 Hello and welcome to this session. In this session, let's discuss first the concept of rational expressions or algebraic fractions. A quotient which appears in the form of m by m is a fraction. So we can say a fraction is any expression which appears in the form of a quotient like m upon n where we have these m and n represent any numbers or any expressions. The fractions having polynomials or you can say rational expressions. Let's see some examples of algebraic fractions or rational expressions. Consider the form of a fraction and it has polynomial and the denominator. So it is called algebraic fraction or rational expression. In the same way consider x square x square minus x plus 1 also a fraction having polynomials in both the numerator and the denominator. So this is algebraic fraction or rational expression. Let's see the quality of algebraic fractions. Then we win two algebraic fractions a upon b and c upon d algebraic fractions. So in these two a denominator or both would have polynomials, fractions that is a upon b and c upon d are equal if and only if we have is equal to dc. The simplification of algebraic fractions, we are given an algebraic fraction. The denominator of that algebraic fraction do not have a common factor. Then we can say that it is the simplest form of the algebraic fraction. An example in which we need to simplify minus 3 upon x square minus 7x plus 2 can see this is an algebraic fraction as it contains the polynomial in the numerator as well as in the denominator. Simplify this algebraic fraction. So this means the numerator and the denominator should not have a common factor. We have a numerator x square minus 7x plus 12. Let's factorize this by splitting the middle term. So it would be equal to minus 4x minus 3x. X common from these two terms and minus 3 common from these two terms we get this is equal to minus 4 the whole minus 3 into x minus 4 the whole. So this is equal to x minus 3 the whole into x minus 4 the whole. Even the denominator can be written as x minus 3 the whole into x minus 4 the whole. The given algebraic fraction x minus 3 upon x square minus 7x plus 12 can be written as x minus 3 upon x minus 3 the whole into x minus 4 the whole. Now as you can see x minus 3 and x minus 3 cancel with each other and so we are left with 1 upon x minus 4 and the denominator do not have a common factor. So simplifying the given algebraic fraction we get 1 upon x minus 4. This is the simplified form of the given algebraic fraction. Now let's discuss the multiplication and the vision of algebraic fractions when necessary and we simplify algebraic fraction. The algebraic fraction is the second fraction. Let's get an example. We need to simplify either by 4x upon x square minus y square. As you can see we need to divide these two algebraic fractions here. So as we know the algebraic fraction we multiply the first fraction by the reciprocal of the second fraction. So this means this is the first fraction. So here we have 16x cube upon x plus y multiplied by the reciprocal of which would be x square minus y square upon 4x. Fractions we need to multiply the numerators and denominators separately and also we can factorize the numerators and denominators in this case like we can say y square plus y the whole into x minus y. Now this is equal to 16x cube multiplied by the numerator of the second fraction which is simplified as x plus y the whole into x minus y the whole and this whole upon x plus y which is the denominator of the first fraction multiplied by the denominator of the second fraction which is 4x. This x plus y cancels with x plus y and 4 4 times a 16 then this 1x cancels with 1x here and here we are left with x is equal to 4x square into x minus y the whole. This we get 4x square into x minus y the whole. Now we should discuss addition and subtraction of algebraic fractions. Addition and subtraction of algebraic fractions is done in the same way as we do in the case of the fractions which involves integers. So in this case also there is the least common multiple or you can say LCD which is most common denominator of the polynomials consider the example in which we have to solve s cube upon cube. This means we need to add these two algebraic fractions find the LCM of the denominator that is p and q. So LC is equal to p cube plus p plus q upon p plus p plus q is equal to the LCM here in the denominator. Now p cube divided by p is 2. So we write q into p plus 2 the whole divided by q is p. So p into p plus 2 the whole is written here plus and the whole upon p cube or we can further write this as p square plus 2 p cube plus q square and the whole upon p cube or you can say whole square upon p cube. Thus fractions given as q upon p plus p plus q upon q plus q the whole square upon the addition of two algebraic fractions is done in the same way we can subtract two algebraic fractions also. The rational expressions are algebraic fractions, the simplification, the multiplication and the vision and the addition and subtraction.