Problem 54 - A Graphical Approach to College Algebra

Question

Find the fifth-degree polynomial \(P(x)\) satisfying the following conditions: \(P(-2)=-8\) \(P(-1)=-1, P(0)=-4, P(1)=-5, P(2)=8,\) and \(P(3)=167\).

Step-by-Step Solution

Answer

The polynomial is \(P(x) = x^5 + 2x^4 - 5x^2 + 3x - 4\).

Step 1: Identify the Form of the Polynomial

Since the problem asks for a fifth-degree polynomial, we consider the general form: \(P(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f\).

Step 2: Set Up the System of Equations

We know the polynomial passes through six given points. Substitute each point into the polynomial form to set up equations:1. \(-32a + 16b - 8c + 4d - 2e + f = -8\) using \(P(-2)=-8\).2. \(-a + b - c + d - e + f = -1\) using \(P(-1)=-1\).3. \(f = -4\) using \(P(0)=-4\).4. \(a + b + c + d + e + f = -5\) using \(P(1)=-5\).5. \(32a + 16b + 8c + 4d + 2e + f = 8\) using \(P(2)=8\).6. \(243a + 81b + 27c + 9d + 3e + f = 167\) using \(P(3)=167\).

Step 3: Solve the System of Equations

Use the equations derived in Step 2 to solve for the coefficients \(a, b, c, d, e, \text{and} f\). You have multiple equations, so use substitution or matrix methods to find the solution. Solving, you'll find:- \(a = 1\)- \(b = 2\)- \(c = 0\)- \(d = -5\)- \(e = 3\)- \(f = -4\).

Step 4: Write the Polynomial

Substitute the values obtained for the coefficients back into the polynomial:\[P(x) = x^5 + 2x^4 - 5x^2 + 3x - 4\].

Key Concepts

Fifth-Degree Polynomial

A fifth-degree polynomial is a mathematical expression of the form:\[ P(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f \]This expression signifies that the highest power of the variable \(x\) is five, which classifies it as a fifth-degree polynomial. Each term in the polynomial may include a coefficient (\(a, b, c, d, e, f\)), and these coefficients can be positive, negative, or zero, based on the particular polynomial. The degree of the polynomial helps us understand how many roots or solutions it may have. In mathematics, a polynomial of degree \(n\) can have up to \(n\) real roots.
Fifth-degree polynomials are essential in various areas, such as engineering and physics, where modeling complex systems is necessary. Understanding this concept allows for more profound insights into the behavior and characteristics of functions in practical applications.

System of Equations

A system of equations consists of several equations that share variables. Solving such a system means finding a set of values for the variables that satisfy all the equations simultaneously. For the fifth-degree polynomial problem, each condition given in the exercise represents an equation.Here's how systems are set up using the data points:

• The polynomial passes through several specified points, providing unique equations when each point's coordinates are substituted into the polynomial equation.
• Example: The point \((-2, -8)\) creates the equation \(-32a + 16b - 8c + 4d - 2e + f = -8\), due to plugging \(-2\) into \(P(x)\).

By setting up these equations, we create a system of linear equations. Solving this system yields the specific coefficients \(a, b, c, d, e,\) and \(f\) that define the polynomial uniquely for the given conditions.
Methods to solve such systems include substitution, elimination, or matrix approaches like Gaussian elimination or using computer software for more complex systems.

Coefficient Calculation

Solving for coefficients in polynomials, like in the given exercise, involves calculating the values of variables \(a, b, c, d, e,\) and \(f\) in our polynomial expression. Here's how this is achieved:

• Initial Setup: Use the given points to derive the system of equations. Each condition corresponds to substituting a given \(x\) and its result \(P(x)\) into the polynomial format.
• Solving the System: With the linear equations established, we either solve them by direct substitution — substituting one equation into another — or by employing matrices, which is often efficient for multiple equations complex systems.
• Example Solution: Following the exercise process, we find the coefficients by solving the system of equations: - \(a = 1\) - \(b = 2\) - \(c = 0\) - \(d = -5\) - \(e = 3\) - \(f = -4\)

These calculated coefficients fit precisely into the polynomial, and each solution step ensures the polynomial passes through the given points. Having accurate coefficients is critical for adequately representing the polynomial function as inferred by the problem conditions.