 Welcome friends. So let us continue with our session on algebraic identities. So so far we have done squares of binomials and trinomials. Now in this session we are going to see what is a cube of a binomial. So before we start we must understand what a binomial is and we already know it. It is x plus for form of x plus y, x minus 2y. So there are two terms in a polynomial then we call that polynomial as a binomial. So now what is a plus b cube? So let us say if you don't know any identities then fundamentally you could have done this a plus b and then multiplied this a plus b with another a plus b and finally third a plus b and then one by one you would have expanded this and then multiplied like that. But now we already have some knowledge of identities. So hence a plus b whole cube can be written as a plus b whole and a plus b whole squared. Isn't it? So power one here is power one and here is power two. How? Because a to the power m into a to the power n is simply a to the power m plus n. By our knowledge of exponents we already know this. Now beautiful this we already know a plus b whole squared. So what will this be? This is a plus b and if I expand this particular identity then it is nothing but a squared plus twice a b plus b squared. Right? Beautiful. Now what? Let's go for the expansion. So what will this be? This will be nothing but a times a squared plus two a b plus b squared and then b this is what we are using is called distributive law. Isn't it? So we are distributing the product. So hence now go for the kill. So what to do? Just multiply. So it is a cube plus two a squared b plus a times b squared. Isn't it? A gets multiplied to all three of them. Similarly here it will be a squared b plus two a b squared plus b cube. Mind you I have written a before b. Some order is always good. Isn't it? So hence this will be the final expansion but now what we see is there are some like terms. So let's club these like terms and like terms. Hence this will finally be a cube plus twice a square b plus three a b squared plus b cube. Now this looks awesome. Why? If you see there's a great pattern which is visible over here. So if you observe the powers of a simply. So the powers of a is a cube here a square. So it is reducing by one every time. Isn't it? And similarly if you see b the powers are increasing one by one. Isn't it? Oh that looks great. In fact if you see later on when you will study binomial theorem you will see even if a plus b is ten you will see this kind of a pattern over there as well. For that matter any power of any integral positive integral power of a plus b you will see similar patterns. Now one thing which is also very you know striking is the sum of the power is three. If you see sum of the powers is three always. Isn't it? Always sum is three. So hence and what is this three? This is this three is coming from this index guys okay. So this is another way to you know keep a check on whether you have written the identity correctly or not. Now and this three also appears here. So you can remember it very easily. So a cube plus thrice a square b plus thrice a b square plus b cube. Very good. Can we manipulate this mathematics? We have been manipulating expressions and you know twisting and turning and seeing it from different perspective. So let us say this was a cube plus three a square b plus three a b square plus b cube. Now if you notice there are common terms in these two. Is it it? So what can this be? This can be written as a plus b cube is equal to a cube plus b cube. So I have clubbed this b cube with this a cube powers together. They are like brothers. Now hence after that I am taking three a b as common and this becomes a plus b. Very very good. So hence if you see a cube plus b cube this one can be expressed as a plus b whole cube. This was on this side and now guys let us take this beautiful looking factor on the other side. It will convert into minus and it will become a plus b. If you didn't understand I would request you to pause the video and understand and then only proceed. So a cube plus b cube is equal to a plus b whole cube minus three a b a plus b. So this particular manifestation of this identity would be useful when let us say they would be giving the value of a plus b and they will be giving the value of a b and they will ask you to find out the value of a cube plus b cube. So please keep this beautiful identity in your mind. Now but we can't really stop here. If you see we can go further. So if you see I can take a plus b is common and what is left within I can see there is a plus b whole square minus three a b. Wow. Now what let us expand the second bracket as well. If you see this will be a squared plus two a b plus b squared isn't it a plus b whole square is this now this has become you know this has gone into our blood now a plus b whole square is a square plus two a b plus b square minus three a b which was there here. So I have just written like that. Now you can see these are like terms again. So we love to combine like terms. So hence it will be a plus b a square minus a b plus b square. Oh my god. We have got another beautiful looking relationship. So a cube plus b cube is equal to a plus b a square minus a b plus b square. Wow. So what did we learn so far in this session? So far we learned identity number one a plus b whole cube is equal to a cube plus three a square b plus three a b square plus b cube. The same a plus b whole cube was manipulated and we wrote a cube plus b cube as a plus b whole cube minus three a b a plus b is it it and the third one which we learned was a cube plus b cube was given as a plus b times a square minus a b plus b square. If you notice we have just factorized sum of two cubes. So hence if someone gives you six cube plus nine cube let's say then you can factorize this using this method is it it. Now look at some very let us look at some variations. Right. What is variation? So let's say if I change b to minus b what will this become? So I can write a my let us say I want to find a minus b whole cube. So what will this be? You already know it that minus b can be written like this. Now it reduces to our known identity and let's expand it. So hence it is a cube plus three a square times minus b plus three a minus b square plus minus b cubed. Isn't it? So hence what is the final result? If you see a cube minus three a square b plus three a b square minus b cube. Okay. So if you see wherever the power of b was odd the sign got changed. Wow. So wherever the power of b was odd the sign got changed. So if you see here the power was odd here the power of b was odd and then sign got changed. So hence this is what we got it. Let us say if it was something like minus a plus b whole cube then what would you do? Wherever the sign of a is odd you can change that. So it will be finally minus a cube plus three a square b then what? Minus three a b square and plus b cube. Isn't it? Very easy. Similarly the other two variants can also be written. So a cube plus b cube so you can treat it as fourth. Fifth would be a cube plus b cube and again wherever the power of b is odd you change that. Sign. So hence it is a minus b whole cube and sorry we are here trying to find out a cube minus b cube. Isn't it? Minus b cube. So hence hence it will be a minus b whole cube minus or rather three a b because power of b here is odd. So you change the sign and here it is a minus b. Beautiful. Now the sixth one is nothing but a cube minus b cube and we are considering this one wherever the power of b is odd because minus is attached to b so wherever the power of b is odd change the sign. Here it is odd. Here it is odd. Here it is even. So you will not change this. Hence it is a minus b and a square plus a b plus b square. Fantastic. So hence guys you can now include these three and these six rather. Identities in the list of your algebraic identities I suggested to you earlier as well. Make a comprehensive list of algebraic identities at one place and whenever you are solving problems keep that list in front of you so that it becomes easier for you to solve the problems. Now I would not recommend you to mug up any identity. So the only way to remember identity is try solving as many problems as you can. That's the best way to learn and remember algebraic identities. Thank you guys.