 Hello and welcome to the session. In this session, we will discuss how to identify zeros of polynomial when suitable factorization is available and how zeros help us in constructing graph of polynomial function. Now let us see what is a polynomial function. Now a polynomial function is n function with a rule that can be written in the form f of x is equal to a n into x raise to the power n plus 0 plus a 4 into x raise to the power 3 plus a 2 into x raise to the power 2 plus a 1 x plus a naught where degree n has to be a whole number a 1, a 2 and so on up to a n are all leading coefficient a n is not equal to 0. And now let us discuss zeros of polynomial. Now consider the polynomial is equal to x cube minus 6 x square plus 10 x minus a. The value of this function or x is equal to 4. We will put x is equal to in the given polynomial is equal to 4 raise to the power 3 minus 6 into raise to the power 2 plus 10 into 4 minus a and this is equal to 64 minus 16 plus 10 into 4 is 14 minus 8 and this is equal to 64 minus 14 is 104 and minus 9 to 6 minus 8 is minus 104 which is equal to 0. This is equal to or we can say that for x is equal to 4 the given that is f of 4 is equal to 0 is a 0 of the given polynomial f of x a polynomial function for which f of x is equal to 0 are called of the graph. It means the graph of the polynomial will intersect. Now let us see the polynomial function if we call it like if we write x cube minus 6 x square plus 10 x minus 8 is equal to 0 then if we solve this equation for x then we obtain plus we find 0 of a function and root of an equation. Now every 0 of a function is also a root of the equation. This number of zeros of a polynomial function now degree of a polynomial of zeros of that function now if the polynomial number of zeros is also 1 if degree of f of x is 2 then number of zeros also 2 similarly if degree is 3 let us discuss an example for this. Now here the polynomial is square plus 10 x minus 8 will be equal to half the variable in the given polynomial. Now here degree of f of x is 3 so number of zeros of the given polynomial function f of x will also be 3. The polynomial now consider the product into x minus 5 the whole when we multiply we get polynomial x minus 10 this means that this polynomial x square minus 3 x minus 10 then the 3 x minus 10 is equal to x plus 2 the whole into x minus 5 the whole which means a polynomial is written as these linear polynomials the given polynomial this means a polynomial x square minus 3 x minus 10. Now here f of x is equal to x square minus 3 x minus 10 x plus 2 and x minus 5 are factors of this polynomial we get 2 is equal to minus 2 whole square minus 3 into minus 2 x minus 10 which is equal to 10 minus 10 and this is equal to 0 thus 2 is equal to 0 this is equal to minus 2 the given polynomial function is equal to 0 this means minus 2 is a 0 of the given polynomial function now x minus 5 is again 0 is equal to 0 of the given polynomial that from factors of the given polynomial zeros are given we can form the factors now if alpha is a 0 of a polynomial alpha will be the factor of that polynomial of the given polynomial beta are its two factors factor of any polynomial then its corresponding these factors are linear when we have to find the zeros or roots of a polynomial function or equation and we are able to factorize it easily by different methods then we write it as a product of linear factors its zeros are to factorize and identify the zeros of a polynomial let us see polynomial factor now here consider a polynomial f of x is equal to 2 x cube minus 6 x square is the smallest part of x that occurs in each term of the polynomial taking 2 x square common from both the terms minus 6 x square upon simplifying this is equal to 2 x square into x minus 3 the whole polynomial function now here you can see the degree of this polynomial is 3 function as a product of linear factors now this is equal to 2 into now x square can be written as x into x x minus 0 into 2 into x minus 0 the whole into x minus 0 the whole into x minus 3 the whole so we have written the given polynomial function as a product of linear factors minus 0 and x 9 of the given polynomial f of x 0 0 and 3 are the zeros of the given polynomial now here you can also see that zeros can be repeated now we can check that 0 0 and 3 are zeros of the polynomial f of x now here you can see if we put x is equal to 0 in the given polynomial function we get f of 0 is equal to 0 in the given polynomial function then again we get f of 3 is equal to 0 so 0 0 and 3 are the zeros of the given polynomial function so how to identify zeros using graph of a polynomial now from the graph we will see that the graph of polynomial will intersect the x axis at the points now consider a polynomial x is equal to x cube minus 9 x this is the graph of the polynomial function f of x is equal to minus 9 x now here you can see that the graph 3 points and here x is equal to minus 3 x is equal to 0 and x is equal to 3 of the given polynomial with 0 3 and minus 3 are the zeros of the given polynomial then minus 3 and x plus 3 will be the 3 factors of the given polynomial and if we multiply the 3 factors that is x minus 0 the whole into x plus 3 the whole into x minus 3 the whole then we get x cube minus 9 x which is the given polynomial function f of x thus x cube minus 9 x is equal to x minus 0 the whole into x plus 3 the whole into x minus 3 the whole now we can check this cube minus 9 x and this will be equal to now taking x common from both the terms this is equal to x into x square minus 9 the whole minus b square is equal to a plus b the whole into a minus b the whole plus 3 the whole 3 the whole this can be written like this 0 minus 3 and 3 are the zeros of the given polynomial then we have learnt how to identify zeros of polynomial when suitable factorization is every graph of polynomial function and this completes our session hope you all have enjoyed the session.