the second time, so squared...Hx + I, over (2x2 + 1)3.1173

And that is the partial fraction decomposition.1186

We haven't figured out what A, B, C, D, E, F, G, H, I are.1187

But we could, if we multiplied, because we know that this thing here is equal to this thing here, once we get those correct constants in.1193

So, we could multiply the left side and the right side by that denominator.1202

And everything would cancel out, and we would be left with this awful, awful massive thing.1206

But we could solve it out, slowly but surely; luckily, all it asked for was to just set things up--1209

set up the partial fraction decomposition, but not solve for these constants.1215

So, happily, we can just see that this is how this breaks down.1219

We break it down this way; we use each of the rules, based on whether it is linear or it is an irreducible quadratic.1222

And they show up the number of times of each of their exponents.1227

Great; the third example: Find the partial fraction decomposition for (x5 + 3x3 + 7x)/(x4 + 4x2 + 4).1232

The first thing to notice is that it has a degree 5 on top and a degree 4 on the bottom, so we start as improper.1241

So, since it is improper, we have to use polynomial division to break it into a format that is proper, first.1248

How many times does x4 + 4x2 + 4 go into this thing here?1254

So, let's set up our polynomial division: 4x2 + 4 goes into x5 +...we have no x4's...1258

plus 3x3...we have no x2's...+ 7x; and we have no constants.1270

OK, how many times does x4 go into x5?1279

It goes in x, because x times x4 gets us x5; and we do it on the other things...+ 4x3 + 4x.1281

So, we subtract all of this; let's distribute that subtraction...1294

x5 - x5 becomes 0--no surprise there.1299

3x3 - 4x3 becomes -x3; 7x - 4x becomes positive 3x.1303

We can now bring down everything else, but as soon as we do that, we realize that there is nothing else to be done here,1311

because how many times does x4 go into 0x4? It goes in 0 times.1324

It can't fit in at all, because it is just a 0 to begin with.1329

So, we found our remainder; our remainder is what remains: our -x3 here, our 3x, and that is it.1331

-x3 + 3x is what remains.1338

That means that what we originally started with, x5 + 3x3 + 7x,1342

all over x4 + 4x2 + 4, is equal to...x came out of it, so x plus what our remainder was,1350

-x3 + 3x, over that original denominator, x4 + 4x2 + 4; great.1366

We need some more room; let's flip to the next page.1377

That is what we figured out: we figured out that what we started with breaks into this thing right here.1380

So, now let's figure out how we can decompose this part on the right.1386

We will work on decomposing this first, but we can't forget (circle it in red, so we don't forget) that x, because that is still part of that function.1391

We can't actually decompose it without putting that back in at the very end.1400

Otherwise, we will have decomposed a different function; we will have decomposed this function right here,1403

but forgotten about what we originally started with, because what we originally started with includes this x.1408

All right, -x3 + 3x...oh, that is one thing I forgot to do last time.1413

We had x4 + 4x2 + 4; I quietly factored that, without ever mentioning that it factors into (x2 + 2)2.1418

I'm sorry about that; hopefully that isn't too confusing.1434

-x3 + 3x...over (x2 + 2)2...that is going to be equal to Ax + B,1437

because it is an irreducible quadratic, divided by x2 + 2, plus Cx + D, over (x2 + 2)2.1446

At this point, we multiply both sides by (x2 + 2)2, (x2 + 2)2, (x2 + 2)2.1458

So, that cancels out here, and we are left with -x3 + 3x =...what happens to the Ax + B?1470

Well, its denominator gets canceled, but then it is still left getting hit with one more x2 + 2.1478

What about Cx + D? Well, its denominator just gets canceled; there is nothing left after (x2 + 2)2 hits,1485

because its denominator was the same thing, so we are just left with Cx + D.1492

So, let's expand that, and let's also put this in a different color, just so we don't get confused by the separation.1495

I am not sure how much that will help us.1503

3x =...Ax times x2 becomes Ax3; Ax times 2 becomes 2Ax.1507

Bx2 + 2B...and we will also bring along + Cx + D.1515

At this point, we can set things up so we can see it a little more clearly by putting everything together that comes in a single form.1524

We have Ax3, and we will leave a blank for the x2's...+ 2Ax.1534

And then, we can write over here + Bx2 + 2B...oops, sorry; that is going to have to leave a blank there, as well...1541

+ 2B...and then + Cx, and then + D; notice how this works.1550

We have our cubes here (degree 3 things here); degree 2 things here...1559

well, not quite "degree," because they are not the whole polynomial,1566

but things with exponent 3 here, exponent 2 here, exponent 1 here, and exponent 0 here.1568

So, now we can compare that over here, where we have exponent 3 here and exponent 1 here.1573

So, all of the things that are going to connect to the -x3, which we can also see as -1 times x3, are just Ax3 here.1578

So, it must be that -1 is equal to A; otherwise, we wouldn't have -x3 in the end.1587

We now know that -1 = A; great.1593

What about 2Bx2? Well, we have 0x2; we know that 0x2 = Bx2.1598

Things of degree 2...well, we could have just written that as 0 = B, because our things of degree 2 have to line up with that number B.1608

So, 0 = B; what about our D here? We can figure out our D.1615

Well, we have a constant...+ 0 here; so it must be that 0 is equal to 2B + D.1622

Well, we already know that B equals 0 here; so that just knocks out; so we are now told that 0 equals D, as well.1632

Finally, we are left figuring out what C is; so we know that 3, here, equals 2A, because there are that many x,1641

because we have 3x and 2Ax and Cx, so it must be that 3 = 2A + C, when we combine them all together.1650

We plug in numbers: 3 = 2 times...what is our A? It is -1, so -1, plus C; so we get 3 = -2 + C, or we add 2 to both sides: 5 = C.1662

All right, so 5 = C; so at this point, we have managed to figure out everything that goes into our decomposition.1684

So, we can write that decomposition out: Ax + B...we have -1 as our A, so -x...what is our B?1692

Our B is 0, so it just isn't anything; x2 + 2, plus Cx + D...our D was 0; our C was 5.1701

So, it is going to be 5x/(x2 + 2)2.1712

And we can't forget that x that was originally there; so x plus that...that is our entire partial fraction decomposition.1719

It breaks down into that one right there.1728

All right, the final example: this one is a rough one, but this is the absolute hardest you would end up coming across in class and a test.1731

So, don't worry: this is basically the top difficulty you will have to deal with, probably.1741

You might see something a little bit harder, but this is really about as hard as it is going to get.1746

All right, so happily, they already factored it for us, and we have (t - 2)2(t2 + 2).1750

So, this is degree 2; this is also degree 2; so that combines to degree 4 on the bottom, degree 3 on the top.1756

So, we are proper; our bottom is already factored; we just need to actually work through the decomposition now.1762

We have -t3 + 8t2 - 6t, over (t - 2)2(t2 + 2).1769

How does that decompose? A over our linear factor, t - 2, plus B over our second version of that linear factor,1781

(t - 2)2, plus Cx + D over our irreducible quadratic of t2 + 2; great.1792

At this point, we multiply both sides by the denominator, and we will get (-t3 + 8t2 - 6t) =1801

A was hit with (t - 2)2 + (t2 + 2), so it is going to cancel out one of those (t - 2)'s,1814

and it will be left being multiplied by (t - 2) and (t2 + 2).1822

Plus...B: (t - 2)2 was in its denominator, so it is going to cancel out all of the (t - 2)2 that hit it, but not the t2 + 2 at all.1828

So, we have t2 + 2.1838

And then finally, plus Cx + D, and it is going to have to go in parentheses, because it was hit by another entire thing.1842

And it is going to get (t - 2)2, but it did cancel out the (t2 + 2) that hit it, because it had that in its denominator.1849

At this point, we want to start simplifying things out.1856

We have A times...what is t - 2 times t2 + 2?1859

We get t3...t times 2 becomes 2t...minus 2t2, minus 4,1863

plus Bt2 + 2B + (Cx + D) times (t - 2)2, which becomes t2 - 4t + 4.1877

All right, move a little bit to the left, so we have more room; it is still the same thing on the left side.1893

We have At3 - 2At2 + 2At - 4A (oops, sorry--I just switched colors there)1899

plus Bt2 + 2B...break onto a new line now, so I can see what is going on...1920

Cx times t2 will become C...oops, I made a mistake right from the beginning.1926

It should not be x; it should be (I just stuck with what I have been used to)...the variable we are dealing with isn't x; it is t here.1932

So, that should have been a t the whole time; sorry about that, but that is the sort of mistake you want to catch on your end, too,1942

because we are dealing with t as our variable.1951

So, while the form was with x, as before, that is because the variable was x before.1955

But now, we are dealing with t as a variable, so the form needs to switch to using t.1958

So, let's work this out: Ct times t2 becomes Ct3; Ct times -4t becomes -4Ct2;1963

Ct times +4 is + 4Ct; + Dt2 - 4Dt + 4D; OK, we have a lot of stuff here.1973

So, we can say that this is (-t3 + 8t2 - 6t) =1985

and we can put this in that form, again, of thing with exponent 3, exponent 2, exponent 1, exponent constant;1993

minus 2At2 + 2At - 4A; the next one is + Bt2 + 2B + Ct3 - 4Ct2 + 4Ct,2000

and then + Dt2 - 4Dt + 4D.2027

So, from this, we will be able to figure out that all of our t3's, our line of t3's here,2036

has to be the same as our line of t3's here.2042

So, we will be able to get things like -1 = A + C.2044

Now, this is a lot of stuff to have to work through; we have 4 different variables;2048

we are going to end up having different simultaneous linear equations that we are going to have to solve through.2053

If you have done much work with simultaneous linear equations, you know that that is going to be kind of a pain to work through.2057

So, at this point, we might think, "Oh, I'm lazy; is there anything clever I can do?"2062

Is there a clever way to work through this--some little trick I could use?2067

Well, if we look back to what we originally had when we multiplied out that denominator,2071

we might think, "Oh, look, there is a t - 2 here; there is a t - 2 here; but there is an absence of t - 2 on our B."2076

So, if we realize that, we could say, "Well, if we plug in t = 2, that would cause..."2086

now, none of this stuff up here is going to be true; but we are going to plug in t = 2;2093

and we are going to realize that, if we plug in t = 2, that causes this to turn to 0, which knocks out our A terms.2099

And it will cause this here to turn to 0, as well, and also knock out our C and our D terms.2105

And we will be left with just B times t2 + 2, where t is 2--only t is 2--2110

and -t3 + 8t2 - 6t, when t is 2.2117

Now, this is true for any t; so that means we can plug in any t we want,2122

and if there is a convenient way to get certain things to disappear by plugging in a cleverly chosen t, it is fair.2127

Anything we can do to help us work through this--anything that makes it easier--is fair.2134

So, in this case, we notice that by plugging in this carefully-chosen t, we can get certain things to disappear.2138

Now, it is true for any t; this equation, this whole thing here, is true in general for any t, before we cross stuff out.2144

But in the specific case where we plug in t = 2, certain things will cancel out, and we will be able to figure things out in a very easy way mathematically.2152

So, we plug this in; and we know that, on our left side, we are going to have -23 + 8(2)2 - 6(2).2161

That equals (sorry, there is not quite a lot of room here; I am going to draw in a line, just so we don't get confused)...2173

That equals B times 22 + 2; OK.2180

So, we have -22 is 8, plus 8(2)2 is 4, so 32, minus 6 times 2 is 12, so - 12, equals B times 4 + 2, or 6.2186

Simplify the left side: -8 - 12 is -20; 32 - 20 gets us 12, equals 6B; we divide both sides by 6, and we get 2 = B.2208

So, by use of some cleverness, we were able to skip having to do four simultaneous linear equations,2221

to where eventually we are effectively going to bring it down to 3.2227

And we will be able to not have to get confused by this most complicated column, where we have four different things all going together.2230

We can just be done with B; B is already figured out, which will be really helpful and make things easier on us later on.2236

We could have solved out those four simultaneous linear equations, because each one will produce things.2242

Remember how the first column produced -1 = A + C; each one of these columns will produce an equation.2247

And from four simultaneous linear equations, solving for four variables, we will be able to do it; it is just kind of a pain.2253

So, we came up with this clever way, and we were able to figure out 2 = B by just happenstance.2259

If we plug in t = 2, it made everything but the B terms disappear, and that just left us with a single equation that was pretty easy to solve.2263

And we figured out that 2 = B; all right, well, let's work through it now.2271

So, we figured out B = 2, and now we have -t3 + 8t2 - 6t =2275

At3 + Ct3 - 2At2 + Bt2 - 4Ct2 + Dt2 + 2At + 4Ct - 4Dt - 4A + 2B + 4D; wow!2282

OK, so at this point, let's put columns to columns; so -t3 goes to our At3 + Ct3.2294

So, it must be that -1 is equal to A + C.2301

We can do this column here; I accidentally cut through that negative sign--hopefully that wasn't too confusing.2309

So, that one here lines up with the -6t; so we have that -6 (let's write it on a separate location) equals 2A + 4C - 4D.2316

They all have t's showing up, so we just divide out the t's, and we are left with this simple linear equation that we can work with.2334

And then, finally, the last one of constants: what is our constant? Our constant is 0.2340

So, we have 0 = -4A + 2B + 4D; for extra credit, let's just see what this would have been.2347

How many t2's did we have? We had 8, so in our difficult-to-read yellow (I'll make it black,2357

just because black is easier to read), 8 = -2A + B - 4C + D.2362

We could have worked that one out, but I wanted you to see what it would have been.2371

But we will actually end up having enough information, because we have three equations and three unknowns;2375

so we have enough with the red, blue, and green things.2379

So, at this point, let's figure out A first; so that means we need to solve for everything that isn't A,2382

so that we can plug them in, get rid of everything else, and just have A.2389

So, on the left side, we have -1 = A + C; so we solve for C, and we get -A - 1 = C.2392

OK; on the green one, we plug in what we know for our B.2405

We have 0 = -4A + 2(2 + 4D); 0 = -4A + 4 + 4D; 0 =...let's divide everything by 4, just to make things easier...-A + 1 + D.2409

So, we move that stuff over, and we get: add A on both sides; subtract by 1 on both sides; A - 1 = D.2430

So, we have A - 1 = D; we have -A - 1 = C; so we can plug this information into this equation, and we will be able to get something that is just using A.2439

-6 = 2(A)...that actually stays around, because it was just A from the beginning; 4...what did we figure out C was?2460

C is the same thing as -A - 1, minus 4 times...what did we figure out D is?2468

It is the same thing as A - 1; simplify that out: -6 = 2A + 4(-A), so we will make that - 4A, minus 4...2475

-4A - 4A...and -4 times -1 becomes + 4; so at this point, we see that we have -4 and +4, so those cancel each other out.2489

We have -6 = 2A - 8A; -6 = -6A; and thus 1 (dividing both sides by -6) = A.2500

With 1 = A in place, we can now easily go back and figure out everything else.2513

We have -A - 1, so negative...plug it in, knowing that it is 1...minus 1, equals C; so we have -2 = C; great.2518

And also, plug in over here; and we have that 1 - 1 = D; it turns out that D is just 0.2533

So, at this point, we have figured out all of the things that we need.2545

We have this set up as (t - 2) + (t - 2)2 + (t2 + 2).2549

It was in the format -t3 + 8t2 - 6t, all over (t - 2)2(t2 + 2).2557

OK, so that is the case; we have that as something over (t - 2).2579

It was A/(t - 2) + B/(t - 2)2 + (Cx + D)/(t2 + 2).2586

OK, at this point, we can actually plug in our numbers; we have that A is 1, so we have 1/(t - 2).2601

B is 2, so we have 2/(t - 2)2, plus...C was -2, so -2x, and 0 was D, so that part just disappears.2608

So, we have -2x/(t2 + 2); oops, once again, I did it again; I got to thinking in terms of x, as opposed to t.2624

But we are using a different variable: -2t/(t2 + 2).2633

And there we are; we have decomposed it into its partial fractions.2638

That is pretty long; it is pretty complicated; but that is pretty much as hard as it gets.2643

As long as you break it down, and then you multiply everything out carefully,2647

and you are careful with all of your algebra and your arithmetic, you can get it into this form right here,2651

where you have this big block compared to these original things, and then you are able to figure out all of these linear equations.2656

And you can possibly be clever and figure out some way to figure out B = 2.2662

But you also can just work through a bunch of linear equations, solving for one thing at a time in terms of the other,2667

plugging them all together, and then solving it out.2673

Sometimes it takes a lot of work; sometimes it goes kind of slowly.2675

But ultimately, if you just keep working at it, you can get it.2678

And it is really, really useful in calculus.2680

I know--I know that you aren't going to be using it immediately.2683

But honestly, this thing makes a problem that would be, otherwise, totally impossible to do, just really easy.2685

So, it is a really handy thing in calculus.2691

All right, we will see you at later--goodbye!2693