 Hello students, let's work out the following problem. It says if x-2 into x plus 3 is the hcf of the polynomials px which is given by x square minus 3x plus 2 into 2x square plus 7x plus a and qx which is given by x square plus 4x plus 3 into 3x square minus 7x plus b, find the values of a and b. Let's now move on to the solution. We are given the polynomial px as x square minus 3x plus 2 into 2x square plus 7x plus e and qx is given by x square plus 4x plus 3 into 3x square minus 7x plus b and we are given that the polynomial x minus 2 into x plus 3 is the hcf of the polynomials px and qx and we know that whenever one polynomial is the hcf of another polynomial that means the first polynomial is the factor of the second polynomial. So from this we can say therefore x minus 2 into x plus 3 is the factor of polynomials px and qx. So this implies p at x is equal to is 0 that is p2 is 0 and similarly p at x is equal to minus 3 is 0 and also q at x is equal to 2 is 0 and q at x is equal to minus 3 is 0. Now using this we will be finding a and b and this we say by using factor theorem. You must mention the theorem which you are using. Now we know that p2 is 0 so we will find p2 and we will equate it to 0. Now px is x square minus 3x plus 2 into 2x square plus 7x plus a so we will put 2 in place of x. So we have 2 square minus 3 into 2 plus 2 into 2x square that is 2 into 2 square plus 7x that is 7 into 2 plus a is equal to 0. So from this we have 4 minus 6 plus 2 into 8 plus 14 plus a is equal to 0. Now 6 minus 6 is 0 so 0 is equal to 0 which is true. Now we have p of minus 3 is 0 so we will put minus 3 in place of x so we have minus 3 square minus 3 into minus 3 plus 2 into 2x square that is 2 into minus 3 square plus 7x that is 7 into minus 3 plus a is equal to 0 and this implies minus 3 square is 9 minus 3 into minus 3 is again 9 plus 2 into 2 into minus 18 7 into minus 3 is minus 21 plus a is equal to 0 and this implies 20 into 18 minus 21 is minus 3 plus a is equal to 0. Now dividing both sides by 20 we have minus 3 plus a is equal to 0 and this implies a is equal to 3. Now we have to find b. Now we know that q at the point 2 is equal to 0 so now we will put x is equal to 2 in the polynomial qx so q2 is given by 2 square that is x square plus 4x that is 4 into 2 plus 3 into 3x square that is 3 into 2 square minus 7x that is 7 into 2 plus b and this is equal to 0 so this implies 4 plus 8 into 3 into 4 is 12 minus 14 plus b is equal to 0 and this implies 7 plus 8 is 15 into minus 2 plus b is equal to 0 and dividing both sides by 15 we have minus 2 plus b is equal to 0 and this implies b is equal to 2. So we have obtained the values of a and b. Now since we have obtained the values of a and b we don't need to find the polynomial q at the point minus 3 but by using the factor theorem we already know that q at x is equal to minus 3 is 0 hence a is equal to 3 and b is equal to 2 is the required solution. That's all for this session. Bye for now. Take care.