 Hello and welcome to the session. In this session we will discuss that rational expressions are closed under addition, subtraction, multiplication and division. Also we will see how to add, subtract, divide and multiply rational expressions. Now in our early session we have discussed that a set of polynomials is closed under addition, subtraction and multiplication. And here we will discuss that a rational expression in which both numerator and denominator are polynomials is closed under different operations. Now we know that a rational expression in those variables. Now here we will discuss rational expressions in one variable. Now a rational expression is of the form f of x upon g of x where g of x is not equal to 0 and f of x and g of x are polynomials. Now where no function is written in the denominator. Now here let us take a polynomial function f of x and here you can see no function is written in the denominator. So we consider the denominator g of x to be 1. So f of x can be written as f of x upon 1. It means it can be written in the form f of x upon g of x thus we can say that every polynomial has rational expression. Now a rational expression can also be of form 1 upon f of x. We have discussed that just like in Degas set of polynomials is closed under addition, subtraction and multiplication. Now we know that every polynomial is a rational expression thus set of rational expressions is also closed under addition, subtraction and multiplication. Now we have to prove that set of rational expressions is closed under addition. So let us consider two rational expressions 3 x plus 12 whole upon 4 x and 2 minus 16 whole upon. Now let us divide 3 x plus 12 whole upon 4 x by 3 whole upon 2. So we have 3 x plus 12 whole upon 4 x by whole divided by x square minus 16 whole upon 2 by whole over our definition of the region by whole into taking the reciprocal of the second x this will be upon x square minus 16 whole. Now let us factorize the numerator of the first rational expression. The numerator of first rational expression is 3 x plus 12. Now taking 3 common from both these terms it will be 3 into x plus 4 whole we can factorize the denominator of second rational expression. Now the denominator of square minus 4 square x minus 4 by whole into x plus by using the identity of this b square is equal to a plus b by whole into a minus b by whole 3 into x plus 4 by whole whole upon 4 x into 2 upon whole by whole into x plus 4 by whole the common factor that is x plus 4 is equal to 3 upon the whole x can be written as 2 into 2 x into 2 upon x minus 4 by whole into 1 is 2 3 upon 2 x into 1 upon x minus 4 by whole which is equal to 4 by whole which is equal to 3 upon minus. So here we can see that dividing two rational expressions again a rational expression. Now we know product of the rational numbers that is a upon b, a upon b is equal to a upon b into c upon b which is equal to ac upon bd. Similarly for multiplication of rational expressions we have first method where the product of the rational expressions is the fraction whose numerator is the product of the denominator is the product of given denominators and we reduce the resulting fraction by factorization if possible to its lowest terms and then in the second method we factorize the polynomials in the two fractions and then we transform any factors common to numerator and denominator and then we multiply by the random numerators and denominators. Now let us discuss these methods with the help of examples. This is an example and here we have to multiply 3 x upon x plus 3 by whole over upon to this method we will multiply the numerators and denominators of the given fractions. So this will be equal to 3 x into which is equal to 3 x into 2 x that is 6 x square x into 6 that is 18 x over upon or in the denominator into x that is 5 x equal to 3 that is 15 x. We reduce this result if possible to its lowest terms. Now factorizing the numerator by taking 6 x common from both the terms in numerator it will be 6 x into x plus 3 by whole. Now in the denominator taking 5 x common from both the terms it will be 5 x into x plus 3 by whole factor that is x plus 3 by whole. So this was method 1 multiply the two rational expressions. The multiplication of these two rational expressions. Now in the second method we will factorize the polynomials in the two fractions if possible x upon x plus 3 by whole into now here in the numerator we will take both the terms and it will be 2 into x plus 3 by whole and this complete half. This is the common factors that is x plus 3 by whole and this is equal to 3 into 2 by whole upon x upon 5. And now let us discuss division of rational expression. Now to divide the two rational numbers or to divide two rational expressions we multiply the result by reciprocal of the divisor and follow the same method as used in multiplication that is upon g of x divided by we will multiply the dividend that is f of x upon g of x by reciprocal of the divisor and here reciprocal of the divisor will be q of x upon as used in multiplication. Now suppose we have to find x plus 1 plus minus 1 by whole divided by x plus 1 by whole upon x by whole. So this will be equal to x plus 1 by whole upon 2x minus 1 by whole into now taking the reciprocal of second rational expression this will be x upon x plus 1 by whole. And now we will cancel the common factor that is x plus 1 by whole and this is equal to x upon 2x minus 1. Now let us discuss how to add and subtract rational expressions. Now we add and subtract the rational expressions just like rational numbers by taking least common multiple of denominators in the numerator as the two fractions must have common denominator and if the denominator is not common then we make the equivalent by multiplying both numerator and denominator by least common multiple of denominators term of denominators is equal to product of the two denominators or the product of all the factors of one of both denominators where the repeated factor is taken only once. Now let us discuss this with the help of an example. Now here we have to add a square upon 2a plus 2 by whole and 3a minus 1 by whole a square minus 1 by whole. Now here denominator of first fraction is 2a plus 2 and denominator of second fraction is a square minus 1. Now on factorizing the denominator of first rational expression this will be equal to now taking two common from both the terms it will be 2 into a plus 1 by whole and denominator of second fraction is a square minus 1 which can be written as a square minus 1 square and this is equal to a minus 1 by whole into a plus 1 by whole and a sum of denominators which will be equal to product of all the factors of one of both denominators. So here a sum of denominators will be equal to 2 into a plus 1 by whole. Now here a plus 1 by whole is the repeated factor so it will be taken only once and the sum of denominators will be equal to 2 into a plus 1 by whole into a minus 1 by whole. Now the denominators of both Now the first fraction is a square upon 2a plus 2 which is equal to a square upon 2 into a plus 1 by whole. Now here for making the denominator equal to the LCM we will multiply the numerator and denominator of this fraction by a minus 1 by whole and this is equal to a square into a minus 1 by whole upon 2 into a plus 1 by whole into a minus 1 by whole. Similarly we will multiply the numerator and denominator of second fraction by 2 and this will be equal to 2 into 3 a minus 1 by whole upon 2 into a plus 1 by whole into a minus 1 by whole. Now for adding these two rational expressions we will have common denominator so this will be equal to now in the denominator we will have 2 into a plus 1 by whole into a minus 1 by whole and in the numerator we will have a square into a minus 1 by whole plus 2 into 3 a minus 1 by whole and this is equal to now a square into a is a cube and a square into minus 1 is minus a square plus 2 into 3 a is 6 a and 2 into minus 1 is minus 2 whole upon 2 into a plus 1 by whole into a minus 1 by whole. So this is the required answer. Now you should note one point that if after adding numerator we can further factorize then we simplify it further and reduce the fraction if possible that is we will simplify the fraction. Now let us discuss addition and subtraction of rational expressions using LCM as product of denominators and for this let us discuss an example where we have to find 3 upon x minus 1 by whole minus x plus 3 whole upon x plus 1 by whole. Now here we will directly take the LCM of denominators as product of denominators which is equal to x minus 1 by whole into x plus 1 by whole. Now here we will need the denominators of both the fractions equivalent to LCM. Now here the first fraction is 3 upon x minus 1 by whole. So here we will multiply the numerator and denominator of this fraction by x plus 1 by whole and this will be equal to 3 into x plus 1 by whole whole upon x minus 1 by whole into x plus 1 by whole. Similarly in the second fraction we will multiply the numerator and denominator by x minus 1 by whole. Now for subtracting these two rational expressions we will subtract the two rational expressions having common denominator. So this is equal to now in the denominator we have x minus 1 by whole into x plus 1 by whole and in the numerator we have 3 into x plus 1 by whole minus x plus 3 by whole into x minus 1 by whole and this complete whole. Now on simplifying in the numerator we get 3x plus 3 minus x square plus x minus 3x plus 3 whole upon x minus 1 by whole into x plus 1 by whole. Now combining the light terms in the numerator we have minus x square plus x plus 6 whole upon and in the denominator x minus 1 by whole into x plus 1 by whole is x square minus 1. So this is the required answer. So in this fraction we have learnt that set of rational expressions is closed under addition, subtraction, division and multiplication and this completes our session. Hope you all have enjoyed the session.