 Hi and welcome to the session. I am Asha and I am going to help you solve the following problem which says verify that x cubed plus y cubed plus z cubed minus 3xyz is equal to half x plus y plus z into x minus y whole square plus y minus z whole square plus z minus x whole square. So to verify this problem we will show that the left hand side is equal to the right hand side. But before that let us first learn the simple identity which says that x cubed plus y cubed plus z cubed minus 3 into x into y into z is equal to x plus y plus z into x square plus y square plus z square minus xy minus yz minus zx. So this identity is a key idea with the help of which we will show that the left hand side is equal to the right hand side. So let us now start with the solution and first we will solve the right hand side which is half into x plus y plus z. In the next packet we have x minus y whole square plus y minus z whole square plus z minus x whole square. Now since we know that a minus b whole square is equal to a square minus 2ab plus b square therefore it can further be written as half into x plus y plus z and now opening x minus y whole square that is x square minus 2 into x into y plus y square with this helper formula and now opening y minus z whole square which is y square minus 2 y z plus z square and lastly on opening z minus x square this is z square minus 2 zx plus x square which is further equal to half into x plus y plus z and in the next packet x square is 2 times so we have 2x square y square is also 2 times so we have 2y square z square is 2 times so 2z square and then we are left with minus 2xy minus 2yz minus 2zx. Now taking two common from the second bracket we have half into x plus y plus z into 2 and what are we left with in the bracket yes x square plus y square plus z square minus xy minus yz minus zx now this two cancels out with 2 and we are left with x plus y plus z into x square plus y square plus z square minus xy minus yz minus zx which is nothing but x cube plus y cube plus z cube minus 3xyz and this is by a key idea and which is nothing but the left-hand side and thus we have proved that the left-hand side is equal to the right-hand side hence verified so this completes the solution hope you enjoyed this session take care have a good day