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].
