 welcome guys so in this session we are going to learn one more algebraic identity and see a special case of that identity okay that this identity is going to be useful because it is going to contain cube terms of a trinomial so let us let us see that identity first and then understand what our conditional identity is okay so the identity goes like this so the identity is a cube plus b cube plus c cube minus 3 a b c is equal to a plus b plus c times a square plus b square plus c square minus a b minus b c minus ca okay so this is the identity right so what is it once again a cube plus b cube plus c cube minus 3 a b c is equal to a plus b plus c a square plus b square plus c square minus a b minus b c minus ca so you can add this identity into the list of identities you have already created so how to prove it prove is very easy so you can take start from RHS actually so you start from RHS and you know what to do you just simply expand it so hence if you see if you open the brackets you will get a cube plus a b squared plus a c squared minus a square b minus a b c minus c a squared correct the next term will be simply b a b and then you have to multiply with this so b a square so what I'll do is I will keep all the like terms together so it is a square b here then b cube so b cube is separate so I'll write b cube separately then b c square b c square so b c square I will write separately okay because there is no b c square here so I will write b c square separately like this okay then what b minus a b square so hence it will be here minus a b square next term will be b times minus b square c so minus b square c is also not there so I will write it here minus b square c okay and then minus a b c so minus a b c will be written over here again the third term is when you start multiplying c with all the terms in the second bracket so c a square or a square c so c square a square is here correct now c b square so c b square is here so plus c b square or b square c then c cubed so c cube was not there so separate I'll have to write and then minus a b c so minus a b c is here so I'll write it here and then minus b c square so minus b c square is here so I'll write minus b c square then minus c square a right c square a so minus c square a so not here here minus c square a correct so now I think we got so we should have got how many terms 1 2 3 6 times 3 18 terms so we should have got 18 so I think we have got 18 so 1 2 so is 1 so it is 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 and 18 so we have got 18 terms okay why 18 because 3 here and for each 3 multiplied by 6 so 18 correct so 18 terms we have got and now you can cancel like terms or whatever you know it's let's say for example a v square and minus a b square can be you know cancelled ac square minus ac square is gone this is this will be gone this will be will be gone this will be gone and finally this will be gone as well so hence if you if you see what is left behind is nothing but a cube plus b cube plus c cube so a cube and then b cube and then c cube and then these three items which is nothing but minus thrice of a bc so what we learn we learn that this a cube plus b cube plus c cube minus 3 a bc can be factored into you can look at it like that also so this can be factored into two terms what all a plus b plus c and then second factor will be a square plus b square plus c square minus a b minus b c minus c a right the corollary to this or the another result will be this that a plus b plus c is always a factor a plus b plus c is always a factor of a cube plus b cube plus c cube minus 3 a bc this is an important result right so example if you take an example so let us say 1 plus 2 plus 3 that is 6 will be a factor of is a factor of 1 cube plus 2 cube plus 3 cube minus 3 into 1 into 2 into 3 let's check what is this this is nothing but 1 plus 8 plus 27 minus 18 right so which is nothing but 9 plus 27 36 36 minus 18 is 18 so clearly 6 is a factor of 18 right so hence similarly you can say 10 plus 11 plus 12 is a factor that is 33 is a factor of is a factor of 10 cube plus 11 cube plus 12 cube minus 3 times 10 times 11 times 12 right this is a very big number and without even doing an actual division I can say 10 plus 11 plus 12 which is nothing but 33 will be a factor of this big number okay so this is the use of this identity now let us come to the point which we are discussing and that is conditional identity so what is conditional identity now what did we learn in this identity we learned that a cube plus b cube plus c cube minus 3 a bc is equal to a plus b plus c so I factored it into two factors a square plus b square plus c square minus a b minus b c minus c a is it it okay now if condition so condition is I'm writing if the condition is if a plus b plus c this on the right hand side becomes 0 this is a condition this will not always be true but should the case be that it is true right that is a plus b plus c is 0 that means let us say you took a is equal to 2 b is equal to 4 c is equal to minus 6 so a plus b c is clearly equal to 0 if this is true then what can I say so hence the given identity will be a cube plus b cube plus c cube minus 3 a b c minus 3 a b c is equal to 0 into a square plus b square plus c square minus a b minus b c minus c a isn't it because a plus b was 0 so I put a plus b plus c is equal to 0 now 0 into something is 0 isn't it so what do we get we get a cube plus b cube plus c cube minus 3 a b c is equal to 0 or a cube plus b cube plus c cube is equal to thrice of a b c right I took the minus 3 a b side a b c on the other side so hence if a plus b plus c is equal to 0 then a cube plus b cube plus c cube is 3 a b c let's check let's check it whether this is true let's check so let me say a is equal to 1 b is equal to 2 and c is equal to minus 3 correct now clearly a plus b plus c is equal to 1 plus 2 minus 3 0 right so what will be a cube plus b cube plus c cube so a cube plus b cube plus c cube is nothing but 1 cube plus b cube b cube is 2 cube plus c cube that is minus 3 cube is equal to 1 plus 8 minus 27 that is equal to 27 minus 9 which is 18 okay minus 18 rather now let us see what is a b c 3 a b c is nothing but 3 times 1 into 2 into minus 3 which is also equal to minus 18 correct so hence it is true that if a plus b plus c is 0 then a cube plus b cube plus c cube will be equal to thrice a b c this is what is called the conditional identity correct so what is the condition condition is a plus b must be equal to 0 if the condition is fulfilled then a cube plus b cube plus c cube is equal to thrice of a b c right so hence if now you you now know right so if you have a cube plus a plus b plus c equals to 0 then you can factorize a cube plus b cube plus c cube as 3 times a times b times c so hence if you see this is another so later when we are discussing factorization you will learn all this but here also you it's better to you know keep this in mind that a plus a cube plus b cube plus c cube can be factored into thrice a b c okay so this is what this conditional identity is all about