 Hello and welcome to the session. In this session, we will discuss just like integers, the polynomials are closed under addition, subtraction and multiplication. And first we define a polynomial and we know that a monomial is a constant or a variable or a product of constants and variables. For example, the algebraic expressions like f, x, fx, 3xy and 2 into x raised to the power 9 are all polynomials or we can say that an algebraic expression containing only one term is called a monomial. Now in expression that is 2x raised to the power 9, 2x the coefficient is the variable and 9 is the power or exponent of x. Now a polynomial can be defined as sum of monomials. Each monomial is a term of polynomial. For example, the expression 2x square plus 3x minus 7 is a polynomial over integers because all its coefficients are integers and in this session we will discuss polynomials over integers in one variable. And now we are going to discuss degree of polynomial, the highest power of the variable in the expression gives degree of polynomial. Now in the expression that is 2x square plus 3x minus 7 the highest power of variable is 2. So degree of this polynomial is 2. Now we should note that any algebraic expression with only one constant term has degree 0. For example, if we have any algebraic expression say 4 then it can be written as 4 into x raised to the power 0 and we can say that the degree of this algebraic expression is 0. And now we are going to discuss the general form of any polynomial expression. A polynomial expression in one variable is given by a naught plus a1x plus a2x square plus a3x cube plus a4x raised to the power 4 plus and so on up to anx raised to the power n where an is not equal to 0. It is a polynomial of degree n which has to be a whole number and a naught a1 up to a n are all coefficients. Now consider the following polynomial expressions over integers and their degrees. And the first expression is x since this expression has only one term x and the power of x is 1. So it has degree 1. And the second expression is 3 into x raised to the power 4 minus 2 into x cube plus x square plus 5x plus 10. And here the highest power of x is 4. So this expression has degree as 4. Now we are going to discuss different types of polynomials based on the number of terms and algebraic expression containing only one term is called a monomial. And we should note that a monomial should not have negative and fractional exponents. For example any expression of the type say 3 into x raised to the power 4, 4x are all monomials and the algebraic expression containing two terms is called a binomial. For example x cube minus x square since this expression has only two terms so this can be called as a binomial expression. Similarly an algebraic expression containing three terms is called a trinomial for example 2x square plus 3x minus 7 since this expression has three terms so it can be termed as a trinomial expression. And similarly an algebraic expression having four or more terms is called a multinomial. Now we will discuss what are light terms when two terms in an expression have same variable and exponents or if both are constant then they are called light terms or similar terms. And now we are going to discuss closure property that is if we perform every operation like addition, subtraction or multiplication on a set of members then the number produced by the operation also belongs to the same set. For example let us consider set of integers now we take any two members from the set of integers let us take 4 and minus 3. Now let us perform an operation of addition for these two members 4 plus or minus 3 which is equal to 4 minus 3 that is 1 which is also an integer. So we say that the circuit integers is closed under addition and if we perform multiplication on this set for this let us take any two members say 2 and minus 5. So if we perform multiplication on these two members we get 2 into minus 5 which is equal to minus of 10 which is also an integer thus we say that set of integers is closed under multiplication. And now we shall discuss closure property for polynomials just like integers the set of polynomials is also closed under operations of addition, subtraction and multiplication. And now we will see how two polynomials now let us consider any two polynomials say 3x plus 2 and 4x cube minus 3x square plus 2x minus 4. Now if we add the two polynomials that is 3x plus 2 plus 4x cube minus 3x square plus 2x minus 4. Now if we read that this first we get 3x plus 2 plus 4x cube minus 3x square plus 2x minus 4. Now we look for the light terms in this expression and we see that 3x and 2x are light terms similarly 2 and minus 4 are light terms. Now we combine all the light terms and write this expression as 3x plus 2x plus 2 minus of 4 plus 4x cube minus of 3x square. And adding these light terms we get 3x plus 2x that is 5x plus 2 minus 4 which is equal to minus of 2 plus 4x cube minus of 3x square. Which is equal to 5x plus of minus of 2 that is minus 2 plus 4x cube minus of 3x square and we can rewrite this expression as 4x cube minus of 3x square plus 5x minus of 2. Which is again a polynomial that is if we add two polynomials then there is some is also a polynomial and thus we say that set of polynomials is closed under addition. And now we will discuss how the set of polynomials is closed under subtraction. Let us take any two polynomials. Let us take the same polynomials that is 3x plus 2 and 4x cube minus of 3x square plus 2x minus 4. And now we subtract the two polynomials that is 3x plus 2 the whole minus of 4x cube minus 3x square plus 2x minus 4 the whole. First we will open the brackets and we get 3x plus 2 minus of plus 4x cube that is minus of 4x cube then minus of minus of 3x square which is equal to plus 3x square. Then minus of plus of 2x that is equal to minus of 2x and then minus of minus 4 which is equal to plus 4. Now again we combine all the light terms and we see that 3x and minus of 2x are light terms and 2 and 4 are light terms. So we write 3x minus of 2x plus 2 plus 4 minus of 4x cube plus 3x square and on solving further we get 3x minus of 2x which is equal to x plus 2 plus 4 that is 6 minus of 4x cube plus 3x square and we can rewrite this expression as minus of 4x cube plus 3x square plus x plus 6 which is again a polynomial that is if we subtract two polynomials then the resultant is also a polynomial thus we say that set of polynomials is closed under subtraction and now we will discuss how to set up polynomials is closed under multiplication. Let us take any two polynomials such as 2x square plus 6x plus 1 and 4x minus 3 and now we multiply the two polynomials that is we multiply each term of the first polynomial with the second polynomial and we write 2x square into the second polynomial that is 4x minus 3 plus 6x into 4x minus 3 the whole plus 1 into 4x minus 3 the whole. Now we will use the property of exponents to solve this expression which says x raise to power a into x raise to power b is equal to x raise to power a plus b and now we get 2x square into 4x plus 2x square into minus of 3 plus 6x into 4x plus 6x into minus 3 plus 1 multiplied by 4x minus 3 which is equal to 4x minus 3 now 2x square into 4x is equal to 2 into 4 that is 8 and we have x square into x which will be equal to x raise to the power 2 plus 1 that is 3 plus 2x square into minus 3 will be equal to minus of 6x square plus 6x into 4x which is equal to 24 that is 6 into 4 is 24 into x into x which is equal to x square plus 6x into minus 3 which is equal to minus of 18x plus 4x minus 3 and now we get 8x cube plus of minus of 6x square that is minus of 6x square plus 25x square plus of minus of 18x that is minus of 18x plus 4x minus 3 now we look for the light terms in this expression and we see that minus of 6x square and 24x square are the light terms and minus of 18x and 4x are light terms now in combining light terms and adding we get 8x cube plus minus of 6x square plus 24x square plus minus of 18x plus 4x minus of 3 which is equal to 8x cube plus minus 6x square plus 24x square that is 18x square plus minus of 18x plus 4x which is equal to minus of 14x minus 3 which can be written as 8x cube plus 18x square plus of minus of 14x that is minus of 14x minus 3 which is again a polynomial that is if we multiply any two polynomials then the resultant is also a polynomial thus the set of polynomials is closed under multiplication and in this session we have learnt that how the set of polynomials is closed under addition, subtraction and multiplication this completes our session hope you enjoyed this session