Which statement describes the graph of this polynomial function?
[tex]f(x)=x^5-6 x^4+9 x^3[/tex]
A. The graph crosses the [tex]$x$[/tex]-axis at [tex]$x=0$[/tex] and touches the [tex]$x$[/tex]-axis at [tex]$x=3$[/tex].
B. The graph touches the [tex]$x$[/tex]-axis at [tex]$x=0$[/tex] and crosses the [tex]$x$[/tex]-axis at [tex]$x=3$[/tex].
C. The graph crosses the [tex]$x$[/tex]-axis at [tex]$x=0$[/tex] and touches the [tex]$x$[/tex]-axis at [tex]$x=-3$[/tex].
D. The graph touches the [tex]$x$[/tex]-axis at [tex]$x=0$[/tex] and crosses the [tex]$x$[/tex]-axis at [tex]$x=-3$[/tex].
Answer (1)
Factor the polynomial: f ( x ) = x 3 ( x − 3 ) 2 .
Identify the roots and their multiplicities: x = 0 (multiplicity 3) and x = 3 (multiplicity 2).
Determine the behavior at each root: crosses at x = 0 and touches at x = 3 .
The graph crosses the x-axis at x = 0 and touches the x-axis at x = 3 .
Explanation
Understanding the Problem We are given the polynomial function f ( x ) = x 5 − 6 x 4 + 9 x 3 and asked to describe its graph's behavior at the x-axis. To do this, we need to find the roots of the polynomial and their multiplicities. The roots tell us where the graph intersects or touches the x-axis, and the multiplicity tells us whether it crosses or touches.
Factoring the Polynomial First, we factor the polynomial function. From the tool, we have f ( x ) = x 3 ( x − 3 ) 2 .
Identifying Roots and Multiplicities Now, we identify the roots and their multiplicities. The factor x 3 gives us a root at x = 0 with multiplicity 3. The factor ( x − 3 ) 2 gives us a root at x = 3 with multiplicity 2.
Determining Graph Behavior Next, we determine the behavior of the graph at each root. Since the root x = 0 has an odd multiplicity of 3, the graph crosses the x-axis at x = 0 . Since the root x = 3 has an even multiplicity of 2, the graph touches the x-axis at x = 3 .
Conclusion Therefore, the graph crosses the x-axis at x = 0 and touches the x-axis at x = 3 .
Examples
Understanding the behavior of polynomial functions at their roots is crucial in various fields. For instance, in engineering, when designing a bridge, engineers need to analyze the stability of the structure, which can be modeled using polynomial functions. The roots of these functions represent critical points, and whether the graph crosses or touches the x-axis at these points indicates different stability conditions. Similarly, in economics, polynomial functions can model market trends, and understanding their roots helps in predicting market behavior and making informed decisions. In computer graphics, polynomial functions are used to create curves and surfaces, and the behavior of these functions at their roots affects the smoothness and appearance of the generated shapes.
