 Hi and welcome to the session. I'm Asha and I'm going to help you solve the following problem which says factorize x cube plus 13 x square plus 32 x plus 20 So let us start with the solution and let the given polynomial be p x So p x is equal to x cube plus 13 x square plus 32 x plus 20 now We shall look for all the factors of 20 of the factors of 20 hour plus minus 1 plus minus 2 plus minus 4 plus minus 5 plus minus 10 and plus minus 20 and now you will replace this x by some of these Factors such that the value of the polynomial comes out to be zero So I'm replacing x by minus one we find that the value of the polynomial comes out to be Zero, let us replace x by minus one. So we have minus one cube Plus 13 into minus one square Plus 32 into minus one plus 20 now minus one whole cube is minus one Plus 30 minus one square is one plus into minus is minus so I'm 32 into minus one is minus 32 plus 20 now On adding the positive terms get 33 and on adding the negative terms. We have minus 33 These two answers out and we have P Replacing x by minus one as zero. So this implies x plus one is a factor of px so dividing px by This plus one and px is x cube plus 13 x square Plus 32 x plus 20 So first my decline x plus one by x square. We get x cube Plus x square Now changing the soil and then simplifying we get 12 x square Taking plus 32 x down then Multiplying x plus one by 12 x we get 12 x square Plus 12 x Again on changing the sign these two answers out and we have 20 x Now taking plus 20 down Then I'm multiplying x plus one by 20 we get 20 x plus 20 Again on changing the signs We get the remainder as zero now since we know Dividend is equal to divisor into quotient plus remainder thus we have Dividend as px divisor is x plus one quotient as x square plus 12 x plus 20 and remainder is zero Now next what we will do is factorize x square plus 12 x plus 20 by splitting the middle term which is 12 x So it can be written as x square Plus 10 plus 2 into x plus 20 That's 10 plus 2 is 12 and the product of 10 plus 2 is 20 Therefore we have x square Plus 10 x plus 2x plus 20 and now taking x common from the first two terms and two common from the last two terms Can further be written as x into x plus 10 Plus 2 into x plus 10 now taking x plus 10 common. We have x plus 10 into x plus 2 and thus x square Plus 12 x plus 20 can be written as x plus 10 into x plus 2 and thus Px which is equal to x plus 1 Into x square plus 12 x plus 20 can be written as x plus 1 And in place of x square plus 12 x plus 20 we will write x plus 10 into x plus 2 That's the given polynomial Px which is x cube plus 13 x square Plus 32 x Plus 20 is equal to x plus 1 Into x plus 2 Into x plus 10. So this completes the solution. Hope you enjoyed this session. Take care and bye for now