 Hello and welcome to the session. In this session we will discuss how to factor out the greatest common monomial factor and we will also learn how to apply zero product property to solve quadratic equations. Let us first define monomial and it is an algebraic expression containing only one term. It can be a constant term or a variable expression for example 2x 3x square y 5 etc. All these are monomials. Next we shall define greatest common factor It is the largest numerical factor that can divide each term. For example, greatest common factor of 12 and 16 is 4. Although 1, 2 and 4 are all the factors of both the numbers as they divide the two numbers, but the largest number that divides all factors 12 and 16 is 4. So greatest common factor is 4. Now we are going to learn about greatest common monomial factor. It is the product of the greatest common variable factor and the greatest common number that divides out the terms in the given expression. Let us consider this example. We have to find its greatest common monomial factor. It is a binomial containing two terms. We can see that 11 is common in both the terms. So 11 is the greatest common factor of these terms. So we take out 11 common from the given binomial and we have 11p minus 11q is equal to 11 into p minus q the whole. Now see there is no term of variable p or variable q that can be taken common from the two terms. So here the greatest common monomial factor is the constant term 11. Thus we follow the following steps to factor out the greatest common monomial factor. And the first step is to find the greatest common number that factors or divides all the given terms in the expression and the greatest common variable factor by choosing the smallest exponent of the variable that occurs in all the terms. And then factor out each term in the expression by the greatest common monomial factor which is the product of the terms obtained in the previous step. Let us consider an example. Factor out the greatest common monomial factor for 15x raised to the power 5 plus 25x cube plus 30x square. Let us first find the greatest common number that factors or divides all the three terms clearly. 5 is the greatest common number that divides 15, 25 and 30. Now we find the greatest common variable by choosing the smallest exponent of the variable that occurs in all the terms. So here 2 is the smallest exponent of variable x that occurs in all the terms. So greatest common variable factor is x square. Thus greatest common monomial factor will be 5 into x square that is 5x square. So we take 5x square common from the three terms or we multiply and divide the given expression by 5x square. Now taking 5x square common from the three terms we get 5x square into 3x cube plus 5x plus 6. So this is the required factorization. Now we will apply 0 product property to solve quadratic equations. It says that if A into B is equal to 0 then either A is equal to 0 or B is equal to 0. It means that at least one of the factors must be 0 or both can also be 0. For example if we have the product of two factors equal to say x plus 4 the whole into x minus 2 the whole is equal to 0 then by 0 product property either x plus 4 is equal to 0 or x minus 2 is equal to 0. This implies x is equal to minus 4 or x is equal to 2. Thus this property enables us to obtain the values of x. Now let us consider the following quadratic equation that is x square plus 2x minus 15 is equal to 0. We have to find the solution of this quadratic. So we first find its factors using factorization and we get x minus 3 the whole into x plus 5 the whole. This implies x square plus 2x minus 15 is equal to 0 can be written as x plus 5 the whole into x minus 3 the whole is equal to 0. Now using 0 product property we get x plus 5 is equal to 0 or x minus 3 is equal to 0. Now solving these equations for x we get x is equal to minus 5 or x is equal to 3. So we have two equations that is x is equal to minus 5 and x is equal to 3. We should note that we can use 0 product property only when the quadratic equation is equal to 0. Thus in this session we have discussed how to factor out the greatest common monomial factor and we have learned how to apply 0 product property to solve quadratic equations. This completes our session. Hope you enjoyed this session.