Question: An anthropologist models cultural adaptation index $ A(x) $ over time $ x $ (in decades) using a cubic polynomial satisfying $ A(1) = 2 $, $ A(2) = -4 $, $ A(3) = 6 $, and $ A(4) = -8 $. Find the remainder when $ A(x)^3 + 2 $ is divided by $ x - 5 $.

["(Note: The question combines a real-world modeling scenario with a polynomial remainder problem, grounded in Intermediate Algebra. The framework is inspired by anthropological data fitting, yet the core is a standard remainder theorem problem with polynomial interpolation.)", "---", "Question: An anthropologist models cultural adaptation index $ A(x) $ over time $ x $ (in decades) using a cubic polynomial satisfying $ A(1) = 2 $, $ A(2) = -4 $, $ A(3) = 6 $, and $ A(4) = -8 $. Find the remainder when $ A(x)^3 + 2 $ is divided by $ x - 5 $.", "---", "Answer:\nTo solve this, we apply the Remainder Theorem, which states that the remainder when a polynomial $ f(x) $ is divided by $ x - c $ is $ f(c) $. Here, we are asked to find the remainder when $ A(x)^3 + 2 $ is divided by $ x - 5 $, so the remainder is:", "$$\nR = A(5)^3 + 2\n$$", "Thus, our task reduces to determining $ A(5) $, given that $ A(x) $ is a cubic polynomial passing through the points:\n$$\n(1, 2),\quad (2, -4),\quad (3, 6),\quad (4, -8)\n$$", "Let $ A(x) = ax^3 + bx^2 + cx + d $. We substitute the four values into this expression to form a system of equations:", "1. $ A(1) = a(1)^3 + b(1)^2 + c(1) + d = a + b + c + d = 2 $\n2. $ A(2) = 8a + 4b + 2c + d = -4 $\n3. $ A(3) = 27a + 9b + 3c + d = 6 $\n4. $ A(4) = 64a + 16b + 4c + d = -8 $", "We solve this system step by step.", "Step 1: Subtract equations to eliminate $ d $\nEquation (2) – Equation (1):\n$$\n(8a + 4b + 2c + d) - (a + b + c + d) = -4 - 2 \Rightarrow 7a + 3b + c = -6 \quad \ ext{(Eq A)}\n$$", "Equation (3) – Equation (2):\n$$\n(27a + 9b + 3c + d) - (8a + 4b + 2c + d) = 6 - (-4) \Rightarrow 19a + 5b + c = 10 \quad \ ext{(Eq B)}\n$$", "Equation (4) – Equation (3):\n$$\n(64a + 16b + 4c + d) - (27a + 9b + 3c + d) = -8 - 6 \Rightarrow 37a + 7b + c = -14 \quad \ ext{(Eq C)}\n$$", "Step 2: Eliminate $ c $\nEq B – Eq A:\n$$\n(19a + 5b + c) - (7a + 3b + c) = 10 - (-6) \Rightarrow 12a + 2b = 16 \Rightarrow 6a + b = 8 \quad \ ext{(Eq D)}\n$$", "Eq C – Eq B:\n$$\n(37a + 7b + c) - (19a + 5b + c) = -14 - 10 \Rightarrow 18a + 2b = -24 \Rightarrow 9a + b = -12 \quad \ ext{(Eq E)}\n$$", "Step 3: Solve for $ a $ and $ b $\nEq E – Eq D:\n$$\n(9a + b) - (6a + b) = -12 - 8 \Rightarrow 3a = -20 \Rightarrow a = -\frac{20}{3}\n$$", "Substitute into Eq D:\n$$\n6\left(-\frac{20}{3}\right) + b = 8 \Rightarrow -40 + b = 8 \Rightarrow b = 48\n$$", "Step 4: Solve for $ c $ using Eq A\n$$\n7a + 3b + c = -6 \Rightarrow 7\left(-\frac{20}{3}\right) + 3(48) + c = -6 \Rightarrow -\frac{140}{3} + 144 + c = -6\n$$", "Convert 144 to thirds: $ 144 = \frac{432}{3} $, so:\n$$\n\left(-\frac{140}{3} + \frac{432}{3}\right) + c = -6 \Rightarrow \frac{292}{3} + c = -6 \Rightarrow c = -6 - \frac{292}{3} = -\frac{18}{3} - \frac{292}{3} = -\frac{310}{3}\n$$", "Step 5: Solve for $ d $ using Eq (1):\n$$\na + b + c + d = 2 \Rightarrow -\frac{20}{3} + 48 - \frac{310}{3} + d = 2\n$$", "Convert 48 to thirds: $ \frac{144}{3} $, so:\n$$\n\left(-\frac{20}{3} - \frac{310}{3}\right) + 48 + d = 2 \Rightarrow -\frac{330}{3} + 48 + d = 2 \Rightarrow -110 + 48 + d = 2\n\Rightarrow -62 + d = 2 \Rightarrow d = 64\n$$", "Thus,\n$$\nA(x) = -\frac{20}{3}x^3 + 48x^2 - \frac{310}{3}x + 64\n$$", "Now compute $ A(5) $:", "$$\nA(5) = -\frac{20}{3}(125) + 48(25) - \frac{310}{3}(5) + 64\n= -\frac{2500}{3} + 1200 - \frac{1550}{3} + 64\n$$", "Combine terms with denominator 3:\n$$\n\left(-\frac{2500 + 1550}{3}\right) + 1200 + 64 = -\frac{4050}{3} + 1264 = -1350 + 1264 = -86\n$$", "So $ A(5) = -86 $, and therefore:", "$$\nR = A(5)^3 + 2 = (-86)^3 + 2\n$$", "Compute $ 86^3 $:\n$ 86^2 = 7396 $, then $ 86^3 = 86 \ imes 7396 $", "Break it down:\n$ 80 \ imes 7396 = 591680 $\n$ 6 \ imes 7396 = 44376 $\nTotal: $ 591680 + 44376 = 636,056 $", "Thus, $ (-86)^3 = -636056 $, so:", "$$\nR = -636056 + 2 = -636054\n$$", "Therefore, the remainder when $ A(x)^3 + 2 $ is divided by $ x - 5 $ is $ \boxed{-636054} $.", "This result reflects how anthropological models, though symbolic of cultural change, still obey rigorous algebraic behavior—especially when using interpolation to predict future trends.", "---", "Keywords: cultural adaptation index, anthropologist, cubic polynomial, finite differences, Remainder Theorem, $ A(x)^3 + 2 $, $ x - 5 $ divisor, Intermediate Algebra, polynomial interpolation, Remainder, $ A(5) $, time series modeling."]