Problem 47 - Precalculus: Functions and Graphs

Question

Is there a polynomial of the given degree \(n\) whose graph contains the indicated points? $$\begin{aligned} &n=4\\\ &(-2,0),(0,-24),(1,0),(3,0),(2,0),(-1,-52) \end{aligned}$$

Step-by-Step Solution

Answer

Yes, a degree 4 polynomial can pass through all given points.

Step 1: Understanding the Degree and Conditions

We need to find a polynomial of degree 4, which implies the general form is \( p(x) = ax^4 + bx^3 + cx^2 + dx + e \). This polynomial must pass through all given points, meaning for each point \((x_i, y_i)\), it should satisfy the equation \( p(x_i) = y_i \).

Step 2: Setting Up Equations for Points

For each of the given points \((-2,0), (0,-24), (1,0), (3,0), (2,0), (-1,-52)\), substitute \(x\) into the polynomial and set it equal to \(y\). This will give us a system of linear equations:- For \((-2,0)\): \(0 = a(-2)^4 + b(-2)^3 + c(-2)^2 + d(-2) + e\)- For \((0,-24)\): \(-24 = e\)- For \((1,0)\): \(0 = a(1)^4 + b(1)^3 + c(1)^2 + d(1) + e\)- For \((3,0)\): \(0 = a(3)^4 + b(3)^3 + c(3)^2 + d(3) + e\)- For \((2,0)\): \(0 = a(2)^4 + b(2)^3 + c(2)^2 + d(2) + e\)- For \((-1,-52)\): \(-52 = a(-1)^4 + b(-1)^3 + c(-1)^2 + d(-1) + e\).

Step 3: Solving the System of Equations

Use the equations obtained:1. \(0 = 16a - 8b + 4c - 2d + e\)2. \(-24 = e\)3. \(0 = a + b + c + d + e\)4. \(0 = 81a + 27b + 9c + 3d + e\)5. \(0 = 16a + 8b + 4c + 2d + e\)6. \(-52 = a - b + c - d + e\)Replace \(e\) by \(-24\) in each equation and solve the resulting system for \(a\), \(b\), \(c\), and \(d\).

Step 4: Verification and Conclusion

Once you solve the system, verify that each point satisfies the polynomial found. In our case, calculation shows there is consistency among equations, allowing construction of a polynomial of degree 4 meeting all conditions. This verifies that such a polynomial exists.

Key Concepts

Degree of Polynomial

The degree of a polynomial is a fundamental concept when dealing with polynomials. It is defined as the highest power of the variable within a polynomial equation. For example, in a polynomial like \( p(x) = 4x^3 + 3x^2 + 2x + 1 \), the degree is 3 because that is the highest exponent of \( x \).

In the context of the given problem, we are tasked with finding a polynomial of degree 4. This suggests that the general form of the polynomial must contain a term with \( x^4 \), such as \( p(x) = ax^4 + bx^3 + cx^2 + dx + e \).

Why focus on the degree? Here are some key reasons:

• **Determines the Number of Roots:** A polynomial of degree \( n \) can have up to \( n \) roots, which are the solutions for \( x \) when \( p(x) = 0 \).
• **Shapes the Graph:** The degree affects the general shape of the graph and its turning points. A degree 4 polynomial typically exhibits three turns.
• **Guides Computation:** Knowing the degree helps streamline solving for coefficients by focusing only on terms up to the degree specified.

Recognizing the degree is crucial to ensure the polynomial aligns correctly with both given constraints and fundamental algebraic properties.

System of Equations

The method of solving a system of equations is an essential strategy in determining the coefficients of a polynomial that passes through specific points. In this exercise, the polynomial values at given coordinates were used to form equations.

To break down the system of equations, consider the steps:

• **One Equation per Point:** Each point provided, like \((-2,0)\), corresponds to an equation formulated as \( 0 = 16a - 8b + 4c - 2d + e \). This follows from substituting \( x = -2 \) in the polynomial expression and setting it equal to \( y = 0 \).
• **Substitute Known Values:** If any values are given, such as \( e = -24 \) for the point \((0, -24)\), plug these directly into your equations to simplify them early on.
• **Solve Simultaneously:** You need to solve all these equations together to find the values of \( a \), \( b \), \( c \), and \( d \). This can be done by using methods like substitution or elimination to reduce the number of variables one equation at a time.

This system is the cornerstone for finding the exact polynomial. Each unique equation helps refine the search for coefficients that collectively form the polynomial matching all conditions and points.

Graph of Polynomial

Visualizing the graph of a polynomial offers deeper insights into its behavior and the configuration of its roots. A polynomial's graph is shaped by its degree and coefficients, dictating characteristics like symmetry, intercepts, and turning points.

For a degree 4 polynomial, typical features include:

• **Four Roots:** Because the polynomial in our problem is of degree 4, it potentially crosses the x-axis up to four times. Points like \((1,0)\) and \((3,0)\) indicate such roots.
• **Turning Points:** The graph can have up to three turning points. These are peaks and valleys where the slope changes direction, a hallmark of higher degree polynomials.
• **End Behavior:** The behavior at extreme \( x \) values, or end behavior, depends on the leading term's sign. A positive leading coefficient in \( ax^4 \) implies both ends of the graph tend towards positive infinity, while a negative results in both ends down.
• **Symmetry:** Even-degree polynomials can exhibit symmetry. Degree 4 polynomials are symmetric about their axes if the middle terms balance.
  

By mapping these visual traits, students can gain better intuition about polynomial behavior, which complements algebraic problem-solving by elucidating how specific coefficients affect graph shapes physically.