 Hello and welcome to the session. In this session, we will discuss how to add, subtract and multiply polynomials. Now in our earlier session, we had discussed that now a polynomial function is denoted by f of x and a polynomial function in one variable is given by f of a n into x raised to power n plus 0 plus a4 into x raised to power 4 plus a3 into x raised to power 3 plus a2x square plus even x plus a0. Where the degree n of the polynomial has to be a whole number and a0, a1 and so on up to a n are all coefficients and the leading coefficient a n is not equals to 0. And now let us discuss addition and subtraction of polynomials in one variable. Now for adding and subtracting, we must keep in mind the following points firstly. For applying equation of addition and subtraction on polynomials, we must keep in mind the like terms and like terms. Now when two terms in an expression have same variables and exponents of both our numbers, then these two terms are called like terms or similar terms. Now if two terms in a given expression are like terms, then the variable and its exponent in these two terms is always saying that the coefficients are different. Now if two terms in an expression have same variables having different exponents, then these terms of terms are called unlike terms. Now let us discuss this example. Now here, see the first expression. Now in this expression there are two terms and they have same variable a is also saying that is 1. So these two terms are called like terms and now we can add these like terms. Now since the coefficients of these two terms are different, that is 3 and 5, so we can add two terms. So it will be 3 plus 5 the whole into a into a, that is you can see the second expression in this, the two terms, these two terms are like terms. Now coefficients of these two terms are different. So we will perform this operation n is 4 the whole into x square which is equal to minus 3 into x square, that is minus 3 x square. Now in the third expression the terms are constant, that is 4 and 6. It means these terms are also like terms which is equal to 10. Now look at the fourth expression here, the variable x is saying that its exponent is different in the two terms, it is 3 and in the second term the exponent is 2 and the unlike terms cannot be added. So this expression will remain. The commutative, associative and distributive properties are true for the polynomial integers and we can use these properties when working with polynomials. Also we should write polynomial in the symbol over of power. It means the term having highest exponent is written first, then the term with certain highest power is written and so on. In the end we write the constant. Now in this example you can see that this polynomial is written in such a way that the term with highest exponent that is 5 and the next higher power that is 3 and in the end we write the constant term of lowest power. Now keeping in mind these points let us start with this example in which we have to add these two polynomials. We open the brackets into y is to power 5 plus 3 y cube minus 1 plus 8 y cube minus 7 y square plus 11. Alternative and distributive properties hold each the positions of terms. We can combine the like terms. So this is equal to our like terms similarly and plus 11. Our like term this is equal to 5 into y is to i cube plus 8 y cube minus 7 y square plus 11 minus 1. So this is equal to 5 into y is to power 5 3 plus 8 the whole into y cube minus 7 y square. So we can directly subtract them and here 11 so this will be plus and further 5 into y is to power 5 plus 11 y cube minus 7 y square. Now here this polynomial is written in the same order of terms that require answer addition. Now we can also add vertically. Now suppose we have to add these two polynomials vertically. Now here we have to add these two expressions vertically. And this we will write the first in different order and that like terms are written just below each other. So here 3 y cube and 8 y cube are like terms below 3 y and then minus 7 y are like terms. So we will write plus 11 below minus 1. Now here you can see that the term with exponents 2 is missing in first expression. These terms vertically where it will be 5 into y is to power 5 there is no term with exponent 5 in the second expression. So where it will be 5 into y is to y cube so it will be 11 y cube. And next it will be minus 7 y square and then plus 11 minus 1 is plus 10. So adding the like terms we have got the result. And now let us learn subtraction of two polynomials minus 7 y square plus 11 y into y is to power 5 plus 3 y cube minus 1. Now here we will also apply an addition. Firstly let us open the brackets. So this is equal to 5 into y is to power 5 plus 3 y cube minus 1 minus 8 y cube into y is to power 5 plus 3 minus 8 the whole into y cube minus 11 will be minus 12 into y is to power 5. Now 3 minus 8 and minus into plus is minus. So this will be minus 5 y cube plus 7 y square minus 12. Subtract the given two polynomials vertically. Here also remember else with the like terms under one another. So we have written the polynomials like this we have to subtract. So we will change the sign of each term of the second polynomial here it is positive so it will be negative. Then here it is negative that is minus 7 y square so it will be positive. Now here it is plus 11 so it will be minus. So here changing the sign of each term of the second expression the new terms will be minus 8 y cube plus 7 y square minus 11. Subtracting it will be 5 into y is to power 5 minus 8 y cube will be minus 5 y cube. Now here it is plus 7 y square and then minus 11 is minus 12. This is the required answer multiplication of polynomials. Now in multiplication we will use product law of exponents. If this is our same then we will add the powers that is x raise to power a into x raise to power b is equal to x raise to power a. We will also use distributive property and according to this property a into v plus c the whole is equal to a v that is a into v plus a c that is a into c. Now for most terms we can also use it means to multiply polynomial by second polynomial. This product first polynomial is x square so we multiply it with the second polynomial we will multiply that is 2x with second polynomial. So this is equal to a into 4x square minus 3x plus 2 the whole plus 2x into 4x square minus 3x plus 2 the whole. Now this is equal to x square into 4x square plus x square into minus 3x plus x square into 2x into 4x square x into minus 3x plus 2x into 2. Now use product law of exponents now this is equal to now x square into 4x square is 4 into x raise to power 2 plus 2 which is 4. So this is equal to 4 into x raise to power 4 now plus into minus is minus into 3x will be 3 into x raise to power 2 plus 1 which is 3. So it is 3 into x raise to power 3 plus x square plus 2 into 4 is now plus into minus is minus and 2x into 3x is 6x square plus 2x into 2 is equal to 4 into x raise to power 4. Now these two are light terms so this will be plus 8 the whole into x cube now here these two are light terms so it will be plus of 2 minus 2x raise to power 4 plus now minus 3 plus 8 so it will be 5x cube minus 4 and minus into plus as minus. So this will be minus 4x square in this session we have learnt how to add subtract and multiply polynomials and this completes our session hope you all have enjoyed the session.