5Question: A climatologist models the annual carbon emissions (in gigatons) as a cubic polynomial $ g(x) $, where $ x $ represents years since 2000. Given $ g(1) = 4 $, $ g(2) = 11 $, $ g(3) = 26 $, and $ g(4) = 57 $, find $ g(x) $.

["# Finding the Cubic Polynomial $ g(x) $: A Climatologist Models Carbon Emissions with Precision", "Understanding environmental trends is crucial in the fight against climate change. One key task is modeling annual carbon emissions, which can be expressed mathematically as polynomials. In this article, we explore how a climatologist determined a cubic polynomial $ g(x) $ that accurately represents annual carbon emissions in gigatons since the year 2000. Using data points from $ x = 1 $ to $ x = 4 $, we’ll derive the explicit form of $ g(x) $.", "## The Nature of Polynomial Modeling in Climate Science", "Modeling greenhouse gas emissions allows scientists to project future climate scenarios and evaluate policy impacts. When data fits a polynomial trend, cubic functions—being flexible and capable of capturing non-linear growth—are often appropriate. Since emissions growth accelerates over time, a cubic function matches both short-term and long-term behavior well.", "## Given Data Points", "We are given:\n- $ g(1) = 4 $\n- $ g(2) = 11 $\n- $ g(3) = 26 $\n- $ g(4) = 57 $", "Assume the cubic model has the form:\n$$\ng(x) = ax^3 + bx^2 + cx + d\n$$", "Our goal is to determine the coefficients $ a $, $ b $, $ c $, $ d $.", "## Setting Up the Equations", "Substitute each $ x $ and $ g(x) $ into $ g(x) $ to form a system of equations:", "1. $ g(1) = 4 $:\n$$\na(1)^3 + b(1)^2 + c(1) + d = 4 \Rightarrow a + b + c + d = 4 \quad \ ext{(Eq. 1)}\n$$", "2. $ g(2) = 11 $:\n$$\n8a + 4b + 2c + d = 11 \quad \ ext{(Eq. 2)}\n$$", "3. $ g(3) = 26 $:\n$$\n27a + 9b + 3c + d = 26 \quad \ ext{(Eq. 3)}\n$$", "4. $ g(4) = 57 $:\n$$\n64a + 16b + 4c + d = 57 \quad \ ext{(Eq. 4)}\n$$", "## Solving the System of Equations", "We solve the system step by step.", "Step 1: Eliminate $ d $\nSubtract Eq. 1 from Eq. 2:\n$$\n(8a + 4b + 2c + d) - (a + b + c + d) = 11 - 4\n\Rightarrow 7a + 3b + c = 7 \quad \ ext{(Eq. 5)}\n$$", "Subtract Eq. 2 from Eq. 3:\n$$\n(27a + 9b + 3c + d) - (8a + 4b + 2c + d) = 26 - 11\n\Rightarrow 19a + 5b + c = 15 \quad \ ext{(Eq. 6)}\n$$", "Subtract Eq. 3 from Eq. 4:\n$$\n(64a + 16b + 4c + d) - (27a + 9b + 3c + d) = 57 - 26\n\Rightarrow 37a + 7b + c = 31 \quad \ ext{(Eq. 7)}\n$$", "Step 2: Eliminate $ c $\nSubtract Eq. 5 from Eq. 6:\n$$\n(19a + 5b + c) - (7a + 3b + c) = 15 - 7\n\Rightarrow 12a + 2b = 8 \Rightarrow 6a + b = 4 \quad \ ext{(Eq. 8)}\n$$", "Subtract Eq. 6 from Eq. 7:\n$$\n(37a + 7b + c) - (19a + 5b + c) = 31 - 15\n\Rightarrow 18a + 2b = 16 \Rightarrow 9a + b = 8 \quad \ ext{(Eq. 9)}\n$$", "Step 3: Solve for $ a $ and $ b $\nSubtract Eq. 8 from Eq. 9:\n$$\n(9a + b) - (6a + b) = 8 - 4 \Rightarrow 3a = 4 \Rightarrow a = \frac{4}{3}\n$$", "Substitute $ a = \frac{4}{3} $ into Eq. 8:\n$$\n6\left(\frac{4}{3}\right) + b = 4 \Rightarrow 8 + b = 4 \Rightarrow b = -4\n$$", "Step 4: Solve for $ c $\nUse Eq. 5:\n$$\n7\left(\frac{4}{3}\right) + 3(-4) + c = 7 \Rightarrow \frac{28}{3} - 12 + c = 7\n\Rightarrow \frac{28 - 36}{3} + c = 7 \Rightarrow -\frac{8}{3} + c = 7 \Rightarrow c = 7 + \frac{8}{3} = \frac{29}{3}\n$$", "Step 5: Solve for $ d $\nUse Eq. 1:\n$$\n\frac{4}{3} - 4 + \frac{29}{3} + d = 4\n\Rightarrow \left(\frac{4 + 29}{3}\right) - 4 + d = 4\n\Rightarrow \frac{33}{3} - 4 + d = 4 \Rightarrow 11 - 4 + d = 4 \Rightarrow 7 + d = 4 \Rightarrow d = -3\n$$", "## Final Polynomial", "Substituting the coefficients:\n$$\ng(x) = \frac{4}{3}x^3 - 4x^2 + \frac{29}{3}x - 3\n$$", "To express it neatly, factor:\n$$\ng(x) = \frac{1}{3}(4x^3 - 12x^2 + 29x - 9)\n$$", "## Verification", "Let’s verify with $ x = 1 $:\n$$\n\frac{1}{3}(4 - 12 + 29 - 9) = \frac{1}{3}(12) = 4 \quad \ ext{✓}\n$$\n$ x = 2 $:\n$$\n\frac{1}{3}(32 - 48 + 58 - 9) = \frac{1}{3}(33) = 11 \quad \ ext{✓}\n$$\n$ x = 3 $:\n$$\n\frac{1}{3}(108 - 108 + 87 - 9) = \frac{1}{3}(78) = 26 \quad \ ext{✓}\n$$\n$ x = 4 $:\n$$\n\frac{1}{3}(256 - 192 + 116 - 9) = \frac{1}{3}(171) = 57 \quad \ ext{✓}\n$$", "All data points are satisfied.", "## Conclusion", "Through systematic polynomial interpolation using real-world climate data, we derived the cubic function\n$$\ng(x) = \frac{4}{3}x^3 - 4x^2 + \frac{29}{3}x - 3\n$$\nthat models annual carbon emissions since 2000. This precise mathematical representation enables scientists to project future emissions trends, inform policy decisions, and support global climate action with greater confidence.", "For climatologists and environmental researchers, transforming raw data into predictive mathematical models is essential—this cubic example exemplifies how polynomial analysis drives impactful environmental science."]