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
You're on a roll — Step 1: Apply the Factor Theorem for the factor (x-1). The Factor Theorem states that if (x-a) is a factor of a polynomial P(x), then P(a) = 0. Given P(x) = 2x^3 + px^2 - x + q and (x-1) is a factor, we set x=1: P(1) = 2(1)^3 + p(1)^2 - (1) + q = 0 2 + p - 1 + q = 0 p + q + 1 = 0 This gives our first equation: p + q = -1 (*) Step 2: Apply the Factor Theorem for the factor (x+2). Given (x+2) is a factor, we set x=-2: P(-2) = 2(-2)^3 + p(-2)^2 - (-2) + q = 0 2(-8) + p(4) + 2 + q = 0 -16 + 4p + 2 + q = 0 4p + q - 14 = 0 This gives our second equation: 4p + q = 14 (**) Step 3: Solve the system of linear equations for p and q. We have the system: 1) p + q = -1 2) 4p + q = 14 Subtract equation (1) from equation (2): (4p + q) - (p + q) = 14 - (-1) 4p + q - p - q = 14 + 1 3p = 15 p = (15)/(3) p = 5 Step 4: Substitute the value of p into equation (1) to find q. Using p + q = -1: 5 + q = -1 q = -1 - 5 q = -6 The values are p = 5 and q = -6. Got more? Send 'em!