At which root does the graph of [tex]f(x)=(x+4)^6(x+7)^5[/tex] cross the [tex]x[/tex]-axis?
Answer (2)
The graph of the function f ( x ) = ( x + 4 ) 6 ( x + 7 ) 5 crosses the x-axis at the root x = − 7 , because it has an odd multiplicity. In contrast, at the root x = − 4 , the graph will touch the x-axis without crossing it due to its even multiplicity.
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Find the roots of the function: x = − 4 and x = − 7 .
Determine the multiplicity of each root: x = − 4 has multiplicity 6 (even), and x = − 7 has multiplicity 5 (odd).
The graph crosses the x-axis at roots with odd multiplicity.
Therefore, the graph crosses the x-axis at − 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. The roots of the function are the values of x for which f ( x ) = 0 .
Finding the Roots The roots of the function are found by setting each factor to zero:
( x + 4 ) 6 = 0 which gives x = − 4
( x + 7 ) 5 = 0 which gives x = − 7
Determining Multiplicity The multiplicity of a root is the exponent of the corresponding factor. The root x = − 4 has multiplicity 6, which is an even number. The root x = − 7 has multiplicity 5, which is an odd number.
Determining Where the Graph Crosses the x-axis If a root has an even multiplicity, the graph touches the x-axis at that root but does not cross it. If a root has an odd multiplicity, the graph crosses the x-axis at that root. Therefore, the graph crosses the x-axis at x = − 7 .
Final Answer 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, like where they cross the x-axis, is crucial in many real-world applications. For instance, in engineering, you might model the stability of a structure using a polynomial. The roots of the polynomial (where the function equals zero) can represent critical points where the structure might fail. Knowing whether the graph crosses the x-axis at these points helps engineers understand if the structure will change its behavior (like going from stable to unstable) at those critical points. Similarly, in economics, polynomial functions can model cost or revenue, and understanding their roots and crossing behavior can help businesses identify break-even points and optimize their strategies.
