 Hello and welcome to the session. In this session we will discuss the rational root theorem. Before moving further, we must know what is the root of a function? Root of a function is that value when plugged in for the variable makes the function equal to 0. Thus roots of a polynomial f of x are values of x such that f of x is equal to 0. Now let us discuss rational root theorem. According to this theorem, let f of x be equal to a n x raised to the power n plus a n minus 1 x raised to the power n minus 1 plus and so on up to a 1 x plus a 0. Where n is greater than or equal to 1, a n is not equal to 0, a 0 is not equal to 0 and all coefficients are integers. Now if p by q is a rational number written in lowest terms that is greatest common divisor of p and q is 1 and if p by q is a 0 of f of x then p is the factor of constant term a 0 and q is the factor of the leading coefficient a n. Now while determining the rational roots of a function, by using this theorem we should follow these steps. For this, first we arrange the polynomial in descending order then we write down all the factors of the constant term and these are the possible values of p. Next we write down all the factors of the leading coefficient and these are the possible values of q. Then we write down the possible values of p by q. Note that the factors can be negative so p by q and minus p by q both must be included. Then we determine the values of p by q for which f of p by q is equal to 0 and these are the rational roots of f of x. Now let us discuss an example. Find all the rational 0s of the function f of x is equal to 2x cube minus 3x plus 1. Let us start with its solution. Now here we are given a polynomial function with n that is 3 and this is greater than or equal to 1. a n the leading coefficient is given as 2 and a 0 is equal to 1 and we see that all the coefficients are integers. So by rational root theorem which says that if p by q is a rational number written in lowest terms and if p by q is a 0 of f of x then p is the factor of the constant term 1 and q is the factor of the leading coefficient 2. Then the possible values of p and q are p is equal to plus minus 1 that is the factor of the constant term and here it is 1 so its factor will be plus minus 1 and q is equal to plus minus 1 and plus minus 2. That is the factors of the leading coefficient that is 2 here so its factor will be plus minus 1 and plus minus 2. So the possible values of rational 0s of f of x that is values of p by q are as follows plus minus 1 upon 1 and plus minus 1 upon 2. Now we have to determine the values of p by q for which f of p by q is equal to 0 and these values will be the rational root of the function f of x. Now let us draw a table of values for x is equal to p by q and f of x. Now here the values of x are as follows and these are 1 minus 1 1 by 2 and minus 1 by 2. Now we shall find the corresponding values of f of x. Now for x is equal to 1 f of x will be equal to 2 into 1 q minus 3 into 1 plus 1 and this is equal to now 2 into 1 q will be 2 into 1 that is 2 minus 3 into 1 is minus 3 plus 1 and this is equal to 3 minus 3 that is 0. For x is equal to 1 f of x is equal to 0 similarly we shall find the other values of f of x for these corresponding values of x. For x is equal to minus 1 f of x is equal to 2 for x is equal to 1 by 2 f of x is equal to minus 1 by 4 for x is equal to minus 1 by 2 f of x is equal to 9 by 4. From this table we see that at x is equal to 1 f of x is equal to 0 so this function has only one rational root that is at x is equal to 1. Thus the given function f of x is equal to 2 x cube minus 3 x plus 1 has one rational root that is at x is equal to 1. Thus in this session we have learnt the rational root theorem. This completes our session. Hope you enjoyed this session.