 Hello and welcome to the session. Let's work out the following problem. It says if x minus 3 into x plus 2 is the GCD of the polynomials px and qx, find the values of a and b. So let's now move on to the solution. Polynomial px is x square minus 2x minus 3 into 2x square plus ax minus 2. Now we'll factorize this x square minus 3x plus x minus 3 into 2x square plus ax minus 2. Now taking x common from the first two terms, we have x into x minus 3 taking plus 1 common from the last two terms we have 1 into x minus 3 into 2x square plus ax minus 2. Now taking x minus 3 common, we have x minus 3 into x plus 1 into 2x square plus ax minus 2. Now we are given that qx is x square plus x minus 2 into 3x square plus bx minus 3. Now we'll factorize this. So we have x square plus 2x minus 1x minus 2 into 3x square plus bx minus 3. Now taking x common from the first two terms, we have x into x plus 2 taking minus 1 common from the last two terms, we have minus 1 into x plus 2 to 3x square plus bx minus 3. Now taking x plus 2 common, we have x plus 2 into x minus 1 into 3x square plus bx minus 3. Now we are given that x minus 3 into x plus 2 is the GCD of the polynomials px and qx. Right? So that means x minus 3 and x plus 2 is the factor of the polynomials px and qx since it's a GCD. So it's a factor of the polynomials px and qx. Therefore p of 3 is 0 p of minus 2 is 0 q of 3 is 0 and q of minus 2 is 0. Now we have p of minus 2 is equal to 0. So p of minus 2 will be minus 2 minus 3 minus 2 plus 1 since px is this. So we'll put x is equal to minus 2 here. Into 2 into minus 2 square plus a into minus 2 minus 2 is equal to 0. So this implies minus 5 into minus 1 into 2 into 4 is 8 minus 2 a minus 2 is equal to 0. Now we have 5 into 8 minus 2 is 6 minus 2 a is equal to 0 dividing both sides by 5. We have 6 minus 2 a is equal to 0. So this implies 6 is equal to 2 a and this implies a is equal to 3. Now we also have that q of 3 is 0. So now we'll put x is equal to 3 in the polynomial qx. So we have q of 3 is 3 minus 1 into 3 plus 2 into 3 into 3 square plus b into 3 minus 3 is equal to 0. So this implies 2 into 5 into 12 plus 3b minus 3 is equal to 0. We have 10 into 12 minus 3 is 9 plus 3b is equal to 0. So this implies 9 plus 3b is equal to 0 dividing both sides by 10. We have this. Now here we have 3 square since we are putting x is equal to 3. It is 3 square. So it is 9 into 3. That is 27. 27 minus 3 is 24. So here also we have 24. So this implies 3b is equal to minus 24. So this implies b is equal to minus 8. So we have a is equal to 3 and b is equal to minus 8 is the answer. So this completes the question and the session. Bye for now. Take care. Have a good day.