Question:** Find the cubic polynomial \( f(x) \) such that \( f(-1) = 4 \), \( f(0) = 1 \), \( f(2) = 10 \), and \( f(3) = 19 \).

["Find the Cubic Polynomial ( f(x) ) That Satisfies Given Points | A Step-by-Step Guide", "When tasked with finding a cubic polynomial ( f(x) ) defined by four known points, the challenge becomes insightful and practical. In this article, we’ll walk through how to determine the unique cubic polynomial ( f(x) = ax^3 + bx^2 + cx + d ) that satisfies the conditions:\n( f(-1) = 4 ),\n( f(0) = 1 ),\n( f(2) = 10 ),\n( f(3) = 19 ).", "This problem is a classic application of interpolation with polynomials and is valuable for students, engineers, and data analysts working with predictive models.", "---", "### What Is a Cubic Polynomial?", "A cubic polynomial has the general form:\n[\nf(x) = ax^3 + bx^2 + cx + d\n]\nwhere ( a, b, c, d ) are real coefficients. Since we have four distinct points, there’s exactly one cubic polynomial that passes through them—assuming the ( x )-values are distinct, which they are in this case.", "---", "### Step 1: Use Given Points to Set Up Equations", "We plug each ( x )-value into ( f(x) ) to generate equations from the conditions.", "Using ( f(0) = 1 ):\n[\na(0)^3 + b(0)^2 + c(0) + d = 1 \Rightarrow d = 1\n]\nSo, ( d = 1 ).", "Now rewrite the polynomial:\n[\nf(x) = ax^3 + bx^2 + cx + 1\n]", "Next, use ( f(-1) = 4 ):\n[\na(-1)^3 + b(-1)^2 + c(-1) + 1 = 4 \\n- a + b - c + 1 = 4 \\n- a + b - c = 3 \quad \ ext{(Equation 1)}\n]", "Use ( f(2) = 10 ):\n[\na(8) + b(4) + c(2) + 1 = 10 \\n8a + 4b + 2c = 9 \quad \ ext{(Equation 2)}\n]", "Use ( f(3) = 19 ):\n[\n27a + 9b + 3c + 1 = 19 \\n27a + 9b + 3c = 18 \quad \ ext{(Equation 3)}\n]", "---", "### Step 2: Solve the System of Equations", "We now solve the system:\n1. ( -a + b - c = 3 )\n2. ( 8a + 4b + 2c = 9 )\n3. ( 27a + 9b + 3c = 18 )", "Start by eliminating variables. Multiply Equation 1 by 2:\n[\n-2a + 2b - 2c = 6 \quad \ ext{(1a)}\n]", "Add (1a) and Equation 2:\n[\n(-2a + 2b - 2c) + (8a + 4b + 2c) = 6 + 9 \\n6a + 6b = 15 \quad \Rightarrow \quad 2a + 2b = 5 \quad \ ext{(Equation 4)}\n]", "Now multiply Equation 1 by 3:\n[\n-3a + 3b - 3c = 9 \quad \ ext{(1b)}\n]", "Add (1b) to Equation 3:\n[\n(-3a + 3b - 3c) + (27a + 9b + 3c) = 9 + 18 \\n24a + 12b = 27 \quad \Rightarrow \quad 8a + 4b = 9 \quad \ ext{(Equation 5)}\n]", "Now solve Equations 4 and 5 together:\nFrom Equation 4: ( 2a + 2b = 5 \Rightarrow a + b = 2.5 ) → ( b = 2.5 - a )", "Substitute into Equation 5:\n[\n8a + 4(2.5 - a) = 9 \\n8a + 10 - 4a = 9 \\n4a = -1 \Rightarrow a = -\frac{1}{4}\n]", "Now find ( b ):\n[\nb = 2.5 - \left(-\frac{1}{4}\right) = 2.5 + 0.25 = 2.75 = \frac{11}{4}\n]", "Now substitute ( a = -\frac{1}{4} ), ( b = \frac{11}{4} ) into Equation 1:\n[\n-\left(-\frac{1}{4}\right) + \frac{11}{4} - c = 3 \\n\frac{1}{4} + \frac{11}{4} - c = 3 \\n\frac{12}{4} - c = 3 \Rightarrow 3 - c = 3 \Rightarrow c = 0\n]", "---", "### Step 3: Write Final Polynomial", "We now have:\n[\na = -\frac{1}{4}, \quad b = \frac{11}{4}, \quad c = 0, \quad d = 1\n]\nThus, the cubic polynomial is:\n[\nf(x) = -\frac{1}{4}x^3 + \frac{11}{4}x^2 + 1\n]\nOr equivalently:\n[\nf(x) = \frac{-x^3 + 11x^2 + 4}{4}\n]", "---", "### Step 4: Verify All Points", "Let’s check the key values:\n- ( f(-1) = -\frac{(-1)^3}{4} + \frac{11(-1)^2}{4} + 1 = \frac{1}{4} + \frac{11}{4} + 1 = 3 + 1 = 4 \quad ✅ )\n- ( f(0) = 1 \quad ✅ )\n- ( f(2) = -\frac{8}{4} + \frac{44}{4} + 1 = -2 + 11 + 1 = 10 \quad ✅ )\n- ( f(3) = -\frac{27}{4} + \frac{99}{4} + 1 = \frac{72}{4} + 1 = 18 + 1 = 19 \quad ✅ )", "All conditions are satisfied.", "---", "### Why This Matters: Interpolation and Polynomial Fitting", "Finding such polynomials is essential in approximation, numerical analysis, and data modeling. Given discrete observations, polynomials allow us to extrapolate, interpolate, or smooth data—useful in physics, economics, and machine learning.", "---", "Conclusion\nThe unique cubic polynomial satisfying ( f(-1) = 4 ), ( f(0) = 1 ), ( f(2) = 10 ), and ( f(3) = 19 ) is:\n[\n\boxed{f(x) = \frac{-x^3 + 11x^2 + 4}{4}}\n]", "Use this formula confidently in future interpolation tasks, and remember: four points define a cubic uniquely!", "---", "### Keywords for SEO Optimization\n- cubic polynomial\n- find polynomial through points\n- interpolation with cubic function\n- solve for coefficients\n- polynomial fitting\n- determine cubic polynomial given values\n- math problem solving cubic\n- algebraic methods for cubic equations", "---", "If you're learning polynomial interpolation, practice with similar problems to master the technique—your ability to fit models precisely will strengthen every analytical project!"]