 Hi and welcome to the session. I am Arsha and I am going to help you with the following question that says use the factor theorem to determine whether gx is a factor of px in each of the following cases. So before solving this problem, let us first learn what does factor theorem say. This theorem says that if px is any polynomial of degree greater than or equal to 1 the real number versus x minus a is a factor of px is equal to 0. Second is p is equal to 0 if x minus a is a factor of px. So the theorem is a key idea that we will be using in this problem to find its solution. Let us now start with the solution. The first one is px is equal to 2x cube plus x square minus 2x minus 1 and we have to check whether gx which is equal to x plus 1 is a factor of px or not. So according to the factor theorem, gx which is equal to x plus 1, it can further be written as x minus of minus 1 is a factor of px if px minus 1 is equal to 0. So let us find the value of px minus 1. If it comes 0, this implies gx is a factor of px. So replacing the variable x by minus 1 in px we have 2 minus 1 whole cube plus minus 1 whole square minus 2 times of minus 1 minus 1 which is equal to 2 minus 1 whole cube is minus 1 plus minus 1 whole square is 1 minus into minus is plus this gives 2 minus 1 whole cube plus 2 minus 1 which is further equal to minus 2 plus 1 plus 2 minus 1. Minus 2 answers out with plus 2 and plus 1 will minus 1 and we get the result as 0. So we have px minus 1 is equal to 0. This implies x minus of minus 1 is a factor of px. This implies x plus 1 is a factor of px and x plus 1 is nothing but gx. So gx is a factor of px. So this completes the first part. Let us now proceed on to the next part where px is equal to x cube plus 3x square plus 3x plus 1 and gx is equal to x plus 2 and we have to check is gx factor of px. According to a key idea gx is a factor of px if px minus 2 is equal to 0 since x plus 2 can be written as x minus of minus 2. So let us now find the value of the polynomial px when x is replaced by minus 2 and we have minus 2 whole cube plus 3 into minus 2 square plus 3 times of minus 2 plus 1. Now simplifying it further minus 2 whole cube is minus 8 plus 3 minus 2 whole square is 4. 3 into minus 2 is minus 6 plus 1 which is further equal to minus 8 plus 12 minus 6 plus 1. Now minus 6 is minus 8 is minus 14 and on adding 12 with 1 we have plus 13. So this comes equal to minus 1. Thus create minus 2 is equal to minus 1 which is not equal to 0. So this implies x minus of minus 2 is not a factor of gx is 2 which is equal to gx is not a factor of px this px. This completes the second part. Now proceeding on to the last part where px is equal to x cube minus 4x square plus x plus 6 and gx is equal to x minus 3. Again here we have to check is gx a factor of px. According to our key idea gx which is equal to x minus 3 is a factor of px p at 3 is equal to 0. So let us find the value of the polynomial px when x is replaced by 3 if it comes equal to 0 this implies 3x is a factor of px otherwise it is not a factor of px. So replacing x by 3 we have 3 whole cube minus 4 into 3 whole square plus 3 plus 6, 3 whole cube is 27 minus 4, 3 whole square is 9, 3 plus 6 is 9. Which is further equal to 27 minus 9 4 is 36 plus 9 and on adding 27 with 9 we get 36 minus 36 this comes equal to 0. And hence the value of the polynomial px at x is equal to 3 is 0. So this implies x minus 3 is a factor of px and x minus 3 is nothing but gx as we can say gx is a factor of px. So this completes the third part let us now write down the answers. In the first part we saw that gx was a factor of px therefore our answer is yes. In the second part gx was not a factor of px and in the third part gx was the factor of px hence our answer is yes. So please do remember the factor theorem while doing these types of problem. Hope you enjoyed this session. Take care and have a good day.