 welcome friends to another session on factorization of polynomials we are going to start with first method and this is something called low hanging fruit right so in any given polynomial FX first of all we need to see whether there are some common factors and we separate it out okay and then automatically the polynomial gets factorized okay what is what is meant by common factors for example let's keep aside algebra for some time let us say we had an arithmetic calculation let us say we had 2 plus 4 plus 6 plus 8 plus 10 okay so this is I have to calculate this so can I do this or do you observe that all these numbers are even so hence even number means 2 will be a factor of it so hence can I write this as 2 into 1 plus 2 into 2 plus 2 into 3 plus 2 into 4 plus 2 into 5 isn't it now all those all these 5 numbers carry 1 2 so hence I can take out 2 as common and we say I can write it as 1 plus 2 plus 3 plus 4 plus 5 this is what is called taking out common so we are using nothing but distributive property distributive law you can say so I'm using distributive law reverse of distributive law actually so hence I can take out comments like that now this was in case of arithmetic similarly in case of algebra also this could be the case for example let us say if I have x plus 1 let us say this is fx fx is x plus 1 plus x plus 1 times x plus 2 okay so again if you see there are you know this there is a common factor here which what is the common factor here if you see the common factor in both the terms here first of all how many terms two terms are there yes it can be simplified into multiple terms but right now the form is there you know in this form there are two factors sorry two terms so hence I can pull x plus 1 as common factor is it it so if I pull x plus 1 whole as a common factor so within brackets what will be left so x plus 1 in the first term is nothing but x plus 1 times 1 isn't it so hence what will be left one here and what will be left in the second one x plus 2 is it it is it now how to check whether we have taken the factors come the common factors correctly or not if you expand it now so it should give back you give you the given polynomial back so x plus 1 if you have to expand this expression you will get x plus 1 plus x plus 1 times x plus 2 which was given here so hence we have taken right common factor so hence this can be further simplified as x plus and this 2 plus 1 can be added so x plus 3 so hence you see just by taking comments I could factorize this now mind you in factorized form there will be something into something into something into something like that okay there will not be any plus or minus sign so this is called fact you know factorized form so this is a factorization right but let's say if you express it like something plus or minus something minus something this is not factorization this is this is not factor what I mean is x square minus 3x plus 2 this is not factored this is there are you know this is not expressed as factors but the same thing can be written as x minus 1 times x minus 2 now both are same but this is not factored so there are I am not showing the factors here right but this is factorization okay so I converted the polynomial and expressed it in terms of its factors product of its factors so this is factorization so similarly any given polynomial can be converted into or could be expressed as product of its factors okay now let us take another example let us say we have x minus 1 this is example another example so let us say we have x minus 1 times x minus 2 and then plus x minus 1 times x minus 3 right so what will be this how do you find out if you see any common factors yes there are this is common factor x minus 1 is common to both so hence I will write x minus 1 common and then within brackets whatever is left so how to find out whatever left here so the first factor you first term you take and divide this by x you know the common factor so what is left only x minus 2 so here it will be x minus 2 and again same thing apply to the next term so if you divide this term by x minus 1 you'll get what x minus 3 so this is how you pull factors common right so hence if you simplify now it will be x minus 1 and x plus x is 2x and minus 2 minus 3 is minus 5 so I this is not this one was not factored but now this is factored from both are same okay another example could be let us take another example and example is let us say we have x times x square plus y square plus z square then another term is y times x square plus y square plus z square and third is z times x square plus y square z square so you can see we have this common factor x square plus y square plus z square so you can write x square plus y square plus z square within brackets x y plus z is it. So, hence I converted a non-factored form into a factored form. Now, many a times what will happen, it will not be that obvious that there is a common factor. So, in that cases, you have to arrange the polynomial in such a way that you can extract factors. For example, the given example itself, if you see, could be, this could be, you know, the example here was let us say, x minus 1 or x square minus 3x plus let's say 2, this plus let's say it's given x square minus 4x plus 3. So, now, it doesn't look like there is a common factor in this form, right? But then, if you, you know, later on we will see splitting the middle term and all that mechanism. So, you will see this is nothing but x minus 1, x minus 2, so I first converted into factors and then this one can be written as x minus 1 and x minus 3. How I am writing this will, you know, deal with this in later sessions. But right now let us say, you have to be aware that this is how it has to be done. And then you take care, you take common x minus 1 and finally you will write x minus 2 plus x minus 3 and you simplify to get x minus 1, 2x minus. Right? So, many a times it will not be that obvious but then after grouping and, you know, arranging, you will get the common factors. So, this is the first type, first or first method. So, first method is, as I tell you, I told you that, you know, it is a low hanging fruit. So, first of all, in any such polynomials if you get, first try to see if you can extract out some common factors and then we can simplify or we can go ahead with further factorization.