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 will examine each factor of the function separately.
1. Root at [tex]\(x = 0\)[/tex]:
- The factor that gives the root [tex]\(x = 0\)[/tex] is [tex]\(x\)[/tex].
- Since this factor is of the form [tex]\(x\)[/tex], the exponent is 1.
- Therefore, the root [tex]\(x = 0\)[/tex] has multiplicity 1.
2. Root at [tex]\(x = -2\)[/tex]:
- The factor that gives the root [tex]\(x = -2\)[/tex] is [tex]\((x + 2)^3\)[/tex].
- Since this factor is of the form [tex]\((x + 2)^3\)[/tex], the exponent is 3.
- Therefore, the root [tex]\(x = -2\)[/tex] has multiplicity 3.
3. Root at [tex]\(x = -4\)[/tex]:
- The factor that gives the root [tex]\(x = -4\)[/tex] is [tex]\((x + 4)^2\)[/tex].
- Since this factor is of the form [tex]\((x + 4)^2\)[/tex], the exponent is 2.
- Therefore, the root [tex]\(x = -4\)[/tex] has multiplicity 2.
4. Root at [tex]\(x = 5\)[/tex]:
- The factor that gives the root [tex]\(x = 5\)[/tex] is [tex]\((x - 5)^4\)[/tex].
- Since this factor is of the form [tex]\((x - 5)^4\)[/tex], the exponent is 4.
- Therefore, the root [tex]\(x = 5\)[/tex] has multiplicity 4.
In summary:
- The root [tex]\(0\)[/tex] has multiplicity 1.
- The root [tex]\(-2\)[/tex] has multiplicity 3.
- The root [tex]\(-4\)[/tex] has multiplicity 2.
- The root [tex]\(5\)[/tex] has multiplicity 4.
