Which statement describes the graph of [tex]$f(x)=4 x^7+40 x^6+100 x^5$[/tex]?
The graph crosses the [tex]$x$[/tex]-axis at [tex]$x=0$[/tex] and touches the [tex]$x$[/tex]-axis at [tex]$x=5$[/tex].
The graph touches the [tex]$x$[/tex]-axis at [tex]$x=0$[/tex] and crosses the [tex]$x$[/tex]-axis at [tex]$x=5$[/tex].
The graph crosses the [tex]$x$[/tex]-axis at [tex]$x=0$[/tex] and touches the [tex]$x$[/tex]-axis at [tex]$x=-5$[/tex].
The graph touches the [tex]$x$[/tex]-axis at [tex]$x=0$[/tex] and crosses the [tex]$x$[/tex]-axis at [tex]$x=-5$[/tex].
Answer (2)
The graph of the function f ( x ) = 4 x 7 + 40 x 6 + 100 x 5 crosses the x-axis at x = 0 and touches the x-axis at x = − 5 . Thus, the correct option is the one stating these points of intersection. The behavior of the graph is determined by the multiplicities of the roots.
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Factor the polynomial: f ( x ) = 4 x 5 ( x + 5 ) 2 .
Identify the roots and their multiplicities: x = 0 with multiplicity 5, and x = − 5 with multiplicity 2.
Determine the behavior at each root: The graph crosses the x-axis at x = 0 (odd multiplicity) and touches the x-axis at x = − 5 (even multiplicity).
State the final answer: The graph crosses the x -axis at x = 0 and touches the x -axis at x = − 5 .
Explanation
Understanding the Problem We are given the function f ( x ) = 4 x 7 + 40 x 6 + 100 x 5 and we want to determine the behavior of its graph at its roots. Specifically, we want to know whether the graph crosses or touches the x-axis at each root. To do this, we need to find the roots of the polynomial and their multiplicities. If a root has odd multiplicity, the graph crosses the x-axis at that root. If a root has even multiplicity, the graph touches the x-axis at that root.
Factoring the Polynomial First, we factor the polynomial f ( x ) . We can factor out 4 x 5 from each term: f ( x ) = 4 x 5 ( x 2 + 10 x + 25 ) Now, we can factor the quadratic expression: f ( x ) = 4 x 5 ( x + 5 ) 2
Finding the Roots and Their Multiplicities Now we find the roots of the polynomial. The roots are the values of x for which f ( x ) = 0 . From the factored form, we can see that the roots are x = 0 and x = − 5 .
The root x = 0 comes from the factor x 5 , so it has multiplicity 5. The root x = − 5 comes from the factor ( x + 5 ) 2 , so it has multiplicity 2.
Determining the Behavior at Each Root Since the root x = 0 has multiplicity 5, which is odd, the graph crosses the x-axis at x = 0 .
Since the root x = − 5 has multiplicity 2, which is even, the graph touches the x-axis at x = − 5 .
Conclusion Therefore, the graph crosses the x -axis at x = 0 and touches the x -axis at x = − 5 .
Examples
Understanding the behavior of polynomial functions at their roots is crucial in various fields, such as engineering and physics. For instance, when analyzing the stability of a system modeled by a polynomial equation, the roots and their multiplicities determine the system's equilibrium points and their stability. A root with odd multiplicity indicates a change in sign, representing a transition point, while a root with even multiplicity indicates a turning point, representing a stable or unstable equilibrium.
