 hello friends welcome to yet another session on algebra and in this session we are going to start something which has troubled lots of students from high school stages to senior secondary level as well and that's factorization right so today we are going to understand factorization of algebraic polynomials okay so factorization so before we start we just let us have a recap of what actually are factors what are factors guys so in a mathematical language we say that a is a factor of B so please note the notations we are I'm saying I'm writing a is a factor of some writing you know I'm just putting a bar like that vertical bar and B so a is a factor of B or B is a multiple of a so this is denoted by this particular notation so hence this is read as a is a factor factor of B so we'll come to what exactly is a factor but a is a factor of B is represented like this or we say a is a divisor divisor of B or this is also read as a divides B or it can be also said as B is a multiple multiple of a so all these statements can be you know summarized like this so it is written like that a divides B okay so what does it mean a is a factor of B means that when B is divided by a the remainder is 0 right a is a factor of B what does it mean that a divides divides B such that such that remainder remainder is 0 then we say that a is a factor of B for example if you see 2 clearly is a factor of 4 is it because 4 upon 2 is 2 and remainder is equal to 0 correct we also say that 2 perfectly divides 4 okay now 3 is a factor of 9 3 is a factor of 15 15 is a factor of 30 right and 30 is a factor of 2700 likewise right so why will we discuss factors we discuss only integers right in this case right a and B are integers now as we are dealing with numbers where we are saying one number is a factor of another number we can also talk about all normals so we say so let us say fx is a polynomial fx and gx are polynomials fx and gx are polynomials right so you know what a polynomial is a polynomial is of the form of for example fx will be something like a n x to the power n plus a n minus 1 x to the power n minus 1 plus so on and so forth a 1 x plus a 0 where where a a n a n minus 1 so on and so forth a 1 a 0 all are real numbers real numbers right and n is a positive integer so you know this is called a polynomial similarly we can have another polynomial gx hx whatever so we say that gx is a factor of fx when when gx when or rather we say when fx leaves a remainder remainder 0 when divided by gx okay that's what we say so when fx leaves a remainder 0 when divided by gx let's take an example so let us say fx is equal to x square minus 3x plus 2 okay this is a quadratic polynomial so fx is a polynomial let us say gx is equal to x minus 1 okay now let us try and divide gx fx by gx so we write x minus 1 this is called a long division method you might have studied it in earlier grades okay so what do we say how to start with it so let's say we I write x so this becomes x square minus x and when you subtract you will get 2x plus 2 this 2 comes down here and then you write what 2 or I'm sorry this will be minus 2 yeah so hence this will be minus 2 right so x minus 2 and hence it is nothing but minus 2x plus 2 and hence you get 0 as remainder so hence if you see x when x square minus 3x plus 2 is divided by x minus 1 remainder is 0 so we say x minus 1 divides x square minus 3x plus 2 okay this vertical bar here means divides okay so hence we say x minus 1 divides x square minus 3x plus 2 or gx is a factor of fx this is right okay moreover you must also understand this that the factor the degree of the factor is lesser than that of the given polynomial what do I mean so if you see fx has a degree has a degree 2 okay so gx the factor gx is a factor right and the possible degree of gx is anything less than 2 right so what do I mean so fx has a degree 2 so gx will be a polynomial gx will be nothing but a polynomial again so factor of fx will be a polynomial polynomial with either degree as 0 1 or 2 the factor of gx can be of degree 0 1 or 2 but never more than 2 never more than 2 in this case more than 2 in this case so if fx has a degree so it's all related to the degree of fx so if fx has degree n gx will have maximum possible degree of gx will also be n okay it cannot go beyond that so please keep that in mind okay now how for example if you take the same x square minus 3x plus 2 if you see 1 is a factor of anything 1 divides x square minus 3x plus 2 or not so if you see what is the degree of 1 here degree of this polynomial is 0 0 degree polynomial 1 is a 0 degree polynomial similarly x minus 1 divides x square minus 3x plus 2 so what is the degree of this polynomial degree is 1 and x square minus 3x plus 2 itself divides itself by 3x plus 2 isn't it what is the degree of this polynomial 2 but you cannot have any polynomial more than degree of more than degree 2 which will be factor of this given polynomial if it was a factor if it was a degree of 3 let us say if the expression is something like this x cube plus 3x square plus 2x minus 2 let us say if gx divides this particular polynomial then gx degree could be either 0 1 2 or 3 but never equal to 4 never equal to 5 or never greater than 3 okay I hope you understood right so this is what factor means so factor what is a factor and we are going to deal with factors only in polynomial so hence if gx divides fx that is gx is a polynomial fx is a polynomial so if when gx is a factor of fx then when we divide fx by gx the remainder will be 0 right this is what a factor means so in the subsequent sessions we are going to understand methods of factorizing a given polynomial so how to convert a given expression into its product of its factors right for what is the what is the meaning meaning is let's say if I write 26 so 26 can be written as 2 into 13 isn't it so 2 and 13 both are factors of 26 similarly if I am writing x square minus 4x plus let's say 4 so we know that x square minus 4x plus 4 can be written as x minus 2 times x minus 2 so x minus 2 is a factor of this thing so hence we will we will now in subsequent sessions we'll see that there will be an fx given fx is nothing but a polynomial and we have to write fx in form of gx times hx times let's say whatever any number of factors such that and and we mean by by this statement we mean that gx is a factor of fx similarly hx is a factor of fx and so on and so forth so basically a polynomial will be there and we need to convert it into product of factors that's what will be the objective of subsequent few sessions