 Hello and welcome to the session. In this session first we will discuss remainder theorem. According to the remainder theorem we have let Px be any polynomial of degree greater than or equal to 1 and let a be any real number divided by the linear polynomial x minus a then the remainder is Pa. Let's try and find the remainder when the polynomial given by Px equal to x to the power 4 plus 2x cube minus 3x square plus x minus 1 is divided by the linear polynomial x minus 2. As you can see Px is a polynomial of degree 4 which is greater than 1. Now 0 of x minus 2 is 2 then P2 is equal to 2 to the power 4 plus 2 into 2 cube minus 3 into 2 square plus 2 minus 1 which is equal to 21 that is we get P2 is equal to 21. Hence by remainder theorem we have that 21 is the remainder when Px is divided by x minus 2. Next we shall discuss factor theorem. It says if Px is any polynomial of degree n greater than equal to 1 and a is any real number then we have x minus a is a factor of Px if Pa is equal to 0 and Pa equal to 0 if x minus a is a factor of Px. We use factor theorem to factorize the polynomials. Consider the polynomial Px equal to x square minus 3x plus 2. Let's factorize this polynomial by factor theorem. First let's see what are the factors for the constant term 2 in this polynomial. So we have factors of 2, r, 1 and 2. Now let's find out what is P of 1. This would be equal to 1 square minus 3 into 1 plus 2. This comes out to be equal to 0. So we say that x minus 1 is a factor of Px. Next we find out P of 2. This is equal to 2 square minus 3 into 2 plus 2. This is again equal to 0. So we say that x minus 2 is a factor of Px. Hence we can write Px that is x square minus 3x plus 2 is equal to x minus 1 multiplied by x minus 2. This is how we factorize the polynomial by factor theorem. We can also factorize the same polynomial by splitting the middle term method in which the middle term which is minus 3 is splitted by writing it as the sum of two numbers whose product would be 1 which is the coefficient of x square multiplied by 2 which is the constant term. Suppose we consider two numbers P and Q. So we have P plus Q should be equal to minus 3 and PQ should be equal to 1 multiplied by 2 which is 2. So using these two we get two numbers P and Q to be equal to suppose P is equal to minus 1 and Q is equal to minus 2. So you can see their sum is minus 3 and their product is 2. Thus we can write x square plus minus 1 minus 2x plus 2 or this could be written as x square minus x minus 2x plus 2 which comes out to be equal to x minus 1 into x minus 2 that is x square minus 3x plus 2 is equal to x minus 1 into x minus 2. So this is how we can factorize a given polynomial by splitting the middle term method also. Next we discuss some algebraic identities. We know that algebraic identity is an algebraic equation that is true for all values of the variables occurring in it. Consider the first identity which is x plus y the whole square that is equal to x square plus 2x y plus y square. Then next is x minus y the whole square is equal to x square minus 2x y plus y square. Then we have x square minus y square is equal to x plus y into x minus y. Then next is x plus a into x plus b is equal to x square plus a plus b into x plus av. Then next identity is x plus y plus z the whole square is equal to x square plus y square plus z square plus 2x y plus 2yz plus 2zx. Then x plus y the whole cube is equal to x cube plus y cube plus 3xy into x plus y. Next is x minus y the whole cube is equal to x cube minus 3x square y plus 3xy square minus y cube or this could be also equal to x cube minus y cube minus 3xy into x minus y. Then we have x cube plus y cube plus z cube minus 3xy z is equal to x plus y plus z multiplied by x square plus y square plus z square minus x y minus y z minus zx. Let's try and expand 2x plus 1 the whole cube. This can be done by using this identity of x plus y the whole cube. Here we shall have x to be replaced by 2x and y to be replaced by 1. So we get 2x plus 1 the whole cube is equal to 2x cube plus 1 cube plus 3 into 2x into 1 into 2x plus 1. This is equal to 8x cube plus 1 plus 6x into 2x plus 1 that is we get this to be equal to 8x cube plus 1 plus 12x square plus 6x. So finally we get 2x plus 1 the whole cube is equal to 8x cube plus 12x square plus 6x plus 1. This is how we can expand any polynomial by using the above identities. This completes the session. Hope you have understood the remainder theorem factor theorem and some algebraic identities.