 Hi and welcome to the session. Today we will learn about polynomials. Let us start with the degree of polynomials. If p of x is a polynomial in x then the highest power of x in the polynomial p of x is the degree of p of x. Let us take an example for this. Suppose we have a polynomial p of x equal to 3 x to the power 4 minus 2 x square plus 7 then here in this polynomial the highest power of x is 4 that means degree of p of x is 4. Now a linear polynomial is a polynomial of degree 1. An example of a linear polynomial is 2 x minus 3. A quadratic polynomial is a polynomial of degree 2. One example of a quadratic polynomial is 2 x square plus 5 x minus 3 upon 5. Also the general form of a quadratic polynomial is a x square plus b x plus c where a, b and c are real numbers with a not equal to 0. And a cubic polynomial is a polynomial of degree 3. One example for this is 2 minus x cube. Now general form of a cubic polynomial is a x cube plus b x square plus c x plus d where a, b, c and d are real numbers not equal to 0. Now let us start with 0s of a polynomial. A real number k is said to be a 0 of a polynomial p of x if p of k is equal to 0. Here for the polynomial p of x equal to 2 x minus 3, 3 upon 2 is the 0 of the polynomial p of x. Now a linear polynomial can have at most 1 0, quadratic polynomial can have at most 2 0s and cubic polynomial can have at most 3 0s. Now let us study the symmetrical meaning of the 0s of a polynomial. The 0s of a polynomial p of x are the x coordinates of the points where the graph p of x intersects the x axis. We have a graph of p of x which intersects the x axis at 2 points whose x coordinates are minus 2 and 2. So the 0s of p of x over here are minus 2 and 2. Similarly in this graph the graph of p of x does not intersects the x axis so here p of x has no 0s. Now let us see the relationship between 0s and coefficients of a polynomial. Let us see a quadratic polynomial first. The general form of a quadratic polynomial is a x square plus v x plus c. Now let us suppose that the 0s of this polynomial are alpha and beta. Then there is alpha plus beta will be equal to minus of coefficient of x upon coefficient of x square that is minus b upon a. Similarly product of 0s that is alpha beta will be equal to constant of upon coefficient of x square that is c upon a. Now let us take one example for this. Suppose we have a polynomial p of x equal to 3 x square minus 2 x plus 5. So some of the 0s that is alpha plus v term will be equal to minus coefficient of x upon coefficient of x square that is minus of minus 2 upon 3 which will be equal to 2 upon 3. And the product of the 0s that is alpha into v term will be equal to constant term upon coefficient of x square that is 5 upon 3. Let us consider a cubic polynomial now. The general term of a cubic polynomial is a x cube plus v x square plus c x plus d. Now let us suppose that the 0s of this cubic polynomial are alpha beta and gamma. So some of the 0s that is alpha plus beta plus gamma will be equal to minus coefficient of x square upon coefficient of x cube which is equal to minus b upon a. And the sum of the product of 0s taken 2 at a time that is alpha beta plus beta gamma plus gamma alpha will be equal to coefficient of x upon coefficient of x cube that is c upon a. Lastly the product of 0s that is alpha beta gamma will be equal to minus of constant term upon coefficient of x cube which will be equal to minus d upon a. Now let us take one example for this. Suppose we have a polynomial p of x equal to 2 x cube plus 5 x square minus 9 x plus 7 then alpha plus beta plus gamma will be equal to minus v upon a that is minus 5 upon 2. And alpha beta plus beta gamma plus gamma alpha will be equal to c upon a that is minus line upon 2. Similarly alpha beta gamma will be equal to minus v upon a that is minus 7 upon 2. So this is the relationship between the 0s and the coefficients of a polynomial. Now let us see what is divisional algorithm for polynomials. Divisional algorithm states that if we are given polynomials p of x and g of x where g of x not equal to 0 then there are 2 polynomials q of x and r of x such that p of x is equal to g of x into q of x plus r of x where is equal to 0 or degree of r of x is less than degree of g of x. Now suppose we have a polynomial p of x equal to x cube minus 7 x square plus 12 x minus 6 and g of x equal to x minus 1 then dividing p of x by g of x we can get the values of q of x and r of x such that p of x that is x cube minus 7 x square plus 12 x minus 6 will be equal to g of x that is x minus 1 into q of x that is x square minus 6 x plus 6 plus r of x that is 0. So this is our divisional algorithm and with this we finish this session hope you must have understood all the concepts related with polynomials goodbye take care and have a nice day.