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.
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Step 1: Apply the Remainder Theorem to set up a system of equations. The Remainder Theorem states that if a polynomial f(x) is divided by (x-a), the remainder is f(a). Given f(x) = (p-1)x^3 + px^2 + qx + r. When f(x) is divided by (x-1), the remainder is 4: f(1) = (p-1)(1)^3 + p(1)^2 + q(1) + r = 4 p-1 + p + q + r = 4 2p + q + r = 5 (Equation 1) When f(x) is divided by (x+1), the remainder is 0: f(-1) = (p-1)(-1)^3 + p(-1)^2 + q(-1) + r = 0 -(p-1) + p - q + r = 0 -p + 1 + p - q + r = 0 1 - q + r = 0 q - r = 1 (Equation 2) When f(x) is divided by (x+2), the remainder is 5: f(-2) = (p-1)(-2)^3 + p(-2)^2 + q(-2) + r = 5 -8(p-1) + 4p - 2q + r = 5 -8p + 8 + 4p - 2q + r = 5 -4p - 2q + r = -3 (Equation 3) Step 2: Solve the system of linear equations for p, q, r. From Equation 2, express q in terms of r: q = r + 1 Substitute q = r+1 into Equation 1: 2p + (r+1) + r = 5 2p + 2r + 1 = 5 2p + 2r = 4 p + r = 2 (Equation 4) Substitute q = r+1 into Equation 3: -4p - 2(r+1) + r = -3 -4p - 2r - 2 + r = -3 -4p - r - 2 = -3 -4p - r = -1 (Equation 5) Add Equation 4 and Equation 5: (p + r) + (-4p - r) = 2 + (-1) -3p = 1 p = -(1)/(3) Substitute p = -(1)/(3) into Equation 4: -(1)/(3) + r = 2 $r = 2 + (1)/(3) =