Algebra 1 Greatest Common Factor & Factor by Grouping

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In this lesson we are going to take a look at the greatest common factor and the technique of factor by grouping.0002

First we will learn how to recognize a greatest common factor for a list of different terms0012

and how we can actually factor out that greatest common factor when you have something like a polynomial.0017

This will lead directly into the technique of factor by grouping, a good way of breaking down a large polynomial.0023

You are going to hear me use that word factor quite a bit in this entire section.0032

You are probably a little curious about what exactly factoring means.0036

Technically it means to write a quantity as a product.0043

We will be breaking down into pieces that are multiplied.0047

In a more practical sense, the way I like to think of factoring is that you are ripping things apart.0051

It is almost like doing the opposite of multiplication.0056

For example if I have 2 × 3 I could put them together and get 6.0059

When I have the number 6 then I can factor it down into those individual pieces again 2 × 3.0066

It is like multiplication but we are going in the reverse direction.0072

Now some expressions and numbers could have many different factors.0077

Let us take the number 12.0082

If you break that one down, you could look at it as 2× 6 and you could continue in breaking down the 6 and 2 × 3.0085

You have factors like 2 would go in there, 4 would go into 12, 3 would be a factor, 6 would be factor.0095

All those things could go into 12.0103

The greatest common factor will be the largest number that will divide into all the numbers present.0110

I will show you this to you twice just to get a better feel for it.0116

Suppose I'm looking at 50 and 75, the greatest common factor would be the largest number that could divide into both of those evenly.0120

5 could divide it into both of those evenly and that would be a pretty good choice, but is not the largest thing that could go in there.0128

The largest thing that could go in would be the number 25.0136

You will see that if I do take them both and divide them by 25, true enough they both go in there evenly.0140

25 is my greatest common factor.0150

We may only do this for pairs but you could do it with an entire list of numbers.0155

In this one I have 12, 18, 26, 32, and again we are looking for the largest thing that divides into all of that.0162

Sometimes a good way to find the largest thing that goes into all of them is just find something that works,0169

maybe something like 2 and end up reducing them bit by bit.0177

If I divide everything in here by 2 I would get 6, 9, 13 and 16.0183

Nothing else will go into all of those since 13 is a prime number.0191

My greatest common factor is 2 in this case.0197

It is not a very big number but it happens to be the largest in there that will go into all for these numbers evenly.0201

In algebra we are not interested in just single numbers we also want to see this process work out when we have lots of variables.0209

Let us see how this works out.0216

Maybe I’m looking at the numbers over the terms -16r⁹, -10r¹⁵, 8r¹².0218

To find the greatest common factor, I’m first going to look at those numbers and see if there is a large number that could go into all of them.0226

I could divide all of them by 2, that would work but actually I think 2 is the largest one.0234

I will say that 2 will divide it into all of these evenly.0242

Divided by 2, divided by 2, divided by 2 and I would get -8, -5, 4.0248

We can also do this thinking about our variables.0258

What was the largest variable raise to the power that we can divide into all of them?0261

Think of using your quotient rule to help out.0266

With this one r⁹ is the greatest thing that I can divide all of them by.0272

I would not have to deal with negative exponents or anything like that.0279

Let us write that over here, r⁹.0284

r⁹ ÷ r⁹ would be a single r⁰ or 1.0288

r¹⁵ ÷ r⁹ = r⁶ and r¹² ÷ r⁹ = r³.0293

Our greatest common factor would be 2r⁹.0303

Be careful when you are dealing with those variables and exponents.0307

Sometimes you will hear greatest common factor in your mind and you think you should go after the greatest exponent.0310

But as we can see in this example, it is not the case.0315

We actually took the smallest exponent because it was simply the greatest thing that could go into all of them.0319

Put that in mind and let us try this again.0327

In this one I have nothing but variables.0330

h⁴ k⁶ h³ k⁶ and h⁹ k².0332

What is the largest power of h that could go into all of them?0340

It is going to be h³.0345

I picked up on that because I can see that looking in the middle one it is the smallest power.0349

Looking at all the k’s, the largest exponent of k that could go into all them would be k².0354

My greatest common factor will be h³ k².0364

Let us see what this will end up reducing down to if I take that greatest common factor.0373

h¹ k⁴ h³ and h³ will cancel each other out.0378

k⁴ I have h⁶ and I have no k that will end up cancelling each other out.0388

Being able to recognize the greatest common factor will help us in the next process.0396

Make sure it is pretty solid.0400

If you can recognize the greatest common factor you can often factor it out of a polynomial.0405

What I mean by factoring out is we are going to place it out front of a pair of parentheses0413

and put the reduced terms inside of that parentheses .0418

3m + 12 this is like looking at 3m and looking at 12, what is the greatest common factor of those two terms?0423

What could potentially go into both of them evenly and what is the largest thing that will do that?0431

The number 3 will go into 3m and the number 3 will also go into 12.0438

3 will be my greatest common factor.0446

I’m going to write that out front on the outside of the parenthesis.0449

What I write on the inside, what happens when I take 3m ÷ 3?0453

I will get just single m and when I take 12 ÷ 3 I will get 4.0458

Here I have taken a polynomial and essentially factored it into a 3 and into another polynomial m + 4.0467

The neat part about this is since factoring is like the opposite of multiplication you can check this by running through the distribution process.0475

If you take 3 and you put it back in here do you get the original answer?0487

You will see that in fact you do get 3m + 12.0492

That is how you know this has been factored correctly.0496

In some instances, just like with numbers when you are looking for the greatest common factor0501

it turns out you would not be able to find greatest common factor to pull from all of the terms or the largest thing will actually be 1 or itself.0506

In cases like that, we say that the polynomial is prime.0513

In other words it does not break down into any other pieces.0517

Let me give you a quick example of a prime polynomial 5x + 7.0520

There is not a number that goes into 5 and into 7 so I would consider that one prime.0526

How about 8x + 9y?0534

Individually those numbers are divisible, but there is not a number that goes in the 8 and 9 it would be a prime polynomial.0542

Let us work on factoring out the greatest common factor from various different polynomials,0555

just so we get lots of good practice with it.0560

Starting with the first one I have 6x⁴ + 12x².0562

Let us hunt down those numbers first.0567

The largest number that would go into both 6 and 12 would be 6.0570

Let us look at those x.0578

What is the largest power of x that would divide into both of them?0581

That have to be x².0586

We will write in the leftovers inside our parentheses 6x⁴ ÷ 6x²=x² and 12x² ÷ 6x² = 2.0592

I factored out the greatest common factor for that polynomial.0611

Let us give this another shot with something has a bunch of terms to it.0615

30x⁶ 25x⁵ 10x⁴.0619

Looking at the numbers 30, 25, and 10, what is the largest thing that could go into all of them, 5 will do it.0625

Looking at our variables the largest variable that could go into all them would be x⁴.0634

It is time to write down what I left over 30 ÷ 5 = 6, x⁶ ÷ x⁴ = x².0645

25x⁵ ÷ 5 = 5, x⁵ ÷ x⁴ =x and 10x⁴ ÷5x⁴ = 2.0659

That one has been factored out.0675

Continuing on, let us get into some of the trickier ones.0678

This one has 3 terms and has multiple variables in there.0681

It has e and t.0685

Let us look at our numbers.0688

What could go into 8, 12 and 16?0689

2 would definitely go into all of them.0696

4 is larger and it would also go into all of them and I think that is largest thing, so we will go with 4.0698

Let us take care of these variables, one at a time.0707

Looking at the e’s, what is the largest power of e that could go into all of them?0710

This will be e⁴.0715

Onto the t’s, the largest exponent of t that would go into all of them is t².0721

Now that we have properly identified our greatest common factor, let us write down what is left over.0732

8 ÷ 4 =2, e⁵ ÷ e⁴ =e and t² ÷ t² =1.0737

16 ÷ 4 = 4, e⁶ ÷ e⁴ = e² and t³ ÷ t² =t.0752

Onto the last one, -12 ÷ 4 = -3, e⁴ ÷ e⁴ = 1 and t⁷ ÷ t² = t⁵.0766

We have our factor polynomial.0790

One last one, I threw this one in so we can deal some fractions.0794

We will first think of what fraction could divide into ¼ and into ¾ ?0800

¼ is the only thing that will do it.0808

What could go into y⁹ and y²?0812

Look for the small exponent that usually helps out.0817

That will be our greatest common factor, let us write what is left over.0823

¼ ÷ ¼ = 1 and y⁹ ÷ y² = y⁷.0828

¾ ÷ ¼ = 3 and y² ÷ y² = 1.0839

At all of these instances you identify your greatest common factor first and simply write your leftovers inside the parentheses.0849

One quick thing that can help out, you can double check your work by simply multiplying your greatest common factor0856

back into those parentheses using your distribution property.0861

If it turns out to be the original problem, you will know that you have done it correctly.0865

In some terms, there is a much larger piece in common that you can go ahead and pull out.0873

Even though it is larger, feel free to still factor it as normal.0880

That is the hard part.0884

To demonstrate this I have 6 × (p + q) –r (p + q).0887

It has that p + q piece again and both of its parts.0894

When I'm looking at the greatest common factor of this first piece and the second piece, it is that p + q.0899

I’m going to take out the p + q as an entire piece and that is my greatest common factor.0910

Inside the set of parentheses I will write what is left over.0919

From the first part there is a 6 that was multiplied by p + q so I will put that in there and there was a -r on the second piece.0924

This looks a little strange, I mean it seems weird that we can take out such a large piece, but unusually that it does work.0933

To convince you I will go ahead and multiply these things back together using foil.0941

What I have here is my first terms would be 6p, outside terms would be –pr.0947

Inside terms 6q and my last terms –qr.0955

Compare this after I use my distribution property on the original.0962

(6p + 6q) – (rp + rq).0970

What you see is that it does match up with the original.0984

There is my 6p, I have my – pr and – rp, there are just in a slightly different order.0988

I have 6q and 6q -qr and I forgot to distribute my negative sign.0995

This should be a -rq and so it is the same term.1007

What you find is that you can pull out that large chunk of p + q and that leads to what is known as factor by grouping.1013

The idea behind factor by grouping is you try and take out the greatest common factor from a few terms at a time.1024

Rather than looking at all of them, just take them in parts.1031

This grades usually a very large piece and you can collect it into two groups.1034

Then you factor within those groups, you take out one of those large pieces.1040

This will allow you to factor the entire polynomial.1045

Sometimes factor by grouping does not seem to work or usually you end up struggling with a little bit.1048

If that is the case, try rearranging the terms and try to factor by grouping one more time.1053

It is a neat process and let us give it a try, when you factor by grouping on p × q + 5q + 2p +10.1060

Watch how this works.1071

Rather than looking at all the terms at once, I’m going to take them two at a time.1073

A part of my motivation for doing that is if I were to look at all the terms all at once, they do not have anything in common.1078

Let us take these first two.1086

What would be the greatest common factor for p × q + 5q.1088

q is in both of the terms so that is my greatest common factor for both of those.1094

What is left over? There is still a p in there and there is still 5.1101

Those are done.1109

Let us look at the next two terms.1111

What is the greatest common factor of just those two.1115

I can see a 2 goes into both of them and then let us see, the only thing left over would be a p + 5.1119

I will factor them into those little individual groups and now notice how I only have two things and they both have a p + 5 in common.1131

I'm going to take out the p + 5 as my greatest common factor of those two terms.1142

What will be left behind will be the q + 2.1151

This will represent my factor polynomials.1156

Let us try this again and see how it could work.1162

I have 2xy + 3y + 2x + 3.1164

Looking at the first two.1170

These have a y in common, let us go ahead and take that out.1174

2x will still be left when I divide 2xy by y and I still have a 3 over here when I divide 3y by y.1187

Looking at the next two terms, it looks like these ones do not have anything in common.1198

I might consider that the only thing that you do have in common is just a 1.1205

I could divide them both by 1.1209

I still have a 2x and I still have a 3.1213

Notice how we have that common chunk in there 2x + 3.1218

We will take that out 2x + 3 and then we will write what is left over just the y +1.1223

We can consider this one factored.1237

In this next example, be on the watch out for some negative signs which could show up as we factor out that greatest common factor.1242

I’m going to take this two at a time.1251

Looking at the first two, they do not have a number in common, but they both looks like can be divided by x².1255

Let us take that out.1262

After taking our x² we would have left over a single x and a 3.1268

Let us look at the next two terms.1280

I can see that if I would go into -5 and 5 would divide into a -15 and they both have the negative sign.1284

That is important because I’m thinking ahead and try to think how I also have this common x + 3 piece.1292

Especially if I’m trying to factor by grouping.1300

The only way that is going to work out is if I take out a greatest common factor that is a -5.1303

What would that does to our left over pieces now?1310

-5x ÷ -5 = x and -15 ÷ -5 = 3.1314

I get those left over pieces like I need to, that same x +3 on the other side.1324

Now I have this I can take out my common piece of x + 3 and I can go ahead and write the leftovers x² – 5.1332

Be on the watch out for certain situations where you need to rearrange things first.1352

In this one I have 6w² - 20x + 15w - 8wx.1357

And it is tempting to just go ahead and jump in there and try and factor.1364

Watch what happens if you do so.1367

First we take the first two and we would look for something common.1370

The only thing I can see that would be common is a 2 goes into 6 and into 20 and they do not have the same variables.1375

That is all I can do.1385

I have 3w² and -10x.1386

Looking at the other terms the only thing that I see common over there is they have a w in common.1393

What would be left over? I still have a 15 and have 8x.1406

That is definitely a problem because these pieces are not the same.1413

I need to do some rearrangement and retry this factor by grouping one more time.1419

Let us go ahead and rewrite this.1433

I will rewrite it as 6w² + 15w - 8wx and we will put that -20x on the very end.1435

Here is a little bit of my motivation for putting in that order.1452

When looking at the w's, notice how this one is a w² and this one is a single w.1456

I have the one is one more in power after the left.1460

And looking at the same w's, here is my w¹, this is like w⁰.1464

It is like I have lined things up according to the w power.1470

Watch how I factor by grouping works out much better.1475

Taking the first two they both have a w in common and I can pull that out.1481

It also looks like I can pullout 3.1485

3w will be like our greatest common factor.1488

What would be left over on the inside?1494

6w² ÷ 3w = 2w and 15w ÷ 3w = 5.1499

I have -8wx -20x, both of those are negative, I think I will go ahead and pull out a -4, that will go into both.1510

They both contain x so we will also take out an x.1523

-8wx ÷ -4x.1532

There are still 2w left in there.1536

-20x ÷ -4 = 5.1540

I can see there is that common piece that I wanted the first time.1546

We can go ahead and factor that out front.1552

2w + 5 and 3w - 4x and now this is completely factored.1556

Let us do one last one, and this is the one where we might have to do a little bit of rearranging just make sure it works out.1575

9xy – 4 + 12x – 3y.1581

What to do here?1587

Let us go ahead and put the things that have x together.1590

9xy + 12x - 3y - 4 and I have just rearranged it.1606

We will look at these first two terms here and take out their greatest common factor.1612

You can see that 3 goes into both of them and they both contain x.1618

It will take both of those out.1624

9x ÷ 3x, 9 ÷ 3 = 3, x ÷ x =1 and we still have a y sitting in there.1628

12 ÷ 3 = 4, x ÷ x = 1.1639

Those two would be gone.1643

That looks pretty nice and it is actually starting to match what I have over here.1646

Notice that the only difference is a negative.1651

I'm going to take out a -1 from both of the terms that not should be able to flip my signs and make it just fine.1655

-3y ÷ -1 = 3y and -4 ÷ -1 = 4.1663

I have that nice common piece that I need.1672

We will go ahead and factor that out .1677

3y + 4 and 3x - 1.1681

Now this is completely factored.1690

When using the technique of factor by grouping take it two at a time and factor those first.1692

Look for your common piece and factor that out and that should factor your entire polynomial completely.1698

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