 Hi and welcome to the session. I am Asha and I am going to help you solve the following problem which says factor is the following using appropriate identities. So starting with the solution, the first one is 9x square plus 6xy plus y square. Now here 9x square can be written as 3x whole square, y square is y whole square and 6xy can be written as 2 into 3x into y, if suppose a is equal to 3x, b is equal to y then 6xy can be written as 2 into 3x into y which is equal to 2 into a into b and we have an identity that says a square plus 2 a b plus b square is equal to a plus b whole square and here in this question we have a as 3x whole square plus 2 a b can be written as 3x into y plus b square and b is y so whole square thus this can be written as the left hand side of this identity and on applying this we have a plus b whole square and a is 3x and b is y. So we have 3x plus y whole square which is 3x plus y into 3x plus y two times and hence on factorizing the given polynomial we have 3x plus y and 3x plus y as factors. So this completes the first part and now proceeding on to the second one where we have to factorize 4y square minus 4y plus 1. So first let us write the identity which we are going to use in this problem to factorize it. Suppose we have a square minus 2 a b plus b square then it is equal to a minus b whole square. So we will try to write down this polynomial in the form of a square minus 2 a b plus b square. So let us start first we have 4y square. Now 4y square can be written as 2y whole square this 1 can be written as 1 whole square and now writing 4y as the product of 2y into 1 where 2. So 4y can be written as 2 into 2y into 1. a is equal to 2y b is equal to 1 and thus 4y square minus 4y plus 1 can be written as 2y whole square minus 3 and 4y can be written as 2 into 2y into 1 plus 1 whole square. So the left hand side of this identity where a is 2y and b is 1 and hence on applying the identity we have 2y minus 1 whole square which is further equal to 2y minus 1 into 2y minus 1. Hence the factors of the given polynomials are 2y minus 1 into 2y minus 1. So this completes the second part and now proceeding on to the third part which is x square minus y square upon 100. So first let us write the identity which we are going to use in this problem to factorize it. Suppose we have a square minus b square and this is equal to a plus b into a minus b. So on writing x square minus y square upon 100 in the form of a square minus b square. So x square can be written as x whole square y square upon 100 can be written as y square upon 10 square which is equal to y upon 10 whole square. Now a is equal to x and b is equal to y upon 10. So x square minus y square upon 100 can be written as x whole square minus y upon 10 whole square. And applying the identity first we have a plus b that is x plus y upon 10. Then we have a minus b that is x minus y upon 10. And hence the factors are x plus y upon 10 into x minus y upon 10. So this completes the third part and hence the solution. So hope you enjoyed this session. Take care and have a good day.