 Hey welcome again friends and we are discussing factorization techniques for algebraic expressions. In this session we are going to understand if there is an expression given in this form a cube plus b cube plus c cube minus 3 a b c then dear friends you can factorize it like a plus b plus c and then another factor is a square plus b square plus c square minus a b minus b c minus c a. We understood the proof for this in the identities chapter. Now we are going to implement it. So let's say you have been given an expression where it is a cube minus 8 b cube minus 64 c cube and minus 24 a b and c. Okay so first of all reduce it into complete cube. So this is a cube you cannot simplify further minus 8 b cube could be written as 2b to the power 3 minus 4 c to the power 3 correct and this can be written as minus 24 can be written as 3 into a into 2b into 4c is it it? So now we are reducing into that form. Also the same thing can be written as a cube plus minus 2b whole cube because minus 2b whole cube is minus 2b cube minus 2b whole cube and minus instead of minus right plus and then take the minus sign inside the bracket. So this remains the same nothing changes minus 3 into a into minus 2b into minus 4c right so nothing goes or nothing changes. So hence this looks familiar expression now what is this this one so can I not factorize it very easily what will that be so this will be a minus 2b minus 4c a plus b plus c then a square is simply a square and then b square in this case will be plus minus 2b whole squared c will be minus 4c whole squared c square and then minus a b is what a into minus 2b right and then here what do I write I write minus 2b into minus 4c and then what next minus 4c into a this is how it would appear okay so this becomes simply a minus 2b minus 4c then becomes a square and this becomes plus 4b square and minus 4 whole square is 16c squared is it this one and then plus there are two negative signs everywhere so minus okay and there was these minus signs also guys so I have left that so please be careful okay so minus a into minus 2b is plus 2ab then there are three minus it becomes minus and 4 to the 8bc and this is plus 4c a okay so hence a minus 2b minus 4c a square plus 4b square plus 16c square 2ab minus 8bc plus 4c these are the two factor factor number one factor number two you cannot reduce it further that's what we learned here okay now let's take up one more example so for example the question is factorize factorize factorize what a cubed minus b cubed plus 1 plus 3ab okay the moment you see that four terms one thing is you know whether we can reduce it into a cube of a binomial but here it appears that since three is only in one term in cube of a binomial you will see through two fact two terms where three is available here it is not so hence the other way which we we have just learned we will be implementing that so it can be very easily written or seen as a cube plus minus b cube plus one cube isn't it three cubes are there so I should get the you know the 3ab part also from there so hence can I not write this as minus 3 into a into minus b into 1 isn't it so now it becomes a cube minus b cube plus c cube minus 3abc isn't it so hence what will that be so it will become now we know the expansion so a minus b plus 1 that is a plus b plus c form and then a square that is a square plus b square so this will be minus b square plus 1 square then minus a b so a times minus b minus bc so minus b times 1 and minus c a so 1 times a right this is the two factors right so this will become a minus b plus 1 times a square plus b square because minus b square is equal to b square plus 1 minus and this minus and this minus becomes plus a so plus a b then plus b and then minus a okay so hence you can rearrange them and right rearrange the terms and write it in a you know arranged manner a square plus b square plus a b all the square terms first higher terms first then linear terms so minus a plus b and then plus 1 so these are the two factors of the given expression right so what is the learning learning is there are you know three terms which can be reduced into cubes and there's a 3abc type term then you can use this identity to factorize