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### Special Factoring Techniques

- By memorizing some special polynomials, you can have quick formulas for factoring. These include
- The difference of squares
- Perfect square trinomials
- The sum and difference of cubes
- Make sure to factor out a greatest common factor before using any of these special formulas.
- Note that there is not a special formula for the sum of squares.
- When using the formula for the sum or difference of cubes, note how the signs follow SOP. This stands for Same Opposite Positive.
- Some of these special formulas could be used more than once in a particular problem.

### Special Factoring Techniques

25x

^{2}− 64

- 25x
^{2}= a^{2} - a = 5x
- 64 = b
^{2} - b = 8

81y

^{2}− 121

- a
^{2}= 81y^{2} - a = 9y
- b
^{2}= 121 - b = 11

49t

^{2}− 225

- a
^{2}= 49t^{2} - a = 7t
- b
^{2}= 15

3m

^{4}− 27m

^{2}

- 3m
^{2}( m^{2}− 9 ) - a
^{2}= m^{2} - a = m
- b
^{2}= 9 - b = 3

^{2}( m + 3 )( m − 3 )

72k

^{4}− 288k

^{2}

- 2k
^{2}( 36k^{2}− 144 ) - a
^{2}= 36k^{2} - a = 6k
- b
^{2}= 144 - b = 12

^{2}( 6k + 12 )( 6k − 12 )

12p

^{4}− 75p

^{2}

- 3p
^{2}( 4p^{2}− 25 ) - a
^{2}= 4p^{2} - a = 2p
- b
^{2}= 25

6n

^{4}− 18n

^{3}− 10n

^{2}+ 30n

- 2n(3n
^{3}− 9n^{2}− 5n + 15) - 2n[ ( 3n
^{3}− 9n ) + ( − 5n + 15 ) ] - 2n[ 3n
^{2}( n − 3 ) − 5( n − 3 ) ]

^{2}− 5 )( n − 3 ) ]

48x

^{4}+ 96x

^{3}− 75x

^{2}− 150x

- 3x( 16x
^{3}+ 32x^{2}− 25x − 50 ) - 3x[ ( 16x
^{3}+ 32x^{2}) + ( − 25x − 50 ) ] - 3x[ 16x
^{3}( x + 2 ) − 25( x + 2 ) ] - 3x[ ( 16x
^{2}− 25 )( x + 2 ) ]

5y

^{4}− 34y

^{3}− 20y

^{2}− 35y

- 5y( y
^{3}− 7y − 4y − 7 ) - 5y[ ( y
^{3}− 7y^{2}) + ( − 4y − 7 ) ] - 5y[ y
^{2}( y − 7 ) − 4( y − 7 ) ] - 5y[ ( y
^{2}− 4 )( y − 7 ) ]

4x

^{6}= 324x

^{2}

- 4x
^{6}− 324x^{2}= 0 - 4x
^{2}( x^{4}− 81 ) = 0 - 4x
^{2}( x^{2}+ 9 )( x^{2}− 9 ) = 0 - 4x
^{2}( x^{2}+ 9 )( x + 3 )( x − 3 ) = 0 - 4x
^{2}= 0x = 0 - x
^{2}+ 9 = 0x^{2}= − 9√{x^{2}} = √{ − 9} (No Solution) - x + 3 = 0x = − 3
- x − 3 = 0x = 3

64g

^{2}− 80g + 25

^{2}

121s

^{2}+ 154s + 49

^{2}

81h

^{2}− 180h + 100

^{2}

20c

^{3}− 60c + 45c

- 5c( 4c
^{2}− 12c + 9 )

^{2}

50e

^{3}− 120e

^{2}+ 72e

- 2e( 25e
^{2}− 60e + 36 )

^{2}

192z

^{3}+ 576z

^{2}+ 432z

- 3z( 64z
^{2}+ 192z + 144 )

^{2}

2y

^{2}− 16y + 32 = 50

- 2( y
^{2}− 8y + 16 ) = 50 - 2( y − 4 )
^{2}= 50 - ( y − 4 )
^{2}= 25 - √{( y − 4 )
^{2}} = ±√{25} y − 4 = ±5 y − 4 = 5y = 9 - y − 4 = − 5y = − 1

3b

^{2}+ 150b + 75 = 147

- 3( b
^{2}+ 50b + 25 ) = 147 - 3( b + 5 )
^{2}= 147 - ( b + 5 )
^{2}= 49 - b + 5 = ±7
- b + 5 = 7b = 2
- b + 5 = − 7b = − 12

16m

^{2}− 48m + 36 = 256

- 4(4m
^{2}− 12m + 9) = 256 - 4( 2m − 3 )
^{2}= 256 - ( 2m − 3 )
^{2}= 64 - 2m − 3 = ±8
- 2m = ±11
- 2m = 11m = [11/2]
- 2m = − 11m = − [11/2]

63a

^{2}+ 336a + 448 = 567

- 7( 9a
^{2}+ 48a + 64 ) = 567 - 7( 3a + 8 )
^{2}= 567 - ( 3a + 8 )
^{2}= 81 - 3a + 8 = ±9
- 3a + 8 = 93a = 1a = [1/3]
- 3a + 8 = − 93a = − 17a = − [17/3]

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Special Factoring Techniques

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Objectives 0:07
- Special Factoring Techniques 0:26
- Difference of Squares
- Perfect Square Trinomials
- No Sum of Squares
- Special Factoring Techniques Cont. 4:03
- Difference of Squares Example
- Perfect Square Trinomials Example
- Example 1 7:31
- Example 2 9:59
- Example 3 11:47
- Example 4 15:09
- Special Factoring Techniques Cont. 19:07
- Sum of Cubes and Difference of Cubes
- Example 5 23:13
- Example 6 26:12

### Algebra 1 Online Course

### Transcription: Special Factoring Techniques

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

*In this lesson we are going take a look at some very special factoring techniques that you want to know.*0002

*Some of these special factoring techniques will be recognizing the difference of squares,*0009

*a perfect square trinomial and the difference and sum of cubes.*0015

*Here is a great pattern to pick up on that can cut out a lot of work for you.*0020

*Recall that when we are multiplying together a few different polynomials that some very special ones came up.*0028

*For example, if you multiply these two together x - y and x + y, the middle term end up dropping out entirely.*0035

*Our first term would be x ^{2}, our outside terms would be x, y, inside terms -xy and last terms –y^{2}.*0046

*The only thing that will remain since these two cancel each other out are just the x ^{2} and the - y^{2}.*0064

*In few other situations you might have had, say x + y the whole thing squared and x - y the whole thing is squared of that one.*0068

*Since we have been working with factoring, we want to look at all 3 of these processes in the other way.*0083

*If we can recognize that we have one of these types of polynomials we will immediately know how it should factor on the other side.*0090

*Think of these as like some nice handy formulas that can help you cut down on the factoring process.*0098

*Since we are looking at these in reverse, we want to look at some key features of these formulas*0108

*to help you recognize them when looking at those polynomials.*0114

*Let us start off.*0117

*When you are looking at the difference of squares here is how you can pick it out.*0119

*One, both of these terms should be squared and notice how there is no middle term.*0124

*One last very important thing is this is the difference of squares, so you should have subtraction.*0138

*You can find all those 3 parts then you know how this will end up factoring.*0146

*This will be x + y and x – y.*0152

*For your perfect square trinomials, you want the first term and the last term to be squared.*0159

*The last term over here should be positive.*0170

*Now comes the tricky part, you want the middle term to be twice what you get when you multiply x and y together.*0176

*It does not matter if it is positive or negative, since you have two different formulas for those.*0187

*But that is what it should turn out to be.*0191

*Now if you can recognize all of those things, then you know how these will break down.*0195

*This one will be x + y ^{2} and that one will be x - y^{2}.*0199

*Be careful when using these quick and easy formulas, note that there is no sum of squares formula.*0207

*It is often very tempting to look at something like x ^{2} + y^{2} and think that it will break down, but be careful it will not.*0216

*Be very keen on picking up these small things that we can save yourself some time when going through that factoring process.*0233

*Let us give it a try.*0244

*I want to go ahead and factor x ^{2} -16^{2}.*0245

*I think this looks like the difference of squares and I will end up rewriting it a little bit so you can tell.*0250

*This is x ^{2} - and I need the other number to be^{2} as well, this would have to be a 4^{2}.*0256

*I'm looking at my two terms of being like an x and 4.*0266

*According to my formula, this will definitely factor.*0271

*Factor into x and x then +4 - 4.*0277

*Let us go ahead and multiply things back together to see why this is the correct factorization.*0288

*We take our first terms, we get x ^{2}, our outside would be - 4x, inside + 4x, and our last terms -16.*0295

*You will notice that those outside and inside terms, one is positive and one is negative, other than they are the same.*0306

*They will end up canceling each other out.*0312

*The only thing left with is just the x ^{2} and the 16 like we should.*0314

*You can be pretty confident that this is the proper factorization when dealing with the difference of squares.*0321

*The other ones that we want to look at are those perfect square trinomials.*0331

*Let us see if we can hunt this one out.*0335

*Looking at it, I can see that my y ^{2} and I want to write the 100 as being a squared number as well.*0337

*y ^{2} + 20y + the number 10^{2}.*0344

*My first term there is a y being squared and my last term is 10 being squared.*0353

*My last term is positive and that is a good sign.*0359

*I need to check just one more thing, is my middle term twice the first term multiplied by the last term?*0363

*2 × y × 10.*0372

*It does not take too much work.*0375

*But if we do multiply these together or maybe rearrange them first, you will see that we do get a 20y like we are supposed to.*0376

*I can use my nice little shortcut formula to go ahead and factor this.*0384

*My first terms will be y and my last terms 10.*0393

*Notice how y + 10 and y + 10 they are exactly the same.*0403

*We will go ahead and we write this as a single term with just a square exponent on it.*0407

*Nice, very quick and easy.*0416

*Let us go ahead and multiply this out to show you that it does work.*0419

*My first terms would be y ^{2}, my outside and inside terms would be exactly the same.*0423

*My last terms, 10 × 10 would be 100.*0431

*The fact that our outside and inside terms are exactly the same means that we will have two of them.*0434

*That is why we need that 2 in our shortcut formula.*0440

*That should make you pretty confident that this is the correct factorization for our perfect square trinomial.*0444

*Let us try some examples to see if we can get better using the special factoring techniques.*0452

*In this first one I have y ^{2} - 81and this looks like it might be the difference of squares.*0457

*I need to write this such that I have my first number squared and the second number squared.*0467

*9 ^{2} is the only thing that will give me 1.*0474

*I can definitely see that it is the difference of squares.*0477

*I have y ^{2} 9^{2} and they are being subtracted.*0480

*This means that I can use my nice shortcut to go ahead and break this down.*0485

*Y - 9 and y + 9.*0490

*Let us try something that has a few more fractions in it.*0500

*This one is x ^{2} - 16/25.*0504

*We want to write this such that we have something squared - something squared.*0508

*Two things that will multiply and give me x ^{2}, that will be x.*0516

*To get that 16/25 I think I we are going to have to use 4/5.*0520

*4 × 4 =16 and 5 × 5 =25.*0526

*Sure enough, this is the difference of squares because both of these are squared and we are taking their difference subtracted.*0532

*Let us write down what we have.*0542

*x - 4/5 and x + 4/5.*0546

*One more example, this one is u ^{2} + 36.*0555

*You can recognize this one as u ^{2} + 6 being squared.*0562

*Be very careful, do not go anywhere beyond that because we cannot use one of our special formulas here.*0568

*Notice how this one we are dealing with a sum and we do not have a special formula for the sum of squares.*0577

*I’m going to write does not factor.*0585

*Be on the watch out for that one.*0591

*If you have the difference of squares you are in good shape.*0593

*If you have the sum of squares then you are stocked.*0595

*Let us try a few more that involved the difference of squares.*0602

*This first one we have a 49x ^{2} - 25.*0608

*So we want to think of it as something squared - something squared.*0613

*Two numbers that will multiply and give me my 49, that must be 7.*0620

*7x is my first one.*0626

*Two numbers that will multiply to 25 must be 5.*0629

*We can write how this one breaks down.*0637

*I have 7x - 5 and 7x + 5.*0641

*One more, here I have two square numbers and we are taking their difference.*0651

*We need to view this as something squared - something squared.*0658

*See 8 × 8 = 64 I will put that in there.*0665

*8p if I square that entire thing I will get 64p ^{2}.*0671

*9q, 9 × 9 will give us 81 and q ^{2} will give us that q^{2}.*0678

*Now we have both of our pieces and we can use our shortcut.*0688

*8p – 9q 8p + 9q and that one is factored completely.*0696

*When using any of the special factoring techniques you should always be on the watch out*0709

*for some common factors that can pull out from the very beginning.*0714

*In these next two examples we will see if we can do just that.*0718

*This one 50 is not a square number nor is 32 but if I look at both of them, they are both divisible by 2.*0722

*I need to take out a common 2 first.*0732

*Let us try that common 2 and see what is left over.*0736

*5 0÷ 2 = 25w ^{2} and 32 ÷ 2 = -16.*0739

*This is looking much better because 25 is a square number and so is 16.*0753

*Let us just focus on those parts and see if we can write this as the difference of squares.*0759

*What would give us 25 must be a 5 ^{2} and 16 that is a 4^{2}.*0767

*I can see how this one will factor.*0777

*5w - 4 and 5w + 4.*0781

*Do not forget that 2 that we took at the very beginning, it still hanging out front during the entire process.*0787

*Go ahead and put it in.*0795

*One more of this difference of squares, but this one is a little bit of a change to it.*0798

*Notice how this one is actually to the 4th power.*0802

*Sometimes you might be able to apply these rules, but you have to start off by looking out as something squared × something squared.*0806

*Let us see what we can do with this one.*0817

*What square would give us a y ^{4}?*0819

*This will be trickier but if you square a y ^{2} that will do it.*0824

*It comes from our rules for exponents because we would end up multiplying the 2’s together.*0829

*What square would give us 81? That would be 9.*0835

*We could use our formula to break this down y ^{2} - 9 and y^{2} + 9.*0841

*It is very tempting to stop right there but actually you can continue this one more step.*0854

*If you look over here you have another difference of squares.*0859

*You have a y ^{2} and 9 can be factored out as 3^{2}.*0863

*Use the formula one more time to break that one down.*0868

*What square would give you a y ^{2}, y and what squared would give you 9 and 3.*0874

*This is y - 3 and y + 3.*0887

*Now remember this one over on the other side is still there, go ahead and write it along with the rest.*0893

*Be careful not to try and apply your formulas to that one because that one is the sum of squares and we do not have a formula for that one.*0900

*Now that we have seen some of those, let us get into our perfect square trinomials.*0912

*The first one I have is x ^{2} - 24x + 144.*0917

*Let us double check to see that this is a perfect square.*0922

*My first term is squared and my last term is squared.*0926

*I'm looking at something like x ^{2} - 24x and what will be squared to get 144, 12^{2}.*0931

*In order for this to work out nicely, I want to make sure that middle term comes from taking 2 multiplied by my first term x and last term 12.*0944

*By combining all that sure enough, you will see that we do get our 24x.*0961

*We are in pretty good shape.*0966

*This is -24 over here and that gives me another clue on how this will break down.*0970

*Break down into x - 12 and x - 12 which of course we just go ahead and package up into one x -12, the whole thing squared.*0978

*It is a very handy formula.*0993

*Let us try the next one.*0995

*18 x ^{3} + 84x^{2} + 94.*0997

*That is quite a big one.*1002

*In order to tackle this one, we definitely want a code for any common factors.*1005

*One thing I can say is one they are all divisible by 2 so that will help break down quite a bit.*1011

*And everything has x in common.*1016

*Let us take those out and see what we have left over.*1023

*18 ÷ 2 = 9x ^{2}, 84 ÷ 2 = 42x and 98 ÷ 2 = 49.*1030

*Let us see if we can use our formula on this.*1059

*Out front I have a 9x ^{2}, on the back I have a 49.*1062

*You want to view this or at least that first one as being like a 3x the whole thing is being squared.*1069

*On the end that is like a 7 ^{2}.*1075

*Check to see that it meshes well with your middle term.*1078

*Can you take your first term 3x and your last term 7 and multiply by 2 to get 42.*1082

*2 × 3 × 7 =42 and there is still an x in there.*1092

*It matches just fine.*1096

*Let us use our formula and break that down.*1098

*That will be 3x + 7.*1106

*We are using the + in here because the 42 is positive*1115

*and the reason why this is a 3x because that is what the first term squared would have to be.*1120

*Do not forget that 2x out front at the very beginning it is still there.*1127

*We will just finally go ahead and condense things since the 3x +7 appears twice.*1131

*I will write this 3x + 7 that whole thing squared with a 2x out front.*1136

*In addition to those special formulas that we have for factoring, there is also some for the sum and difference of cubes.*1149

*We have not done a lot of factoring with cubes so these are important in order to break these types of problems down.*1158

*For the sum and difference of cubes, they break down using these two formulas.*1165

*I think these are a little bit unusual because we look at what they factor into, they factor in some rather large polynomials.*1170

*You have this x - y, which is not too big, but then over here we have x ^{2} + xy + y^{2}.*1178

*Let me first convince you that it is how it should factor by taking those large polynomials and multiplying them together.*1186

*I’m going to do this using one of my tables.*1202

*I will write the terms of one polynomial along the top and let me write the other one along the side.*1204

*I’m using the first one here.*1214

*To fill in this box we will multiply x × x ^{2} = x^{3}, x^{2} × y = -x^{2}y.*1218

*I have x ^{2} y then - xy^{2} then xy^{2} then – ý^{3}.*1228

*When I combine these terms I can see that all have a single x ^{3},*1242

*but that my x ^{2} y will end up canceling each other out since they are different in sign.*1251

*The same thing will happen with my xy ^{2}, they are different in signs so they will cancel each other out as well.*1257

*I just have one little lonely y ^{3} on the end, - ý^{3}.*1264

*You can see that if I put these back together I do get the difference of cubes.*1270

*These two formulas will help us factor them out.*1276

*Another thing is it can be very difficult trying to remember these formulas.*1281

*First try and identify what is being cubed, this x and y, because you will see that they show up in your formula in that first polynomial.*1286

*Just x and y.*1295

*They show up in the second polynomial as well, x and y with them being multiplied in the middle.*1297

*It even works for the sum of cubes.*1306

*You have your x and y just as they are and you have your x, xy and y.*1310

*On the outside of the second one, these will always be squared.*1317

*And one last thing that will help you get these two formulas down is look at how the signs are related.*1323

*If you are dealing with the difference of squares in the first polynomial will have exactly the same sign.*1330

*If you have the sum of cubes in the first polynomial again will have the exact same sign.*1337

*The next sign present is opposite of what you used originally.*1346

*If you had a negative over here, use the opposite now it is positive.*1351

*Same thing applies on this one.*1355

*If you use the positive, now this was going to be negative.*1357

*For the last sign, this will always be positive.*1362

*Here is how you can remember what the signs will be.*1368

*Always start off with the same one as in your original then you will have opposite signs.*1371

*Then the last one will always be positive.*1380

*Same, opposite, positive.*1384

*Let us see if we can use the sum and difference of cubes to help us factor up.*1387

*We are going to use this on 8 ^{3} - 8 and 27r^{3} + 8.*1394

*The very first thing you do is see what two things are being cubed.*1402

*I have x and then 2 ^{3} would give me my 8.*1411

*This will factor into a smaller polynomial and a larger one.*1419

*The values that I put in here will be an x and 2, x ^{2} x × 2 and 2^{2}.*1427

*I have put them the same as they are, then I have my x ^{2}.*1440

*I have them multiplied together and I have my 2 ^{2}.*1444

*We do cleanup this a little bit by putting some signs.*1449

*Same, opposite, positive.*1454

*This is almost a completely factored.*1460

*I will just go ahead and end up rewriting this because we usually like to put our coefficients in front of our variables.*1464

*2x rather than writing it as x times two and we should write this as 4 rather than 2 ^{2}.*1474

*Now that one is factored.*1480

*You can see they do take a little bit more work but it does get the job done.*1482

*Let us try this other one.*1489

*What cubed + what cubed.*1492

*What number cubed would give us 27?*1497

*That would have to be a 3.*1502

*We know that 3r is our first number in there.*1504

*What cubed could give us an 8?*1507

*I think we saw it before, that must have been 2.*1509

*This will breakdown into a smaller polynomial and a much larger one.*1513

*Let us write down our numbers.*1519

*3r and 2, 3r ^{2} 3r × 2 and 2^{2}.*1522

*Let us go ahead and add the signs to this.*1535

*Same, opposite and positive.*1538

*Now this one is almost done.*1545

*We just need to clean it up a little bit.*1547

*3r + 2 3 ^{2} would be 9, 9r^{2} is here.*1551

*-3 × -2 = 6r and 2 ^{2} would be 4.*1559

*This one is factored completely.*1567

*Let us do this one more time using the sum or the difference of cubes.*1574

*Since it does take a little bit of practice to figure out what pieces are being cubed and where all of those pieces need to go.*1578

*We need to recognize for this first one, something cubed + something cubed.*1586

*Looks like my first one, and let us see what cubed would give me 64?*1596

*That would have to be 4.*1601

*I’m thinking of breaking this down into one small piece and one larger piece.*1605

*We will go ahead and write down what these pieces are.*1612

*I have k, 4k ^{2}, k × 4 and 4^{2}.*1614

*Now that we have all of our pieces, let us go ahead and put in our signs.*1626

*Same, opposite, and positive.*1631

*Do not forget this last step where we go ahead and clean everything up.*1639

*k + 4k ^{2} - 4k + 16.*1644

*Another thing that you may sometimes be tempted to do is sometimes you look at the second one*1655

*and seems like you should be able to factor it in some sort of way.*1662

*However, this is as far as it goes.*1666

*Feel free to just leave it as it is.*1669

*Let us try one last one.*1676

*This one is 27x ^{3} – 64y^{3}.*1677

*Something cubed + something cubed.*1683

*What cubed would give us a 27?*1691

*That must be a 3 and x ^{3} would give and x^{3}.*1694

*To get a 64 this must be a 4y.*1700

*Make sure we have our negative sign in there.*1706

*We have that breakdown into a smaller one and a much larger one.*1711

*The pieces are 3x and 4y.*1720

*Be very careful as you put in those pieces of the much larger one.*1724

*Remember we have 3x ^{2}, we have 3x × 4y.*1729

*We have 4y all of that squared.*1735

*One common mistake I see with these is many people only square just the y.*1738

*But it is the entire thing that means to be squared.*1744

*The 4 and y.*1746

*I will put in some signs.*1751

*Same, opposite, and positive.*1753

*One last step, let us go ahead and clean it up.*1761

*3x – 4y now I have the entire thing 3x being squared.*1765

*That will be 9x ^{2} 3 × 4 would be 12, so 12 xy and 4y^{2} is 16y^{2}.*1773

*This one is factored completely.*1793

*That definitely get familiar with the special formulas they can save you lots of time and works.*1796

*You do not have to go through as much.*1801

*Remember to look for those key patterns when using these formulas.*1803

*Always remember we do not have a sum of squares formula, so watch out for that one.*1806

*Thank you for watching www.educator.com.*1812

2 answers

Last reply by: Professor Eric Smith

Mon Dec 2, 2013 8:41 PM

Post by Mirza Baig on November 23, 2013

Why do we always write as square like x^2+16 as x^2+ 4^2 Why ?????