At which root does the graph of [tex]f(x)=(x+4)^6(x+7)^5[/tex] cross the [tex]x[/tex]-axis?
A. -7
B. -4
C. 4
D. 7
Answer (1)
Find the roots of the function: x = − 4 and x = − 7 .
Determine the multiplicity of each root: multiplicity of x = − 4 is 6, and multiplicity of x = − 7 is 5.
If the multiplicity is odd, the graph crosses the x-axis; if even, it touches the x-axis.
The graph crosses the x-axis at x = − 7 because its multiplicity is odd. − 7
Explanation
Understanding the Problem We are given the function f ( x ) = ( x + 4 ) 6 ( x + 7 ) 5 and asked to find the root where the graph crosses the x-axis.
Finding the Roots The roots of the function are the values of x for which f ( x ) = 0 . Thus, the roots are x = − 4 and x = − 7 .
Determining Multiplicity The multiplicity of a root is the number of times the corresponding factor appears in the factored form of the function. For the root x = − 4 , the factor ( x + 4 ) appears with an exponent of 6, so the multiplicity of the root x = − 4 is 6. For the root x = − 7 , the factor ( x + 7 ) appears with an exponent of 5, so the multiplicity of the root x = − 7 is 5.
Determining Where the Graph Crosses If the multiplicity of a root is odd, the graph crosses the x-axis at that root. If the multiplicity of a root is even, the graph touches the x-axis at that root but does not cross it. Since the multiplicity of the root x = − 4 is 6, which is even, the graph touches the x-axis at x = − 4 but does not cross it. Since the multiplicity of the root x = − 7 is 5, which is odd, the graph crosses the x-axis at x = − 7 .
Final Answer Therefore, the graph of f ( x ) = ( x + 4 ) 6 ( x + 7 ) 5 crosses the x-axis at the root x = − 7 .
Examples
Understanding the behavior of polynomial functions, such as where they cross the x-axis, is crucial in many real-world applications. For instance, in engineering, determining the roots of a function can help identify critical points in a system's behavior, like resonance frequencies in a mechanical structure or stability points in a control system. Similarly, in economics, understanding where a cost function crosses the x-axis can indicate the break-even point for a business. These concepts provide a foundation for more advanced modeling and analysis in various fields.
