This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Apply the Remainder Theorem for the first condition. According to the Remainder Theorem, if f(x) is divided by x-a, the remainder is f(a). For division by x+1, we have a = -1. The remainder is given as 5. So, f(-1) = 5. Substitute x = -1 into f(x) = 2x^3 + px^2 + qx + 1: 2(-1)^3 + p(-1)^2 + q(-1) + 1 = 5 -2 + p - q + 1 = 5 p - q - 1 = 5 p - q = 6 (*) Step 2: Apply the Remainder Theorem for the second condition. For division by x-2, we have a = 2. The remainder is given as 5. So, f(2) = 5. Substitute x = 2 into f(x) = 2x^3 + px^2 + qx + 1: 2(2)^3 + p(2)^2 + q(2) + 1 = 5 2(8) + 4p + 2q + 1 = 5 16 + 4p + 2q + 1 = 5 4p + 2q + 17 = 5 4p + 2q = 5 - 17 4p + 2q = -12 Divide by 2: 2p + q = -6 (**) Step 3: Solve the system of linear equations. We have two equations: 1. p - q = 6 2. 2p + q = -6 Add equation (1) and equation (2) to eliminate q: (p - q) + (2p + q) = 6 + (-6) 3p = 0 p = 0 Step 4: Substitute the value of p back into one of the equations to find q. Using equation (1): p - q = 6 0 - q = 6 -q = 6 q = -6 The values are p=0 and q=-6. The values of p and q are p=0, q=-6. That's 2 down. 3 left today — send the next one.