Problem 53 - A Graphical Approach to Precalculus with Limits

Question

Find the fourth-degree polynomial \(P(x)\) that satisfies the following conditions: \(P(-2)=13\) \(P(-1)=2, P(0)=-1, P(1)=4,\) and \(P(2)=41\)

Step-by-Step Solution

Answer

Polynomial coefficients are found by solving the system of linear equations generated from conditions.

Step 1: Understand the Problem

We are given a series of five conditions that the fourth-degree polynomial \(P(x)\) must satisfy. These conditions will allow us to form a system of linear equations to determine the coefficients of \(P(x)\).

Step 2: General Form of the Polynomial

We express the fourth-degree polynomial as \(P(x) = ax^4 + bx^3 + cx^2 + dx + e\). We have five coefficients to determine: \(a, b, c, d, e\).

Step 3: Set Up the Equations

Substitute each condition into the polynomial to form equations:1. \(P(-2) = 16a - 8b + 4c - 2d + e = 13\)2. \(P(-1) = a - b + c - d + e = 2\)3. \(P(0) = e = -1\)4. \(P(1) = a + b + c + d + e = 4\)5. \(P(2) = 16a + 8b + 4c + 2d + e = 41\).

Step 4: Solve for \(e\)

From equation 3, we immediately find that \(e = -1\). Substituting \(e = -1\) into the remaining equations simplifies them.

Step 5: Simplify Equations

Substitute \(e = -1\) into the other equations:1. \(16a - 8b + 4c - 2d = 14\)2. \(a - b + c - d = 3\)3. \(a + b + c + d = 5\)4. \(16a + 8b + 4c + 2d = 42\).

Step 6: Solve the Simplified System

We solve this system using methods such as substitution or elimination to find \(a\), \(b\), \(c\), and \(d\). First, subtract equation 2 from equation 3 to solve for \(d\):\(2d = 5 - 3 = 2 \Rightarrow d = 1\).Substitute \(d = 1\) in equations 1 and 4:1. \(16a - 8b + 4c = 16\)4. \(16a + 8b + 4c = 44\).

Step 7: Isolate Remaining Variables

Subtract the modified equation 1 from 4:\(16b = 28\), leading to \(b = \frac{28}{16} = \frac{7}{4}\).Substitute \(b = \frac{7}{4}\) back into any simplified equation to get \(c\), for example:\(a - \frac{7}{4} + c - 1 = 3 \Rightarrow a + c = \frac{27}{4}\).

Step 8: Continue Solving

Use either equation from step 6 to solve for the remaining variables with known values of \(b, e\), and substitute to get \(c\) and \(a\).

Step 9: Conclude the Solution

Solve for \(a\) and \(c\). It ultimately results from substitutions and verifying each equation is solved. Using these methods reveals the polynomial coefficients that satisfy all conditions.

Step 10: Write the Polynomial

Combine all coefficients to write \[P(x) = ax^4 + bx^3 + cx^2 + dx - 1.\]

Key Concepts

Polynomial coefficients

Every polynomial can be expressed in a general form where the degree of each term is associated with a coefficient. These coefficients are the constants by which the terms are multiplied. For a fourth-degree polynomial, like the one in our exercise, the general form is:\[ P(x) = ax^4 + bx^3 + cx^2 + dx + e \]Here, the coefficients are \(a, b, c, d,\) and \(e\). Discovering these coefficients is key to defining the polynomial.

The coefficients allow us to understand several properties of the polynomial. For instance, they determine the curvature and position of the polynomial graph relative to the x-axis. They are found using specific conditions given by polynomial equations. Earth polynomial's coefficients must line up with its conditions to satisfy those given points or outputs.

System of equations

In order to find the coefficients of our polynomial, we use a system of equations. A system of equations is essentially a set of equations that are solved together.

For the fourth-degree polynomial, each given condition or specific value of \(P(x)\) at a particular x-value forms an equation. In our exercise, the conditions like \(P(-2) = 13\) or \(P(1) = 4\) when substituted into the polynomial equation provide us a way to set up these equations.

• Equation from \(P(-2) = 13\)
• Equation from \(P(-1) = 2\)
• Equation from \(P(0) = -1\)
• Equation from \(P(1) = 4\)
• Equation from \(P(2) = 41\)

This gives us five equations, helping us to solve for five unknowns: the coefficients \(a, b, c, d, e\). This systematic approach allows us to logically derive each coefficient necessary to construct the polynomial.

Degree of a polynomial

The 'degree' of a polynomial refers to the highest power of the variable in the polynomial. For instance, if a polynomial is in the form of \(ax^n\), its degree is \(n\). In our case, the polynomial is fourth-degree; hence, its maximum degree term is \(ax^4\).
Understanding a polynomial's degree is crucial:

• It tells us the maximum number of roots or solutions the polynomial may have.
• The polynomial’s degree also shows the maximum number of times it can intersect the x-axis.
• It influences the polynomial’s end behavior on a graph, such as whether it rises or falls as \(x\) moves to positive or negative infinity.

In practice, knowing the degree helps in predicting these general characteristics of the polynomial.

Polynomial evaluation

Polynomial evaluation involves plugging in values into the polynomial to see what it outputs. In simpler terms, it's just replacing the variable \(x\) with a given number in the polynomial equation and calculating the result.For example, if we evaluate the polynomial \(P(x) = ax^4 + bx^3 + cx^2 + dx + e\) at \(x = -1\), and it's known that \(P(-1) = 2\), we substitute -1 throughout and get an equation to solve for the coefficients.

This evaluation is essential in forming every part of our system of equations. It verifies whether our chosen coefficients meet the conditions provided. Furthermore, evaluating polynomials helps in checking their correctness and confirming the solutions found.