Determine the multiplicity of the roots of the function [tex]$k(x)=x(x+2)^3(x+4)^2(x-5)^4$[/tex].
0 has multiplicity $\square$
-2 has multiplicity $\square$
-4 has multiplicity $\square$
5 has multiplicity $\square$
Answer (2)
The multiplicity of a root is determined by the exponent of its corresponding factor in the polynomial.
The factor for the root 0 is x 1 , so its multiplicity is 1.
The factor for the root -2 is ( x + 2 ) 3 , so its multiplicity is 3.
The factor for the root -4 is ( x + 4 ) 2 , so its multiplicity is 2.
The factor for the root 5 is ( x − 5 ) 4 , so its multiplicity is 4.
Therefore, the multiplicities are 1 , 3 , 2 , 4 .
Explanation
Understanding the Problem We are given the function k ( x ) = x ( x + 2 ) 3 ( x + 4 ) 2 ( x − 5 ) 4 and we need to find the multiplicity of the roots 0, -2, -4, and 5. The multiplicity of a root is the exponent of the corresponding factor in the polynomial.
Multiplicity of root 0 For the root 0, the factor is x , which can be written as ( x − 0 ) 1 . The exponent of this factor is 1. Therefore, the multiplicity of the root 0 is 1.
Multiplicity of root -2 For the root -2, the factor is ( x + 2 ) , which is raised to the power of 3. Therefore, the multiplicity of the root -2 is 3.
Multiplicity of root -4 For the root -4, the factor is ( x + 4 ) , which is raised to the power of 2. Therefore, the multiplicity of the root -4 is 2.
Multiplicity of root 5 For the root 5, the factor is ( x − 5 ) , which is raised to the power of 4. Therefore, the multiplicity of the root 5 is 4.
Final Answer In summary:
The root 0 has multiplicity 1.
The root -2 has multiplicity 3.
The root -4 has multiplicity 2.
The root 5 has multiplicity 4.
Examples
Understanding the multiplicity of roots is crucial in various fields, such as physics and engineering. For instance, when analyzing the stability of a system, the multiplicity of the roots of the characteristic equation determines the system's behavior near equilibrium points. A root with a higher multiplicity indicates a stronger influence on the system's stability. Similarly, in signal processing, the multiplicity of poles in a transfer function affects the system's response to different frequencies. By understanding the multiplicity of roots, engineers can design more robust and efficient systems.
The multiplicity of the roots for the function k ( x ) = x ( x + 2 ) 3 ( x + 4 ) 2 ( x − 5 ) 4 is as follows: 0 has multiplicity 1, -2 has multiplicity 3, -4 has multiplicity 2, and 5 has multiplicity 4.
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