 welcome friends to another session on factorization of polynomials in the previous session we had discussed a very important theorem in polynomials called factor theorem and we are going to see the use or the application of that factor theorem in this question while solving this question okay so what was factor theorem by the way so factor theorem if you remember was this that if fx is a polynomial okay if fx is divided divided by ax plus b okay linear factor that is linear divisor why linear because the power is 1 so if fx is divided by ax plus b then the remainder you can find out the remainder without actually dividing it remainder is nothing but f of minus b upon a that's what we discussed in the previous session okay now fx is in this case px px is given as x to the power 4 minus 3x square plus 2x plus 1 this is a px and our linear divisor is x minus 1 divisor let us say gx is x minus a x minus 1 so what is a and b if you compare these two right this is of that form ax plus b so clearly a is 1 b is minus 1 correct so minus b by a is simply 1 okay so therefore remainder how to find out remainder a remainder will be simply f of or in this case p because the name of the polynomial is given to be p so p1 okay p1 means what I have to find out the value of value of px at x equals to 1 this is what is meant by p1 so let's find out p1 so p1 will be how much 1 to the power 4 minus 3 into 1 squared plus 2 into 1 plus 1 okay so if you calculate this one 1 minus 3 plus plus 1 which is equal to simply 1 that means if you divide this px this expert this polynomial by x minus 1 you don't need to perform actual division you will get the remainder as 1 only right so you can check by actually dividing it and then see that the remainder actually comes out to be 1 okay if you want we can check it here also so let's try to divide this given expression by long division method though we have studied though we have studied synthetic division as well but just for those people who have not learned it in our sessions the long division method right so this is x cube x 3 power 4 minus you will get what x cube correct now this goes but this is power 3 and this is 2 so what will happen this is subtraction so x cube minus 3 x square will come down right the powers were different now what plus x squared you'll get x cube and then minus x squared if you subtract you'll get minus 2x square plus 2x comes down right then what you see minus 2x here becomes minus 2x square and then plus 2x so hence goes right so what is left one only right so the remainder is 1 this one comes down 1 right so hence if you see quotient is this much remainder is this but we didn't require to actually perform the division you could have got this by applying factor sorry remainder theorem only which says that if the effects is divided by ax plus b then remainder is simply f of minus b by a value of the polynomial at x equals to minus b by a