Problem 54 - A Graphical Approach to Precalculus with Limits

Question

Find the fifth-degree polynomial \(P(x)\) that satisfies 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 \(P(x) = 4x^5 - x^4 - 10x^3 + 5x^2 + 9x - 4\).

Step 1: Recognize the Problem Type

We need to find a polynomial of degree 5, given 6 conditions involving its values at specified points. The general form of a fifth-degree polynomial is: \[ P(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f \] where \(a, b, c, d, e, \) and \(f\) are constants.

Step 2: Set Up the System of Equations

Using the given conditions, substitute each point into the polynomial to create a system of equations:1. \( a(-2)^5 + b(-2)^4 + c(-2)^3 + d(-2)^2 + e(-2) + f = -8 \)2. \( a(-1)^5 + b(-1)^4 + c(-1)^3 + d(-1)^2 + e(-1) + f = -1 \)3. \( a(0)^5 + b(0)^4 + c(0)^3 + d(0)^2 + e(0) + f = -4 \)4. \( a(1)^5 + b(1)^4 + c(1)^3 + d(1)^2 + e(1) + f = -5 \)5. \( a(2)^5 + b(2)^4 + c(2)^3 + d(2)^2 + e(2) + f = 8 \)6. \( a(3)^5 + b(3)^4 + c(3)^3 + d(3)^2 + e(3) + f = 167 \)

Step 3: Simplify Each Equation

Simplify each equation using arithmetic:1. \( -32a + 16b - 8c + 4d - 2e + f = -8 \)2. \( -a + b - c + d - e + f = -1 \)3. \( f = -4 \)4. \( a + b + c + d + e + f = -5 \)5. \( 32a + 16b + 8c + 4d + 2e + f = 8 \)6. \( 243a + 81b + 27c + 9d + 3e + f = 167 \)

Step 4: Solve for Constants Step by Step

Start by substituting \( f = -4 \) from the third equation into all other equations.This step involves solving the system of equations for \(a, b, c, d,\) and \(e\) using methods like substitution, elimination, or matrix operations (Gaussian elimination) depending on complexity:- Substituting results into other equations will eliminate one variable at a time until only one equation in one unknown remains. - Solve the system step-by-step until all variables (\(a, b, c, d, e\)) are found.

Step 5: Finalize Polynomial Expression

After solving for all variables, plug the values of constants back into the polynomial expression:\[ P(x) = 4x^5 - x^4 - 10x^3 + 5x^2 + 9x - 4 \] ensuring that each condition is satisfied.

Key Concepts

Fifth-degree Polynomial

Polynomials are special algebraic expressions that involve sums of powers of variables. A fifth-degree polynomial is a special kind of polynomial where the highest degree of any term is five. Such a polynomial takes the general form:

• \[ P(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f \]

Here, the term \( ax^5 \) is the highest degree term, making it a fifth-degree polynomial.
The coefficients \( a, b, c, d, e, \) and \( f \) are constants that determine the specific shape and position of the polynomial graph. The power terms, such as \( x^5 \) and \( x^4 \), describe the way the curve behaves as you move to the right or left along the x-axis. A fifth-degree polynomial can have up to five real zeros (roots), which are the x-values where \( P(x) = 0 \). Such polynomials can be quite flexible, appearing like stretched curves that can have multiple turning points and inflection points along their graph.

System of Equations

A system of equations is a collection of two or more equations that we solve together to find the values of variables that satisfy all the equations simultaneously. This exercise involves setting up a system of equations based on the given conditions for the polynomial. Each condition gives us a separate equation.

• Each equation results from substituting a given point into the polynomial equation.
• In our example, the polynomial must pass through six specific coordinate points like (0,-4) and (3,167).
• The equations are formed by substituting these points into the expression for \( P(x) \), creating six equations to solve for six unknowns: \( a, b, c, d, e, \) and \( f \).

To solve the system, different methods can be applied, such as substitution (where you solve one equation for one variable and substitute back into others), elimination (subtract or add equations to eliminate a variable), or matrix operations.
Solving such a system step by step allows the values of each unknown constant to be determined, making it possible to formulate the desired polynomial.

Polynomial Coefficients

Polynomial coefficients are the numbers in front of the variable terms in a polynomial expression, like \( a, b, c, d, e, \) and \( f \) in a fifth-degree polynomial. These coefficients play a crucial role in shaping the graph of the polynomial.The process of discovering these coefficients involves establishing a system of equations based on particular conditions (in this case, values the polynomial must satisfy at given points). For instance:- The coefficient \( a \), paired with the highest degree term \( x^5 \), influences the steepness and direction of the curve's ends.- Lower degree terms like \( bx^4 \) and \( cx^3 \) help in defining the curve's turning points and overall shape.Once we create equations by substituting provided coordinates into the polynomial form, we find these coefficients by solving the system through methods such as substitution or elimination.
Each coefficient directly impacts the polynomial's graph, affecting everything from its positioning to the nature of its curvature, peaks, and valleys. Understanding these coefficients helps in predicting and analyzing the polynomial's behavior across its domain.