Find the best answers to your questions at Westonci.ca, where experts and enthusiasts provide accurate, reliable information. Join our Q&A platform and get accurate answers to all your questions from professionals across multiple disciplines. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
To determine the multiplicity of the roots of the function [tex]\( k(x) = x(x+2)^3(x+4)^2(x-5)^4 \)[/tex], we need to look at the exponents of the factors in the function. Here’s a step-by-step explanation:
1. Identifying Factors and Their Exponents:
- The function [tex]\( k(x) \)[/tex] has factors [tex]\( x \)[/tex], [tex]\( (x+2)^3 \)[/tex], [tex]\( (x+4)^2 \)[/tex], and [tex]\( (x-5)^4 \)[/tex].
- The exponent of each factor indicates the multiplicity of the corresponding root.
2. Determining Roots and Their Multiplicities:
- [tex]\( x = 0 \)[/tex]: The factor [tex]\( x \)[/tex] appears with an exponent of 1. Therefore, the multiplicity of the root [tex]\( 0 \)[/tex] is 1.
- [tex]\( x = -2 \)[/tex]: The factor [tex]\( (x+2) \)[/tex] appears with an exponent of 3. Therefore, the multiplicity of the root [tex]\( -2 \)[/tex] is 3.
- [tex]\( x = -4 \)[/tex]: The factor [tex]\( (x+4) \)[/tex] appears with an exponent of 2. Therefore, the multiplicity of the root [tex]\( -4 \)[/tex] is 2.
- [tex]\( x = 5 \)[/tex]: The factor [tex]\( (x-5) \)[/tex] appears with an exponent of 4. Therefore, the multiplicity of the root [tex]\( 5 \)[/tex] is 4.
3. Final Answer:
- [tex]\( 0 \)[/tex] has multiplicity [tex]\( 1 \)[/tex].
- [tex]\( -2 \)[/tex] has multiplicity [tex]\( 3 \)[/tex].
- [tex]\( -4 \)[/tex] has multiplicity [tex]\( 2 \)[/tex].
- [tex]\( 5 \)[/tex] has multiplicity [tex]\( 4 \)[/tex].
Thus, the multiplicities of the roots are:
- [tex]\( 0 \)[/tex] has multiplicity [tex]\( 1 \)[/tex].
- [tex]\( -2 \)[/tex] has multiplicity [tex]\( 3 \)[/tex].
- [tex]\( -4 \)[/tex] has multiplicity [tex]\( 2 \)[/tex].
- [tex]\( 5 \)[/tex] has multiplicity [tex]\( 4 \)[/tex].
1. Identifying Factors and Their Exponents:
- The function [tex]\( k(x) \)[/tex] has factors [tex]\( x \)[/tex], [tex]\( (x+2)^3 \)[/tex], [tex]\( (x+4)^2 \)[/tex], and [tex]\( (x-5)^4 \)[/tex].
- The exponent of each factor indicates the multiplicity of the corresponding root.
2. Determining Roots and Their Multiplicities:
- [tex]\( x = 0 \)[/tex]: The factor [tex]\( x \)[/tex] appears with an exponent of 1. Therefore, the multiplicity of the root [tex]\( 0 \)[/tex] is 1.
- [tex]\( x = -2 \)[/tex]: The factor [tex]\( (x+2) \)[/tex] appears with an exponent of 3. Therefore, the multiplicity of the root [tex]\( -2 \)[/tex] is 3.
- [tex]\( x = -4 \)[/tex]: The factor [tex]\( (x+4) \)[/tex] appears with an exponent of 2. Therefore, the multiplicity of the root [tex]\( -4 \)[/tex] is 2.
- [tex]\( x = 5 \)[/tex]: The factor [tex]\( (x-5) \)[/tex] appears with an exponent of 4. Therefore, the multiplicity of the root [tex]\( 5 \)[/tex] is 4.
3. Final Answer:
- [tex]\( 0 \)[/tex] has multiplicity [tex]\( 1 \)[/tex].
- [tex]\( -2 \)[/tex] has multiplicity [tex]\( 3 \)[/tex].
- [tex]\( -4 \)[/tex] has multiplicity [tex]\( 2 \)[/tex].
- [tex]\( 5 \)[/tex] has multiplicity [tex]\( 4 \)[/tex].
Thus, the multiplicities of the roots are:
- [tex]\( 0 \)[/tex] has multiplicity [tex]\( 1 \)[/tex].
- [tex]\( -2 \)[/tex] has multiplicity [tex]\( 3 \)[/tex].
- [tex]\( -4 \)[/tex] has multiplicity [tex]\( 2 \)[/tex].
- [tex]\( 5 \)[/tex] has multiplicity [tex]\( 4 \)[/tex].
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.