 Hi and welcome to the session. I am Asha and I am going to help you solve the following problem that says factorize 2y cube plus y square minus 2y minus 1. So let us start with the solution and let p y be the given polynomial which is 2y cube plus y square minus 2y minus 1. Now factors of minus 1 are plus minus 1. So we will replace this y by these factors such that the value of the polynomial comes out to be 0. So in observing we find that on replacing y by 1 we get 2y whole cube plus y square minus 2 into 1 minus 1 which is equal to 2 plus 1 minus 2 minus 1. 2 cancels up with minus 2 and 1 with minus 1. So we are left with 0. So we have when y is replaced by 1 we get the value of the polynomial as 0. So this implies y minus 1 is a factor of p y on dividing p y by y minus 1 p y is 2y cube plus y square minus 2y minus 1. So first multiplying y minus 1 by 2y square we get 2y cube minus 2y square. Now changing the signs these 2 cancel out and here we have 3y square minus 2y. Now multiplying y minus 1 by 3y we have 3y square minus 3y and now changing the signs these 2 cancels out and we have plus y and now taking 1 down we have minus 1 and then multiplying y minus 1 by 1 we get y minus 1 and thus the remainder is 0. The dividend is equal to divisor into quotient plus remainder. So we must have p y equal to divisor which is y minus 1 into quotient that is 2y square plus 3y plus 1 plus 0. Now our next step is to factorize 2y square plus 3y plus 1 as the product of binomials by splitting the middle term which is 3y. So we have 2y square plus 3y can be written as 2y plus 1 by plus 1 and now taking 2y common from the first 2 terms and 1 common from the last 2 terms we have 2y into y plus 1 plus 1 into y plus 1. Now taking y plus 1 common we have 2y plus 1 and thus y square plus 3y plus 1 can be written as y plus 1 into 2y plus 1 and hence p y which is equal to y minus 1 into 2y square plus 3y plus 1 can be further written as y minus 1 into y plus 1 into 2y plus 1 and thus factorizing the given polynomial which is 2y cube plus y square minus 2y minus 1 we get the factor as y minus 1 into y plus 1 into 2y plus 1. So this completes the solution hope you enjoyed it take care and have a good day. Thank you.