Given that Ax3 + Bx2 + x + 6 is exactly divisible by (x + 1)(x – 2), find the value of Aand the value of B.

Sure, here is the step by step solution:

1. Given that the polynomial Ax^3 + Bx^2 + x + 6 is exactly divisible by (x + 1)(x – 2), it means that if we substitute x = -1 and x = 2 into the polynomial, it should equal to zero.

2. Substituting x = -1 into the polynomial, we get: A*(-1)^3 + B*(-1)^2 - 1 + 6 = 0 -A + B - 1 + 6 = 0 -A + B + 5 = 0 A = B + 5 ---- (equation 1)

3. Substituting x = 2 into the polynomial, we get: A*(2)^3 + B*(2)^2 + 2 + 6 = 0 8A + 4B + 2 + 6 = 0 8A + 4B + 8 = 0 2A + B + 2 = 0 B = -2A - 2 ---- (equation 2)

4. Now we have two equations (equation 1 and equation 2) and two unknowns (A and B). We can solve these equations simultaneously to find the values of A and B.

5. Substituting equation 2 into equation 1, we get: A = (-2A - 2) + 5 A = -2A + 3 3A = 3 A = 1

6. Substituting A = 1 into equation 2, we get: B = -2*1 - 2 B = -4

So, the value of A is 1 and the value of B is -4.