 Hi friends so let's carry on with our series of lectures on algebraic identity so far we have discussed you know algebraic identities with the binomial terms for example we talked about a plus b whole square is a square plus b square plus twice a b n then we also discussed about a plus b cube and all that right so here basically we discussed identities related to binomials what is a binomial binomial is an algebraic expression carrying only two terms right so these are all binomials now we are going to discuss trinomials and especially the square of a trinomial right so if you can see here a plus b plus c is a trinomial because there are three terms three terms algebraic expressions are called trinomials all right why because try means three now three terms now what is this identity so what will be the expression here let us try to use our previous knowledge of algebraic identities using binomials to find this out so if you see a plus b plus c can be written as a plus b plus c isn't it within brackets right now a plus b plus c whole square can be written as within brackets a a plus b plus c and this is a whole square now if you club these two together and treat this as let us say capital A and treat this as capital B so within brackets actually the identity is something like this isn't it where a is equal to small a and capital V is equal to b plus c right now we know this expansion and it's written over here so hence it is nothing but a square plus twice a b plus b square isn't it so let's use the same identity here so what do we get we get a square plus two times a times b plus c plus b plus c whole squared correct so let's simplify it by opening all the brackets and multiplication so hence it is two times a b plus two times a c plus b square plus two b c plus c squared isn't it so b plus c square again can be expanded using the same identity now if I keep all the square terms together we will get a square plus b square plus c square plus two a b plus two b c plus two c a this is what the identity we will get at the end right so hence what do we see a plus b plus c whole square is a square plus b square plus c square plus two a b plus two b c plus two c a isn't it so keep this in your mind now the next question is we got a plus b plus c whole square isn't it but what if if it is something like a minus b plus c whole square right so if you see b is not a positive quantity here it is a negative quantity so answer is pretty simple wherever you find b you can replace it b minus b so this particular identity can be written as a plus minus b plus c whole square isn't it so wherever you see b in this identity you change that to minus b so hence we will get a square plus minus b square plus c square plus two times a times minus b plus two times minus b times c plus two c a if you notice I have just written down this one again with b replaced with minus b now you simplify it is nothing but a square plus b square because square of a negative term is also a square positive square now this will become minus two a b minus two a b and minus two b c plus two c a so you don't need to remember all these different varieties we'll see others as well for example a plus b minus c whole square can be written as a plus b plus minus c whole square so you can replace c by minus c in the same identity and you will get a square plus b square plus minus c whole square plus two times a b plus two times b times minus c plus two times minus c times a and simplifying you'll get a square plus b square plus c square plus twice a b minus two b c minus two c a so wherever in the twice a twice terms wherever two is there wherever you also see c you replace it by minus sign that's that's what we have observed here the square terms are not touched only where there is a c it will get replaced by minus c can you see that right another could be let us say it is minus a plus b plus c now you must be knowing already so it will be nothing but minus a whole squared plus b square plus c square plus two times minus a times b plus two times b c plus two times c times minus a is it so if you see this is nothing but a square plus b square plus c square minus twice a b plus two b c minus two c a right so all these varieties different variations you can tackle now you know there could be it could be all minus a minus b minus c as well or it could be terms like a minus b minus c whole square now again you don't need to mug up the formula what you need to do is simply replace b by minus b and c by minus c so hence it is simply minus b square plus minus c square plus twice a minus b plus twice b minus c plus twice a minus c times a hence the final expression a square plus b square plus c square minus twice a b minus two b c and sorry this is minus two a b there is minus b over here as well so hence it plus two a b minus two b sorry not plus two a b it will be minus two a b only plus two b c and minus two c a is it so hence there could be a variety of it you just need to remember only one this one and as I told you wherever you see a minus sign you replace it by you replace b by minus b or c by minus c or a by minus a and you'll get the desired identity now if you see how many terms are there in this trinomial expansion there are six terms six terms in this trinomial trinomial trinomial expansion now if you see let us first again write this once again a plus b whole square is nothing but a square plus b square plus c square plus two a b plus two b c plus two c a isn't it now if let us say c is equal to zero if c is equal to zero what will happen l h s will be a plus b plus zero whole square which is nothing but a plus b whole square only and let us see if the r h s matches with our known identity so hence r h s will be a square plus b square plus zero square plus two a b plus two b into zero plus two into zero into a which is nothing but a square plus b square plus twice a b so if you see the trinomial identity is reduced to the binomial identity isn't it so hence this is another way to look at it so we will be solving problems on the expansion of trinomial identity in the next session