At which root does the graph of [tex]f(x)=(x-5)^3(x+2)^2[/tex] touch the [tex]x[/tex]-axis?
A. -5
B. -2
C. 2
D. 5
Answer (2)
The graph of the function f ( x ) = ( x − 5 ) 3 ( x + 2 ) 2 touches the x -axis at the root x = − 2 , since it has an even multiplicity of 2. The root x = 5 has an odd multiplicity and crosses the x -axis. Therefore, the chosen option is − 2 .
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Find the roots of the function f ( x ) = ( x − 5 ) 3 ( x + 2 ) 2 , which are x = 5 and x = − 2 .
Determine the multiplicity of each root: x = 5 has multiplicity 3, and x = − 2 has multiplicity 2.
Recall that the graph touches the x-axis at roots with even multiplicity.
Conclude that the graph touches the x-axis at x = − 2 , so the answer is − 2 .
Explanation
Understanding the Problem We are given the function f ( x ) = ( x − 5 ) 3 ( x + 2 ) 2 and asked to find the root at which the graph of f ( x ) touches the x -axis. This means we need to find the root where the graph is tangent to the x -axis, rather than crossing it.
Finding the Roots The roots of the function are the values of x for which f ( x ) = 0 . Thus, the roots are x = 5 and x = − 2 .
Determining Multiplicity The multiplicity of a root is the number of times the corresponding factor appears in the factored form of the polynomial. For the root x = 5 , the factor ( x − 5 ) appears with a power of 3, so the multiplicity of the root x = 5 is 3. For the root x = − 2 , the factor ( x + 2 ) appears with a power of 2, so the multiplicity of the root x = − 2 is 2.
Connecting Multiplicity to Graph Behavior If a root has an even multiplicity, the graph of the function touches the x -axis at that root. If a root has an odd multiplicity, the graph of the function crosses the x -axis at that root.
Identifying the Touching Root Since the root x = 5 has multiplicity 3 (odd), the graph crosses the x -axis at x = 5 . Since the root x = − 2 has multiplicity 2 (even), the graph touches the x -axis at x = − 2 . Therefore, the graph of f ( x ) touches the x -axis at the root x = − 2 .
Examples
Understanding the behavior of polynomial functions, such as where they touch or cross the x-axis, is crucial in many real-world applications. For instance, in engineering, when designing a bridge, engineers need to analyze the load distribution, which can be modeled by polynomial functions. The points where the function touches the x-axis (roots with even multiplicity) might represent stable points, while crossing points (roots with odd multiplicity) could indicate areas of instability. Similarly, in economics, polynomial functions can model market trends, and understanding their roots and behavior helps in predicting economic stability or volatility.
