Problem 19 - Algebra for College Students

Question

For Problems \(1-22\), graph each of the polynomial functions. $$ f(x)=(x-2)(x-1)(x+1)(x+2) $$

Step-by-Step Solution

Answer

Graph is a symmetric curve crossing x-axis at -2, -1, 1, and 2, rising at both ends.

Step 1: Identify the Roots

To find the roots of the polynomial function, set each factor equal to zero: \[(x-2) = 0, \quad (x-1) = 0, \quad (x+1) = 0, \quad (x+2) = 0\]The roots are x = 2, x = 1, x = -1, and x = -2. These are the points where the graph will intersect the x-axis.

Step 2: Determine the End Behavior

This is a fourth-degree polynomial with positive leading coefficient, which implies that as \(x \to \infty\) or \(x \to -\infty\), the function \(f(x)\) also goes to \(\infty\). Thus, the graph will rise on both ends.

Step 3: Identify Additional Characteristics

Because all roots are of multiplicity 1, the graph will cross the x-axis at each root. Also, since the polynomial is symmetric about the y-axis, we can conclude that the function is even, which means the graph is symmetric with respect to the y-axis.

Step 4: Plot the Roots and Additional Points

Start by plotting the points for which \(f(x) = 0\). Then, calculate the function values at some points between and beyond these roots to understand the shape. For example, calculate \(f(0), f(3), f(-3)\).\[f(0) = (0-2)(0-1)(0+1)(0+2) = 4\]\[f(3) = (3-2)(3-1)(3+1)(3+2) = 20\]\[f(-3) = (-3-2)(-3-1)(-3+1)(-3+2) = 20\]

Step 5: Sketch the Graph

Using the information gathered - the roots, end behavior, and symmetry - sketch the polynomial function. The curve will intersect the x-axis at x = -2, -1, 1, 2 and rise towards infinity as \(x\) moves beyond these points. Ensure the graph is symmetric with respect to the y-axis.

Key Concepts

Roots of Polynomial

The roots of the polynomial function are the values of \( x \) where the function equals zero. For the given polynomial \( f(x)=(x-2)(x-1)(x+1)(x+2) \), we can find the roots by setting each factor equal to zero:

• \((x-2) = 0 \Rightarrow x = 2 \)
• \((x-1) = 0 \Rightarrow x = 1 \)
• \((x+1) = 0 \Rightarrow x = -1 \)
• \((x+2) = 0 \Rightarrow x = -2 \)

The roots \( x = -2, -1, 1, \) and \( 2 \) are where the graph of \( f(x) \) intersects the x-axis. Each root corresponds to a factor of the polynomial. These points are essential for plotting the graph as they indicate potential changes in the direction of the curve, also known as turning points.

End Behavior

The end behavior of a polynomial function describes how the function behaves as \( x \) approaches infinity or negative infinity. For the function \( f(x)=(x-2)(x-1)(x+1)(x+2) \), which is a fourth-degree polynomial with a positive leading coefficient:

• As \( x \to \infty \), \( f(x) \to \infty \).
• As \( x \to -\infty \), \( f(x) \to \infty \).

This means that as \( x \) moves towards positive or negative infinity, the graph will rise. This upward trend on both ends of the graph is typical for even-degree polynomials with positive leading coefficients.

Multiplicity

Multiplicity refers to how many times a root is repeated in the polynomial. In \( f(x)=(x-2)(x-1)(x+1)(x+2) \), each factor appears exactly once, meaning each root has a multiplicity of 1. Consequently:

• The graph will cross the x-axis at each root.
• There's no flattening or stall where it crosses at roots \( x = -2, -1, 1, \), and \( 2 \).

Higher multiplicity roots generally cause the graph to touch the x-axis and turn back rather than crossing it. Since all roots here are of multiplicity 1, the behavior is straightforward and robust in indicating a crossing.

Symmetry

Symmetry in polynomial functions helps to understand their visual representation without excessive plotting. The function \( f(x)=(x-2)(x-1)(x+1)(x+2) \) is symmetric about the y-axis:

• It is an even function.
• The graph will mirror itself across the y-axis.

This reflection symmetry makes it easier to plot the graph, knowing it will be identical on both sides of the y-axis. Checking for symmetry involves replacing \( f(x) \) with \( f(-x) \) and confirming the equality \( f(x) = f(-x) \). If true, the function is even, as in this case.

Graphing Polynomials

Graphing a polynomial involves putting together all gathered insights about it. For \( f(x)=(x-2)(x-1)(x+1)(x+2) \), we already know:

• The roots and x-axis intersections at \( x = -2, -1, 1, \), and \( 2 \).
• End behavior with the graph rising towards \( +\infty \) as \( x \) approaches \( \pm\infty \).
• Symmetry with respect to the y-axis.

We begin by plotting the roots and additional key points such as \( f(0), f(3), \) and \( f(-3) \), which gives us values to construct the curve. Calculating values for such points helps shape our graph accurately, reflecting symmetry and behavior at those crossing points. The final sketch marries these aspects into a coherent visual representation.