 Hello and welcome to the session. In this session we discuss the concept of HCF and LCM of polynomials. First let's discuss the HCF of polynomials. By HCF you mean highest common factor. So HCF of polynomials would be the highest common factor of the polynomials. Now HCF that is the highest common factor of two polynomials is the polynomial of the highest degree greatest numerical coefficient which divides both the polynomials exactly. So when we are given two polynomials and we are supposed to find the highest common factor or the HCF of the two polynomials then the HCF of the two polynomials would be that polynomial which has the highest degree and the greatest numerical coefficient and it divides both the polynomials exactly. We can write the highest common factor as HCF as GCF also which is the greatest common factor. So the highest common factor and the greatest common factor is one and the same thing. Now we discuss the HCF of monomials. We know that monomials are the polynomials having only one term. In finding the HCF that is the highest common factor of monomials we will follow certain steps. The first step would be the HCF that is the highest common factor of the numerical coefficients of the given monomials. Then in the next step we find each of the variables to all monomials. Now consider the variables which are not common. Then in the next step we multiply the reverse step one common factor or you can say the HCF of the given monomials. Let us now consider an example in which we have to find the monomials y2 z to the power 5 then the next monomial is 35 x to the power 10 y square z to the power 10. So the third monomial as power 4 y to the power 5 z to the power 5. So we have to find the HCF of the given three monomials. From the first step first we will find the HCF of the numerical coefficients of the given monomials. We will find HCF of the numerical coefficients which are 5, 35 and 40. Now let us find the HCF of these three numbers. 5 can be written as 5 multiplied by 1, 35 can be written as 5 multiplied by 7 and 40 can be written as 5 multiplied by 8 or you can say 2 raised to the power 3. Now as you can see that 5 is the number that divides all the three numbers exactly. So we can say that HCF of the three numbers 5, 35 and 40 is given by 5. Next step we find the highest power of each of the variables common to all monomials. The highest power of the variable x common to all the three monomials would be as you can see that x square is common to all the three monomials. Then the highest power of the variable y would be y square since it is common to all the three variables. Then the highest power of the variable z which is common to all the three variables is z to the power 5. So if we multiply the results of the previous two steps so x square into y square into z to the power 5 this is what we obtain after multiplying the results of the two previous steps and so we can say that this is the highest common factor or you can say the HCF of the three monomials. So this is the answer. Next we shall discuss to find the HCF of polynomials which can be factorized. For this again we will follow certain steps. First of all we factorize the given polynomials of the numerical coefficients. We find of the given polynomials we multiply the results and three when polynomials. An example to find the HCF of two polynomials given as plus 7x plus 6 factorize the given polynomials. So consider the polynomial. In the middle term we get x square plus x we get x into x plus 1 the whole plus 6 into x plus 1 the whole. Or this could be written as the whole into the given polynomial x plus 1 the whole into x plus 60 whole. Let this be result one. Consider the second polynomial which is x cube minus. Let us now factorize this polynomial. x is common to all the three terms of this polynomial. So we take x common. So here we have x into x square minus the whole. We can split the middle term of this polynomial. So this is equal to x into sxx minus 6 the whole and we get this is equal to x into x into x plus 1 the whole minus 6 into x plus 1 the whole. So we get this is equal to plus 1 the whole into x minus 60 whole. Therefore the given polynomial x cube minus 5x square minus 6x is equal to x into x plus 1 the whole into x minus 60 whole. Let this be result two. Now after we have factorized the two polynomials we will find the set of common factors of the two polynomials. And as you can see the x plus 1 is common to both the polynomials. We have the common factor of the two polynomials. Therefore we can say that the hcf of the given two polynomials. So this is how we find the hcf of the two polynomials which can be factorized. Let's discuss the LCM of polynomials. By LCM we mean this common of polynomials as the nearest degree all the numerical coefficients will be divisible by the given polynomials. We discuss how to find the LCM of monomials for this of the numerical coefficients of all the monomials. Then in the next step the common multiples of the variables of the variables is the variable which is exactly divisible by each of the given variables. Next step we multiply the results and the products would be monomials. Now consider an example in which we have to find the LCM of the monomials 5x squared y2 z to the power 5 45x to the power 10 y squared z to the power 12 and 40x to the power 4 y to the power 5 into z to the power 5. Now in this case the first step would be to find the LCM of the numerical coefficients of all the monomials. Now in all the three given monomials as you can see that the numerical coefficients are 5, 35 and 40. The LCM 35 on finding the LCM of the numbers 5, 35 and 40 we get that its LCM would be equal to 5 into 7 into 8 which is equal to 280. Next we find the common multiples of the variables of the given monomials and see in all the three monomials we have the variables x, y and z to the power of something. So common multiple x to the power 10 and x to the power 4 would be equal to exactly divisible by each of these. Then the common multiple of y2 y squared y to the power 5 would be equal to y to the power 5 as again this would be exactly divisible by each of these. Then we have the common multiple z to the power 5 z to the power 12 would be equal to z to the power of 12 be exactly divisible by each of these. Then next we multiply the results obtained in these two steps. So we get 280 into x to the power 10 into y to the power 5 into z to the power 12 and this is the LCM given. So when we have to find the LCM of the monomials we would follow these steps. Next we find the LCM of the monomials which can be factorized. For this we would follow certain steps. In the first step we factorize the given polynomials. Then in the next step the remaining and the product of the common factor would be the LCM of the polynomials. That is now considered two polynomials x squared plus 7x 5x squared minus 6x and we need to find the LCM of these two polynomials. Factorize each of these polynomials which we have already done while finding the HCM. So x squared plus 7x plus 6 can be written as x plus 60 whole into x plus 1D whole. Let this be result 3 then the other polynomial x cube minus 5x squared minus 6x can be factorized as x plus 1D whole into x minus 60 whole. Let this be result these polynomials we will find out the common factors. So the common factors two polynomials would be equal to x plus 1. So we write here x plus 1. Maining factors and the C factors which are not common. The two polynomials would be equal to x into x minus 6x plus 60 whole. The given is would be equal to the product of the common factors and the remaining factors. So this would be equal to x into x plus 1D whole into x minus 60 whole into find the LCM of the given two polynomials. In the session we have understood how to find the LCM and HCM of the polynomials.