 Hi and welcome to the session. I am Asha and I am going to help you solve the following problem that says factorize x cube minus 3x square minus 9x minus 5. So let us start with the solution and let the given polynomial be denoted by px. So px is equal to x cube minus 3x square minus 9x minus 5. We shall now look for all the factors of minus 5 and some of the factors of minus 5 are plus minus 1 and plus minus 5 and now we will replace this x by these values of the factors of minus 5 such that we get the value of this polynomial as 0. Now observing one by one we find that on replacing x by 5 we get the value of the polynomial as 0. So let us check 5 raised to the power 3 minus 3 into 5 raised to the power 2 minus 9 into 5 minus 5. Now 5 raised to the power 3 is 125 minus 3 5 raised to the power 2 is 25 minus 9 into 5 is 45 and thus in the next step we have 125 this is 75 minus 45 minus 5 which is further equal to 125 and on adding all these negative terms we get minus 125 which is further equal to 0. Thus on replacing x by 5 we get the value of the polynomial as 0. So this implies x minus 5 is a factor of px and so on dividing px which is x cube minus 3x square minus 9x minus 5 by x minus 5 first on multiplying x minus 5 with x square we get x cube minus 5x square. Now changing sign these two cancels out and we are left with 2x square taking minus 9x in the denominator then we have plus 2x and on multiplying plus 2x square x minus 5 we get 2x square minus 5 into 2x minus 10x. Again on changing signs these two cancels out and this comes out to be x minus 5. Now multiplying x minus 5 with 1 we get x minus 5 and on changing the signs we get the remainder as 0. Now since dividend is equal to divisor into quotient plus remainder and our dividend is px divisor is x minus 5 and on dividing px by x minus 5 we get the quotient as x square plus 2x plus 1 and the remainder is 0. Now x square plus 2x plus 1 can be written as x square plus x plus x plus 1 by splitting the middle term. Now taking x common from the first two terms and one common from the last two terms it can further be written as x into x plus 1 plus 1 into x plus 1. Now taking x plus 1 common we have x plus 1 and here we are left with x and here plus 1. So we have px is equal to x minus 5 and x square plus 2x plus 1 can be written as x plus 1 into x plus 1 and thus the given polynomial x cube minus 3x square minus 9x minus 5 is equal to x plus 1 into x plus 1 into x minus 5. So this completes the second part. Hope you enjoyed it. Take care and have a good day.