 Hi and welcome to the session. I am Asha and I am going to help you solve the following problem which says, expand each of the following by using suitable identities. So first let us learn the identity with the help of which we will expand the following given six problems. The square of u plus v plus w is equal to u square plus v square plus w square plus 2 times of u into v plus 2 times of v into w plus 2 times of u into w. So this identity is a key idea with the help of which we will expand the following given problems. Let us now start with the solution and the first one is x plus 2y plus 4z whole square. Now comparing it with the left hand side of this identity, we find here that u is equal to x, v is equal to 2y and w is equal to 4z. And when applying this identity, first we have u square at x square plus v square, v is 2y so 2y whole square plus w square, w is 4z so 4z whole square plus 2 times of u into v, u is x and v is 2y plus 2 times of v into w, v is 2y and w is 4z plus 2 times of u into w. So u is x and w is 4z. Now x whole square is x square plus 2y whole square is 4y square plus 4z whole square is 16z square plus 2 into 2 is 4, x into y plus 2 to the 4, 4 for the 16 yz plus 2 for the 8 xz. And hence the expansion of the first one which is x plus 2y plus 4z whole square is x square plus 4y square plus 16z square plus 4xy plus 16yz that is xz. So this completes the first part. Let us now proceed on to the next one where we have to expand 2x minus y plus z whole square. Now again comparing it with the left hand side of the identity here we find that u is equal to 2x, v is equal to minus y and w is equal to z. And that is an applying identity this can forever written as first we have u square that is 2x whole square plus v square minus y whole square plus w square that is z square. Then we have plus 2 into u into v that is 2 into 2x into minus y plus 2 into v into w, v is minus y and w is z plus 2 times of u into w that is 2x into z. Now 2x whole square is 4x square minus y whole square is y square plus z square plus into minus is minus and 2 into 2 is 4xy again plus into minus is minus. So minus 2y say lastly plus 2 to the 4 exit and hence on expanding 2x minus y plus z whole square we get 4x square plus y square plus z square minus 4xy minus 2y z plus 4x z. So this completes the second part. Now proceeding on to the third part here we have to expand minus 2x plus 3y plus 2z whole square. Again on comparing it with the left hand side of the identity we find here that u is equal to minus 2x v is equal to 3y and w is equal to 2z. Now on applying the identity this can forever written as u square that is minus 2x whole square plus v square that is 3y whole square plus w square which is 2z whole square. After that we have 2u into v that is minus 2x into 3y plus 2 times of v into w that is 3y into 2z plus 2 times of u into w that is minus 2x into 2z. Now minus 2x whole square is 4x square 3y whole square is 9y square plus 2z whole square is 4z square and plus into minus is minus so we have minus 2 to the 4, 4 to the 12xy plus on multiplying 2 with 3 we get 6 and 6 into 2 is 12 so we have 12xy z and lastly again plus into minus is minus so we have minus 2 to the 4, 4 to the 8 x z and hence when expanding minus 2x plus 3y plus 2z whole square we get 4x square plus 9y square plus 4z square minus 12xy plus 12xy z minus 8x z. So this completes the third part and now proceeding to the fourth part here we have to expand 3a minus 7p minus c whole square. Again on comparing it with the left hand side of the identity here we find that u is equal to 3a, v is equal to minus 7p and w is equal to minus c and on applying the identity this can further be written as u square that is 3a whole square plus v square minus 7b whole square plus w square which is minus c whole square plus 2 times of u into v 3a into minus 7p plus 2 times of v into w that is minus 7b into minus c plus 2 times of u into w that is minus 3a into minus c. Now 3a whole square is 9a square plus minus 7b whole square is 49b square minus c whole square is c square plus into minus is minus and 3 twos are 6, 6 into 7 is 42a into b minus into minus is plus and 7 into 2 is 14bc and again plus into minus so we have 3 twos are 6ac and thus on expanding 3a minus 7b minus c whole square we get 9a square plus 49b square plus c square minus 42ab plus 14bc minus 6ac. So this completes the fourth part and now proceeding on to the fifth part which is minus 2x plus 5i minus 3z whole square. On comparing it with the left hand side of the identity here we find that u is equal to minus 2x v is equal to 5y and w is equal to minus 3z and on applying the identity this can be written as u square that is minus 2x whole square plus v square that is 5y whole square plus w square which is minus 3z whole square plus 2 times of u into v that is minus 2x into 5y plus 2 times of v into w that is 5y into minus 3z plus 2 times of v into w which is minus 2x into minus 3z. Now let us expand this on simplifying minus 2x into minus 2x we get minus plus 4x square 5y whole square is 25y square minus 3z whole square is 9z square plus into minus is minus and 2 into 2 is 4 4 into 5 is 20xy plus 2 into minus is minus 2 5s are 10 10 3s are 30 yz and minus into minus is plus 2 to the 4 4 3s are 12 exit and thus the product of minus 2x plus 5y minus 3z with its surface 4x square plus 25y square 9z square minus 20xy minus 30 yz plus 12 exit so this completes the fifth part and now proceeding on to the last part where we have to expand 1 upon 4a minus 1 upon 2b plus 1 whole square. Now again on comparing it with the left hand side of the identity here we find that u is equal to 1 upon 4a v is equal to minus 1 upon 2b and w is equal to 1. So let us now apply the identity first we have u square is 1 upon 4a whole square plus v square which is minus 1 upon 2 into b whole square plus w square which is 1 square plus 2 times of u into v that is 1 upon 4 into a into minus 1 upon 2b plus 2 times of v into w minus 1 upon 2 into b and w is 1 plus 2 times of u into w so u is 1 upon 4a and w is 1. Now 1 upon 4a whole square is 1 upon 16a square plus minus 1 upon 2 whole square is 1 upon 4 and b square is b square plus 1 this took answers out with 2 and plus into minus is minus so we have minus 1 upon 4a into b. Again this took answers out with this 2 and plus into minus is minus so with b into 1 is b and 2 is a 2, 2 is a 4 so we have plus 1 upon 2a and hence by expanding 1 upon 4a minus 1 upon 2b plus 1 whole square we get 1 upon 16a square plus b square upon 4 plus 1 minus a into b upon 4 minus b plus a upon 2. So this completes the last part and hence the session so hope you enjoyed it please do remember the identities while solving are expanding these type of problems take care and have a good day.