 Hello and welcome to the session. In this session, we will discuss polynomial identities and find the squares of sum and difference using them and here we will prove following identities. First identity is x plus y whole square is equal to x square plus 2x y plus y square and second identity is x minus y whole square is equal to x square minus 2x y plus y square then third identity is x square minus y square is equal to x plus y the whole into x minus y the whole and the full identity is the plus y square whole square is equal to x square minus y square whole square plus 2x y whole square and in this session we will discuss these polynomial identities. Now let us discuss squares of sum and difference. Now in our earlier session we have learnt how to multiply two polynomials and here we see that products of some pairs of polynomials that is polynomials having two terms follow a specific pattern. One such pattern is square of the sum that is x plus y whole square which can be written as x plus y the whole into x plus y the whole. Now we can see the pattern from the following diagram here we have taken a square that is this bigger square whose side is at length x plus y now let us name it as a v c d so area of square a b c d is equal to which is equal to x plus y whole square in the diagram we can see that area of square a b c d is composed of a square of side x that is this blue portion then two rectangles of sides x and y that is this pink portion and y portion and a square of side y that is this yellow portion so area of the square with side x is x square then area of the rectangle with sides x and y is x y similarly area of the rectangle is again x y and area of the square with sides y square these figures in square a b c d then we will write the square a b c d so on adding the areas of all these figures this will be equal to x square plus x y plus x y plus y square which is equal to x plus x y will be 2 x y plus y square per a b c d that is x plus y whole square is equal to some of the areas of m v square a b c d which is equal to x square plus 2 x y y square plus 2 x y now let us solve it as we are becoming let us open the product by multiplying the two binomials here multiplying each term of first binomial we have plus y by whole plus y into x plus y by whole and this is equal to plus y x plus y square which is equal to plus now multiplication is commutative so y x can be written as y square the right terms we have y plus x y will be so this will be x square plus 2 x y plus y square thus square of the sum that is equal to x plus y whole square is equal to 2 x y plus y square now if we replace y by minus y in this above identity of square then v x plus of minus y whole square is equal to x square plus 2 of minus y whole square minus y will be x minus y whole square is equal to identity difference now let us discuss an example f plus 5 v so here I will use this identity that is x plus y whole square is equal to x square plus 2 x y plus y square take the first term that is 3 a square and the second term that is 5 v as y we will apply this identity 5 v whole square will be equal to that is 3 a square whole square plus into x that is 3 a square into y that is 5 v plus y square that is 5 v whole square now whole square will be equal to 9 into a is to that is a is to power 4 the first term is 9 into a is to power 4 now 2 into 3 into 5 is 13 into v will be a v whole square will be the sum will be equal to of these two terms will be equal to whole into x minus y the whole now here we can use a diagram to find the pattern now here let us consider a square go to side square that is x square now from this square we have to cut another square of this square that is this smaller square is equal to side square that is y square the area we subtract the area of the remaining portion will be equal to area of the square with side x with side y so area of the remaining portion will be x square minus now here we can see that this remaining area has 2 rectangles minus y and the second rectangle with sides x minus y minus y the whole the smaller rectangle will be a into x minus y the whole the required left area is area of the remaining portion will be equal to of the areas of these two rectangles that is equal to x into x minus y the whole plus y into x minus y the whole which is equal to now here we can see x minus y the whole is a common factor so taking it common we have x minus y the whole in the area minus y square minus y the whole into x plus y the whole and this is the required identity now let us solve it algebraically we will multiply each term of the first polynomial with the second polynomial so this will be equal to x into x minus y the whole plus y into x minus y the whole which is equal to now x into x is x square then x into minus y is minus x y into x is yx and as product is commutative so yx can be written as xy then y of minus y is minus y square now combining the like terms these terms will be cancelled with each other and this is equal to x square minus y square thus we have x square minus y square is equal to x plus y the whole into x minus y the whole now let us discuss an example here we have to solve this 11 the whole into 3a plus 11 the whole it is the product of sum and difference of two same terms so here we will apply this identity now here let us take 3a plus sy and here we have the whole into x plus y the whole so applying this identity this will be equal to 3a whole square minus y square that is 11 and this is equal to now 3a whole square is equal to 9a square minus 11 square is 121 and this is the required answer identities in factorizing now suppose we have a polynomial x square minus 8 to 1 and we have to factorize it so we can write it as x square minus square minus y square is equal to x minus y the whole into x plus y the whole minus 9 the whole into x plus 9 the whole so here we have factorized the given polynomial just derive this result now here let us start with the left hand side x square plus y square whole square we use this identity that is the identity so this will be equal to x square whole square into x square into y square whole square now this is equal to now whole square is equal to x raise to power 2 into 2 that is x raise to power 4 plus square y square plus now y square whole square is y raise to power 4 now let us start with the right hand side now here for opening the brackets that is for solving x square minus y square whole square that is the identity of square of difference in this identity x square minus y square whole square will be x square whole square minus 2 y square plus y square whole square plus now 2 x square whole square is equal to 4 x square y square now this is equal to x raise to power 4 now combining the left hand plus 4 x square y square will be plus 2 whole square is y raise to power 4 thus left hand side is equal to right hand side is equal to x square minus y square whole square plus 2 is equal to x plus y whole square minus 4 x y now replacing y by minus y in this identity we have minus y whole square is equal to x plus of minus y minus 4 into x into minus y and this implies x plus y whole square is equal to x minus y whole square we have discussed polynomial identities and this completes the session hope you all have enjoyed the session