 Hello and welcome to the session. Let's work out the following problem. It says find the gcd and lcm of the following polynomials Let's now move on to the solution We have to find the gcd that is greatest common divisor and lcm of these two polynomials lcm is the least common multiple now let px be the polynomial 18 into xq minus 9x square plus 8x Now 18 can be written as 2 into 9 taking x common We'll first factorize this polynomial into irreducible factors So we have x into x square minus 9x plus 8 now again 9 can be written as 3 into 3 Into x now we'll factorize this quadratic equation. So we have x square minus 8x minus x plus 8 Now again 2 into 3 into 3 can be written as 3 to the power 2 into x now here Taking x common from the first two terms. We have x into x minus 8 taking minus one common From last two terms. We have minus 1 into x minus 8 Now again 2 Into 3 to the power 2 into x Taking x minus 8 common. We have x minus 8 Into x minus 1. So we have factorized this polynomial into irreducible factors now we find the irreducible factors of the polynomial qx which is 24 into x square minus 3 x plus 2 now 24 can be written as 8 into 3 and This is a quadratic equation in x and now we'll factorize this equation So we have x square minus 2x minus x plus 2 Now 2 can it can be written as 2 into 2 into 2 into 3 Into taking x common from the first two terms. We have x into x minus 2 Taking minus one common from the last two terms. We have minus 1 into x minus 2 Now 2 into 2 into 2 can be written as 2 to the power 3 into 3 Taking x minus 2 common here. We have x minus 2 into x minus 1 So this is 2 to the power 3 into 3 into x minus 2 into x minus 1 now we find the gcd That is the greatest common multiple greatest common divisor and gcd is the product of all common irreducible factors now here we see that One of the two is common. So we have 2 into here one of the three is common so 2 into 3 and Here we have x minus 1 as Common factor. So we have into x minus 1 so gcd is 2 into 3 into x minus 1 that is 6 into x minus 1 now we find LCM LCM is the least common multiple and LCM is the product of all irreducible factors of px and qx raised to the highest exponent now here 2 has the highest exponent as 3 So in LCM we'll have 2 to the power 3 and here 3 to the power 2 2 is the highest exponent of 3 here. So we have 3 to the power 2 into x minus 8 Into x minus 1 into x minus 2 right so this is 2 to the power 3 is 8 3 to the power 2 is 9 into x minus 8 into x minus 1 into x minus 2 now 8 into 9 is 72 Also, we have one of the factor as x. So we'll be having x also and you have 72 into x Into x minus 8 into x minus 1 into x minus 2 So gcd is 6 into x minus 1 and LCM is 72 x into x minus 8 into x minus 1 into x minus 2 So this completes the question and the session. Bye for now take care. Have a good day