 Welcome friends so let's move ahead with our problem solving sessions and in this session we are going to discuss problems like factorize x to the power 8 minus y to the power 8 and this could be categorized as factorization based on difference of two squares so the thing to be remember is difference of two squares so whenever you see such problems this particular methodology could be adopted so you see this is so you will ask me okay you know there is power 8 and you're saying difference of two squares but if you see carefully it can be converted into square so what do we know about x to the power 8 we can express this particular thing as x to the power 4 whole square isn't it and minus y to the power 4 whole square now obviously we have never learned any identity which has you know powers of 8 so hence it has to be either reduced to powers of 3 or 2 that's what we have learned in algebraic identities isn't it so reducing 8 into power of 3's looks a little unlikely or you know that will not make much of a sense so but we can very much reduce it to powers of 2 so this is how it can be reduced to power of 2 right x to the power 4 whole square minus y to the power whole square so now again if you look closely these two these two items if you see here this one and this one right it is appearing to be like a square minus b square where a is what a is simply x to the power 4 and b is y to the power 4 so hence when a square minus b square is the thing so you can you know that this can be expressed as a minus b times a plus b this is another type of identity we have learned already so let's do that so hence it will be nothing but x to the power 4 minus y to the power 4 times x to the power 4 plus y to the power 4 isn't it so see we have got two factors automatically now but this process can be continued again if you look closely here this can be written as x squared squared minus y squared squared so I'm again changing it to powers of 2 now there will not be any advantage of changing x to the power 4 plus y to the power 4 into powers of 2 because this cannot be reduced further you cannot really factorize it because there is no such identity which says which talks about a square plus b square is factorization we don't have such identity yeah so what next so you know we can write this as again x square minus y squared times x square plus y square and x to the power 4 y to the power 4 right so we have reduced it to three factors now again if you see this can be further reduced so what can you say about x square minus y squared you can say x minus y x plus y then x square plus y square then x to the power 4 plus y to the power 4 so we got four factors now we can't really reduce any further so this would be our factorization isn't it so x minus y x plus y x square plus y square and yes if you want you can reduce this one by some trick yeah you can still reduce it with some trick what is that trick so if you see I can we can let me do it separately then now what you can say x to the power 4 plus y to the power 4 can be written as x to the power 4 plus y to the power 4 and can I complete this square if I complete this square I'll get 2x squared y squared minus 2x squared y squared now many of you would be asking why do you need to do this now if you look closely guys this will be reduced to x squared squared plus y squared squared plus 2x squared y squared now if you have you would have noticed by now this looks like a square plus b square plus 2ab right and hence for precisely for that reason I introduced I added 2x square plus 2x square y square but since you are adding something to a given expression to compensate for that condition you have to subtract it as well so I subtracted it as well so what is the advantage the advantage is nothing but if you see this will reduce to x square plus y square whole square minus 2x square y square right so these three terms together will be converted into this term and the minus 2x square y square term will be like that now again if you see I am I can write this as x square plus y square square minus root of 2xy whole square so hence if you see here if I write here root 2 can be written as root 2 square isn't it and now all the these three square powers can be taken as taken outside and we written like this isn't it so hence further what will happen so again if you see this is like a square minus b square so hence guys this can be reduced further to x square plus y square times sorry not times x square plus y square minus root 2xy and x square plus y square plus root 2xy is it fine so if you see I could express x to the power four plus y to the power four as well like that so this can be so earlier I said that x to the power four plus y to the power four cannot be reduced but actually just for correction it can be reduced like this isn't it so this can be reduced like that and again I got two factors but if you leave it at this juncture as well it would not be wrong so we could factorize this so for that matter you can you know if you see you can reduce root you know you can reduce x square plus y square similarly but then we don't need to get to that level okay so this is how you have to so basically what is the learning learning is x to the power eight minus y to the power eight so it has to be you know reduced to difference of two squares form and then you can factorize