Which statement describes the graph of [tex]f(x)=-4 x^3-28 x^2-32 x+64[/tex]?
The graph crosses the [tex]x[/tex]-axis at [tex]x=4[/tex] and touches the [tex]x[/tex]-axis at [tex]x=-1[/tex].
The graph touches the [tex]x[/tex]-axis at [tex]x=4[/tex] and crosses the [tex]x[/tex]-axis at [tex]x=-1[/tex].
The graph crosses the [tex]x[/tex]-axis at [tex]x=-4[/tex] and touches the [tex]x[/tex]-axis at [tex]x=1[/tex].
The graph touches the [tex]x[/tex]-axis at [tex]x=-4[/tex] and crosses the [tex]x[/tex]-axis at [tex]x=1[/tex].
Answer (1)
Factor the polynomial f ( x ) = − 4 x 3 − 28 x 2 − 32 x + 64 to find its roots.
Determine the roots and their multiplicities: x = 1 (multiplicity 1) and x = − 4 (multiplicity 2).
Since x = 1 has odd multiplicity, the graph crosses the x-axis at x = 1 .
Since x = − 4 has even multiplicity, the graph touches the x-axis at x = − 4 . The graph touches the x-axis at x = − 4 and crosses the x-axis at x = 1 .
Explanation
Understanding the Problem We are given the function f ( x ) = − 4 x 3 − 28 x 2 − 32 x + 64 and asked to describe its graph in terms of where it crosses or touches the x-axis. Crossing the x-axis indicates a root with odd multiplicity, while touching indicates a root with even multiplicity.
Factoring the Polynomial First, we factor the polynomial to find its roots. We can factor out a -4: f ( x ) = − 4 ( x 3 + 7 x 2 + 8 x − 16 ) . By trying integer roots that divide -16, we find that x = 1 is a root since 1 3 + 7 ( 1 ) 2 + 8 ( 1 ) − 16 = 1 + 7 + 8 − 16 = 0 . Also, x = − 4 is a root since ( − 4 ) 3 + 7 ( − 4 ) 2 + 8 ( − 4 ) − 16 = − 64 + 112 − 32 − 16 = 0 .
Finding the Factors Since we know that x = 1 is a root, ( x − 1 ) is a factor. Dividing x 3 + 7 x 2 + 8 x − 16 by ( x − 1 ) gives x 2 + 8 x + 16 . We can factor this quadratic as ( x + 4 ) 2 . Therefore, f ( x ) = − 4 ( x − 1 ) ( x + 4 ) 2 .
Determining Multiplicities and Behavior The roots of the polynomial are x = 1 and x = − 4 . The root x = 1 has multiplicity 1, which is odd, so the graph crosses the x-axis at x = 1 . The root x = − 4 has multiplicity 2, which is even, so the graph touches the x-axis at x = − 4 .
Conclusion Therefore, the graph touches the x-axis at x = − 4 and crosses the x-axis at x = 1 .
Examples
Understanding the behavior of polynomial functions, like determining where they cross or touch the x-axis, is crucial in many real-world applications. For instance, in engineering, these points can represent critical stability thresholds in a system. Imagine designing a bridge; the roots of a polynomial equation describing the bridge's load capacity could indicate the maximum weight it can bear before failure. Similarly, in economics, understanding the roots and turning points of a cost function can help businesses optimize production levels to minimize expenses and maximize profits. By analyzing the function f ( x ) = − 4 x 3 − 28 x 2 − 32 x + 64 , we can predict its behavior and apply similar principles to analyze more complex systems in various fields.
