For question number 4: Given that [tex]x=-\frac{1}{5}[/tex] and [tex]x=4[/tex] are zeros for the following function [tex]f(x)=\left(x+\frac{1}{5}\right)^4(x-4)^3[/tex], determine the multiplicity for each zero. State whether the graph crosses the [tex]x[/tex]-axis or touches the [tex]x[/tex]-axis and turns around, at each zero.
[tex]f(x)=\left(x+\frac{1}{5}\right)^2(x+8)^5[/tex]
A. [tex]\frac{1}{5}[/tex], multiplicity 2, touches the [tex]x[/tex]-axis and turns around;
8, multiplicity 5, crosses the [tex]x[/tex]-axis.
B. [tex]\frac{1}{5}[/tex], multiplicity 2, crosses the [tex]x[/tex]-axis;
8, multiplicity 5, touches the [tex]x[/tex]-axis and turns around
C. [tex]-\frac{1}{5}[/tex], multiplicity 2, touches the [tex]x[/tex]-axis and turns around; -8, multiplicity 5, crosses the [tex]x[/tex]-axis.
Answer (2)
The zeros of the function f ( x ) = ( x + 5 1 ) 2 ( x + 8 ) 5 are x = − 5 1 and x = − 8 .
The multiplicity of the zero x = − 5 1 is 2, so the graph touches the x-axis and turns around.
The multiplicity of the zero x = − 8 is 5, so the graph crosses the x-axis.
The correct answer is: − 5 1 , multiplicity 2, touches the x -axis and turns around; − 8 , multiplicity 5, crosses the x -axis.
Explanation
Understanding the Problem We are given the function f ( x ) = ( x + 5 1 ) 2 ( x + 8 ) 5 . We need to determine the multiplicity of each zero and whether the graph crosses or touches the x-axis at each zero.
Finding the Zeros The zeros of the function are the values of x that make f ( x ) = 0 . These occur when x + 5 1 = 0 or x + 8 = 0 . Thus, the zeros are x = − 5 1 and x = − 8 .
Determining Multiplicity The multiplicity of a zero is the exponent of the corresponding factor in the function. For x = − 5 1 , the factor is ( x + 5 1 ) , and its exponent is 2. Therefore, the multiplicity of the zero x = − 5 1 is 2.
Determining Multiplicity For x = − 8 , the factor is ( x + 8 ) , and its exponent is 5. Therefore, the multiplicity of the zero x = − 8 is 5.
Determining Graph Behavior If the multiplicity of a zero is even, the graph touches the x-axis and turns around at that zero. If the multiplicity is odd, the graph crosses the x-axis at that zero. Since the multiplicity of x = − 5 1 is 2 (even), the graph touches the x-axis and turns around at x = − 5 1 . Since the multiplicity of x = − 8 is 5 (odd), the graph crosses the x-axis at x = − 8 .
Final Answer Therefore, the zero x = − 5 1 has multiplicity 2, and the graph touches the x-axis and turns around at this point. The zero x = − 8 has multiplicity 5, and the graph crosses the x-axis at this point.
Examples
Understanding the multiplicity of zeros is crucial in various fields. For instance, in engineering, when analyzing the stability of a system, the zeros of a transfer function determine the system's response. Even multiplicities indicate stable behavior, while odd multiplicities can suggest instability or oscillations. In physics, analyzing wave functions often involves identifying zeros and their multiplicities to understand the behavior of particles in quantum systems. This concept also extends to economics, where understanding the roots of polynomial models can help predict market trends and stability.
The zeros of the function are − 5 1 with multiplicity 2, where the graph touches the x-axis and turns around, and − 8 with multiplicity 5, where the graph crosses the x-axis. The chosen option is C.
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