 And so in the previous lecture, we looked at distributing over some expression and now we're going to do the reverse of that We are going to factor expressions So we're going to start with an expression and we're going to break it down into its constituent factors So our discussion is going to be all about factory Factoring so last time as mentioned we talked about Expanding so we would have No certain expressions that we want to expand this is just the reverse of that We want to factor we want to go from something that we've expanded by using distribution We want to now break it apart into its constituent parts again So let's put a little line right under factoring And the first thing that we are going to talk about is we're going to talk about polynomials Polynomials So let's just define what a polynomial is and I'm going to write out a polynomial just in terms of variables So there are many ways that you can write this, but let's have a Subscript in and then we're going to have x to the power in Plus now I'm going to just decrease this n by 1 so n minus 1 so that I have x to the power n minus 1 as well Plus we're going to go x 8 and then the subscript 2 minus 1 and then x to the power in Minus 2 and then we're going to go all the way And so there's our little ellipses as we've had before and then I'm going to have a sub 2 x squared Plus a sub 1 x to the power 1 and we usually don't put that and lastly we have a sub 0 and We have x to the power 0 which is this one so we leave it there So that's an odd sort of notation One thing that we do always want to have is that a sub n is not equal to 0 So we have these little subscripts just to denote the fact that these are Different values these are different variables and because we're using a a a all the time and we run out of letters for the alphabet It's a very easy thing just to put subscripts there just to denote that that is a different number from that Or at least a different variable a different variable a different variable a different variable The x remains the same that's the same variable and we just raising it to a certain power So if we start right at the back that would be x to the power 0 x to the power 1 x squared all the way And you can count up until you get to x to the power n so let's have a little example So let's have an example Let's say 3 x to the power 4 plus 2 x to the power 3 Plus let's have x squared. Let's have plus 2 x. Let's have plus 4 And now you can see what's happening here here is my a sub 0 value. That's just 4 There's my a sub 1 which is 2 my a sub 2 which is just 1 so a sub 1 is 2 a sub 2 is 1 Here's a sub 3. That's a 2 and there's a sub 4 which is 3 and Then I just have corresponding to these subscripts and remember that does nothing to the number That's just denoting that these are all different values Later on we'll call them coefficients. We'll have a look at that But that just corresponds to the power that we raise the variable by so that will be an example of a polynomial and usually we write this descending order of this variable x And we do like this first coefficient a sub n in this instance that would be a sub 4 not to be equal to 0 if that was 0 This term doesn't really exist because 0 times anything is just 0 And then we just would have started right there at the a sub 3 So that's important for us to know and so the subscript notation i that's really what we call We call it a counter and this instance we started counting at 0 and 1 2 3 and we go all the way up to n And that's just a counter that we you know helps us keep track of these things But it just denotes these different values. So let's just define this idea of a coefficient So coefficient is this constant that goes in front of The variable that we're raising to a certain power So all the a sub 4 a sub 3 a sub 2 a sub 1 or Whatever my n value is just in decreasing order I'm going of course just write it the other way around if I wanted to you'll see many different ways as I mentioned But these are all our coefficients and we usually use a or b or c summing to that order denoting that these will be You know constant values And that is the way that we denote them and those are just those coefficients I'm going to call them a sub i And that's why I wrote the i there. These are all my i's now. I say n minus 1 n But i just means it's this counter that we're talking about this little counter at the bottom the subscript does nothing to what the number really is Just denoting that i can be 0. There's a 0 We're denoting i can be 1. There's my a sub 1 which in this instance is a 2 etc. So those are just going to be my coefficients We define something called the degree Define something called the degree of the polynomial by the way Let's just keep track of all of these. There is where I defined my polynomial. So let's keep that one in green And here is why I defined what a coefficient is. So please keep track of that one for me Now we're going to be calling talking about the degree of a polynomial. So let's say degree of a polynomial polynomial And so by degree of a polynomial what we mean is we definitely have to have that a subscript n is not equal to 0 Then n is the degree of the polynomial And so let's look at an example very quickly. Let's go back to this example where we had 3 x to the power 4 plus 2 x to the power 3 Plus x squared plus 2 x plus 4 and indeed there's my n That is the highest power that we went to as far as the variables power is concerned Here we had x to the power 0. That's just one here. We had x to the power 1 x squared x cube x to the power 4 so the degree The degree of this polynomial is equal to 4. That's a fourth degree polynomial So let's put a another little green marker there. We have to find something called the degree of a polynomial Now there's a couple of polynomials of various degrees that we give names to And so the first one that we have is called the quadratic polynomial. So let's write that quadratic polynomial Polynomial and let's make a column just for the degrees. That's a polynomial that's of degree 2 So we have something like x squared plus 4 x plus 3. Next thing we're going to have a cubic polynomial and that will be of degree 3 and something like let's put 2 x cube Plus 4 x squared minus 3 x plus 2 We also get a higher order quartic A quartic polynomial so quadratic even though it says quad we don't mean 4 by that a quartic Is degree 4 and something like let's make it x to the power 4 minus 3 now. Can I have that? Yes Because what we are saying in this instance instance is that x to the power 3 has a coefficient of 0 x to the power 2 has a coefficient of 0 And then x has a coefficient of 0 and then x to the power 0 has its coefficient 3 That's still the highest Power that we have there and that makes it a fourth order. Then we also get a quintic And there are names for higher order, but I think we can just stop there That'll be a polynomial of order 4 and so let's make it 3 to the power 5. I should say x to the power 5 Plus now let let me do this. Let me write 0 x to the power 4 Plus 0 x to the power 3 Plus let's make it x squared minus x plus 2 and you can see those will be 0 So this will just be equal to 3 x to the power 5 Plus x squared minus x plus 2 And so you need and have these other powers because they coefficient might be 0 as long as this what we call a leading coefficient So this is going to be my leading coefficient. Let's put a Align it that's called my leading Coefficient Now that leading coefficient as long as that is not equal to 0 in this instance in this Example that we ace up 5 But that's not equal to 0 if that was 0 that would be 0. That would be zero and we'd end up right there There'll be a second degree polynomial So it is the highest Of these powers of our variable as long as the coefficient is not equal to 0 and the leading is going to be that one That's attached to the variable that is raised to the highest power So let's just make that another little definition of ours Now we're going to start factoring now that we have our definitions out of the way Let's factor some something that are you know fairly common to do. Let's say factor common Elements, so what is common about certain terms in an expression? Common elements and so we're going to start with something very simple And we're going to start with something we learned last week distribution So imagine I had x and I multiplied that by x minus 3 Now by distribution, that's going to be x times x and x times x minus 3 So let's have a look at that that's going to be x times x Which means x times x plus the x times the negative 3 So there'll be a negative 3 actually times the x That's a 1. That's a 1. I'm multiplying the bases are the same I can add those two x 1 plus 1 And then the positive times the negative would be negative 3x And this would just be x squared minus 3x. So just a little simple revision of last week So this is going to be x times x minus 3 And what we mean by factoring is instead of going in this direction We're now going to go in this direction. We're going to go from x squared minus 3x to its component Terms you could say here or at least these parts of this expression So there's one part of it and there's the second part. So we are factoring it. These are two factors of our expression here. This expression has two terms And if we factorize it we end up with these two factors And so we're just going to go in reverse So how would we go about if we are given this example problem? Let's do an example problem Let's say we have x squared minus 3x And we want to factorize it and as I say this Example is something about common elements. What is common between these terms in this expression? And so let's just rewrite it as we had it right there. That would be x times x minus 3 times x So if I look at these two different terms in this expression, what is common to both of them? Well, there is certainly an x there and there's an x there This x here does not have a counterpart in this term. There's only a single x there So if I look at what is common between these two There's only a single x that is common between these two and we say we take out that x This is a common term that we use we take out that x we factor that x out What is left? Well, if I've taken this one out all that is left is here is an x If I took that one out all that is left there is a negative 3 and now I've gone from This expression with its two terms and I have factorized it into its two Factors those are the two components that make up this expression And that is what we mean by taking out or factoring out a common element here x was a common element between these two Now let's look at another example because it's not always the variable. That is the common factor Sometimes it's the constant. That's a common factor. So let's look at this example. I have 6 x squared plus 9 y Let's factorize that but first of all, let's just Write it out in its constituent parts now. Remember we said that every integer is the product of prime numbers So let's have a look quickly at 6 I'm going to divide it by its smallest the smallest prime, which is 2 and 2 indeed does divide 6 divides What we mean by that is there's no remainder if I say 6 divided by 2 that's equal to 3 and the remainder is 0 So 2 now I've got a 3 left And 3 is a prime and so That is it 6 would be equal to 2 times 3 those are both primes and that is prime factorization of the number 6 Let's look at 9 We cannot divide 2 into 9 because that'll be 4 with a remainder of 1. So that doesn't work the next smallest prime is 3 That's great. If I say 9 divided by 3, I've got 3 left Now I'm going to divide 3 by the smallest prime, which is 2 doesn't work Next smallest prime is 3. Yes, it does divide 3 does divide 3 without a remainder And there we see the prime factorization of 6 and 9 And now if I bring these down, let's have a look at these What is the only common factor? So remember if I brought these down, we'll have the least common multiple But that's not what we want to do here We want to know what is common between these two and certainly if I look here Let's draw a little box here. It is the 3. It's the only thing that is common Among these two so 3 would be a common factor I just as we factorize this expression here We factorize this expression here We factorizing the constants and the only common one between them would be the 3 So I can definitely factor out a 3 Now look, there's 2x's and there's a y. There's nothing common between those two terms as far as x's and y are concerned So there's nothing I can take out. So that would be it So if I say 6 divided by 3, well there's a remainder of or there it goes in 3 goes into 6 twice There's no remainder and then I have x squared and it goes into 9 3 times So I'll have plus 3 y and now I factored this expression into its factors The only common factor that I can take out between 6x squared and 9 y is the 3 That's the only thing that's common between those two terms So as I mentioned that we can also think about Constants not only variables that can be taken out as common factor Now let's have look at a Much nicer example because that'll bring everything home 2 1 6 Let's make 2 1 6 x plus. Let's make it 450 y And now can I factorize this now first of all we have to think Let's think about the variables. There's a x here and a y there There's nothing common as far as the variables are concerned between those two. So again, we left with The two constants. So can we think about prime factorization as far as these two are concerned? So there are a couple of laws and you can see them if you do look at the notes The description to the notes that would be available down in the description You can look at those notes and there's a couple of laws here The easiest one to look at is to say that if we take The largest number divided by the smallest number and we look at the remainder of that division Then we're going to look at only the remainder and the smallest of those two numbers The remainder and the smallest of those two numbers So that's a very easy way a nice way to go about that and let's Just do a little bit of long division now. This is of course just reviewing algebra We're not going to do much arithmetic yet. So let's just remind ourselves how to do long division So that's 450 divided by 216 And if we do this long division, what is left? Well, first of all, let's look at 216 In 450 that does not really go. It does not go into four. It doesn't go into 45 But it does go into 450 and it goes into 450 exactly twice So six times two is 12 I carry the one Two times one is two plus the one would be a three and two times two is four And now I'm going to do subtraction And here we go I'm going to borrow there. So that becomes a four that becomes a 10. So I'm left with eight And I've borrowed there so that becomes a one And I have a remainder of 18. So we can say 450 divided by 216 that is going to equal two with a remainder of 18 And now we say the following if we look at these Common multiples that we can take out and we're going to call those This this the largest of these common multiples that we can take out We're going to talk about this remainder and the smallest of these two numbers So there's my remainder is 18. Let's do prime factorization of that 18. Can it divide two? Yes, it can That's the smallest prime that leave me with a nine Can nine be divided by two? No. What's the next prime is three? Yes, it can be divided by three So there's my three. I've got three left. Can it be divided by two? No next prime is three Yes, it can be divided by three So that's what I have 18 if I do prime factorization. There'll be two times three times three Two times three is six six times three is 18. There we go. Let's look at 216 That's going to be a little bit more difficult. Can it divide two? Yes, it can If I divided by two, what do I have left 108? Can 108 be divided by two? Yes, it can And if I divided by two, I am left with 108 divided by two. That'll be 54 Can 54 be divided by two? Yes, it can. So there's another two There's another two And now I'm left with 27 27 on the other hand Can be divided into three And that leaves me with nine, which is three times three And there we have this prime factorization of 216 Now what I want to do is I just want to write this in a slightly better way Let's do a two And a three and a three And so here I'm going to have a two and a two and a two And a three and a three and a three There we go. That was the prime factorization of 18. That was the prime factorization of 216 And let's look at what is common between these two If I wrote it in nice columns, look at this. We have that two is common there We have that there's one three that is common. We have another three that is common and that's it This two does not have a mate. This one doesn't have and that three doesn't have so we've got two times three times three And two times three is six six times three is 18. So indeed 18 Is common to both of these it is common to both of these we call that the greatest common divisor So if I take 18 and I divided by 216 I take 18 divided by 450 that is going to be the largest of all the numbers All the possible numbers that I can get that will divide both 216 and 450 So let's do 216 and I'm going to divide that into 18 At least I'm going to divide 18 into 216 And you can do your long division or use a calculator. You'll see that is 12 And if I take 18 and I divide it into 450 if I divide it into 450 I'm going to get 25 So use your calculator and see that that happens. So see there's no remainder with these There's absolutely no remainder and so By doing this prime factorization writing things neatly in order I can just see what is common amongst those two and if I multiply that out That is going to give me something called the greatest common divisor That's the greatest number that divides both of those And so if you can you can read the notes and you can go through one of these techniques that gives us this number So 18 is going to be my common factor And I know that it's going to be 12 times x plus 25 times y And that is the way the only way that I can properly factor 216 x plus 450 y is discovering this greatest Common divisor the greatest number that will divide both 216 and 450 And as I say if you want to do this by hand and not use something like python to do this for you Which is very very easy. This is the way that I would go about it Take the smallest number divided into the biggest number so 450 divided by 216 and look at what the remainder is Now you're going to take the smallest of these two numbers And the remainder and you're going to do between those two you're going to do prime factorization And then you're going to see what is common between those two There's a two that is common, but there's no two there. There's no two there There's one three one three, but there's no three there. So these were the only common values you multiply them out You're going to get 18 and that is the greatest common divisor Of 216 and 450. Let's continue our discussion and we're going to do one more example Now this example is going to be easier because as I mentioned The last one was kind of the most difficult one you will get when you want to factorize Or at least get the greatest common divisor of two integers And the larger they get I really want to suggest that you take a look at the python videos And learn how to do that using a computer language. That's just so much easier So let's do this one. Let's have six x squared Plus nine times x times y. We want to factor those So let's factor each of these six when we do prime factorization would be two times three That's a prime. That's a prime two times three is six. I can't factor this any further x squared can be factored. I can write x times x And then I have a plus there Nine can be factored into three times three. That's the prime factorization of nine Then I have a times x and I have a times y. Let's look at what is common between these two These two Terms there's a two on this side. There's no two there. There's a single three there. There's a three there There's an x here and there's an x there But there's no other x If I look at this three, yes, it has its corresponding three on the side, but the other three doesn't have A partner on the side There's the x's they have a partner, but this y doesn't have a partner So what is common between those two are the three and the x and I can take those out as my common factor What is left behind? Well, if I've taken the three and the x out the three and the x what is left on this side? Well, two times x Two times x and here we have three times y three times y and we have factorized Now once again, if you do distribution, you're just going to get back six x squared plus nine x y So you can always check your work And so those would be examples of just taking a common factor And factoring that out of an expression Now the next thing I want to talk to you about is factoring So let's write factoring Let's do quadratic quadratic expressions In quadratics, those are second order polynomials So let's have a sub two x squared plus a sub one x Remember there's a little one there and a sub zero and there's an x sub zero But that's just one and we also have to have that a sub n is not equal to zero if And by a sub n here, we mean the two a sub two If that was zero and this term would fall away and then be a first order polynomial but it's a second order polynomial because This a sub n which is a sub two in our instance for quadratic equations or second order polynomials would you know, you can't have that zero So let's consider an example. So we want to factor the following. Let's have x squared Plus let's do five x and let's do plus six So what do we have here? It's second order My a sub two which is right there or a sub n right there. That's just equal to one So it's not zero I have my variable the highest power would be there's also an x sub zero And they remember would be two this makes it a second order polynomial or a quadratic expression Now quadratic expressions. They are always going to have two factors And because we do distribution, that'll be an x and that'll be an x And we just have to figure out which number goes here and which number goes there And through distribution, I mean x times x is x squared So we're going to have the x squared there But if I have a look at this constant times that constant, I better get to six So how can I get six if I multiply two numbers because I'm going to have that number multiplied by that number How can I get a six? Well, let's have a look I can do six times one I can do negative six times negative one. I can do one times six I can do negative one times negative six I can do two times three. I can do negative two times negative three I can do three times two And I can do negative three times negative two all of those will give me six So this is not prime factorization. I'm just looking at if I multiply two integers How do I get to six? Well, those would be the integers that I can multiply with each other to give me six And so let's see if we plug these in on this side whether we get To this expression. So let's try plus six and plus one Because I'm going to get x squared there and I'm going to get plus six there But what about this five x? So what we have to consider is those multiplications that distribution So here I'm going to get x plus Six x and that equals seven x and that's not what I want. I want five x So I know I can remove this one From my list my list of possibilities Now exactly the same thing think about is going to happen if I have negative six and negative one If I have negative six and negative one, I'm still going to have six at the end negative six times negative one gives me positive six, but I'm going to have negative Six x and negative one x which is going to give me negative seven x that one's also not going to work And the same is going to apply to those two And so let's start with the two by threes. Let's have a look at those And I'm going to still have an x there. I'm going to have an x there and let's have a look Let's start with two and three. So I'm going to have positive two and positive three Now I'm still going to have x times x is x squared. So I get that one I have two times three which is six. So let's look at multiplying these two So I'm going to have x times three that's going to be three x And two times x that's going to be positive two x and learn. Behold if I add three apples And another two apples I get five apples and that's exactly what I want And so x plus two And x plus three is going to be The factorization of that second order polynomial and you can check on that Now look at these the way that I've you know done my Distributions, but remember we usually do it this way those two and these two But that's the same as Doing it this way around nothing. That's that's no different those two ways of thinking about it So I'm still going to have x squared plus x times Three that's three x and here two plus six two times x that's positive two x And then two times three which is six. So they right there in the middle. I still have my three x and two x So it doesn't matter which way around I do this I still get the same solution and that gives me five x which is exactly what I want So this was going to be the selection And by the way, uh two and three and three and two where I put the three in the three there in the two there That was also going to work for me. We have the commutative property of multiplication So I could have x plus two times x plus three or x plus three times x plus two So this one was also going to work for me Let's do another one the more of these you do as always The better it's going to be for you mathematics is not a Sport a spectator sport. It's a participation sport. Let's do x squared minus Let's do minus x. Let's do minus six And so it is a polynomial I'm going to have two factors and I'm the first ones are going to be x times x to get me x squared Now I just have to think how to get negative six. What do I multiply to give me negative six? Well, this time around let's start with saying negative six times positive one or six times negative one Or negative one times six or one times negative six And then I can also have negative two times three or two times negative three I can get three times two and I can also well That's going to be let's make this negative three times two and three times negative two All the all these eight multiplications are all going to give me negative six So let's start with that first one. Let's do a negative six And they're a positive one positive one So let's have a look at this again. Remember, I'm going to get my x squared here and I'm going to get my negative six from there Let's see what happens with these two x times positive one. That's going to give me x And here I'm going to have negative six x minus six x that gives me minus five x That equals minus five x. That's not what I want. I want a negative x So surely this first one does not work at all. That one does not work at all And let's think about some of the others which ones are going to work for us Well, let's try the second one. Let's try a six a positive six and a negative one Do you think that one's going to work? If I have positive six, I'm going to have a positive six x minus six that's going to give me Positive six minus one that gives me a positive five x. It's certainly still not what I want And you can work through all of these The six times ones With the different signs. None of them are going to work Let's try this one negative two and three So let's go for this one x And an x let's try negative two And positive three now you can immediately see that's not going to work We need a negative one and if we look at this if we have Positive three x minus two x that leaves us with a positive x. That's not what we want So this one's not going to work Either and the one that's going to obviously work is this one. Let's have a look at that one Let's do x plus two and x minus three Now let's just do our normal distribution And let's go straight for that and you'll see it does work indeed So x times x would be x squared I have negative three x That's the x times the negative three two times x is positive two x and as we had Plus two times negative three that is going to give us negative six So we do have our x squared and we do have our negative six. That's where we always start It's about this middle term and indeed negative three x positive two x gives me negative x and I have x squared minus x minus six, which is exactly my problem And so this would be the correct one That would definitely be the correct one Now, of course there is commutativity. So I could also have x minus three x plus two But Sydney you just need to write it in one, you know one of the solutions here and that absolutely works So if you look carefully We have the two factors we have x and x then we think about this value here That would be our a sub zero in this instance would be a negative six Yeah, our a sub zero was Positive six and we just have to think how can we get that that gives us these last two terms But there's many ways to get negative six many Multiplication of two integers is going to give me negative six and I have to choose the right combination Such that if I think about these two terms the outside and the inside distribution That if I add them that I get what I want, which in this instance was a negative one Now I need not have that my leading coefficient Is equal to just one. Let's have a look at this example. I'm going to have two x squared plus seven x Plus three Now that is a wholly different beast and as much as my leading coefficient here Is not equal to one This is still a second order polynomial or quadratic expression And so let's see if we can factor this. We know this is a polynomial. So we're going to definitely have two terms But now I have to get two x when I multiply these first two terms when I multiply these two terms I must get two x squared. So certainly this has to be a two x and that one is an x such that if I do that multiplication I get I did two x squared, which is two x squared now I have to think how can I get positive three positive three would be my a sub zero term as positive three So let's think how we can get this we can get this as three times one We can get this as negative three times negative one. We can get this as one times three We can get this as negative one times negative three So certainly as far as integers go that would be the multiplication Now it's not always going to work out like this. We might You know get Fractions that we need to consider but those would be more advanced cases in this lecture We're only thinking about You know a systematic way of going about factoring these polynomial expressions So let's start with this first one. That would be a positive three and a positive one So positive three and a positive one And so if I do this multiplication there this distribution three times one gives me a positive three That's correct and the two x times x gives me the two x squared Let's have a look at what happens to my inner term if I do that distribution So two x times positive one that's going to give me two x plus three x plus three x that gives me Five x and definitely that is not going to work for me the three times one Let's remove it That is not going to work So let's choose Negative three and negative one. So I'll go through the motions. I'm going to get two x there and I'm going to get an x there Let's go negative three and negative one So I am no I'm definitely going to get the first term and I'm definitely going to get the last term That's how I set it up. So I'm going to get two x squared and I'm going to get a positive three So let's look at this middle term. I'm going to get two x times negative one That's negative two x And negative three times x that gives me negative three x and that gives me negative five x and that's not what I want I want positive seven x so definitely Those two we're not going to work. Let's have a look at let's have a look at this next On my list. I'm still got my two x And I've still got my x and now I've got positive one And positive three Now again two x times x going to give me two x squared the one times three is going to give me three So I certainly do have the first term and the last term. Let's again look at this middle term Yeah, I'm going to have six x plus one x gives me seven x six x Plus one x that equals seven x and that's exactly what I want the seven x So my solution is indeed two x plus one Times x plus three And you can go about that just in the way that we've always gone about it when we do this distribution But doing it this way or this way is exactly the same. You're going to get Two x squared plus six x plus one x plus three and the six x plus one x gives me a seven x So i'm going to get two x squared plus seven x plus three And I've got the right solution. There's my factorization Of this quadratic expression As I said before we definitely cannot get enough of these. So let's do another example. I'm going to do two x squared minus five x minus 12 Now again, that's a more difficult one because I don't have a leading coefficient that is one It's still a second order polynomial or quadratic expression So I know that I'm going to get these two terms And there's my two terms now. I have to think about how do I get negative 12 Now there are many ways to get negative 12 and you've got to start I think after doing quite a few of these start doing this in your head And you've got to think if there are two things that I multiply by each other to get 12 Which one of those pairs is going to get me close to a negative five and certainly if I think About six and two well six and two six times two is 12 But the difference between six and a two is a four. So that's not going to really help That's not really going to help me What if I think about a three and a four? Now I have to remember that I also have to bring this two into it And that's what makes life very difficult I'm starting to think though that if I mix a three and a four A three let's think about it three and a four But I've also got to multiply one of them by a two And if I multiply this by a two I get eight And from eight and three yes, if I could get eight A negative eight and a positive three that'll give me two five So I think The three and the fours are perhaps going to work for me. So to get negative 12 Let's think about it. The first thing that I can think about is three times negative four Let's try those ones positive three And negative four Now that's exactly what I had in mind because I said there's my eight and there's my three And there is a way to use eight and three to get to five Especially negative five if I had a negative eight and a positive five that Is going to a positive three that's going to get me a negative five and there I have My negative eight and my positive three There we go negative eight x Positive three x is going to give me negative five x So this is absolutely the solution So let's have a look at that just to make sure that that works two times x two x times x is going to give me two x squared And now I'm going to have the negative eight x I'm going to have the positive three x and I'm going to have the negative 12 So I'm going to have two x squared minus five x minus 12 So you have to start thinking in your head You know, how can I get close to negative five? What of these numbers if I have a two there and I'm thinking of getting negative 12 What of the numbers that I multiply to give me negative 12? Can I possibly use to get me to a negative five? And the six and the two wasn't really going to work for me But definitely the three and the four if I could get to this stage as I mentioned Now you can still write out all of the ways that you can get to negative 12 And you can go through all of them But after you've done a few of these I want you just to start thinking it through in your head without having to write down All the possible values in the beginning though Please write them all out And let's do one more example and let's make this one. Let's make it 40 x squared Let's do minus 40 x. Let's do minus 240 And that's a still a second order polynomial degree two polynomial or quadratic expression My leading coefficient here is not zero and the highest power of my variable x is two And what I can't see here is immediately that I can factor out one of these I can't factor out You know looking at these coefficients a 40 a negative 40 and a 240 What if I do it just to take 40 out as a common factor? I'm left with x squared minus x and minus six x squared minus x minus six if I were to distribute 40 Into this expression here. I'm definitely going to get 40 x squared minus 40 x minus 240 and now all that is left is x squared minus x minus six We have seen that before we're going to get 40 There's going to be my x term. There's going to be an x now. I just have to think how can I get To negative six such that I can't think of getting to a negative one and certainly that is going to be A negative three and a positive two That's going to give me the negative six, but negative three x positive two x gives me negative x And that was a simple simple example Where you just have to think about first simplifying your polynomial by looking at what is common Amongst these common amongst these coefficients And we're looking for the greatest common divisor So 40 goes into 40 once 40 goes into 40 once and 40 goes into 246 times and I just have to remember what the signs are Now that is very exciting. But now let's move on to building, you know, the building blocks in a later chapter We're going to build on what we have now, but I want to talk to you about the difference Difference between squares And so we talk about these as perfect squares and what we have is x squared minus a squared And that's a square and that's a square and a difference Now remember what we are talking about here is A x squared plus Let's do b x plus c Before I did a sub two x squared plus a sub one x sub, you know to the power one plus a sub zero But you may also see this in the textbooks a sub two is this a a sub one is b a sub zero c The subscript is just about, you know making you aware of the fact that these are different values different coefficients But what we have here is that this term right here is equal to A squared and with a negative out front a sub one here is equal to zero Times x and here we have that a sub one is equal to one and here I have an x squared So this is what we mean by x squared. That's a zero term minus a squared Just so happens that this plus a sub zero is equal to negative a squared. There we go negative a squared So the middle term there is not represented And these are very very easy to factor And they are easy to factor in as much as we're going to have x minus a and x plus a Why is that so? Well x times x it's my x squared And there's my negative a times positive a there's my negative a squared But look what happens to these two terms. I have positive ax negative ax And if I add positive ax plus negative ax, I get zero x and so that middle term is gone Let's have a look at let's have a look at an example. Let's have x squared minus 25 Now is that the difference between squares most definitely? But please do remember this is x squared plus zero x minus 25 It's just that that coefficient there is zero. So we don't even put it there And let's rewrite this I can rewrite this as x squared minus five squared because five squared Five squared if I just look at that five squared is 25 And there's a square and there's a square. So these are going to be exceedingly easy to do There's my x minus five and my x positive five So i'm taking the negative of this number and the positive of this number There's my five there's my five there's my five the negative and the positive version of that Because if I multiply negative five times positive five I get negative 25 But yeah, I have positive five x Minus five x those cancelled out and we left with zero Let's do another example in this example. I'm going to do 25 x squared. Let's make it minus 81 y squared Is this the difference between squares? Yes, because I can write 25 is five squared x squared minus 81 I can write as nine squared y squared And now I just have to remember If I have a to the power n Times b to the power n that is just going to equal a times b to the power n So I have the same power. There's an n. There's an n. There's a two and a two So I can just multiply these two by each other that will be a five x And I'm squaring that minus and the same is going to happen here I've got nine y and I'm squaring that Now I'm thinking of this as one term. This is one term I have the square of this term minus the square of that term and lo and behold I can do what we've had Way up here So I just have to remember that it looks different from this But it is still the square of something minus the square of something the square of something minus the square of something So this first one is five x is my first something and now I have to have minus The other one and five x Plus the other one And so there's my one term and there's my other term Whereas here I hit my one term and my other term here that we go It's going to be this whole term minus this whole term and this whole term plus that whole term So you can clearly see here the middle term if I look at these two They are going to cancel out and all I'm left with are these two terms and if you multiply them out You're going to get 25 x squared minus 81 y squared Good. Let's have a look at some more. There is the difference Difference between cubes And so what we're going to have here is something like let's make it a cubed minus b cubed And this would be a if this was an x. I would think of this as being a third order polynomial And there is a very nice way that we can factorize this. This is going to be a minus b And then I'm going to have a squared plus a b plus b squared And this second term cannot be factored and we can't have any factorization of the second term So let's have a look at an example. Let's have something like let's do eight x to the power three minus 27 Now once again, remember this is eight x to the power three plus zero x squared plus zero times x minus 27 And so that term is zero that term is zero and I'm just left with the difference between these two And so when we write it like this do not get confused The only thing we're trying to show here is that we have something that is cubed minus something else that is cubed This instance i'm writing an example as a polynomial. It's a third degree polynomial. This leading coefficient is not zero So it's a degree three or third order or a Cubic expression now. Let's rewrite this. I've got two to the power three is eight times x to the power three minus three to the power three Three times three is nine nine times three is 27 And I just have to remember just as we had before If I have any number to the power n and another number to the power n and I'm multiplying those I can rewrite this as a times b to the power n. So I'm going to do the same here. I've got two x To the power three minus three to the power three. Can you see I have the difference between two cubes the difference between two cubes And this instance if I just look at a's and b's There's my a is going to be two x And my b is going to equal three Positive three because that negative is right there. I'm just looking at the b part and that is my three now I can look at this So I'm going to say a minus b which is going to be two x Minus b which is negative three And then I'm going to have this term So I'm going to have a squared that's going to be four x squared Plus a times b. So plus Two x times three that gives me six x and then b squared which is three squared, which is positive nine And that is the factorization of eight x cube minus 27 And once again this part I cannot factorize There's no way that I can write this as You know two factors Now there's a sum or this let's do sum of And let's make that sum of let's just say cubes And in other words, I have a to the power three plus b to the power three And it turns out there's a nice way to factorize this as well It's going to be a plus b instead of the negative And here we're going to have a squared minus a b plus b squared Once again, this cannot be factored any further So let's look at a nice example. Let's do the same. Let's do 27. Let's do 27 x to the power three plus eight Can I rewrite this? Yes 27 is three to the power three Times x to the power three plus eight can be written as two to the power three I remember the rule that I had there so I can write this as three times x To the power three plus two to the power three And what do I have? I have a cube plus another cube exactly what I have there so in this instance a is going to equal three x And b is going to equal two And so now I can just use this factorization So a plus b would be three x plus two And I have a squared would be three x squared three squared is nine x squared is x squared Minus a times b minus three x times two that's six x And then plus b squared so plus and b is two Two squared is four and once again We can't factorize that and that would be the solution Factoring this sum of cubes. So look at the difference between these two cubes and the sum of these Look at how these are done and please do not get confused Between there's a negative b there negative 27 So the negative is there and I just have to look at the 27 all on its own So that's going to be three cubed And now the very last bit of this foundational work in factorization is we're going to think about factoring rational expressions Factoring rational expressions. Now. What do we mean by rational expression? What are you going to have is a polynomial? polynomial divided by a polynomial so Let's have a look at an example. Let's have two x squared Plus seven x. Let's make it plus six And we're going to divide that into x squared minus four Divide into that. I should say x squared minus four So that is our example. I'm taking a polynomial divide by another polynomial And let's do some factorization. Let's factorize the numerator and we're going to factorize the denominator So this is another technique or the last technique for this lecture that I want you to be aware of So definitely here we can have a two x and here we can have an x And we have to think about how do we get how on earth do we get to seven? Now, hopefully if you can you know if you think about it Let's think if we if we have somehow if we have a three Getting to a six three times two if we have a three And we have a four And I can make a seven from a three and a four specifically if I have plus three there And I have plus two there so that I have positive six. There's my positive six, but if I look at four x plus three x I get to seven x So once again, you can write all these down go through them one by one But start thinking about doing these in your head Here I have x squared Minus two squared that's the difference between two squares And I know how to do this that is going to be x minus two x plus two do remember Not use the four not use the four here. You have to write that four as A square and the negative stands all on its own So I just want to think about this four writing it somehow as a square and of course two squared Just this all on its own two squared is equal to four now I have the difference between two squares So here I can put x minus two and x plus two And we'll remember from any kind of simple algebra Is that if I have Multiplication numerator multiplication in the denominator He has one term multiplied by another one expression multiplied by another expression One expression multiplied by one expression I can think about starting to cancel and lo and behold I have an x plus two and an x plus two So I can think about if I have x plus two in the numerator and exactly the same thing in the denominator Those can cancel and I'm left with one over one, which is just one So those two have canceled and all I'm left with are these two So that's going to be two x plus three And I'm going to have oops. I'm going to have x minus two So I've gone from a difficult Expression here to something that's a much simpler by making use of factorization So what we've seen in this last example for this lecture is How can I make factorization work for me? How can it be useful? And here we've seen a beautiful example of how it can be useful This would be difficult and this would be a lot easier to do