 Hello and welcome to the session. In this session we discussed the following question which says, find the LCM of the polynomials 15xq minus 75x square plus 90x and 6 into x to the power 4 minus 18xq minus 108x square. Let's move on to the solution now. So we are given the two polynomials and we need to find the LCM of the two polynomials. We have the first polynomial 15xq minus 75x square plus 90x. First of all we will write this polynomial in a factorized form. Now this could be further written as, now since 15x is the common factor to all the three terms of this polynomial. So we write here 15x and in the bracket we are left with x square minus 5x plus 6. Let us now factorize the polynomial x square minus 5x plus 6. So this could be written as x square minus 3x minus 2x plus 6 and this whole multiplied by 15x. So we get this is equal to 15x into x into minus 3 minus 2 into x minus 3. So we get this is equal to 15x into x minus 3 the whole multiplied by x minus 2. So we have the polynomial 15xq minus 75x square plus 90x is equal to 15x into x minus 3 the whole multiplied by x minus 2. So this is the factorized form of this polynomial. Now we consider the second polynomial which is 6x to the power 4 minus 18xq minus 108x square. We write this in a factorized form. Now 6x square is a common factor to all the three terms of this polynomial. So 6x square into x square minus 3x minus 18. Now we factorize further the polynomial x square minus 3x minus 18. So here we have x square minus 6x plus 3x minus 18. So we get this is further equal to 6x square into x into x minus 60 whole plus 3 into x minus 60 whole. So we have this is equal to 6x square into x minus 6 the whole multiplied by x plus 3. Thus we get the polynomial that is the second polynomial 6x to the power 4 minus 18xq minus 108x square is equal to 6x square into x minus 6 the whole multiplied by x plus 3. So this is the factorized form of this polynomial. Now we will find the LCM of the two polynomials. In the LCM we would include the factors that are common to the given polynomials. Like when you consider the factorized forms of the two polynomials we find that no factor is common to both these polynomials except x. So we write here x and now we find out the LCM of the numerical factors which is 6 and 15. Now 3 2 times is 6 and 3 5 times is 15 2 1 times is 2 and 5 1 times is 5. So we have the LCM of 15 and 6 is equal to 3 into 2 into 5 which is equal to 30. So we write here 30 with x. Now next in the LCM we will also include the remaining factors that are not common to the two polynomials which is x since we have taken just one x one x was left in this polynomial then this multiplied by x minus 3 multiplied by x minus 2 multiplied by x minus 6 multiplied by x plus 3. So we get this is equal to 30 x square multiplied by x minus 3 multiplied by x minus 2 multiplied by x minus 6 multiplied by x plus 3. Thus we have got the LCM of the two given polynomials. So final answer is 30 x square multiplied by x minus 3 multiplied by x minus 2 multiplied by x minus 6 and this whole multiplied by x plus 3. This is our final answer. This completes the session. Hope you have understood the solution of this session.