Problem 10 - Precalculus : Building Concepts and Connections

Question

Determine the multiplicities of the real zeros of the function. Comment on the behavior of the graph at the \(x\) -intercepts. Does the graph cross or just touch the \(x\) -axis? You may check your results with a graphing utility. $$g(x)=x^{3}(x+2)(x-3)$$

Step-by-Step Solution

Answer

The roots of the function \(g(x)\) are \(x=0\), \(x=-2\), and \(x=3\) with multiplicity of 3, 1 and 1 respectively. At each x-intercept, the graph crosses the x-axis.

Step 1: Write down the given function

The function provided is \(g(x)=x^{3}(x+2)(x-3)\). This function is already factored.

Step 2: Find the roots of the function

The roots of the function \(g(x)\) are the values of \(x\) that make \(g(x)=0\). This is obtained by setting each factor to zero, thus \(x=0\), \(x=-2\), and \(x=3\).

Step 3: Determine the multiplicity of each root

The multiplicity of a root is the power of the corresponding factor. As we can see, \(x=0\) has a multiplicity of 3 as the related factor \(x^{3}\) is raised to the power 3. \(x=-2\) and \(x=3\) each have a multiplicity of 1.

Step 4: Analyze the behavior at each x-intercept

The graph of a polynomial function will cross the x-axis at a root if the root has odd multiplicity and will touch (but not cross) the x-axis at a root if the root has even multiplicity. Since all our roots have odd multiplicities, the graph will cross the x-axis at these intercepts.

Key Concepts

Real Zeros

Real zeros of a polynomial function are the values of the variable for which the function equals zero. For the polynomial function \( g(x) = x^3(x+2)(x-3) \), finding the real zeros involves setting each factor equal to zero and solving for \( x \).

In this case, the real zeros are:

• \( x = 0 \)
• \( x = -2 \)
• \( x = 3 \)

These zeros are where the graph of the function touches or crosses the x-axis. Real zeros of a polynomial are significant as they reveal the points of intersection with the x-axis, which are critical for graph sketching and understanding the function's behavior at those points.

Graph Behavior

The behavior of the graph of a polynomial function near its zeros provides insight into how the function behaves as the input variable approaches these points of interest. Near each \(x\)-intercept, the graph can either cross the x-axis, indicating a root of odd multiplicity, or merely touch it before changing direction, indicating a root of even multiplicity.

In our example, the function \( g(x) = x^3(x+2)(x-3) \) shows that:

• At \( x = 0 \), there is a crossing due to its odd multiplicity of 3.
• At \( x = -2 \) and \( x = 3 \), the graph also crosses, each having an odd multiplicity of 1.

Understanding the graph's behavior near these points helps with plotting and interpreting the function's overall trend.

X-Intercepts

X-intercepts are the points where the function graph meets the x-axis, i.e., where the value of the function is zero. These intercepts are synonymous with the real zeros of the polynomial. For the function \( g(x) = x^3(x+2)(x-3) \), the x-intercepts are:

• \((0, 0)\)
• \((-2, 0)\)
• \((3, 0)\)

Each of these points is where the graph crosses the x-axis because each zero has an odd multiplicity. Recognizing x-intercepts is crucial for graphing as it guides gap positioning along the x-axis.

Roots Multiplicity

The multiplicity of a root of a polynomial function reflects how many times a particular root repeats. It determines how the graph behaves at the x-intercept.

For the polynomial \( g(x) = x^3(x+2)(x-3) \):

• The root \( x = 0 \) has a multiplicity of 3 (from the factor \( x^3 \)), signaling a crossing at this intercept.
• Roots \( x = -2 \) and \( x = 3 \) each have a multiplicity of 1, which indicates that the graph will cross the x-axis at these points too.

A higher, even multiplicity implies a "touch" without crossing, whereas an odd multiplicity typically implies a crossing. Understanding multiplicity helps predict and explain the shape and trend of the graph near each intercept.