 welcome friends once again to another session on mathematics and today we are going to start another series of sessions on algebraic identities and their applications okay so we will also see how to expand various algebraic expressions powers of expressions and then we will also understand how to factorize right so we will understand two things one is there is an algebraic expression so so we will learn under this topic primarily we will be learning what expansion how to expand expand a given expression so let us say there is an algebraic expression I am assuming you all know what an expression is but just to recap an expression is nothing but a combination of various terms right and how to combine the terms using plus or minus sign that's what an expression is about so let us say 2x minus 3y is there and let us say we need to find out its second power right then how to expand them so such kind of things we will be learning another thing let's say 4x minus 5y to the power 4 or let us say a minus 2b to the power 3 and things like that or let us say it seems like x minus y times x plus 3y things like that so how to expand these right so this is one and then second thing is factorization exactly the reverse process factorization so let us say if there is an expression given let's say x square plus 3x plus 2 so this is a polynomial now how to factorize it that means how to convert this into this form x minus a times x minus b or something like that right so you can clearly differentiate one is to convert the factors back into expressions you know and the other one is factorization means how to convert a given expression and express it as factors of multiple expressions that's what we are going to learn right so we will start with algebraic identities okay so let's understand what is an algebraic identity before that if the name suggests algebraic identities that means there must be identities which are not algebraic isn't it right so let us you know understand both these terms one by one so what is an identity identity so identity as the name suggests is something which doesn't change it stays identical doesn't it stays identical so hence identity in mathematics is nothing but is an equality so basically it's an equation it's an equation or an equality which which which is true true or which holds or that holds holds for all values of the variable all values of the variables right that's what an identity is so let us say if I say x plus 3 is equal to 5 now here clearly x is equal to 2 right is the only only value which holds only value x equals to 2 is the only value which which satisfies satisfies x plus 3 is equal to 5 isn't it but if I say but if let's say x plus 1 whole square is x square plus 2x plus 1 let us say this is another equality here this particular equality or equation is true true for any value of x which is a variable over here isn't it so hence this is an example of an identity right identity so hence true for true for all values of the variables but this is a plain and simple equation it's not an identity this one is not an identity not an identity I hope you understood the difference between identity and a generic equation so identities are also equations but equations not necessarily our identities okay so this is the definition of identity now what we are going to concentrate is on algebraic identities that is all the identities in algebra related to algebra but there can can be things like trigonometric trigonometric identities as well trigonometric identities also exist also exist for example if you are aware of basic trigonometry then we say that sine square theta plus cosine square theta right is always one where theta is an angle so this is called a trigonometric identity right so whatever be the value of theta this relationship is not going to be affected so it is independent of theta so hence it is trigonometric identities right but here we are going to focus only on algebraic identities and let's do a quick recap of whatever identities which we have covered in the previous grades okay so let's do a quick recap of all the identities so I'm going to jot them down one by one and it's always a good practice to store or let's say write these identities in a piece of paper and affix it in front of your study table or something like that so that you are in constant touch with these identities so let's start so let us say a plus b and this is the first identity which would have learned and it is nothing but a square plus two a b plus b square many students have this issue of writing a plus b whole square as simply a square plus b square so this is not true so hence please be very very careful a plus b whole square it's pronounced like that a plus b whole square is a square plus 2 a b plus b square next one is you simply replace b by minus b if I replace b by minus b on the left hand side wherever there is b on the right hand side I will replace that with minus b as well so hence you can see what is the identity now it is nothing but a square minus 2 a b plus b square correct this is the second one third one is let us say now I have a minus b and a plus b so a minus b a plus b if you expand it so I will also give you a few examples of expansion so that you know you are familiar with what exactly this means so hence it is nothing but I multiply a by a so I get a square then I multiply a by plus b so you'll get a plus a b then you multiply minus b by a so it is minus b a and then minus b square so this disappears so hence it is left with a square minus b square right third one what is the fourth one now these were all about squaring the identities so let us say if instead of two terms I had three terms within the square so what will this be this will be nothing but a square plus b square plus c square plus twice of a b plus twice of b c plus twice of c a right now there could be multiple varieties and if you know various types of this identity itself for example let us say if this is a minus b plus c whole square so how to do it so you know the trick wherever there is plus b replace it by minus b simply so hence on the right hand side what will it what will it become a square plus minus b square plus c square plus 2 into a into minus b plus 2 into minus b times c plus 2 c a right and then you can simplify further and it is nothing but a square minus plus b square sorry plus c square minus 2 a b minus 2 b c plus 2 c a so if you notice the square terms are not affected by the minus sign only where b appears only once that is the power of b is once the sign will change correct this is the fourth identity fifth one now we'll go to cube so if you do a plus b whole cube what is it a cube plus 3 a square b plus 3 a b square plus b cube this is the identity many people write this as a cube plus b cube so you're combining a cube plus b cube and then taking 3 a b common from here so you write 3 a b and then a plus b this is another form of the same identity right then sixth one you can guess you can simply change plus b to minus b and what will happen wherever there are odd powers of b they will change so hence it will be nothing but plus 3 a square minus b so it is a cube plus 3 a square minus b plus 3 times a times minus b squared plus minus b cubed right simplifying you get a cube minus 3 a square b plus 3 a b square minus b cube so there are a few things to notice over here if you see the signs are altering alternating so plus becomes minus then becomes plus then becomes minus or if you notice only where the powers of b were odd the sign changed so for example here it was cube so sign changed here it was only one so sign changed isn't it so please bear that in mind going to the next one what is the next identity so next identity is let's say a cube plus b cube which can directly come out from this identity itself if you manipulate it and let us say if you see how how do I get a cube plus b cube it's nothing but a plus b whole cube according to the above identity is a cube plus b cube plus 3 a b a plus b isn't it right so hence what can I say about this so hence what you can do is take a cube plus b cube on one side and rest of the terms on the other side so you can write a cube plus b cube is equal to a plus b whole cube minus 3 a b a plus b so I'm taking this term whole term on this side right so in the right hand side if you see you can take a plus b common actually so what is left within a plus b whole squared minus 3 a b isn't it so a plus b was taken as common so what does it become so let a plus b be as it is and now you know the identity a plus b whole square is a square plus b square minus plus 2 a b sorry plus 2 a b minus 3 a b isn't it right so hence what do I get a plus b and if you simplify you'll get a square 2 a b and minus 3 a b will give you minus a b and plus b square correct so this is the identity so if you see you could factor if you notice carefully this was a sum of two terms and you could express that as a factor or a product of two factors isn't it so this is one of the tricks to factorization which anyways will be learning later on so if you didn't understand go through it slowly what I did was I simply change the side of 3 a b a plus b and then took a plus b common and a plus b whole square was a square plus b square plus 2 a b and minus 3 a b was already there I did the manipulation I got a plus b a square minus a b plus b square similarly you can try this as a as an exercise a cube minus b cube will be simply a minus b times a square plus a b plus b square so if you see this minus sign changes here and this minus sign changes here as well so wherever b was you know have to replace b by minus b and you'll get the identity right okay now there are a few more identities which we'll take up in the next session