 Hello and welcome to the session. Let us discuss the following question. It says, find the GCD and LCM of the following polynomials. So let's now move on to the solution and let px be the polynomial 24 into x cubed plus 9x squared plus 20x. Now we have to find the GCD and the LCM of these two polynomials. GCD is the product of all the common factors of px and the polynomial qx raised to the least exponent. Now we need to find irreducible factors of this. That means we need to factorize this polynomial. Now 24 can be written as 8 into 3. Now from here we take x common. So we have x into x squared plus 9x plus 20. Now 8 can be written as 2 to the power 3 into 3 into x into. Now here we need to factorize this quadratic equation. So we have x squared plus 5x plus 4x plus 20x common from the first two terms here. We have x into x plus 5 taking plus 4 common from the last two terms here 4 into x plus 5, 2 cubed into 3 into x into. Taking x plus 5 common we have x plus 5 into x plus 4. So we have factorized the polynomial bx as 2 cubed into 3 into x into x plus 5 into x plus 4. Now we will factorize the polynomial qx which is 28 into x to the power 4 plus x cubed minus 12x squared. It can be written as 7 into 4 and from here taking x squared common we have x squared into x squared plus x minus 12. 7 into 4 can be written as 2 squared into x squared into. Now we need to factorize this quadratic equation. We have x squared plus 4x minus 3x minus 12. We have 7 into 2 squared into x squared. Now taking x common from the first two terms we have x into x plus 4 taking minus 3 common from the last two terms we have minus 3 into x plus 4. Again we have 7 into 2 squared into x squared taking x plus 4 common we have x plus 4 into x minus 3. Here 7 into 2 squared into x squared into x plus 4 x minus 3. Reducible factors of px and qx raised to the least exponent. Now we see that x4 is one of the common irreducible factor. Then x to the power 1 is one of the common factor. 2 to the power 2 is the common factor. The product of all common irreducible factors of both the polynomials raised to the least exponent. Now here least is 2 to the power 2. 2 to the power 3 is greater than 2 to the power 2. And here x to the power 1 is less than x to the power 2. So this is 2 squared into x into x plus 4. Now this is equal to 2 squared is 4, 4x into x plus 4. Now we have to find LCM. LCM is the product of all irreducible factors. LCM is the product of all irreducible factors of both the polynomials raised to the highest exponent. Now here factors of px are 2 to the power 3 into 3 into x into x plus 5 into x plus 4. So it is 2 to the power 3 into 3 into since we have to take the highest exponent. So we need to take x square. Here one of the factor of the polynomial qx is 7. So we need to multiply it with 7 also. Then we have x plus 4. Then we have x plus 5 and then we have x minus 3. So this is equal to 168 into x square x minus 3 into x plus 4 into x plus 5. Hence the GCD of the polynomials is 4x into x plus 4 and the LCM is 168 into x square to x minus 3 into x plus 4 into x plus 5. So this completes the question and the session. Bye for now. Take care. Have a good day.