Problem 101 - Precalculus with Limits

Question

In Exercises 99-102, use a system of equations to find the cubic function \(f(x) = ax^3 + bx^2 + cx + d\) that satisfies the equations. Solve the system using matrices. \(f(-2) = -7\) \(f(-1) = 2\) \(f(1) = -4\) \(f(2) = -7\)

Step-by-Step Solution

Answer

The solution requires performing matrix operations and will yield the coefficients of the cubic function a, b, c, and d.

Step 1: Formulate the System of Equations

By substituting the x-values into the cubic function, we get the following system of equations:1. \( a(-2)^3 + b(-2)^2 + c(-2) + d = -7 \)2. \( a(-1)^3 + b(-1)^2 + c(-1) + d = 2 \)3. \( a(1)^3 + b(1)^2 + c(1) + d = -4 \)4. \( a(2)^3 + b(2)^2 + c(2) + d = -7 \)This simplifies to:1. \( -8a + 4b - 2c + d = -7 \)2. \( -a + b - c + d = 2 \)3. \( a + b + c + d = -4 \)4. \( 8a + 4b + 2c + d = -7 \)

Step 2: Write the System of Equations as a Matrix

The system can be written as the following augmented matrix:\[ \left[ \begin{array}{cccc|c} -8 & 4 & -2 & 1 & -7 \ -1 & 1 & -1 & 1 & 2 \ 1 & 1 & 1 & 1 & -4 \ 8 & 4 & 2 & 1 & -7 \end{array} \right] \]

Step 3: Solve the Matrix

We can solve the system using Gaussian elimination or another method convenient for you. The solution, after simplifying the matrix, will give us the values of a, b, c and d respectively.

Key Concepts

Solving Cubic Equations

Understanding how to solve cubic equations is a fundamental aspect of algebra that can challenge students. A cubic equation, typically written as \( ax^3 + bx^2 + cx + d = 0 \), contains an \( x^3 \) term and has up to three real roots. The strategy to solving these equations often requires reducing them to a system of linear equations through substitution or through factoring, if possible.

In our exercise, we were given four values of the cubic function \( f(x) = ax^3 + bx^2 + cx + d \), allowing us to write four equations that hold true for specific values of \( x \). We used these equations to solve for the coefficients \( a, b, c, \) and \( d \). This is a practical approach when the roots of the cubic equation are known or can be determined from given points. It's important to recognize that each point on the curve of a cubic function provides valuable information that can be translated into an equation, contributing to a solvable system that reveals the function's form.

Using Matrices in Algebra

Employing matrices to solve systems of equations is a powerful technique in algebra, especially with larger systems where traditional substitution and elimination methods can be cumbersome. A matrix is essentially a rectangular array of numbers, and when used correctly, it can greatly simplify operations involving linear equations.

The key is to set up an augmented matrix that includes all coefficients of the variables in one section and the constants of each equation in an adjoining section. After the augmented matrix is established, as seen in our exercise, we can perform operations such as row swapping, multiplication, or addition to modify the original matrix into a form that makes it easier to extract the solution, called reduced row-echelon form. Understanding how to manipulate matrices is crucial to this method and provides an efficient path to find the values of multiple variables simultaneously.

Gaussian Elimination Method

Gaussian elimination, named after the mathematician Carl Friedrich Gauss, is a systematic method for solving systems of linear equations. It's a sequence of operations performed on the augmented matrix representing the system, with the goal of transforming it into an upper triangular matrix.

The process typically involves three types of row operations: (1) swapping the positions of two rows, (2) multiplying a row by a non-zero scalar, and (3) adding or subtracting the multiples of one row to another row. These operations are used to systematically eliminate variables from equations, making them simpler to solve.

Once the matrix is in upper triangular form, the process of back-substitution starts. This means solving for the variable in the last row first and then working upwards to solve for the remaining variables. In our exercise, after the Gaussian elimination method simplifies the matrix, we can find the values of \( a, b, c, \) and \( d \) which define our original cubic function. This method is reliable and works for any number of variables and equations, as long as the system has a unique solution.