5Question: A pharmacologist models drug concentration over time using a cubic polynomial $ g(x) $ such that $ g(1) = 3 $, $ g(2) = -1 $, $ g(3) = 4 $, and $ g(4) = 10 $. Find $ g(0) $.

["Finding $ g(0) $: A Pharmacologist’s Model Using Cubic Interpolation", "When modeling complex biological processes like drug concentration over time, pharmacologists often rely on polynomial interpolation to fit observed data points. In this article, we explore a cubic polynomial $ g(x) $ that models drug concentration at time $ x $, satisfying the conditions:", "$$\ng(1) = 3, \quad g(2) = -1, \quad g(3) = 4, \quad g(4) = 10\n$$", "We are tasked with determining $ g(0) $, the predicted drug concentration at time $ x = 0 $.", "---", "### Step 1: General Form of the Cubic Polynomial", "Since $ g(x) $ is a cubic, we write:", "$$\ng(x) = ax^3 + bx^2 + cx + d\n$$", "We substitute the known values to form a system of equations:", "1. $ g(1) = a(1)^3 + b(1)^2 + c(1) + d = a + b + c + d = 3 $\n2. $ g(2) = a(8) + b(4) + c(2) + d = 8a + 4b + 2c + d = -1 $\n3. $ g(3) = a(27) + b(9) + c(3) + d = 27a + 9b + 3c + d = 4 $\n4. $ g(4) = a(64) + b(16) + c(4) + d = 64a + 16b + 4c + d = 10 $", "This yields the system:", "$$\n\begin{aligned}\n(1) &\quad a + b + c + d = 3 \\n(2) &\quad 8a + 4b + 2c + d = -1 \\n(3) &\quad 27a + 9b + 3c + d = 4 \\n(4) &\quad 64a + 16b + 4c + d = 10 \\n\end{aligned}\n$$", "---", "### Step 2: Solving the System of Equations", "We subtract equations to eliminate $ d $:", "- Subtract (1) from (2):\n $$\n (8a + 4b + 2c + d) - (a + b + c + d) = -1 - 3 \Rightarrow 7a + 3b + c = -4 \quad \ ext{(5)}\n $$", "- Subtract (2) from (3):\n $$\n (27a + 9b + 3c + d) - (8a + 4b + 2c + d) = 4 - (-1) \Rightarrow 19a + 5b + c = 5 \quad \ ext{(6)}\n $$", "- Subtract (3) from (4):\n $$\n (64a + 16b + 4c + d) - (27a + 9b + 3c + d) = 10 - 4 \Rightarrow 37a + 7b + c = 6 \quad \ ext{(7)}\n $$", "Now subtract (5) from (6):", "$$\n(19a + 5b + c) - (7a + 3b + c) = 5 - (-4) \Rightarrow 12a + 2b = 9 \Rightarrow 6a + b = \frac{9}{2} \quad \ ext{(8)}\n$$", "Subtract (6) from (7):", "$$\n(37a + 7b + c) - (19a + 5b + c) = 6 - 5 \Rightarrow 18a + 2b = 1 \Rightarrow 9a + b = \frac{1}{2} \quad \ ext{(9)}\n$$", "Now subtract (8) from (9):", "$$\n(9a + b) - (6a + b) = \frac{1}{2} - \frac{9}{2} \Rightarrow 3a = -4 \Rightarrow a = -\frac{4}{3}\n$$", "Substitute $ a = -\frac{4}{3} $ into (8):", "$$\n6\left(-\frac{4}{3}\right) + b = \frac{9}{2} \Rightarrow -8 + b = \frac{9}{2} \Rightarrow b = \frac{9}{2} + 8 = \frac{25}{2}\n$$", "Now use (5) to solve for $ c $:", "$$\n7a + 3b + c = -4 \Rightarrow 7\left(-\frac{4}{3}\right) + 3\left(\frac{25}{2}\right) + c = -4\n$$", "$$\n-\frac{28}{3} + \frac{75}{2} + c = -4\n$$", "Find common denominator (6):", "$$\n-\frac{56}{6} + \frac{225}{6} + c = -4 \Rightarrow \frac{169}{6} + c = -4 \Rightarrow c = -4 - \frac{169}{6} = -\frac{205}{6}\n$$", "Now use (1) to solve for $ d $:", "$$\na + b + c + d = 3 \Rightarrow -\frac{4}{3} + \frac{25}{2} - \frac{205}{6} + d = 3\n$$", "Common denominator is 6:", "$$\n-\frac{8}{6} + \frac{75}{6} - \frac{205}{6} + d = 3 \Rightarrow \frac{-138}{6} + d = 3 \Rightarrow -23 + d = 3 \Rightarrow d = 26\n$$", "---", "### Step 3: Evaluate $ g(0) $", "Since $ g(x) = ax^3 + bx^2 + cx + d $, then:", "$$\ng(0) = d = \boxed{26}\n$$", "---", "### Conclusion", "By fitting a cubic polynomial through key time points, pharmacologists can accurately model drug concentration dynamics. Using systematic elimination and substitution, we found that the concentration at time $ x = 0 $ is precisely $ \boxed{26} $. This approach exemplifies the power of cubic interpolation in predicting biological behavior from discrete measurements.", "For researchers and students alike, such models bridge data and insight—turning observations into actionable knowledge.", "---", "Keywords: drug concentration, cubic polynomial, pharmacology modeling, interpolation, $ g(x) = ax^3 + bx^2 + cx + d $, $ g(0) $, polynomial fitting, scientific computation."]