 Hi and welcome to the session. I am Asha and I am going to help you with the following question which says, find the remainder when x cube plus 3x square plus 3x plus 1 is divided by the following linear polynomials. So, before finding the remainder, let us first learn what does remainder theorem say. According to this theorem, if Px is any polynomial of degree greater than or equal to 1, the real number is divided by the linear polynomial x minus a then the remainder is equal to PA. This remainder theorem is the key idea which we are going to use in this problem to find the remainder of the following polynomial. Let us now begin with the solution. The given polynomial Px which is equal to x cube plus 3x square plus 3x plus 1 and the first part is x plus 1 which can further be written as x minus of minus 1. Now, when Px is divided by x plus 1 x minus of minus 1 then remainder theorem is equal to P minus 1 which is equal to minus 1 whole cube plus 3 times of minus 1 whole square plus 3 into minus 1 plus 1 which is further equal to minus 1 whole cube is minus 1 plus 3 into minus 1 whole square is 1 plus 3 into minus 1 is minus 3 plus which is further equal to minus 1 plus 3, minus 3 plus 1 and now minus 3 cancels out with plus 3 and minus 1 cancels out with plus 1 and we get 0. Thus remainder is equal to 0. And now proceeding on to the second part which is x minus half. We have to find the remainder when px which is equal to x cube plus 3x square plus 3x plus 1 is divided by x minus half. So by a key idea the remainder will be equal to p at half. Now replacing x by half in px we have half whole cube plus 3 into half whole square plus 3 times of half plus 1. Now half whole cube is 1 upon 8 plus half whole square is 1 upon 4 plus 3 upon 2 plus 1 and now taking LCM. LCM of 8, 4 and 2 is 8 and on dividing 8 by 8 we get 1. Therefore on multiplying 1 with 1 we have 1 plus on dividing 8 by 4 we get 2. Therefore multiplying the numerator 3 with 2 we get 6 plus on dividing 8 by 2 we get 4. So multiplying the numerator 3 with 4 we get 12 plus 1 can be written as 1 upon 1 and on dividing 8 by 1 we get 8. So multiplying the numerator 1 with 8 we get 8 which is further equal to on adding the numerators we have 27 and in the denominator we have 8. Thus the remainder is equal to 27 upon 8 which completes the second part and now proceeding on to the third part which is x and it can be written as x minus 0 and we are required to find the remainder when px which is equal to x cube plus 3x square plus 3x plus 1 is divided by x minus 0. Then the clear idea remainder will be equal to px0. Now on replacing x by 0 in px we have 0 whole cube plus 3 times of 0 square plus 3 into 0 plus 1 which is equal to 0 plus 0 plus 0 plus 1 which is further equal to 1 and hence the remainder when px is divided by x is equal to 1. So this completes the fourth part and now proceeding on to the fourth part which is x plus pi. Now this can be written as x minus of minus pi since minus into minus is equal to plus and we have to find the remainder when px which is the given polynomial x cube plus 3x square plus 3x plus 1 is divided by x minus of minus pi. Then by the clear idea a remainder will be equal to px minus pi and on replacing x by minus pi in px we have minus pi whole cube plus 3 times of minus pi whole square plus 3 times of minus pi plus 1. Now minus pi whole cube is minus pi whole cube plus 3 times of minus pi whole square is pi square and 3 into minus pi is minus 3 pi plus 1. Thus when px is divided by x plus pi a remainder is equal to minus pi cube plus 3 pi square minus 3 pi plus 1. So this completes the fourth part and now proceeding on to the last part which is 5 plus 2x. Now this can be written as 2x plus 5 and first we need to find the 0 of this linear polynomial which is 2x plus 5 which we get on equating it to equal to 0. This implies on dividing both sides with 2 we have 2x upon 2 plus 5 upon 2 is equal to 0 upon 2. This is by dividing both sides by 2. So this gives x plus 5 upon 2 is equal to 0 or x minus of minus 5 upon 2 is equal to 0. So this is the linear polynomial which will divide the given polynomial that is let px be the given polynomial which is equal to x cube plus 3x square plus 3x plus 1 and according to the key idea remainder will be equal to p at minus 5 upon 2. Now on replacing x by minus 5 upon 2 we have minus 5 upon 2 whole cube plus 3 times of minus 5 upon 2 whole square plus 3 times of minus 5 upon 2 plus 1. Now minus 5 whole cube is minus 125 upon 2 whole cube is 8 plus 3 times of minus 5 whole square is 25 and 2 whole square is 4 plus 3 into minus 5 is minus 15 upon 2 plus 1. Now taking LCM of 8, 4 and 2 LCM is 8 and on dividing 8 by 8 we get 1 so multiplying the numerator minus 125 with 1 gives minus 125 then we have in the numerator 25 into 3 75 and on dividing 8 by 4 we get 2 so multiplying the numerator with 2 and we have plus sign and minus 15 in the numerator and on dividing 8 by 2 we get 4 so multiplying the numerator with 4 then we have plus 1 which can be minus 1 upon 1 and on dividing 8 by 1 we get 8 so multiplying the numerator with 8 which is further equal to minus 125 on multiplying 75 by 2 we get 150 plus on multiplying minus 15 with 4 we get minus 60 plus 8 upon 8. Now first let us add both these negative terms which gives minus 185 and on adding 150 with 8 we get plus 158 in the denominator we have 8 which is further equal to minus 27 upon 8 hence remainder when the polynomial px is divided by 5 plus 2x is equal to minus 27 upon 8. This completes the last part.