 Hello and welcome to the session. In this session we will discuss approximation of real roots of polynomials. Here we will use the intermediate value theorem to approximate real zeros of polynomial functions. Now, intermediate value theorem states that if f of x is a polynomial function and f of a and f of b have different signs, then there is at least one value c between a and b such that f of c is equal to zero. Or you can say when you have a polynomial function and one input value causes the function to be positive and the other negative, then there has to be at least one value in between them that causes the polynomial function to be zero. The fact behind this is zero separates positives from negatives. So, when the function moves from positive to negative or vice versa, then it must hit a point in between that goes through zero. Now, let us discuss an example. Show that the function f of x is equal to x raised to the power four minus four x cube plus twelve has a real zero between one and two. Use the intermediate value theorem to find an approximation for this zero to the nearest tenth. Now, let us start with its solution. When finding functional values, you can either use synthetic division or directly plug the number into the function. Since we want to know the functional value in this problem, so we directly plug x value into the function. So, let us find the value of the function at x is equal to one and x is equal to two. So, f of one will be equal to one raised to the power four minus four into one cube plus twelve. That is, we have put the value of x as one in this function. Now, this is equal to one minus four into one plus twelve. That is, one minus four into one is four plus twelve. This is equal to nine. Also, f of two will be equal to two raised to the power four minus four into two cube plus twelve. And this is equal to sixteen minus four into eight plus twelve. This is further equal to sixteen minus thirty-two plus twelve and this is equal to minus four. Now, we have got the value of f of one as nine and f of two is equal to minus four. And we can see that there is a sign change between f of one is equal to nine and f of two is equal to minus four. Then, according to intermediate value theorem, there is at least one value between one and two. That is, a zero of the polynomial function. Now, checking functional values at intervals of one tenth for a sign change. And we see that f of one point one is equal to eight point one four zero one. f of one point two is equal to seven point one six one six. Now, f of one point three is equal to six point zero six eight one. Similarly, f of one point four is equal to four point eight six five six. f of one point five is equal to three point five six two five. f of one point six is equal to two point one six nine six. f of one point seven is equal to zero point seven zero zero one. And f of one point eight is equal to minus zero point eight three zero four. Now, here we see that again we have a sign change. So, it will be x is equal to one point seven or x is equal to one point eight. Now, we want to find the zero to the nearest tenth. And we cannot necessarily go by which functional value is closer to zero. Further, let us subdivide this interval that is from one point seven to one point eight and calculate functional values at intervals of one hundredths for a sign change. So, now we find out f of one point seven one and it is equal to zero point five five zero three. f of one point seven two is equal to zero point three nine eight three. Now, f of one point seven four is equal to zero point zero nine four three. Similarly, f of one point seven five is equal to minus of zero point zero five eight six. Here again, we have a sign change between successive hundreds. This means there is a zero between one point seven four and one point seven five. So, the required zero to nearest tenth would be one point seven. So, we have used the intermediate value theorem to find an approximation for this zero to the nearest tenth. This completes our session. Hope you enjoyed this session.