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 + 8 = 0 8A + 4B = -8 2A + B = -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 1 into Equation 2, we get: 2(B + 5) + B = -2 2B + 10 + B = -2 3B = -12 B = -4

6. Substituting B = -4 into Equation 1, we get: A = -4 + 5 A = 1

So, the values of A and B are 1 and -4 respectively.