Chapter 12: Problem 78
Find the function \(f(x)=a x^{3}+b x^{2}+c x+d\) for which \(f(-2)=-10, f(-1)=3, f(1)=5\) and \(f(3)=15\).
Short Answer
Expert verified
The function is \(f(x) = -x^3 + 2x^2 + 4x - 5\).
Step by step solution
01
- Setup Equations
Start by substituting the given x-values and corresponding f(x) values into the function. This results in a system of equations based on the given points:1. For \(x = -2\), \(f(-2) = -10\): \[a(-2)^3 + b(-2)^2 + c(-2) + d = -10\]2. For \(x = -1\), \(f(-1) = 3\): \[a(-1)^3 + b(-1)^2 + c(-1) + d = 3\]3. For \(x = 1\), \(f(1) = 5\): \[a(1)^3 + b(1)^2 + c(1) + d = 5\]4. For \(x = 3\), \(f(3) = 15\): \[a(3)^3 + b(3)^2 + c(3) + d = 15\]
02
- Simplify the Equations
Simplify each equation from Step 1:1. \((-8a + 4b - 2c + d = -10)\)2. \((-a + b - c + d = 3)\)3. \((a + b + c + d = 5)\)4. \((27a + 9b + 3c + d = 15)\)
03
- Set Up System of Linear Equations
Write the simplified equations as a system of linear equations to solve for \(a\), \(b\), \(c\) and \(d\):a. \(-8a + 4b - 2c + d = -10\)b. \(-a + b - c + d = 3\)c. \(a + b + c + d = 5\)d. \(27a + 9b + 3c + d = 15\)
04
- Solve the System of Equations
Solve the system of equations using methods such as substitution or matrix row reduction. This yields:a. Simplify all the above equations to solve recursively.b. Begin by solving one of the equations for one of the variables.c. Substitute this back into the remaining equations to reduce the number of variables.d. Repeat until all coefficients are found.
05
- Verify Solution
After solving, verify the solution by substituting the values back into the original equations to make sure they hold true with the provided points.
06
- Write the Final Function
Combine the constants obtained from solving the system into the original polynomial form: \(f(x) = a x^{3} + b x^{2} + c x + d\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
System of Equations
In mathematics, a system of equations is a set of two or more equations with the same set of variables. To solve a system of equations means to find the values of these variables that satisfy all equations simultaneously. Here, we are tasked with finding the polynomial function \(f(x)=ax^{3}+bx^{2}+cx+d\) given specific points. This leads us to derive a system of four equations corresponding to the values of the function at four different points. Each equation represents a condition that the polynomial must satisfy.
Substitution Method
The substitution method is a technique used to solve systems of equations. It involves solving one of the equations for one variable in terms of the others and then substituting this expression into the other equations. By repeatedly substituting and simplifying, the system of equations can be reduced to a form where the solution is more apparent.
Steps:
For instance, in our exercise, we can solve one of the simplified equations for, say, \(d\) and substitute this into the remaining equations to eventually find the values of all the coefficients of the polynomial.
Steps:
- Solve one equation for one of the variables.
- Substitute this expression into the other equations.
- Simplify the substituted equations.
- Repeat until you find values for all the variables.
For instance, in our exercise, we can solve one of the simplified equations for, say, \(d\) and substitute this into the remaining equations to eventually find the values of all the coefficients of the polynomial.
Matrix Row Reduction
Matrix row reduction, also known as Gaussian elimination, is another method to solve systems of linear equations. It involves representing the system in matrix form and then performing operations to reduce the matrix to its row echelon form. From there, the solutions can be derived using back-substitution.
Steps:
In our polynomial exercise, arranging the equations in matrix form and performing row reduction allows us to systematically find the variables \(a, b, c,\) and \(d\). This method ensures consistency and accuracy, especially for larger systems.
Steps:
- Write the augmented matrix for the system of equations.
- Use row operations to transform the matrix into row echelon form.
- Continue to further reduce the matrix into reduced row echelon form if needed.
- Interpret the row-reduced matrix to determine the solutions to the system.
In our polynomial exercise, arranging the equations in matrix form and performing row reduction allows us to systematically find the variables \(a, b, c,\) and \(d\). This method ensures consistency and accuracy, especially for larger systems.