 Hello friends, welcome again to another session on polynomials and Moving forward with our sessions the series of sessions Now we have reached at a point where we can discuss remainder theorem and factor theorem So in the previous few sessions use you you'd have seen our preparation to come to this level and then now In this particular session, we are going to understand this in detail So if you have not seen the previous videos I would request you to go back and see the division method the synthetic division method and then the immediate previous video of where we have explained the The very basic Equation or relation where the dividend was equal to quotient times Deviser plus the remainder, right? So we are going to use this particular relation to understand remainder theorem now So if you'd have if you recall in the previous Previous session we discussed something of the sort that FX is a polynomial in X and It was being divided by a divisor a linear expression X minus C linear polynomial wherein we got a quotient GX and a constant remainder Rx now I'm Purposefully mentioning Rx just to make you you know feel that all these expressions which we are dealing with are all polynomials in X Okay, so variable is X So whether it is a constant or it is a polynomial if it is a constant the degree of that constant is zero You know that But at the end of the day we can say that all of them are polynomials in X, right? So this is what the relation was which we were discussing the other day and And now it's time to do a little bit of more analysis on this Other things which we learned in the previous session was that degree of FX and degree Degree of FX. I am writing like that degree of FX was equal to degree of Degree of X minus C which was one isn't it Which was one plus degree of Degree of GX that is the quotient. This is what we learned In the last last session right degree of GX Correct is what we learned and we also learned that degree of Rx in This case, it's a constant but generally speaking degree of Rx In this case Rx is constant You know we saw that when you divided it with a linear polynomial the remainder is always constant We discussed this in the previous session But generally speaking degree of Rx the remainder expression or remainder polynomial will be less than degree of the divisor polynomial Okay, so in this case the divisor was X minus C So hence since the degree of X minus C is how much guys, you know this this is linear Polynomials if this is one and less than one the only possibility is zero negative numbers. Anyways, I'm not allowed in polynomials So degree of Rx will be zero and hence it is it was a constant term correct. This is what we learned Now what is remainder theorem guys? What does remainder theorem mean now remainder theorem is a mechanism to find out the remainders When a particular polynomial FX is divided by a linear polynomial X minus C again I'm writing it so that it becomes, you know solidly you know Stablished in your mind. So you must remember this that remainder theorem is a Is a mechanism. So let me write it as RP in short is a mechanism mechanism or way or method mechanism to find out find out remainders Remainders when when Or not remainders remainder right remainder remainder Only one remainder will be found out in one division when a polynomial Poly No male again, please remember we are talking about polynomials only when a polynomial FX is divided divided by a linear linear polynomial polynomial X minus C Okay, where C is constant X is the variable right is a mechanism to find out remainder when a polynomial FX is divided by a linear polynomial X minus C or Rather we can generate it generalize it. Sorry instead of writing a X minus C. Let's write it as Any linear linear polynomial will be of the form of a X plus B, right? This is a linear polynomial. So what is Remainder theorem remainder theorem is a mechanism to find out remainder when a polynomial FX is divided by a linear polynomial AX plus B without in brackets you can write without actual without actual division. So you don't need to divide it Okay, you don't need to perform the division to find out the remainder you can do it just by using this theorem and let us see how So, let us say if if FX is divided by AX plus B the The quotient is let's say GX. Okay, so we're saying let us say let us say that that when when when FX when FX then FX FX is divided by divided by FX is divided by AX plus B Okay, then quotient is GX and remainder is Rx Okay, let us say you're dividing you're not actually performing the division But you just divide you're saying that if you divide it, you'll get these then Can we not say that by whatever we discussed in the previous sessions and earlier as well that FX will be equal to AX plus B the divisor into the quotient plus the remainder Is it it? Can we say that? Yes, we can say that right now now That means can I now say that FX is equal to A times X plus B by A So GX plus Rx, can I say that? Yes, we can so if you see this thing is same as AX plus B So I've just taken a common purposefully why what what is the purpose will you will come to know a little while later? Okay, now this is the What is this basically? This is an equation in X everywhere There will be polynomials and expressions in X. Isn't it now what happens if X is equal to minus B upon A If let's say for one specific value of X, which is minus B by A What will you see? You will see FX. So instead of X. I can write minus B by A. So F of minus B by A Okay, so you now know how to find out the value of a polynomial at any given value of the variable So we have dealt with in previous sessions. You can go and check that out as well So now if X is minus B by A I'm interested in finding the value of the entire polynomial both LHS and RHS. What will happen? So LHS I wrote in the RHS you have to write A and instead of X You have to just deploy a minus B by A and there was a B by A already Okay, then this is G and minus B by A and Plus I'm generally saying R whatever it is. Let's say R remainder, right? You can say minus B by A as well, but then I'm just saying whatever it is. It is a remainder at the end of the day Okay, remainder. So what do you see guys in this case? If you see this is coming out to be zero Minus B by A plus B by A is zero. Zero multiplied by all of the factors will Whatever is the value of G of minus B by A and A if you see this is becoming zero So you will be left with minus B by A f of minus B by A is equal to R Right, so don't we get the remainder guys? So we just got remainder remainder remainder when when FX is divided divided by AX plus B Right remainder when FX is divided by AX plus B simply f of minus B by A Right, this is what is called remainder theorem not special case Cases special case. What is the special case? special case is let's say a is equal to 1 and B is equal to minus C. Okay, and then we'll get back to our initial, right? So in that case AX plus B will be AX plus B simply X minus C right, so hence Friends if you want to divide FX by X minus C let's say the Caution is GX and the remainder is R You can write X You can leave X as well because anyways that is going to be a constant Right on constant terms of RX. So hence what will happen? So R X R will be simply f of minus B by A Minus B was our what C and a is one so f of C. So simply R is f of C Correct. Is that okay? So this is the this is what is called remainder theorem. Let's take an example and Then you know, you will be more comfortable. So if you remember in the previous Session, we had one division Okay, and this was the division. So FX was in that case FX was X cube, let's take an example. I'm taking an example now FX is X cube plus X square plus 2x plus 3 let's say this was the polynomial and you are dividing it by X plus 2 Okay, you are dividing it by X plus 2. So hence you wrote X cube plus X square plus 2x plus 3 if you remember in the previous session, we did this if you if you remember this was X plus 2 and the quotient was X square minus X plus 4 and The revender was minus 5. This is what we had got the previous You know previous session, this is what FX it was This was X plus 2. This is GX and This is R Okay, and this was the divisor linear divisor Okay, now let's say this was obtained by long division method by long or the synthetic division method Which we saw long or basically by division method. We actually divided divided it Okay, but then let's say we didn't divide then what will be the remainder According to the remainder theorem. So X plus 2 if you if you see this, what is the value of A here? A is 1 and B is 2. Is it it? So the remainder will be simply F of Minus B by A is the remainder, isn't it? If you remember Just check above. We just saw that remainder is F of minus B by A. What are B and A? If you are dividing it by AX plus B, then you know, what is A and what is B? A is coefficient of X B is the constant term So in this case X plus 2. So what is A? 1. What is B? 2 So hence R should be equal to F of minus B by A. Let's check So that means I have to find out the value of F at minus B by A So mine that means F of minus 2 B by A is minus 2 minus B by A minus 2, right? So I have to basically find out F of minus 2 Now what was FX? FX was X cubed plus X squared plus 2X plus 3 Isn't it? That means we have to just simply find out F minus 2 and this will be the remainder, right? This is what it is saying. Reminder is F of minus 2 So let's find F of minus 2. So minus 2 to the power 3 plus minus 2 squared plus 2 times minus 2 plus 3 Okay, so this is minus 8 plus 4 minus 4 Plus 3 and you you see we are getting minus 5, right? Which matches with our long division Reminder, isn't it? So this is a beautiful theorem that without dividing you can find out the remainder on if for any given division, isn't it? Once again, right? So let's say if you are dividing it by if you're dividing FX by AX plus B Okay, then simply remainder is F of minus B by A, right? And for most of the cases it will be very simplified if you are dividing FX by X minus C, let's say for this form then then remainder is simply F of C because if you see minus B by A in this case if you try to compare it by compare it with AX plus B What is A guys? A in this case is 1 and B minus C So minus B by A is simply C, right? So hence R is minus B by A or F of C. So both these forms you remember This is a general form and this is a specific form when the coefficient of X is 1 and X minus C is what you're using to divide FX, okay? This is remainder theorem. We'll see some problems on this to make it more clear. I hope this theorem is understood. This is a very vital theorem. You're going to utilize it multiple number of times till your algebra journey is on. Thanks for watching the video.