 Hello and welcome to another session on polynomials and we have been discussing about operations on polynomials in the last few sessions. We have talked about multiplication that too with method called method of detached coefficient. Then we learned two methods on division as well. One is method of detached coefficient and the other one was synthetic division. So you can check those videos to have a recap of the methods because that method is going to be used and we will be requiring division techniques to proceed further in this course. So I have written down a very common you know relationship or an equation which you have been studying for some time now and whenever you performed a division you always had this kind of relationship in the background where a dividend is equal to divisor here the spelling of divisor so this is divisor this is i. So divisor into quotient plus remainder correct this is a very general a very common equation or a relation which we have been using for some time. Now this holds not only for numbers but it also holds for algebraic expressions and especially polynomials because polynomials is anyways one variety of algebraic expressions so it definitely holds for polynomials as well and here is where your understanding of this relation and its implications will be very very important for understanding theorems like remainder theorem and factor theorem henceforth. So let us understand with an example first. So I have taken a polynomial x to the power 4 minus 3x square this one is 3x square sorry this is 3x square plus 2x plus 5 and I want to divide this polynomial by x minus 1 so my divisor is x minus 1 and let us say the quotient after division we get as q and r is the remainder. So since we are equipped with the synthetic division so we are going to use that method to find out the quotient and remainder so what was it? We had to first write down all the coefficients and with all either for the complete polynomial so there is no x cube you know term here so hence the coefficient is 0 hence I have mentioned that 0 then minus 3 is for x square plus 2 for x and plus 5 and we learned in the previous session that if you want to divide it by x minus 1 write 1 here then you can start dividing first of all write down this one downward like this and then then multiply this to 1 so 1 into 1 is 1 write that product here add the column so you will get 1 again again multiply this by 1 so you get 1 again write this one here add the column you will get minus 2 then again multiply this by 1 so and put it here so minus 2 so you'll get 0 then 1 times 0 is 0 and add and put it here write down 0 and the remainder is or the last term is 5 so what are the remainder and quotient so quotient is nothing but you can see you can now attach the coefficients back it is x to the power 0 then this is x to the power 1 x squared and x cube right so quotient will be x cube plus x squared minus 2x that's it and what is remainder the last term which is left is remainder r is 5 now understand and this and you know notice few things if you see degree of let's write degree of everything degree of dividend dividend what was the degree of dividend if you look carefully it's 4 is it highest degree highest power term right now here degree of degree of divisor was 1 it is a linear term correct 1 degree of quotient what did we get degree of quotient we get this as 3 and degree of degree of r is 0 there is no x term over there isn't it constant term so constant polynomial has a degree 0 now if you observe something here this 4 is simply 1 plus 3 okay so wherever divisor times the quotient right divisor times the quotient the degree of that polynomial and degree of the dividend is same this is something interesting so please note this you can take many other examples for example i'll i'll i'll give you another example without actually dividing it but if you divide let's say on dividing on dividing x cube plus x square plus 2x plus 3 you can check yourself but i'm just giving you the result by x plus 2 if you do this and if you want to or let's do one thing let's divide it quickly so how to do that complete polynomial coefficient detached coefficients right here so 1 1 2 3 correct and here i will write minus 2 why because i have to express this as x minus minus 2 x minus something right now again quickly if i divide some 1 comes directly here minus 2 times 1 is minus 2 right minus 2 add minus 1 you'll get minus 1 then minus 2 times minus 1 is plus 2 add keep it here add you get 4 then you do this and then minus 8 so write it here minus 8 add you'll get minus 5 okay so what are the q q is simply x square minus x plus 4 okay and r is minus 5 if you're not thorough with this process please check the previous video you'll get uh you know understanding of how synthetic division works so hence what do i get i get this expression again that x cube plus x square plus 2x plus 3 this was your dividend divisor was x plus 2 and quotient was x square minus x plus 4 and this is minus 5 the remainder now if you check again what is this here the degree what is the degree of this dividend degree is 3 this one degree is sorry 1 and this degree this is a quadratic expression so our polynomial so degree is 2 again if you check 3 is equal to 1 plus 2 and it will be like that why because if you now if you talk like this for example if you open this bracket if you really want to multiply and check whether it comes to lhs or not so the first term which you will get is x times x square isn't it and that is the highest degree you can expect because all other multiplication terms will have lesser power isn't it so x multiplied by or lesser powers will give you lesser powers of x check okay so this is the highest power after opening this bracket will be x cubed isn't it right so hence what do we learn from this we are learning something very very fundamental and that whenever fx let's say fx is a polynomial so we learned how to represent fx so fx is this one let's say this one dividend fx is divided by x minus let's say alpha alpha is just a constant okay or for your benefit let me write this as c x minus c where c is a constant so you're dividing fx by x minus c then you will get a quotient gx okay or let's use the word letters qx becomes easier to understand qx plus rx okay or better still we'll use small letters that's the convention so let's use the convention only so small qx and rx now what is qx qx is an expression or polynomial in x and what is rx rx is a expression in x again but in this case if you notice if you are dividing by a linear term if this is linear term if you divide by linear term or linear expression the remainder is always constant term constant term why because if this contained another x let's say let's say in this case if it was minus 5x then you could have gone for one more round of division isn't it you could you could have divided it further isn't it so hence you will see that the remainder will always be having a degree lesser than the divisor okay so please understand this so in this case degree of rx degree of rx will always be one degree of sorry always be zero not one zero if degree of degree of divisor is one okay please remember this so this is very fundamental fundamental behavior of polynomials when you divide polynomials it behaves like this isn't it i hope this is clear to you because again once again i'll i'll try to explain if fx is a dividend and you are dividing by x minus c let's say x minus c why i have taken minus c and not plus c it will become evident the moment we discuss factor theorem and remainder theorems but if you write like that fx equals to x minus c times qx plus rx what is very very clear thing which is very clear is let's say if this degree is n whatever it is two three five six seven eight whatever if the degree is n here the degree is one this degree is one and clearly qx will have a degree of n minus one okay this is what if you see it if this was three it is one it is three minus one here also the above example it was four it's one it's four minus one three right and the degree of the remainder is always zero always zero degree of remainder is always zero everywhere everywhere it is zero now this zero is happening only because only because the divisor is linear okay if there is if the divisor is linear then only remainder will have a zero degree if the divisor itself is nonlinear that means if it is of degree two three and whatever then rx or the remainder could be anything any of now could be of any degree lesser one less than that of the degree of divisor always remember so once again if i'm writing fx is let's say equal to gx now the divisor i'm writing as gx divisor could be anything and you get a quotient qx and this is r of x so all our expressions in x right fx is our dividend gx is the divisor qx is the quotient and rx is the remainder rx could be a remainder of x containing x as well you can see okay now in that case what do we learn we learn that gx right the degree of gx of gx let me write it properly once again so or you know you can say degree of degree of fx generalizing you can say is equal to degree of degree of gx plus degree of qx this will always hold also degree of rx will be less than degree of gx the divisor these two important points you please remember okay so degree of fx is degree of gx plus degree of qx we learned from the example and rightly so because if you open the brackets that's how it was justified if you open the bracket and start multiplying these two terms you will say that you know this one here is getting added up to this two right so it becomes three so highest degree of gx and highest term or highest power term of qx when multiplied together will give you the degree of fx or right this is how it was established and the second thing is the remainder will always have a degree lesser than that of the divisor why because if not then you can go for one more division right it's like you know when in the remainder is more than the divisor you can go for one more division is it so you and the end process of division is only when the remainder which comes is lesser than the divisor in our general division of numbers but in case of polynomials we take care of the degrees right so if the degree of the remainder is less than the degree of the divisor then we stop dividing till then we keep on dividing that's how the fundamental goes now once you are aware of this relationship or once you are you know thorough and very clear with this then we can understand remainder theorem and factor theorem very very easily let's look at it in the next session