 hello friends welcome again and let's continue in the discussion on algebraic identities just to give you a quick recap this is what we learned in the previous session so we were working on identities and we saw these identities a plus b whole square a minus b whole square and we learned that whenever you have to come up with an identity with minus b you just replace plus b with minus b and we also saw a plus b plus c whole square then a plus b whole cube a minus b whole cube and then a cube plus b cube and a cube minus b cube so these were the identities which we learned in the previous session you can go through the previous session to get a quick recap now let's continue our journey with identities now the next one which I'm going to give you is this so this is identity number 9 just to continue with the number which we had so we had done with 8 1 now this is the ninth one and again in this case I am going to give you this identity a cube plus b cube plus c cube and minus 3 abc so if you see this particular thing can be factored into a plus b plus c and then a square plus b square plus c square minus ab minus bc minus ca so this is the identity and now if you see each of the terms here has a degree 2 isn't it each of them has a degree 2 here it is 1 so eventually you will get a degree 3 expression right so this is what is this identity now there is a special case of this identity and that is let us say if if a plus b plus c is somehow 0 it's given so what will happen this will become a cube plus b cube plus c cube minus 3 abc is equal to 0 into whatever is there in the bracket is it it so what is it simply a 3 a to the power 3 plus b cube plus c cube minus 3 abc will be equal to 0 and hence we can write a cube plus b cube plus c cube is equal to 3 abc isn't it so this is a very important result but when a plus b plus c is 0 only then otherwise no okay so now let us move on to the next one what is the next one guys next one is let us say x plus a times x plus b now this will be simply x squared plus a plus b within brackets times x plus ab and you can do it very easily by just expanding you will see this is the result now there could be multiple you know varieties of this you keep changing sign of let us say a and b so let us say if I have x minus a x plus b so what is the result result is nothing but x square then simply you replace a by minus a nothing much x plus sorry in this case it will be minus a times b right if it is let us say x plus a x minus b so it will be nothing but x squared plus what say a minus b x mine and then plus a times minus b simply right and then the one final one is x minus a times x minus b is equal to x squared plus minus a minus b x plus minus a times minus b simply correct you can simplify and then find the results like that this is another identity and if it was late let's say one more it's like x plus a if I have three factors like that x plus b and x plus c then what will happen is it is nothing but x cube so clearly it is a degree 3 expression so x cube plus x square and here it is a plus b plus c okay so if you notice here it was only a plus b here it is a plus b plus c then plus a b plus b c plus c a times x and plus a b c okay so there are four terms in this so this one is another identity okay so all these 11 identities you must remember and we will be using these identities in as I told you in things like expansion and factorization so let us take some examples let us say I have to the question is question is expand so now we are taking up expansion so expand expand what 3x plus 2y whole squared right so if you notice here I can treat this as a and treat this as b and go for the first identity what was that a plus b whole square was nothing but a square plus twice a b plus b square so let us fit a and b in this into this so a was nothing but 3x whole square plus 2 times 3x times 2y plus 2y whole squared so hence it is 9x squared plus 12 x y is it it plus 4 y squared right so this is how we expand so if you see what was this this was nothing but 3x plus 2y times 3x plus 2y so whose result is or whose expansion is this let me take another example another example is question is find root 2 times x minus 3 y square so if you now see this one is a here your a this one is b and it is a minus b whole square okay so what is it if it is nothing but a squared minus 2 a b plus b squared let us say if you don't want to mug up all these formulae so you just need to remember one a plus b whole square and what is it a square plus 2 a b plus b square and now you could have treat treated a minus b whole square can be if let's say if you don't want to remember this so you can be written as a plus minus b whole square is it and then put it back into this so what will you get you'll get a square plus 2 times a times minus b plus minus b whole square which is equal to a square minus 2 a b plus b square isn't it so hence but it is better if you remember this so that you reduce your time while solving it now let's use this identity so it will be root 2x whole squared minus since it is a minus sign here so minus 2 times root 2x times 3 y plus 3 y whole squared just don't just write 3 y square it is 3 y whole square so hence what will it be it would be 2x squared minus 6 root 2 x y plus plus 9 y square so this is the expansion okay so we use two identities here next is let us say next question is x minus 2 x plus 3 so if you see this is nothing but which one I did which identity it is it is this one x square plus a plus b x plus a b right so here what is a a is minus 2 isn't it a is minus 2 and b is 3 so let me write the identity once again x plus a x plus b is equal to x square plus a plus b x plus a b isn't it so here a was minus 2 so let's write x squared plus minus 2 plus 3 x plus minus 2 times 3 isn't it so what is it nothing but x squared then this is minus 2 plus 3 is 1 so plus x and minus 6 okay so this is how you use identities to expand okay very good so we will be taking some more questions and examples in the next session thank you