 Hello friends welcome again to another session on factorization techniques So we will move ahead and so far we have seen splitting the middle term and taking commons and other methods in this methodology We will Take up, you know Algebraic expressions which are expressible as the sum or difference of two cubes. What do I mean? Let's say if you have expressions like a cubed plus d cubed Okay, so in the previous, you know a series of lectures we have Seen how do we use algebraic identities to express these? expressions, so if you see a cube plus b cube is simply a plus b times a Square minus a b plus b square to know more about this you can always check the the chapter on Algebraic identities and you will understand that a cube plus b cube is a plus b times a square minus a b plus b square and a cubed minus b cubed can be expressed as a Minus b and then a square plus a b plus b square, isn't it? So these are all called algebraic identities Algebraic identities to know more about these identities you can always check the you know series on Algebraic identities. It's in the same coursework right algebraic identities. Okay now what? So if I know this any such expression now in this form will be easily factorizable Let's take example example one. Okay What is example? Let's take example one here if let's say I have a Cube plus 64 Okay, and this is an algebraic expression and I have to factorize this. How do I factorize? Very simple. So this could be written as a cube and if you notice 64 is nothing but 4 cubed, right? Which fits in into this scheme. So what will be what will it be? So it will be simply a plus 4 times a square minus a times 4 plus 4 Square right if you see a here b is 4 if you take b as 4 in this case in this case If you take this b here compare this be here that b comes out to be 4 isn't it now if you You know adopt that formula you will get this expression and this is how you factorize it So hence, what is it a plus 4 times a square minus 4a plus? 16 this will be the factorization. Okay, let's take another example Let's say we have another example is let's say we have 2a plus 1 cubed minus a minus 1 Cubed okay, this is the algebraic expression again 2a plus 1 whole cubed minus or other. Let's take plus first a minus 1 whole cubed again If you look at this entity, this is nothing about it is of this form a cube Plus b cube right it looks like this where a clearly is 2a plus 1 and b is a minus 1 so I can adopt again the same Identity to factorize it this one. So what will it be if I call this as a and this as b then it will be simply a Plus b then a square minus a b plus b square This is what the identity says and now let's deploy and then our job is done So it is 2a plus 1 plus a minus 1 This is the first term and then here it will be 2a plus 1 whole square minus What is a again 2a plus 1? What is capital B? a minus 1 and then finally a minus 1 whole Squire right again. So this was a I just simply deployed all the values as they are so a a This b comes here and this b goes here Like that. I hope this is clear and now the job is only to simplify So, you know how to combine like terms and simplify this so this 2a plus this a will become 3a right and then this one and this minus one will get cancelled. So it's only 3a here, isn't it? Now, let's go inside the second factor now, you know the identity that a plus b whole square is Is a square plus twice a b plus b square, isn't it? So, let's use this identity to expand this term. So it is at a plus b whole square form So if you expand it, you will get 4a squared plus 2 times 2a times 1 Plus 1 squared minus. What is this if you? Yeah, if you if you do this opening up that will be nothing but 2a into a is 2a squared Then it is a plus 2a and then it is a minus a and then this is plus 1 and then this is a square Minus 2a plus 1 using the identity if a Minus b whole square will be a square minus 2a b plus b square, right? So we using this I have expanded the second term now You just need to collate all the like terms together and simplify So if you see this is nothing but 4a square minus 2a square is 2a square and then plus 1a square at Wait, so it will give you 3a square first of all then let's collate all the a term So this is 4a 4a plus 2a is 6a minus a is 5a and Then minus 2a is 3a again, right? So this is this is 3a again You can check this one is 4a Then plus 2 6a minus 1 that is 5a minus 2a that is 3a, right? So I written I've written 3a now the constant terms if you see 1 plus 1 plus 1 so it is 3 Okay So this is what it is right and then you can take three common from the next term as well So you can see three is here is here is here so three common taken as a square plus a plus 1 So it becomes and there is a of course here. So it is 9a a square plus a plus 1 this is how we factorized a very you know frightening or let's say dreading Difficult-looking expression. Thank you