Solution: Let $ f(x) = ax^3 + bx^2 + cx + d $. Use the given values to form a system of equations:
["Title: Solving for Coefficients in a Cubic Function Using Given Conditions", "In advanced mathematics, understanding how to determine the coefficients of a cubic polynomial using known values is a foundational skill. Let ( f(x) = ax^3 + bx^2 + cx + d ) represent a cubic function. When practical applications impose specific constraints—such as known function values at particular points—these conditions can be formulated into a system of linear equations to solve for the unknowns ( a ), ( b ), ( c ), and ( d ). This article explains how to construct this system using given data and solve the resulting equations using standard algebraic techniques.", "---", "### The General Form of a Cubic Function", "The function\n[\nf(x) = ax^3 + bx^2 + cx + d\n]\nis a cubic polynomial, where ( a ), ( b ), ( c ), and ( d ) are unknown constants. We aim to determine these coefficients using known function values.", "---", "### Forming a System of Equations", "Suppose we are given three distinct points ( (x_1, y_1) ), ( (x_2, y_2) ), and ( (x_3, y_3) ) through which the cubic passes. Substituting each point into the polynomial yields:", "[\n\begin{cases}\nf(x_1) = a x_1^3 + b x_1^2 + c x_1 + d = y_1 \\nf(x_2) = a x_2^3 + b x_2^2 + c x_2 + d = y_2 \\nf(x_3) = a x_3^3 + b x_3^2 + c x_3 + d = y_3 \\n\end{cases}\n]", "This system of three equations with four unknowns is underdetermined—means there are infinitely many solutions in general. However, if a fourth condition is provided—such as the derivative at a point, or a fourth point—we obtain a full system to uniquely determine ( a ), ( b ), ( c ), and ( d ).", "For example, if we know the derivative at ( x = 1 ), i.e., ( f'(1) = e ), then\n[\nf'(x) = 3ax^2 + 2bx + c \Rightarrow f'(1) = 3a + 2b + c = e,\n]\nwhich adds a fourth equation.", "---", "### Example: Constructing and Solving the System", "Let us illustrate with concrete numbers. Suppose:\n- ( f(0) = 2 )\n- ( f(1) = -1 )\n- ( f(-1) = 4 )\n- ( f'(1) = 3 )", "Using the general form, we substitute the points:", "1. ( f(0) = d = 2 )\n2. ( f(1) = a(1)^3 + b(1)^2 + c(1) + d = a + b + c + d = -1 )\n3. ( f(-1) = a(-1)^3 + b(-1)^2 + c(-1) + d = -a + b - c + d = 4 )\n4. ( f'(1) = 3a(1)^2 + 2b(1) + c = 3a + 2b + c = 3 )", "Now substitute ( d = 2 ) into equations 2 and 3:", "- Eq. (2): ( a + b + c + 2 = -1 \Rightarrow a + b + c = -3 )\n- Eq. (3): ( -a + b - c + 2 = 4 \Rightarrow -a + b - c = 2 )\n- Eq. (4): ( 3a + 2b + c = 3 )", "We now have the system:", "[\n\begin{cases}\na + b + c = -3 \quad \ ext{(i)}\\n-a + b - c = 2 \quad \ ext{(ii)}\\n3a + 2b + c = 3 \quad \ ext{(iii)}\\n\end{cases}\n]", "Add (i) and (ii) to eliminate ( a ) and ( c ):", "[\n(a + b + c) + (-a + b - c) = -3 + 2 \Rightarrow 2b = -1 \Rightarrow b = -\frac{1}{2}\n]", "Substitute ( b = -\frac{1}{2} ) into (i):", "[\na - \frac{1}{2} + c = -3 \Rightarrow a + c = -\frac{5}{2} \quad \ ext{(iv)}\n]", "Now substitute ( b = -\frac{1}{2} ) into (iii):", "[\n3a + 2(-\frac{1}{2}) + c = 3 \Rightarrow 3a - 1 + c = 3 \Rightarrow 3a + c = 4 \quad \ ext{(v)}\n]", "Now solve (iv) and (v):", "From (iv): ( c = -\frac{5}{2} - a )\nSubstitute into (v):", "[\n3a + (-\frac{5}{2} - a) = 4 \Rightarrow 2a - \frac{5}{2} = 4 \Rightarrow 2a = \frac{13}{2} \Rightarrow a = \frac{13}{4}\n]", "Then\n[\nc = -\frac{5}{2} - \frac{13}{4} = -\frac{10}{4} - \frac{13}{4} = -\frac{23}{4}\n]", "---", "### Final Coefficients", "Thus, the unique cubic satisfying the given conditions is:", "[\nf(x) = \frac{13}{4}x^3 - \frac{1}{2}x^2 - \frac{23}{4}x + 2\n]", "This demonstrates how forming a system of equations from known function values allows us to determine the coefficients—even for underdetermined systems, when a fourth condition closes the system.", "---", "### Conclusion", "Solving for coefficients of a cubic polynomial using given function values is a powerful technique rooted in linear algebra. By forming a system from constraints—whether function values or derivative information—we can uniquely determine ( a ), ( b ), ( c ), and ( d ), enabling precise modeling in engineering, physics, and data analysis. This approach underpins interpolation, curve fitting, and symbolic computation.", "For further learning, explore Gaussian elimination and matrix methods to efficiently solve systems arising from polynomial fitting.", "---", "Keywords:\ncubic polynomial, solving for coefficients, system of equations, polynomial interpolation, algebra, function modeling, math education, cubic function $ f(x) = ax^3 + bx^2 + cx + d $, derivative conditions, underdetermined systems."]
