Solution: Let $ f(x) = ax^3 + bx^2 + cx + d $. We use the given values to form a system of equations:
["Title: Solving Cubic Equations: Building a System of Equations with Given Points", "Meta Description:\nLearn how to construct a system of equations for a cubic polynomial $ f(x) = ax^3 + bx^2 + cx + d $ using known function values. This foundational method helps solve real-world problems in engineering, physics, and data science.", "---", "### Introduction", "Working with cubic polynomials is essential in fields like data modeling, curve fitting, and optimization. When you're given specific values of $ f(x) $ at particular points, the next critical step is forming a system of equations to determine the unknown coefficients $ a, b, c, d $.", "In this article, we explore how to construct a system of equations from given data points using the general form of a cubic:", "$$\nf(x) = ax^3 + bx^2 + cx + d\n$$", "Using known values $ f(x_i) = y_i $, we derive a set of linear equations. Here’s a step-by-step solution and explanation of this essential technique in algebra.", "---", "### Step 1: Understand the Problem", "Suppose we know four distinct $ x $-values: $ x_1, x_2, x_3, x_4 $, with corresponding $ f(x_i) = y_i $. The goal is to determine the unknown coefficients $ a, b, c, d $.", "Each $ f(x_i) $ gives an equation:\n$$\ny_i = a x_i^3 + b x_i^2 + c x_i + d\n$$\nThis yields a system of four linear equations with four unknowns.", "---", "### Step 2: Formulating the System of Equations", "Using four data points $ (x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4) $, substitute into the cubic:", "$$\n\begin{cases}\na x_1^3 + b x_1^2 + c x_1 + d = y_1 \\na x_2^3 + b x_2^2 + c x_2 + d = y_2 \\na x_3^3 + b x_3^2 + c x_3 + d = y_3 \\na x_4^3 + b x_4^2 + c x_4 + d = y_4 \\n\end{cases}\n$$", "This system is linear in $ a, b, c, d $, and can be written in matrix form as:", "$$\n\begin{bmatrix}\nx_1^3 & x_1^2 & x_1 & 1 \\nx_2^3 & x_2^2 & x_2 & 1 \\nx_3^3 & x_3^2 & x_3 & 1 \\nx_4^3 & x_4^2 & x_4 & 1 \\n\end{bmatrix}\n\begin{bmatrix}\na \ b \ c \ d\n\end{bmatrix}\n=\n\begin{bmatrix}\ny_1 \ y_2 \ y_3 \ y_4\n\end{bmatrix}\n$$", "This design matrix is structured for Gaussian elimination, Cramer’s rule, or matrix inversion.", "---", "### Step 3: Solving for Coefficients", "Once the system is formed, solvability depends on the determinant of the coefficient matrix. For a non-redundant set of $ x_i $, the determinant is non-zero if $ x_1, x_2, x_3, x_4 $ are distinct, guaranteeing a unique solution.", "Using methods like substitution, elimination, or matrix algebra, we solve:", "- $ a = \dfrac{\ ext{determinant of matrix with first column replaced by } [y_1, y_2, y_3, y_4]}{\ ext{det of coefficient matrix}} $\n- Follow similarly for $ b, c, d $", "---", "### Real-World Applications", "This system of equations is not just theoretical—here’s how it applies:", "- Engineering: Fit cubic curves to stress vs. displacement data\n- Economics: Model nonlinear cost or demand functions\n- Data Science: Perform polynomial regression on scattered data\n- Physics: Analyze motion trajectories requiring cubic displacement modeling", "---", "### Summary", "Given a cubic function $ f(x) = ax^3 + bx^2 + cx + d $, forming a system of equations from known values $ f(x_i) = y_i $ enables accurate coefficient determination. The four known points yield a solvable system of linear equations, ready for analytical or computational solution.", "Mastering this method empowers accurate modeling and insight extraction from real-world data across multiple disciplines.", "---", "### Further Reading", "- Polynomial interpolation in numerical methods\n- Using Vandermonde matrices for polynomial fitting\n- Solving underdetermined or ill-conditioned systems in regression", "---", "Keywords: cubic polynomial, system of equations, polynomial interpolation, cubic function coefficients, $ f(x) = ax^3 + bx^2 + cx + d $, solving real-world data models, linear algebra in algebra, cubic curve fitting."]
